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GIFT  OF 


THE 

CIVIL  ENGINEER'S  POCKET-BOOK. 


MR.  TRAUTWINE'S  ENGINEERING  WORKS. 


The  Field  Practice  of  Laying  out  Circular  Curves  for  Rail- 
roads. By  JOHN  C.  TRAUTWINE,  Civil  Engineer.  Eleventh 
edition,  enlarged  and  rewritten.  I2mo,  morocco,  tuck,  gilt  edge,  $2  50 

A  New  Method  of  Calculating  the  Cubic  Contents  of  Exca- 
vations and  Embankments  by  the  aid  of  Diagrams ;  together  with 
Directions  for  Estimating  the  Cost  of  Earthwork.  By  JOHN  C. 
TRAUTWINE,  Civil  Engineer.  10  steel  plates.  Seventh  edition, 
completely  revised  and  enlarged.  8vo,  cloth, 2  oo 

Civil  Engineer's  Pocket-Book  of  Mensuration,  Trigonometry, 

Surveying,  Hydraulics,  Hydrostatics,  Instruments  and  their 
adjustments,  Strength  of  Materials,  Masonry,  Principles  of 
Wooden  and  Iron  Roof  and  Bridge  Trusses,  Stone  Bridges  and 
Culverts,  Trestles,  Pillars,  Suspension  Bridges,  Dams,  Railroads, 
Turnouts,  Turning  Platforms,  Water  Stations,  Cost  of  Earthwork, 
Foundations,  Retaining  Walls,  etc.,  etc.,  etc.  In  addition  to 
which  the  elucidation  of  certain  important  Principles  of  Construc- 
tion is  made  in  a  more  simple  manner  than  heretofore.  By  JOHN 
C.  TRAUTWINE,  C.E.  I2mo,  695  pages,  illustrated  with  about 
700  wood-cuts.  Morocco,  tuck,  gilt  edge.  Twentieth  thousand, 
revised  and  corrected, 5  °° 

Any  of  the  above  books  will  be  sent  to  any  part  of  the  United  States 
or  Canada  on  receipt  of  list  price. 

Send  money  in  Registered  Letter,  P.  O.  Order,  or  Check. 

E.  CLAXTON  &  CO.,  PUBLISHERS, 

No.  930  MARKET  STREET, 

PHILADELPHIA,  PA. 


THE 


CIVIL  ENGINEER'S  POCKET-BOOK, 

OP 

Mensuration,  Trigonometry,  Surveying,  Hydraulics,  Hydrostatics, 
Instruments  and  their  Adjustments,  Strength  of  Materials, 
Masonry,  Principles  of  Wooden  and  Iron  Eoof  and 
Bridge  Trusses,  Stone  Bridges  and  Culverts, 
Trestles,  Pillars,  Suspension  Bridges,  Dams, 
Railroads,  Turnouts,  Turning-Plat- 
forms, "Water  Stations,  Cost  of 
Earthwork,  Foundations, 
Retaining  Walls, 
Etc,,  Etc,,  Etc. 

IN  ADDITION  TO  WHICH  THE  ELUCIDATION  OF  CERTAIN 

IMPORTA-NT  PRINCIPLES  OF  CONSTRUCTION  IS 

MADE  IN  A  MORE  SIMPLE  MANNER 

THAN  HERETOFORE. 

BY  JOHN  C.  TRAUTWINE,  C.E., 

AUTHOR  OF   "  A   NEW  METHOD  OF  CALCULATING  THE  CUBIC  CONTENTS 

OF  EXCAVATIONS   AND   EMBANKMENTS,"    "THE   FIELD   PRACTICE 

OF  LAYING  OUT  CIRCULAR  CURVES  FOR  RAILROADS,"  ETC. 

ILLUSTRATED  WITH  690  ENGRAVINGS  FROM  ORIGINAL  DESIGNS, 
[wenitdh 

REVISED    AND    CORRECTED. 

PHILADELPHIA: 

E.  CLAXTON    &    COMPANY. 

LONDON :  TRUBNER  &  CO. 

1883. 


Entered,  according  to  Act  of  Congress,  in  the  year  1882,  by 

JOHN  C.  TRAUTWINE, 
in  tne  Office  of  the  Librarian  of  Congress  at  Washington. 


1LECTROTTPED  BY  J.  FAG  AN  &  SON,  PHILADELPHIA. 


THE  AUTHOR 

i$i 

TO   THE   MEMORY    OF    HIS    FRIEND, 

THE  LATE 

BENJAMIN  H.  LATROBE,  Esq., 

CIVIL  ENGINEER. 


50 


PUBLISHEES'  ANNOUNCEMENT. 


THE  publishers  have  much  pleasure  in  issuing  this  twentieth  thou- 
sand of  Mr.  Trautwine's  popular  book.  Although  originally  written 
for  civil  engineers  only,  the  vast  amount  of  practical  information  con- 
densed into  it  has  made  it  equally  a  favorite  with  contractors,  builders, 
and  machinists ;  besides  leading  to  its  adoption  as  a  text-book  in  many 
educational  institutions. 

In  proof  of  the  estimation  in  which  the  work  is  held  by  competent 
professional  authorities,  we  cite  the  following: 

Mr.  Alfred  P.  Boiler,  C.  E.,  in  his  "  Iron  Highway  Bridges,"  says : 
"  For  a  wonderfully  clear  and  elaborate  discussion  of  force,  strains,  etc., 
as  well  as  upon  the  subject  of  Trusses  arid  Strength  of  Materials,  free 
from  all  technicalities,  the  learner  is  referred  to  Mr.  Trautwine's  l  En- 
gineer's Pocket-Book/  a  work  that  should  be  the  corner-stone  of  every 
engineer's  library." 

Mr.  George  L.  Vose,  C.  E.,  Professor  of  Civil  Engineering  in  the 
Massachusetts  Institute  of  Technology,  at  Boston,  in  his  "  Civil  En- 
gineering," says :  "  Mr.  Trautwine's  '  Civil  Engineer's  Pocket-Book '  is, 
beyond  all  question,  the  best  practical  manual  for  the  engineer  that  has 
ever  appeared." 

Mr.  Thomas  M.  Cleemann,  C.  E.,  in  his  "  Railroad  Engineer's  Prac- 
tice," remarks  that  "he  considers  that  no  other  book  has  appeared 
which  supplies  so  well  a  constant  want  of  the  engineer,  at  all  stages  of 
his  career,  as  Trautwine's  *  Engineer's  Pocket-Book.' " 

Many  other  writers  refer  to  the  book  in  similar  terms,  beside  showing 
their  appreciation  of  it  by  extracting  freely  from  it ;  and  the  profes- 
sional periodicals  of  the  country  generally  have  expressed  their  ap- 
proval of  it.  Although  Mr.  Trautwine  has  for  twenty  years  abandoned 
the  active  pursuit  of  his  profession,  the  publishers  trust  that  he  will 
continue  to  feel  sufficient  interest  in  this  book  to  enable  them  to  add  to 

its  value  in  future  editions. 

iv 


PREFACE  TO  FIRST  EDITION. 


QHOULD  experts  in  engineering  complain  that  they  do  not  find 
^  anything  of  interest  in  this  volume,  the  writer  would  merely 
remind  them  that  it  was  not  his  intention  that  they  should.  The 
book  has  been  prepared  for  young  members  of  the  profession ;  and 
one  of  the  leading  objects  has  been  to  elucidate  in  plain  English,  a 
few  important  elementary  principles  which  the  savants  have  envel- 
oped in  such  a  haze  of  mystery  as  to  render  pursuit  hopeless  to  any 
but  a  confirmed  mathematician. 

Comparatively  few  engineers  are  good  mathematicians ;  and  in  the 
writer's  opinion,  it  is  fortunate  that  such  is  the  case ;  for  nature  rarely 
combines  high  mathematical  talent,  with  that  practical  tact,  and 
observation  of  outward  things,  so  essential  to  a  successful  engineer. 

There  have  been,  it  is  true,  brilliant  exceptions ;  but  they  are  very 
rare.  But  few  even  of  those  who  have  been  tolerable  mathematicians 
when  young,  can,  as  they  advance  in  years,  and  become  engaged  in 
business,  spare  the  time  necessary  for  retaining  such  accomplish- 
ments. 

Nearly  all  the  scientific  principles  which  constitute  the  founda- 
tion of  civil  engineering  are  susceptible  of  complete  and  satisfactory 
explanation  to  any  person  who  really  possesses  only  so  much  element- 
ary knowledge  of  arithmetic  and  natural  philosophy  as  is  supposed 
to  be  taught  to  boys  of  twelve  or  fourteen  in  our  public  schools.* 

*  Let  two  little  boys  weigh  each  other  on  a  platform  scale.  Then  when  they 
balance  each  other  on  their  board  see-saw,  let  them  see  (and  measure  for  themselves) 
that  the  lighter  one  is  farther  from  the  fence-rail  on  which  their  board  is  placed,  in 
the  same  proportion  as  the  heavier  boy  outweighs  the  lighter  one.  They  will  then 
have  learned  the  grand  principle  of  the  lever.  Then  let  them  measure  and  see  that 
the  light  one  see-saws  farther  than  the  heavy  one,  in  the  same  proportion;  and  they 
will  have  acquired  the  principle  of  virtual  velocities.  Explain  to  them  that  equatity 
of  moments  means  nothing  more  than  that  when  they  seat  tb.emse.lyes  $t  their  meMK 


VI  PREFACE. 

The  little  that  is  beyond  this,  might  safely  be  intrusted  to  the 
savants.  Let  them  work  out  the  results,  and  give  them  to  the  engi- 
neer in  intelligible  language.  We  could  afford  to  take  their  words  for 
it,  because  such  things  are  their  specialty ;  and  because  we  know  that 
they  are  the  best  qualified  to  investigate  them.  On  the  same  princi- 
ple we  intrust  our  lives  to  our  physician,  or  to  the  captain  of  the 
vessel  at  sea.  Medicine  and  seamanship  are  their  respective  special- 
ties. 

If  there  is  any  point  in  which  the  writer  may  hope  to  meet  the 
approbation  of  proficients,  it  is  in  the  accuracy  of  the  tables.  The 
pains  taken  in  this  respect  have  been  very  great.  Most  of  the  tables 
have  been  entirely  recalculated  expressly  for  this  book ;  and  one  of 
the  results  has  been  the  detection  of  a  great  many  errors  in  those  in 
common  use.  He  trusts  that  none  will  be  found  exceeding  one,  or 
sometimes  two,  in  the  last  figure  of  any  table  in  which  great  accuracy 
is  required.  There  are  many  errors  to  that  amount,  especially  where 

ured  distances  on  their  see-saw,  they  balance  each  other.  Let  them  see  that  the  weight 
of  the  heavy  boy,  when  multiplied  by  his  distance  in  feet  from  the  fence-rail 
amounts  to  just  as  much  as  the  weight  of  the  light  one  when  multiplied  by  his  dis- 
tance. Explain  to  them  that  each  of  the  amounts  is  in  foot-pounds.  Tell  them  that 
the  lightest  one,  because  he  see-saws  so  much  faster  than  the  other,  will  bump 
against  the  ground  just  as  hard  as  the  heavy  one ;  and  that  this  means  that  their 
momentum*  are  equal.  The  boys  may  then  go  in  to  dinner,  and  probably  puzzle  their 
big  lout  of  a  brother  who  has  just  passed  through  college  with  high  honors.  They 
will  not  forget  what  they  have  learned,  for  they  learned  it  as  play,  without  any  ear- 
pulling,  spanking,  or  keeping  in.  Let  their  bats  and  balls,  their  marbles,  their 
swings,  &c,  once  become  their  philosophical  apparatus,  and  children  may  be  taught 
(really  taught)  many  of  the  most  important  principles  of  engineering  before  they 
can  read  or  write. 

It  is  the  ignorance  of  these  principles,  so  easily  taught  even  to  children,  that  con- 
stitutes what  is  popularly  called  "  THE  PRACTICAL  ENGINEER  ;  "  which,  in  the  great 
majority  of  cases,  means  simply  an  ignoramus,  who  blunders  along  without  knowing 
any  other  reason  for  what  he  does,  than  that  he  has  seen  it  done  so  before.  And  it 
is  this  same  ignorance  that  causes  employers  to  prefer  this  practical  man  to  one  who 
ia  conversant  with  principles.  They,  themselves,  were  spanked,  kept  in,  &c,  when 
boys,  because  they  could  not  master  leverage,  equality  of  moments,  and  virtual  velo- 
cities, enveloped  in  x's,  p's,  Greek  letters,  square-roots,  cube-roots,  &c,  and  they 
naturally  set  down  any  man  as  a  fool  who  could.  They  turn  up  their  noses  at  science, 
not  dreaming  that  the  word  means  simply,  knowing  why.  And  it  must  be  confessed 
that  they  are  not  altogether  without  reason  ;  for  the  savants  appear  to  prepare  their 
books  with  the  express  object  of  preventing  purchasers,  (they  have  but  few  readers,) 
from  learning  why. 


PREFACE.  Vll 

the  recalculation  was  very  tedious,  and  where,  consequently,  interpo- 
lation was  resorted  to.  They  are  too  small  to  be  of  practical  import- 
ance. He  knows,  however,  the  almost  impossibility  of  avoiding  larger 
errors  entirely  ;  and  will  be  glad  to  be  informed  of  any  that  may  be 
detected,  except  the  final  ones  alluded  to,  that  they  may  be  corrected 
in  case  another  edition  should  be  called  for.  Tables  which  are  abso- 
lutely reliable,  possess  an  intrinsic  value  that  is  not  to  be  measured  by 
money  alone.  With  this  consideration  the  volume  has  been  made  a 
trifle  larger  than  would  otherwise  have  been  necessary,  in  order  to 
admit  the  stereotyped  sines  and  tangents  from  his  book  on  railroad 
curves.  These  have  been  so  thoroughly  compared  with  standards 
prepared  independently  of  each  other,  that  the  writer  believes  them 
to  be  absolutely  correct. 

In  order  to  reduce  the  volume  to  pocket-size,  smaller  type  has  been 
used  than  would  otherwise  have  been  desirable. 

Many  abbreviations  of  common  words  in  frequent  use  have  been 
introduced,  such  as  abut,  cen,  diag,  hor,  vert,  pres,  &c,  instead  of 
abutment,  center,  diagonal,  horizontal,  vertical,  pressure,  &c.  They 
can  in  no  case  lead  to  doubt;  while  they  appreciably  reduce  the 
thickness  of  the  volume. 

Where  prices  have  been  added,  they  are  placed  in  footnotes.  They 
are  intended  merely  to  give  an  approximate  or  comparative  idea  of 
value ;  for  constant  fluctuations  prevent  anything  farther. 

The  addresses  of  a  few  manufacturing  establishments  have  also 
been  inserted  in  notes,  in  the  belief  that  they  might  at  times  be  found 
convenient.  They  have  been  given  without  the  knowledge  of  the 
proprietors, 

The  writer  is  frequently  asked  to  name  good  elementary  books  on 
civil  engineering ;  but  regrets  to  say  that  there  are  very  few  such  in 
our  language.  "  Civil  Engineering,"  by  Prof.  Mahan  of  West  Point ; 
"  Roads  and  Railroads,"  by  the  late  Prof.  Gillespie ;  and  the  "  Manual 
for  Railroad  Engineers,"  by  George  L.  Vose,  Civ.  Eng,  and  Professor 
of  Civil  Engineering  in  Bowdoin  College,  Brunswick,  Maine,  are  the 
best.  The  last,  published  by  Lee  &  Shepard,  Boston,  1873,  is  the 
most  complete  work  of  its  class  with  which  the  writer  is  acquainted. 


Vlll  PREFACE. 

Many  of  Weale's  series  are  excellent.  Some  few  of  them  are 
behind  the  times ;  but  it  is  to  be  hoped  that  this  may  be  rectified  in 
future  editions.  Among  pocket-books,  Haswell,  Hamilton's  Useful 
Information,  Henck,  Molesworth,  Nystrom,  Weale,  &c,  abound  in 
valuable  matter. 

The  writer  does  not  include  Rankine,  Moseley,  and  "Weisbach, 
because,  although  their  books  are  the  productions  of  master-minds, 
and  exhibit  a  profundity  of  knowledge  beyond  the  reach  of  ordinary 
men,  yet  their  language  also  is  so  profound  that  very  few  engineers 
can  read  them.  The  writer  himself,  having  long  since  forgotten  the 
little  higher  mathematics  he  once  knew,  cannot.  To  him  they  are  but 
little  more  than  striking  instances  of  how  completely  the  most  simple 
facts  may  be  buried  out  of  sight  under  heaps  of  mathematical  rubbish. 

Where  the  word  "  ton "  is  used  in  this  volume,  it  always  means 
2240  Ibs,  because  that  is  its  meaning  in  U.  S.  law. 

JOHN  C.  TRAUTWINE. 

PHILADELPHIA,  Nov.  13, 1871. 


REMAEKS  ON  THIS   EDITION. 


Q  EVERAL  slight  alterations  and  additions  have  been  made.  Among 
O  the  most  important  are  those  on  the  Transit  Instrument ;  Turnouts ; 
Mr.  Eliot  C.  Clarke's  tables  of  strengths  of  Concrete  Beams,  and  of 
Cement  Mortar,  both  on  page  508 ;  the  Fritz  &  Say  re  Splice  Plate 
for  rails,  on  page  395 ;  the  rate  at  which  rain  water  reaches  a  sewer, 
on  page  566 ;  Mr.  C.  L.  Gates'  table  of  strength  of  Built  Iron  Pil- 
lars, on  page  233 ;  comparison  of  Gordon's  and  Hodgkinson's  strengths 
of  pillars,  on  page  242 ;  and  alterations  in  five  of  the  eight  Gordon 
rules  for  the  strength  of  Iron  Pillars,  pages  221  to  223. 

Hitherto  wrong  divisors  have  been  used  by  myself  and  many  others, 
in  some  of  the  Gordon  formulas.  Thus,  in  Rule  1,  400  should  be  600 ; 
in  Rule  2,  3000  should  be  4500  ;  in  Rule  3,  533.3  should  be  800 ;  in 
Rule  5,  200  should  be  300 ;  in  Rule  7,  266.7  should  be  400.  We  have, 
therefore,  altered  the  RULES  and  their  EXAMPLES  in  this  edition ; 
but  have  not  been  able  to  alter  any  of  the  TABLES  in  time,  except 
the  most  important  one,  on  page  232,  which  has  also  been  enlarged. 
The  reader  had  better,  therefore,  insert  a  caution  "  not  correct "  at 
Tables,  pages  224  to  229.  The  old  ones  on  pages  230,  231,  are  correct. 
The  SOLID  METAL  AREAS,  however,  at  the  foot  of  ALL  these  tables, 
are  correct,  and  therefore  the  breaking  load  for  any  column  con- 
tained in  them,  may  now  be  found  by  multiplying  its  area  by  the 
corresponding  number  in  the  Table,  page  232. 

We  may  add  that  all  the  five  old  rules,  now  altered,  erred  on  the 
safe  side;  the  loads  given  by  the  new  rules  being  greater  in  all  cases. 

IT  MUST  BE  BORNE  IN  MIND  that  the  divisors  in  the  so-called 
GORDON'S  FORMULAS  for  hollow  columns,  whether  cylindrical  or 
square,  are  based  upon  Rankine's  rules,  p.  235,  for  finding  the 
square  of  the  Radius  of  Gyration  for  THIN  columns.  WHAT  is  A 
THIN  COLUMN  ?  It  is  usual  to  apply  the  rules  to  columns  with 
thicknesses  up  to  J  of  the  outer  diam. ;  and  in  view  of  this  fact,  we 
would  respectfully  ask  the  attention  of  experts  to  the  question 
WHETHER  THIS  PRACTICE  is  SAFE.  We  have  our  doubts  on  the 
subject,  but  hope  they  are  unfounded,  inasmuch  as  we  believe  the 
rules  are  so  applied  even  by  experts  of  high  mathematical  attain- 
ments, as  well  as  by  engineers  generally. 


ERRATA 

IN    THE    PRECEDING    EDITION    (THE    17TH   THOUSAND),  AND    SOME 
OTHERS,   BUT    CORRECTED    IN    THIS. 

SEE  ALSO  REMARKS  ON  THIS  EDITION,  for  others. 

P.  219.— On  the  15th  and  16th  lines  of  Example  1,  for  5  read  2.67. 
P.  222.— Near  the  middle,  for  Rule  5,  read  Rule  1. 
P.  223. — Near  the  middle,  for  foregoing  wrought  read  foregoing  cast. 
P.  235. — In  the  small  Fig.  of  a  thin  hollow  rectangle,  change  c  to  a,  and 

a  to  c. 

P.  236.— Upper  half,  in  four  places  change  3200  to  4800;  also  in  the 
formula  for  channel  iron,  change 

total  area  area  fl  X  area  web 

to 


4  X  sq  of  total  area  4  X  sq  total  area 

also  all  the  loads  for  CAST  iron  in  the  table  are  too  small. 

P.  266. — 27  lines  from  bottom,  for  strain  =  c  h  read  strain  =  e  h. 

P.  315. — 3d  line,  for  three  read  two. 
8  lines  from  bottom,  omit  1. 

P.  333.— 22d  line,  for  15  read  5. 

P.  336.— First  three  letters  of  5th  line  above  Rem  1,  for  com  read  c m  o. 

P.  417. — 10  lines  from  foot,  for  prod  read  quot. 

P.  437.— Near  top,  for  2.072  and  22.812,  read  2.074  and  22.814. 

P.  495.— 21st  line,  for  8.143  -f  read  8.143  X.    Also,  on  the  4th  line  of 
table,  omit  the  *  after  .5,  and  for  the  same  .5,  read  .707. 

P.  590.— Art.  7.     For  Rem  3,  p.  463,  read  Rem  2,  p.  462. 

Art.  8.     In  rule  and  formula,  for  (H  defl)2  read  H  (defi2). 
This  will  make  length  of  chain  588.3  ft,  instead  of  591.1. 

P.  637.— The  tables  of  loads  for  CAST  iron  are  too  small. 


CONTENTS. 


For  a  full  reference  to  the  Contents  in  detail,  see  Index,  page  677. 

PAGE 

MENSURATION 13 

PLANE  TRIGONOMETRY 39 

SQUARES,  CUBES,  AND  KOOTS 48 

GEOMETRY 61 

ARITHMETIC 69 

WEIGHTS  AND  MEASURES 73 

LAND  SURVEYING 90 

SINES  AND  TANGENTS,  &c 102 

CONTOUR  LINES 147 

DIALLING 150 

PAPER 151 

THE  LEVEL 152 

THE  ENGINEER'S  TRANSIT 157 

THE  THEODOLITE , 162 

THE  Box  OR  POCKET  SEXTANT 163 

THE  COMPASS 164 

LEVELLING  BY  THE  BAROMETER 167 

PENDULUMS * 172 

SOUND 173 

STRENGTH  OF  MATERIALS 174 

STRENGTH  OF  IRON  AND  WOODEN  PILLARS,  WITH  FULL  TABLES.  221 

TRUSSES  FOR  HOOFS  AND  BRIDGES 243 

TRESTLES 307 

THERMOMETERS 309 

STONEWORK 310 

FOUNDATIONS 313 

COST  OF  DREDGING 329 

RETAINING- WALLS 331 

STONE  BRIDGES,  CULVERTS,  ARCHES 341 

BOARD  MEASURE .' 357 

TABLES  OF  WEIGHTS  OF  BARS,  BOLTS,  PIPES,  &c,  &c 362 

SPECIFIC  GRAVITY 383 

KAIL  JOINTS,  CHAIRS,  &c ..  , 390 

xi 


Xll  CONTENTS. 

PAGE 

TURNOUTS 397 

RAILROADS 409 

TABLES  OF  LEVEL  CUTTINGS 420 

TURNTABLES 429 

WATER  STATIONS 432 

COST  or  EARTHWORK 435 

CENTRE  OF  GRAVITY , 442 

MECHANICS.    FORCE  IN  RIGID  BODIES.  ...  . 443 

CENTRIFUGAL  FORCE 494 

MORTAR,  BRICKS,  CEMENT,  CONCRETE,  &c 496 

PLASTERING 509 

SLATING .... 510 

SHINGLES........ 512 

PAINTING .... 512 

GLASS,  AND  GLAZING — 514 

WATER ... 515 

BAIN 518 

SNOW , ,.,,...  519 

AIR.  —  ATMOSPHERE 519 

WIND 520 

EVAPORATION,  FILTRATION,  AND  LEAKAGE 521 

HYDROSTATICS 521 

HYDRAULICS 534 

DAMS 583 

SUSPENSION  BRIDGES 588 

FRICTION 597 

TRACTION 603 

ANIMAL  POWER 605 

CHORDS  TO  A  KADIUS  1,  FOR  PROTRACTING 608 

GLOSSARY  OF  TERMS 615 

APPENDIX.. 630 

SHEARING  OF  BEAMS 642 

OPEN  AND  CLOSED  BEAMS 644 

K  UTTER' s  FORMULA 650 

VELOCITIES  IN  SEWERS 652 

RIVETS  AND  RIVETING 653 

CENTERS  FOR  ARCHES 665 

For  a  full  reference  to  the  Contents  in  detail, 
see  Index,  page  677. 


MENSURATION.- 


PARALLELOGRAMS. 

Square.  Rectangle.         Rhombus.  Rhomboid. 


s  s  s  s 

A  PABAI.LKLOGRAM  is  any  figure  of  four  straight  sides,  the  opposite  ones  of  which  are  parallel.  There 
are  but  four,  as  in  the  above  figs.  lu  the  square  aud  rhombus  all  the  four  sides  are  equal  ;  in  the 
rectangle  and  rhomboid  only  the  opposite  ones  are  equal.  In  any  parallelogram  the  four  angina 
*miunt  to  four  right  angles,  or  360°;  aud  any  two  diagonally  opposite  angles  are  equal  to  each  other; 
hence,  having  one  angle  given,  the  other  three  can  readily  be  found.  In  a  square,  or  a  rhombus,  a 
diag  divides  each  of  two  angles  into  two  equal  parts;  but  in  the  two  other  parallelograms  it  does  not. 

To  find  the  area  of  any  parallelogram. 

Multiply  any  side,  as  S,  by  the  perp  height,  or  dist  p  to  the  opposite  side.   Or,  multiply  together 
two  sides  and  nat  sine  of  their  included  angle. 
The  diag  a  b  of  any  square  is  equal  to  one  side  mult  by  1.41421  ;  and  a  side  is  equal  to 


The  side  of  a  square  equal  in  area  to  a  given  circle,  is  equal  to  diam  X  .886227. 

The  side  of  the  greatest  square,  that  can  be  inscribed  in 
a  diven  circle,  is  equal  to  diam  X  .707107. 

The  side  of  a  square  mult  by  1.51967  gives  the  side  of  an  equi- 
lateral triangle  of  the  same  area.  All  parallelograms  as  A 
and  C,  which  have  equal  bases,  a  q,  aud  equal  perp  heights  n 
c,  have  also  equal  areas;  aud  the  area  of  each  is  twice  that  of  a  tri- 
angle having  the  same  base,  and  perp  height.  The  area  of  a 
square  inscribed  in  a  circle  is  equal  to  twice  the  square  of  the 
ra.l. 

In  every  parallelogram,  the  4  squares  drawn  on  its  sides  have  a  united  area  equal  to  that  of 
the  two  squares  drawn  ou  its  2  diags.  If  a  larger  square  he  drawn  on  the  diag  a  b  of  a  smaller 
square,  its  area  will  be  twice  that  of  said  smaller  square.  Either  diag  of  any  parallelogram 
divides  it  into  two  equal  triangles,  and  the  2  diags  div  it  into  4  triangles  of  equal  areas.  The  two 
diags  of  any  parallelogram  divide  each  other  into  two  equal  parts.  Any  line  drawn  through 
the  center  of  a  diag  divides  the  parallelogram  into  two  equal  parts. 

Remark  1.—  The  area  of  any  fig  whatever  as  B  that  is  enclosed  by  four  straight 
lines,  may  be  found  thus  :  Mult  together  tue  two  diags  a  m,  n  it  ;  and  Mie  uat  sine  of  the  least  angle 
a  o  b  ;  or  n  o  m,  formed  by  their  intersection.  Div  the  product  by  2.  This  is  useful  in  land  surveying, 
when  obstacles,  as  is  often  the  case,  make  it  difficult  to  measure  the  sides  of  the  fig  or  field  ;  while  it 
may  be  easy  to  measure  the  diags  ;  and  after  finding  their  point  of  intersection  o,  to  measure  the  re- 
quired angle.  But  if  the  fig  is  to  be  drawn,  the  parts  o  a,  ob,  on,  o  m  of  the  diags  must  also 
be  measd. 

Kem.  2.  —  The  sides  of  a  parallelogram,  triangle,  and  many  other  figs  may  be 
found,  when  only  the  area  and  angles  are  given,  thus  :  Assume  some  particular  one  of  its 
sides  to  be  ot  the  length  1  ;  aud  calculate  what  its  area  would  be  if  that  were  the  case.  Then  as  the 
sq  rt  of  the  area  thus  found  is  to  this  side  1,  so  is  the  sq  rt  of  the  actual  given  area,  to  the  corre- 
sponding actual  side  of  the  fig. 

TRIAXGLES. 


We  speak  here  of  plane  triangles  only  ;  or  those  having  straight  sides.  Since  the  area  of  any 
triangle  is  equal  to  halt  that  of  a  parallelogram  which  has  the  same  base  and  perp  height,  therefore, 

To  find  the  area  of  any  triangle,  having  its  base  and  perp 
height, 

Mult  its  base  S  by  its  height,  or  perp  dist  p  to  the  opposite  angle;  and  div  the  prod  by  2.  For 
an  equilateral  one,  when  its  perp  height  is  not  known;  Square  one  side;  and  mult  the 
square  by  the  decimal  .433013.  Any  side  may  be  assumed  as  the  base  of  a  triangle;  but  the  perp 
height  must  always  be  measured  from  the  side  so  assumed  ;  to  do  which,  the  side  must  sometimes  be 
prolonged,  as  in  Fig  E;  but  the  prolongation  is  not  to  be  considered  as  a  part  of  the  base. 

Th.e  perp  height  of  an  equilateral  triangle  is  equal  to  one  side  X  .866025.  Hence  one  of 
its  sides  is  equal  to  the  perp  height  div  by  .866025  or  to  perp  height  X  1.1547.  Or.  to  find  a  side. 
mult  the  s>a  rt  of  its  urea  by  1.51967.  The  side  of  an  equilateral  triangle,  mult  bv  .65H037  =  side  of  a 
square  of  tBP  same  area;  or  div  by  1.34677  it  gives  the  diam  of  a  circle  of  same  a'rea. 

To  find  area,  having  two  sides,  and  the  included  angle. 

Mult  together  t^e  tw<>  sides,  and  the  nat  sine  of  the  included  angle ;  div  by  2. 

Ex.    Sides  650 Yl  and  980  ft;  included  angle  69°  20'.    By  the  table  we  find  the  nat  sine,  .9356; 


13 


14 


MENSURATION. 


Having:  the  three  sides. 

4<ld  t,hem  together,;  div.  the.  sum  by  2;  from  the  half  sum,  subtract  each  side  separately;  mult 
the  btlf  sum  and  the  tore"  remainders  continuously  together  ;  take  the  sq  rt  of  the  prod. 

Ex.  —  ThVo  side3  -  2C,  30,  40  Tt.    Here  20  +  30  +  40  =  90  ;  and    ^    =  45.        And  45  —  20  =  25  ; 

45  —  30  =  15;  and  45  —  40  =  5.     And  45  X  25  X  15X5  =  84375;  and  the  sq  rt  of  84375  is  290.47  sq  ft, 
area  reqd. 

Having  one  side  and  the  two  adjacent  angles. 

Add  the  2  angles  together  ;  take  the  sum  from  180°  ;  the  rem  will  be  the  angle  opp  the  given  side. 
Find  the  nat  sine  of  this  angle  ;  also  find  the  nat  sines  of  the  other  angles,  and  mult  them  together. 
Then  as  the  nat  sine  of  the  single  angle,  is  to  the  prod  of  the  nat  sines  of  the  other  2  angles,  so  is 
the  square  of  the  given  side  to  double  the  reqd  area. 

Having  the  three  angles,  d,  b  and  o,  and  the  area  ;  to  find  any  side,  as  d  o. 

(Sine  of  d  X  sine  of  o)  :  sine  of  6  :  :  twice  the  area  :  square  of  do. 

Having:  the  three  angles,  and  the  pern 
height,  a  b. 

Find  the  nat  sines  of  the  three  angles  ;  mult  together  the  sines  of  the  angles 
d  and  o  ;  div  the  sine  of  the  angle  b  by  the  prod  ;  mult  the  quot  by  the  square 
of  the  perp  height  o  b  ;  div  by  2.  See  Remark  2,  after  Parallelograms. 

A  triangle  Is  equilateral  when  all  its  sides  are  equal,  as  A  ;  Isosceles 
c  when  only  two  sides  are  equal,  as  B  ;  scalene  when  all  the  sides  are  unequal, 
as  C,  D,  and  E  ;  acute-angled  when  all  its  angles  are  acute,  or  each  less  than 
90°,  as  A,  B,  and  (J  ;  right-angled  when  it  contains  a  right  angle,  as  D  ;  ob- 
tuse-angled when  it  contains  an  obtuse  angle,  or  one  greater  than  90°,  as  E. 
All  the  three  angles  of  any  triangle  are  equal  to  two  right 
angles,  or  180°;  therefore,  if  we  know  two  of  them,  we  can  find  the  third  by 
subtracting  their  sum  from  180°.  All  triangles  which  have  equal  bases, 
and  equal  perp  heights,  have  also  equal  areas  ;  thus  the  areas  of  a  to  c,  a  w  d  and 
awe,  are  equal  to  each  other.  The  area  of  any  triangle  is  equal  to  half 
that  of  any  parallelogram  which  has  an  equal  base,  and  an  equal  perp  height.  The 
areas  of  triangles  which  have  equal  bases,  but  diff  perp  heights,  are  to 
each  other  as,  or  in  proportion  to,  their  perp  heights;  thus  the  triangle  a  «;  n, 
n,  equal  to  but  one-half  that  (s  e)  of  the  three  other  triangles,  but  with  the 

89  and  68, 


with  a  perp  height       ,     m 

same  base  a  w,  has  also  but  half  the  area  of  either  of  those  others. 


Bee  pages  8 


TRAPEZOIDS. 


A  trapezoid  a  c  n  m,  is  any  figure  with  four  straight  sides,  only  two  of  which,  as  ae  and  n  m,  are 
parallel. 

To  find  the  area  of  any  trapezoid. 

Add  together  the  two  parallel  sides,  a  c  and  m  n ;  mult  the  sum  by  the  perp  dist  *  I  between 
them ;  div  the  prod  by  2.  See  the  following  rules  for  trapeziums,  which  are  all  equally  applicable 
to  trapezoids ;  also  see  Remarks  after  Parallelograms. 

TRAPEZIUMS. 


A  trapezium  o  b  c  o,  is  any  fig  with  four  straight  sides,  of  which  no  two  are  parallel. 

To  find  the  area  of  any  trapezium,  having  given  the  diag- 
{><>,  or  a  c,  between  either  pair  of  opposite  angles;  and  also 
the  two  perps,  n,  n.,  from  the  other  two  angles. 

Add  together  these  two  perps ;  mult  the  sum  by  the  diag;  div  the  prod  by  2. 

Having  the  fotir  sides;  and  either  pair  of  opposite  angles, 

as  n  b  c9  noc  ;  or  b  a,  o,  and  b  c  o. 

Consider  the  trapezium  as  divided  into  two  triangles,  in  each  of  which  are  given  two  sides  anfthe 
included  angle.  Find  the  area  of  each  of  these  triangles  as  directed  under  the  preceding  hea^»  Tri- 
angles," and  add  them  together. 

Having  the  four  angles,  and  either  pair  of  opposite  sides. 

Begin  with  one  of  the  sides,  and  the  two  angles  at  its  ends.  If  the  sum  of  these  two  n^ie,  exceeds 
180°,  subtract  each  of  them  from  180°.  and  make  use  of  the  rems  instead  of  the  angl^  themselves. 
Then  consider  this  side  and  its  two  adjacent  angles  (or  the  two  rems,  as  the  case  ;nav  ^  as  tbomj 
of  a  triangle;  and  find  its  area  as  directed  for  that  case  under  the  preceding  he^  ..  Triangle."  Do 


MENSURATION. 


.15 


the  same  with  the  other  giveu  side,  and  its  two  adjacent  angles,  (or  their  rems,  as  the  case  may  be.) 
Subtract  the  least  of  the  areas  thus  found,  from  the  greatest ;  the  rem  will  be  the  reqd  area. 

Having  three  sides ;  and  the  two  included  angles. 

Mult  together  the  middle  side,  and  one  of  the  adjacent  sides ;  mult  the  prod  by  the  nat  sine  of  their 
included  angle ;  call  the  result  a.  Do  the  same  with  the  middle  side  and  its  other  adjacent  side, 
and  the  nat  sine  of  the  other  included  angle ;  call  the  result  b.  Add  the  two  angles  together  ;  find 
the  diff  between  their  sum  and  180°,  whether  greater  or  less  ;  find  the  nat  sine  of  this  diflf;  mult 
together  the  two  given  sides  which  are  opposite  one  another ;  mult  the  prod  by  the  nat  sine  just  found ; 
call  the  result  c.  Add  together  the  results  a  and  b  ;  then,  if  the  sum  of  the  two  given  angles  is  lest 
than  180°,  subtract  c  from  the  sum  of  a  and  6 ;  halftbe  rem  will  be  the  area  of  the  trapezium.  But 
if  the  sum  of  the  two  given  angles  be  greater  than  180°,  add  together  the  three  results  a,  b,  and  c; 
half  their  sum  will  be  the  area. 

Having  the  two  diagonals,  and  either  angle  formed  by  their 
intersection. 

See  Remarks  after  Parallelograms,  p  13. 

In  railroad  measurements 

Of  excavation  and  embankment,  the  trapezium 
I  in  u  o  frequently  occurs ;  as  well  as  the  two  5-sided 
figures  I  m  n  o  t  and  Imno  s;  in  all  of  which  m  n 
represents  the  roadway  ;  rs,  r  c,  and  r  t  the  center- 
depths  or  heights ;  I  u  and  o  v  the  side-depths  or 
heights,  as  given  by  the  level ;  I  m  and  n  o  the  side- 
slopes. 

The  same  general  rule  for  area  applies  to  all  three 
of  these  figs ;  namely,  mult  the  extreme  hor  width 
u  v  by  half  the  center  depth  r  s,  r  c,  or  r  t,  as  the 
case  may  be.  Also  mult  one  fourth  of  the  width  of 
roadway  m  n,  by  the  sum  of  the  two  side-depths  I  u 
and  o  v.  Add  the  two  prods  together ;  the  sum  is  the 
reqd  area.  This  rule  applies  whether  the  two  side-  U  m  r  Tl  V 

elopes  m  I  and  n  o  have  the  same  angle  of  inclination  or  not.  In  railroad  work,  etc.,  the  mid- 
way  hor  width,  center  depth,  and  side  depths  (see  foot  of  p  33)  of  a  prismoid  are  respectively  =  The 
half  sums  of  the  corresponding  end  ones,  and  thus  can  be  found  without  actual  measurement. 

POLYGONS. 

0      b 


All  straight-sided  figs  of  more  than  4  sides,  are  called  polygons ;  if  all  the  sides  and  angles  are 
equal,  it  is  a  regular  polygon ;  if  not,  irregular.  Of  course  their  number  is  infinite ;  but  those 
in  most  common  use  are  the  four  shown  above. 


Number 
of 
Sides. 

Name  of  Polygon. 

Areas. 

Outer 
Radii. 

Sides. 

Angle  con- 
tained between 
two  sides. 

Angle  at 
center  of 
circle. 

3 
4 

Equilateral  triangle. 
Square. 

.4330 
1. 

.5774 
.7071 

1.0000 
1.0000 

60° 
90° 

120° 
90° 

5 

6 
7 
8 
9 
10 
11 
12 

Pentagon. 
Hexagon. 
Heptagon. 
Octagon. 
Nonagon. 
Decagon. 
Undecagon. 
Dodecagon. 

1.7205 
2.5981 
3.6339 
4.8284 
6.1818 
7.6942 
9.3656 
11.1962 

.8507 
1. 
1  .1524 
1.3066 
1.4619 
1.6180 
1.7747 
1.9319 

1.0000 
1.0000 
1.0000 
1.0000 
1.0000 
1.0000 
1.0000 
1.0000 

108° 
120° 
128°  34'.29 
135° 
140° 
144° 
147°  16'.36 
150° 

72° 
60° 
51°  25'.71 
45° 
40° 
36° 
32°  43'.64 
30° 

To  find  the  area  of  any  regular  polygon. 

Mult  together  one  of  its  sides,  a  b ;  the  perp  p  drawn  from  the  center  of  the  fig  to  the  center  of  itt 
side ;  and  the  number  of  its  sides.  Div  the  prod  by  2.  Or,  square  one  of  its  sides ;  and  mult  sai4 
square  by  the  number  in  the  foregoing  table,  in  the  column  of  areas. 

Having  a  side  of  a  regular  polygon,  to  find  the  rad  of  a 
circumscribing  circle. 

Mult  the  side  by  the  corresponding  number  in  the  column  of  radii. 

Having  the  rad  of  a  circumscribing  circle,  to  find  the  side 
of  the  inscribed  regular  polygon. 

Divide  the  given  rad  by  the  corresponding  rad  in  the  table. 

Side  of  an  octagon  =  o  o  X  .41421.    Of  a  hexagon  =  o  o  X  .57735. 


16 


MENSURATION. 


To  find  the  area  of  any  irregular  poly 
goii,  a  n  b  c  m. 

Div  it  into  triangles,  as  a  n  b,  a  m  c,  and  a  b  c ;  in  each  of 
which  find  the  perp  dist  o,  between  its  base  a  I,  a  c,  or  I  c;  and 
the  opposite  angle  ?i,  m,  or  a;  mult  each  base  by  its  perp  dist; 
add  all  the  prods  together ;  div  by  2. 

To  find  the  area  of  a  long1  irregular 
fig,  as  a  b  c  d. 

Between  its  two  ends  a  6,  and  c  d, 


space  off  equal  dists,  (the  shorter  they  are  the  more  accurate  will  be  the  result,)  through  which 
draw  the  intermediate  parallel  lines  1,  2,  3,  &c,  across  the  breadth  of  the  fig.  Measure  the  lengths 
of  these  intermediate  lines,  and  add  them  together;  to  the  sum  add  oue  half of  the  two  end  breadths 
a  b  and  c  d.  Mult  the  entire  sum  by  one  of  the  equal  spaces  between  the  parallel  lines.  The  prod 
will  be  the  area.  This  rule  answers  as  well  if  either  one  or  both  the  ends  terminate  in  points,  as  at  m 
and  n.  In  the  last  of  these  cases,  both  a  b  and  c  d  will  be  included  in  the  intermediate  lines ;  and 
half  the  two  end  breadths  will  be  0,  or  nothing. 

To  find  the  area  of  a  fig  whose  outline  is  extremely 
irregular. 

Draw  lines  around  it  which  shall  enclose  within 
them  (as  nearly  as  can  be  judged  by  eye)  as  much 
space  not  belonging  to  the  fig,  as  they  exclude  space 
belonging  to  it.  The  area  of  the  simplified  fig  thus 
formed,  being  in  this  manner  rendered  equal  to  that 
of  the  complicated  one, may  be  calculated  by  dividing 
it  into  triangles,  &c.  By  using  a  piece  of  fine  thread, 
the  proper  position  for  the  new  boundary  lines  may 
be  found,  before  drawing  them  in.  Small  irregular 
\  areas  may  be  found  from  a  drawing,  by  laying  upon 
*  it  a  piece  of  transparent  paper  carefully  ruled  into 
small  squares,  each  of  a  given  area,  say  10,  20.  or 
100  sq  ft  each;  and  by  first  counting  the  whole 
squares,  and  theu  adding  the  fractions  of  squares. 

CIRCLES. 

A  circle  is  the  area  included  within  a  curved  line  of  such  a  character  that  every  point  in  it  is 
equally  distant  from  a  certain  point  within  it,  called  its  center.  The  curved  line  itself  is  called  the 
circumference,  or  periphery  of  the  circle ;  or  very  commonly  it  is  called  the  circle. 

To  find  the  circumference. 

Mult  diam  by  3.1416,  which  gives  too  much  by  only  .148  of  an  inch  in  a  mile.  Or,  as  113  is  to  355 
so  is  diam  to  circumf;  too  great  1  inch  in  186  miles.  Or,  mult  diam  by  3^;  too  great  by  about  1 
part  in  2485.  Or,  mult  area  by  12.566,  and  take  sq  root  of  prod.  Or,  use  table  p  18  or  p  675.  The 
Greek  letter  IT,  also  p,  is  used  by  writers  to  denote  this  3.1416;  and  p2  =  9.86960. 

To  find  the  diam. 

Div  the  oircumf  by  3.1416  ;  or,  as  355  is  to  113,  so  is  circumf  to  diam ;  or,  mult  the  circumf.  by  7: 
and  div  the  prod  by  22,  which  gives  the  diam  too  small  by  only  about  one  part  in  2485;  or,  mult  the 
area  by  1.2732;  and  take  the  sq  rt  of  the  prod;  or  use  the  following  table  of  circles. 

The  diam  is  to  the  circumf  more  exactly  as  1  to  3. 14159265. 

To  find  the  diam  of  a  circle  equal  in  area  to  a  given  square. 

Mult  one  side  of  the  square  by  1.12838. 

To  find  the  rad  of  a  circle  to  circumscribe  a  given  square. 

Mult  one  side  by  .7071 ;  or  take  H  the  diag. 

To  find  the  side  of  a  square  equal  in  area  to  a  given  circle. 

Mult  the  diam  by  .88623. 

To  find  the  side  of  the  greatest  square  in  a  given  circle. 

Mult  diam  by  .7071.  The  area  of  the  greatest  square  that  can  be  inscribed  in  a  circle  is  equal  to 
twice  the  square  of  the  rad.  ThediamX  by  1.3468  gives  the  side  of  an  equilateral  triangle  of  equal  area. 

Having  the  chord  and  rise  of  an  arc,  to  find  radius. 

Square  half  the  chord.  Divide  by  rise.  Add  rise.  Divide  by  2.  This  rule  applies  also  to  area 
greater  than  the  semicircle ;  as  does  also,  Radius  =  square  o'f  chord  of  half  arc  -±-  twice  rise  of 
whole  arc. 

Having  the  chord,  and  rad,  to  find  the  rise. 

Square  the  rad  ;  also  square  half  the  chord ;  take  the  last  square  from  the  first ;  take  sq  rt  of  the 
rem ;  and  subtract  it  from  the  rad,  if  rad  is  greatest ;  but  if  not,  add  it  to  rad. 


MENSURATION. 


17 


Having:  the  racl,  and  rise,  to  find  the  chord. 

From  rad  subtract  rise  (or  from  rise  subtract  rad  if  rise  is  greatest),  square  the  rem;  also  square 
the  rad ;  from  this  last  square  take  the  first ;  take  sq  rt  of  rem  ;  and  mult  it  by  2. 

Having  the  rise  of  arc,  and  diam  of  circle,  to  find  the  chord. 

From  the  diam  take  the  rise  ;  mult  the  rem  by  the  rise ;  take  sq  rt  of  prod  and  mult  it  by  2. 

Above  rule  applies  whether  the  arc  is  greater  or  less  than  a  semicircle. 

Half  the  chord  of  an  arc,  if  div  by  the  rad  of  the  circle,  will  give  the  uat  sine  of  half  the  angle 
gubtended  by  the  whole  chord,  or  arc. 

Table  of  Minutes  and  Seconds  in  decimals  of  a  Degree. 


Min. 

Deg. 

Min. 

Deg. 

Min. 

Deg. 

Sec. 

Deg. 

Sec. 

Deg. 

Sec. 

Deg. 

1 

.016666 

21 

.350000 

41 

.683333 

1 

.0002781 

21 

.005833 

41 

.011389 

2 

.0:{3:333 

22 

.366666 

42 

.700000 

2 

.000556 

22 

.006111 

42 

.011667 

3 

.050000 

23 

.383333  I  43 

.716666 

3 

.000833 

23 

.006389 

43 

.011944 

4 

.066666 

24 

.400000  i  44 

.733333 

4 

.001111 

24 

.006667 

44 

.012222 

5 

.083333 

25 

.416666 

45 

.750000 

5 

.001389 

25 

.006944 

45 

.012500 

6 

.100000 

26 

.433333  ! 

46 

.766666; 

6 

.001667 

26 

.007222 

46 

.012778 

7 

.  11(5666 

27 

.450000 

47 

.783333; 

7 

.001944 

27 

.007500 

47 

.013056 

8 

.133333 

28 

•466666 

48 

.800000 

8 

.002222 

28 

.007778 

48 

.013333 

9 

.150000 

29 

.483333 

49 

.811)666 

9 

.002500 

29 

.008056 

49 

.013611 

10 

.166666 

30 

.500000 

50 

.833333 

10 

.002778 

30 

.008333 

50 

.013889 

11 

.183333 

31 

.516666 

51 

.850000 

11 

.003056 

31 

.008611 

51 

.014167 

12 

.200000 

32 

.533333 

52 

.866666 

12 

.003333 

32 

.008889 

52 

.014444 

13 

.216666 

33 

.550000 

53 

.883333 

13 

.003611 

33 

.009167 

53 

.014722 

14 

.233333 

34 

.566666 

54 

.900000 

14 

.003889 

34 

.009444 

54 

.015000 

15 

.250000 

35 

.583333 

55 

.916666 

1  15 

.00*167 

35 

.009722 

55 

.015278 

1ft 

.266666 

36 

.600000 

56 

.933333 

16 

.004444 

36 

.010000 

56 

.015556 

17 

.283333 

37 

.616666 

57 

.950000 

17 

.004722 

37 

.010278 

57 

.015833 

18 

.300000 

38 

.633333 

58 

.966666 

18 

.005000 

38 

.010556 

58 

.016111 

19 

.316666 

39 

.650000 

59 

.9S3333 

19 

.005278 

39 

.010833 

59 

.016389 

•20 

.333333 

40 

.666666 

60 

1.000000 

20 

.005556 

40 

.011111 

60 

.016667 

of  the  outer  one  ;  and 


To  find  the  area  of  a  circle. 

Square  the  diam;  mult  this  square  by  .7854;  or  more  accurately  by  .78539816;  or  square  the  cir- 
cumf  ;  mult  this  square  by  .07958  ;  or  more  accurately  by  .07957747  ;  or  mult  half  the  diam  by  half  the 
circumf  ;  or  refer  to  the  following  table  of  areas  of  circles.  Also  area  =  sq  of  rad  X  3.1416. 

The  area  of  a  circle  is  to  the  area  of  any  circumscribed  straight-sided  fig,  as  the  circumf  of  th<» 
circle  is  to  the  circumf  or  periphery  of  the  fig.  The  area  of  a  square  inscribed  in  a  circle,  is  equal  to 
twice  the  square  of  the  rad.  Of  a  circle  in  a  square,  -  square  X  .7854. 

It  is  convenient  to  remember,  in  rounding  off  a  square  corner  a  b  c,  by  a  quarter  of 
a  circle,  that  the  shaded  area  a  b  c  is  equal  to  about  1.  part  (correctly  .2146)  of  the 
•whole  square  abed. 

To  find  the  breadth  of  a  circular  ring-. 

Having  its  area,  and  the  diam  of  its  outer  circle.  Find  the  area  of  the  whole  circle. 
From  it  take  the  area  of  the  ring.     Mult  the  rem  by  1.2732.     Take  the  sq  rt  of  the 
prod.    This  sq  rt  will  be  the  diam  of  the  inner  circle.     Take  it  from  the  diam 
div  the  rem  by  2,  for  the  reqd  breadth. 

To  find  the  length  of  a  circular  arc. 

Find  the  chord  a  &  of  half  the  arc  ;  *  and  mult  it  by  8.  From  the  prod  take 
the  chord  a  c  of  the  whole  arc  ;  div  the  rem  by  3.  This  is  a  close  approximation 
tor  flat  arcs.  It  always  gives  them  a  little  too  short.  When  greater  accuracy  is 
required,  add  as  follows,  for  semicircles,  -yx^0  Part;  rise  •£* 

*i*;  rise  T3<y>  irh;  rise  *>  -5-7?;  rise  i>  Twn  rise  *• 

refer  to  the  two  tables  of  arcs  on  pages  21  and  23. 

REMARK.—  It  may  frequently  be  of  use  to  remember,  that  in  any 
arc  bos,  not  exceeding  29°,  or  in  other  words,  whose  chord  b  s  is  at 
least  sixteen  times  its  rise,  the  middle  ordinate  a  o,  will  be  one 
half  of  a  c,  quite  near  enough  for  many  purposes  ;  6  c  and  s  c  being 
tangents  to  the  arc.  t  And  vice  versa,  if  in  such  an  arc  we  make  o  c  equal 
ao,  then  will  c  be,  very  nearly,  the  point  at  which  tangents  from  the 
ends  of  the  arc  will  meet.  Also  the  middle  ordinate  n,  of  the  half  arc 
o  6.  or  o  *,  will  be  approximately  J4  of  a  o,  the  mid  ord  of  the  whole 
arc.  Indeed,  this  last  observation  will  apply  near  enough  for  many 
approximate  uses  even  if  the  arc  be  as  great  as  45°  ;  for  if  in  that  case  we  take  %of  o  a  for  the  ord  n, 
n  will  then  be  but  1  part  in  103  too  small  ;  and  therefore  the  principle  may  often  be  used  in  drawings, 
for  finding  points  in  a  curve  of  too  great  rad  to  be  drawn  by  the  dividers  ;  for  in  the  same  manner,  y± 
of  n  will  be  the  mid  ord  for  the  arc  n  b  or  no;  and  so  on  to  any  extent.  On  p  434  will  be  found 

a  table  by  which  the  rise  or  middle  ord  of  a  half  arc  can  be 

obtained  with  greater  accuracy  when  required  for  more  exact  drawings. 

*  To  find  the  chord  of  half  the  arc,  add  together  the  square  of  the  rise  ;  and  the  square  of  half  th« 
•pan  ;  and  take  the  sq  root  of  the  sum. 
t  At  29°  o  c  thus  found  will  be  but  about       Part  to°  short.  t  .0002777778. 


18 


MENSURATION. 


TABL.E  OF  CIRCLES. 

Circumferences  or  areas  intermediate  of  those  in  the  table,  may  be  found  by  simple  arithmetics 
proportion.  The  diameters,  &c,  are  in  inches ;  but  it  is  plain  that  if  the  diauis  are  takeu  as  feet^ 
yards,  &c,  the  other  parts  will  also  be  in  those  same  measures.  See  p  6?7.  No  errors. 


Diam. 
Ins. 

Circumf. 
Ins. 

Area. 
Sq.  Ins. 

Diam. 
Ins. 

Circumf. 
lus. 

Area. 
Sq.  Ins. 

Diam. 
Ins. 

Circumf. 

Area. 
Sq.Ins. 

Diam. 
Ins. 

Circumf. 
lus. 

Area.. 
Sqlns. 

1-64 

.049087 

.00019 

'>.    1A 

10.9956 

9.6211 

10^ 

31.8086 

80.516 

19  H' 

60.4757 

291.04 

1-32 

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9-16 

11.1919 

9.967b 

34 

32.2013 

82.516 

% 

60.8G84 

294.83 

3-64 

.147262 

.  00173 

% 

11.3883 

10.321 

32.5940 

84.541 

$6 

61.2611 

298.65 

1-16 

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11-16 

11.5846 

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V 

32.9867 

86.590 

% 

61.6538 

302.49 

3-32 

.291524 

.00890 

% 

11.7810 

11.045 

% 

33.3794 

88.664 

% 

(•2.04J5.J 

306.35 

.392699 

.01227 

13-16 

11.9773  1    11.416 

33.7721 

90.763 

% 

62.4392 

310.24 

5-32 

.490874 

.01917 

% 

12.1737 

11.793 

• 

34.1618 

92.886 

20. 

62.8319 

314.16 

3-16 

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.0*761 

15-16 

12.3700 

12.177 

11.* 

34.5575 

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i£ 

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99.402 

64.0100 

320.05 

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sj 

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13.1554 

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¥ 

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367/28 

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39.6626 

125.19 

N 

68.3296 

371.54 

H 

1.96350 

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70.2931 

393/20 

25-32 

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%'   5.10509 

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y* 

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% 

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476/26 

11-16    5.30144 

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20.4204 

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15-16     6.08684 

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500.  74 

•*          !    6.28319 

3.1416 

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51.0509 

207.39 

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1-16    6.47953 

3.3410 

i/ 

22.7765 

41.282 

51.4436 

210.60 

80.1106 

510.71 

^     6.67588 

3.546S 

2 

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42.718 

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51.8363 

213.82 

% 

80.5033 

515.72 

3-16     687223 

3.7583 

34 

23.5619 

44.179 

% 

217  08 

H 

80.8960 

520.77 

y±     7.06858 

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52l6'217 

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4.2000 

24.3473 

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IX 

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546.35 

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551.55 

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% 

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240.53 

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3/£ 

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84.8-230 

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y3     9.03208 

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i,.. 

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14  10.2102 

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B/ 

30.2378 

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\ 

58.9049 

276.12 

% 

87.5719 

610/27 

5-16  10.4065 

8.6179 

3x 

30.6305 

74662 

'y 

59.2976 

279.81 

28. 

87.9646 

615.75 

%'  10.6029 

8.9462 

•y 

31.0232 

76.589 

19. 

59.6903 

283.53 

1A 

88.3573 

621/26 

M6,  10.7992 

9.2806 

10. 

31.4159 

78.540 

X 

60.0830 

287.27 

34 

88.7500 

626.80 

1 

MENSURATION. 


19 


TABLE  OF  CIRCLES  — (Continued.) 


Diam. 

Circumf. 

Area. 

Diam. 

Circumf. 

Area. 

Diam. 

Circumf. 

Area. 

Diam. 

Circumf. 

Area. 

Ins. 

Ins. 

Sq.  lus. 

lus. 

IllS. 

Sq.  Ins. 

Ins. 

lus. 

Sq.  lus. 

lus. 

Ins. 

Sq.Ins. 

28% 

89.1427 

632.36 

38. 

119.3»1 

1134.1 

«K 

149.618 

1781.4 

57^ 

179.856 

2574.2 

« 

89.5354 

637.94 

X 

119.773 

1141.6 

% 

150.011 

1790.8 

% 

180.249 

2585.4 

% 

89.9281 

643.55 

y* 

120.166 

1149.1 

y» 

150.404 

1800.1 

% 

180.642 

2596.7 

34 

90.3208 

649.18 

% 

120.559 

1156.6 

48 

150.796 

1809.6 

y« 

181.034 

2608.0 

K 

90.7135 

654.84 

y* 

120.951 

1164.2 

H 

151.189 

1819.0 

H 

181.427 

2619.4 

29. 

91.1062 

660.52 

% 

121.344 

1171.7 

151.582 

1828.5 

% 

181.820 

2630.7 

H 

91.4989 

666.23 

% 

121.737 

1179.3 

151.975 

1837.9 

58. 

182.212 

2642.1 

i/ 

91.8916 

671.96 

% 

122.129 

1186.9 

i^ 

152.367 

1847.5 

H 

182.605 

2653.5 

% 

92.2843 

677.71 

39. 

122.522 

1194.6 

M 

152.760 

1857.0 

M 

182.998 

2664.9 

X 

92.6770 

683.49 

X 

122.915 

1202.3 

H 

153.153 

1866.5 

183.390 

2676.4 

% 

93.0697 

689.30 

H 

123.308 

1210.0 

y» 

153.545 

1876.1 

y% 

183.783 

2687.8 

H 

93.4624 

695.13 

% 

123.700 

1217.7 

49. 

153.938 

1885.7 

% 

184.176 

2699.3 

A 

93.8551 

700.98 

X 

124.093 

1225.4 

H 

154.331 

1895.4 

% 

184.569 

2710.9 

30. 

94.2478 

706.86 

% 

124.486 

1233.2 

H 

154.723 

1905.0 

K 

184.961 

2722.4 

U 

94.6405 

712.76 

H 

124.878 

.  1241.0 

% 

155.116 

1914.7 

59. 

185.354 

2734.0 

3 

95.0332 

718.69 

% 

125.271 

1248.8 

% 

155.509 

1924.4 

M 

185.747 

2745.6 

% 

95.4259 

724.64 

40. 

125.664 

1256.6 

155.902 

1934.2 

h 

186.139 

2757.2 

K 

95.8186 

730.62 

H 

126.056 

1264.5 

% 

156.294 

1943.9 

186.532 

2768.8 

96.2113 

736.62 

126.449 

1272.4 

% 

156.687 

1953.7 

y*. 

186.925 

2780.5 

% 

96.6040 

742.64 

% 

126.842 

1280.3 

50. 

157.080 

1963.5 

% 

187.317 

2792.2 

J* 

96.9967 

748.69 

J^ 

127.235 

1288.2 

H 

157.472 

1973.3 

% 

187.710 

2803.9 

31. 

97.3894 

754.77 

N 

127.627 

1296.2 

157.865 

1983.2 

% 

188.103 

2815.7 

M 

97.7S21 

760.87 

H 

128.020 

1304.2 

% 

158.258 

1993.1 

60. 

188.496 

2827.4 

y 

98.1748 

766.99 

% 

128.413 

1312.2 

y* 

158.650 

2003-0 

H 

188.888 

2839.2 

8 

98.5675 

773.14 

41. 

128.805 

1320.3 

H 

159.043 

2012.9 

X 

189.281 

2851.0 

M 

98.9602 

779.31 

H 

129198 

1328.3 

159.436 

2022.8 

H 

189.674 

2862.9 

% 

99.3529 

785.51 

y* 

129.591 

1336.4 

•  % 

159.829 

2032.8 

% 

190.066 

2874.8 

« 

99.7456 

791.73 

% 

129.983 

1344.5 

51. 

160.221 

2042.8 

190.459 

2**6.6 

% 

100.138 

797.98 

y* 

130.376 

1352.7 

M 

160.614 

2052.8 

% 

190.852 

2898.6 

82 

100.531 

804.25 

% 

130.769 

i:;no.8 

M 

161.007 

2062.9 

% 

191.244 

2910.5 

K 

100.924 

810.54 

% 

131.161 

1369.0 

H 

161.399 

2073.0 

61. 

191.637 

2922.5 

M 

101.316 

816.86 

% 

131.554 

1:57;.'.' 

U 

161.792 

2083.1 

H 

192.030 

2934.5 

% 

101.709 

823.21 

42. 

131.947 

1385.4 

% 

162.185 

2093.2 

M 

192.423 

2946.5 

8 

102.102 

829.58 

H 

132.310 

1393.7 

H 

162.577 

2103.3 

% 

192.815 

2958.5 

% 

102.494 

835.97 

% 

132.732 

1402.0 

% 

162.970 

2113.5 

% 

193.208 

2970.6 

N 

102.887 

842.39 

h 

133.125 

1410.3 

52. 

163.363 

2123.7 

% 

193.601 

2982.7 

% 

103.280 

848.83 

H 

133.518 

1418.6 

tf 

163.756 

2133.9 

% 

193.993 

2994.8 

83. 

103.673 

855.30 

133.910 

1427.0 

164.148 

2144.2 

H 

194.386 

3006.9 

H 

104.065 

861.79 

H 

134.303 

1435.4 

% 

164.541 

2154.5 

62. 

194.779 

3019.1 

N 

104.458 

868.31 

% 

134.696 

1443.8 

y% 

164.934 

2164.8 

K 

195.171 

3031.3 

% 

104.851 

874.85 

43. 

135.088 

1452.2 

% 

165.326 

2175.1 

y* 

195.564 

3043.5 

^ 

105.243 

881.41 

y» 

135.481 

1460.7 

H 

165.719 

2185.4 

% 

195.957 

3055.7 

N 

105.636 

888.00 

% 

135.874 

1469.1 

% 

166.112 

2195.8 

M 

196.350 

3068.0 

?4' 

106.029 

894.62 

% 

136.267 

1477.6 

53. 

166.504 

2206.2 

% 

196.742 

30H0.3 

Ji 

106.421 

901.26 

H 

136.659 

1486.2 

N 

166.897 

2216.6 

H 

197.135 

3092.6 

B4 

106.814 

907.92 

% 

137.052 

1494.7 

M 

167.290 

2227.  0 

% 

197.528 

3104.9 

K 

107.207 

914.61 

H 

137.445 

1503.3 

8 

167.683 

2237.5 

63 

197.920 

3117.2 

107.600 

921.32 

% 

137.837 

1511.9 

y<a 

168.075 

2248.0 

M 

198.31H 

3129.6 

% 

107.992 

928.06 

44 

i:i».230 

1520.5 

% 

168.468 

2258.5 

198.706 

3142.0 

i^ 

108.385 

934.82 

^ 

138.623 

1529.2 

H 

168.861 

2269.1 

% 

199.098 

3154.5 

^g 

108.778 

941.61 

139.015 

1537.9 

% 

169.253 

2279.6 

K 

199.491 

3166.9 

% 

09.170 

948.42 

% 

139.408 

1546.6 

54. 

169.646 

2290.2 

% 

199.884 

3179.4 

% 

09.563 

955.25 

y% 

139.801 

1555.3 

w 

170.039 

2300.8 

H 

200.277 

3191.9 

*5 

09.956 

962.11 

% 

140.194 

1564.0 

I/ 

170.431 

2311.5 

X 

200.^)9 

3204.4 

N 

0.348 

969.00 

% 

140.586 

1572.8 

% 

170.824 

2322.1 

64. 

2UI.UB-2 

3217.0 

N 

0.741 

975.91 

% 

140.979 

1581.6 

H 

171.217 

2332.8 

H 

20i^5 

3229.6 

% 

1.1,34 

982.84 

45. 

141.372 

1590.4 

171.609 

2343.5 

M 

201.^7 

3242.2 

« 

1.527 

989.80 

H 

141.764 

1599.3 

% 

172.002 

2354.3 

% 

202.240 

3254.3 

1.919 

996.78 

M 

142.157 

1608.2 

y» 

172.395 

2365.0 

X 

202.633 

3267.5 

^ 

2.312 

1003.8 

142.550 

1617.0 

55 

172.788 

2375.8 

H 

203.025 

3280.1 

Ji 

2.705 

1010.8 

LZ 

142.942 

1626.0 

w 

173.180 

2386.6 

% 

203.418 

3292.8 

*> 

3.097 

1017.9 

R£ 

143.335 

1634.9 

M 

173.573 

2397.5 

% 

203.81  1 

3305.6 

H 

3.490 

1025.0 

% 

143.728 

1643.9 

H 

173.966 

2408.3 

65. 

204.204 

3318.3 

J4 

13.883 

10321 

% 

144.121 

1652.9 

y* 

174.358 

2419.2 

M 

204.596 

3331.1 

N 

114.275 

1039.2 

46. 

144.513 

1661.9 

N 

174.751 

2430.1 

N 

204.989 

3343.9 

3^ 

114.668 

1046.3 

H 

144.906 

1670.9 

H 

175.144 

2441.1 

% 

205.382 

3856.7 

% 

115.061 

1053.5 

y*. 

145.299 

1680.0 

y» 

175.536 

2452.0 

y* 

205.774 

3369.6 

% 

115.454 

1060.7 

% 

U5.691 

1689.1 

56 

175.929 

2463.0 

% 

206.167 

3382.4 

% 

115.846 

1068.0 

N 

146.0K4 

1698.2 

K 

176.322 

2474.0 

M 

206.560 

3395.3 

17 

16.2:59 

1075.2 

N 

146.477 

1707.4 

N 

176.715 

2485.0 

% 

206.952 

3408.2 

^ 

6.632 

1082.5 

« 

146.869 

1716.5 

% 

177.107 

2496.1 

66. 

207.345 

3421.2 

H 

7.024 

1089.8 

% 

147.262 

1725.7 

% 

177.500 

2507.2 

fc 

207.738 

3484.2 

% 

7.417 

10W7.1 

47 

147.655 

734.9 

% 

177.893 

2518.3 

y\ 

208.131 

3447.2 

¥ 

7.810 

1104.5 

H 

148.048 

744.2 

% 

178.285 

2529.4 

% 

208.523 

3460.2 

8.202 

1111.8 

M 

148.440 

753.5 

% 

178.678 

2540.6 

y* 

208.916 

3473.2 

% 

118.596 

1119.2 

% 

148.833 

762.7 

57. 

179.071 

2551.8 

209.309 

3486.3 

% 

1.18.988 

1126.7 

X 

149.226 

772.1 

H 

179.463 

2563.0 

% 

209.701 

3^99.4 

20 


MENSURATION. 


TABLE  OF  CIRCLES  —  (Continued.) 


Diam. 

Circumf. 

Area. 

Diam. 

Circumf. 

Area. 

iam. 

Dircumf. 

Area. 

Diam. 

Circumf. 

Area. 



83^ 

262.716 

92. 

6647.6 

210.094 

3512.5 

75J4 

236.405 

4447.4 

5492.4 

289.027 

67. 

210.487 

3525.7 

236.798 

4462.2 

H 

263.108 

5508.8 

289.419 

66657 

210.879 

3538.8 

xti 

237.190 

4477.0 

263.501 

5525.3 

IX 

289.812 

6683.8 

34 

211.272 

3552.0 

K 

237.583 

4491.8 

84. 

263.894 

5541.8 

H 

290.205 

6701.9 

9i 

211.665 

3565.2 

% 

237.976 

4506.7 

H 

264.286 

5558.3 

y% 

290.597 

6720.1 

IX 

212.058 

3578.5 

% 

238.368 

4521.5 

34 

264.679 

5574.8 

Y 

290.990 

6738.1 

% 

212.450 

3591.7 

76. 

238.761 

4536.5 

% 

265.072 

5591.4 

% 

291.383 

6756.4 

x4 

212.843 

3605.0 

239.154 

4551.4 

y% 

265.465 

5607.9 

K 

291.775 

6774.7 

213.236 

3618.3 

IX 

239.546 

4566.4 

% 

265.857 

5624.5 

93. 

292.168 

6792.9 

68. 

213.628 

3631.7 

&/ 

239.939 

4581.3 

H 

266.250 

5641.2 

292.561 

6811.2 

214.021 

3645.0 

y* 

240.332 

4596.3 

266.643 

5657.8 

x4 

292.954 

6829.5 

IX 

214.414 

3658.4 

% 

240.725 

4611.4 

85. 

267.035 

5674.5 

% 

293.346 

6847.8 

su 

214.806 

3671.8 

H 

241.117 

4626.4 

267.428 

5691.2 

y% 

293.739 

6866.1 

1Z 

215.199 

3685.3 

241.510 

4641.5 

lx 

267.821 

5707.9 

6X 

294.132 

6884.5 

g 

215.592 

3698.7 

77. 

241.903 

4656.6 

zi 

268.213 

5724.7 

H 

294.524 

6902.9 

215.984 

3712.2 

X 

242.295 

4671.8 

y 

268.606 

5741.5 

% 

294.917 

6921.3 

xl 

216.377 

3725.7 

242.688 

4686.9 

% 

268.999 

5758.3 

94. 

295.310 

6939.8 

69. 

216.770 

3739.3 

y» 

243.081 

4702.1 

% 

269.392 

5775.1 

M 

295.702 

6958.2 

217.163 

3752.8 

y* 

243.473 

4717.3 

T/ 

269.784 

5791.9 

296.095 

6976.7 

34 

217.555 

3766.4 

243.866 

4732.5 

86. 

270.177 

5808.8 

% 

296.488 

6995.3 

217.948 

3780.0 

ax 

244.259 

4747.8 

270.570 

5825.7 

296.881 

7013.8 

TX 

218.341 

3793.7 

% 

244.652 

4763.1 

•y 

270.962 

5842.6 

% 

297.273 

7032.4 

K 

218.733 

3807.3 

78. 

245.044 

4778.4 

% 

271.355 

5859.6 

% 

297.666 

7051.0 

219.126 

3821.0 

r^ 

245.437 

4793.7 

c 

271.748 

5876.5 

% 

298.059 

7069.6 

TX 

219.519 

3834.7 

34 

245.830 

4809.0 

xl 

272.140 

5893.5 

95. 

298.451 

7088.2 

70. 

219.911 

3848.5 

246.222 

4824.4 

fix 

272.533 

5910.6 

H 

298.844 

7106.9 

220.304 

3862.2 

H     246.615 

4839.8 

Jf 

272.926 

5927.6 

M 

299.237 

7125.6 

x4 

220.697 

3876.0 

%\    247.008 

4855.2 

87. 

273.319 

5944.7 

g 

299.629 

7144.3 

sz 

221.090 

3889.8 

%     247.400 

4870.7 

273.711 

5961.8 

300.022 

7163.0 

rx 

221.482 

3903.6 

y» 

247.793 

4886.2 

ix 

274.104 

5978.9 

xi 

300.415 

7181.8 

&x 

221.875 

3917.5 

79. 

248.186 

4901.7 

£ 

274.497 

5996.0 

a/ 

300.807 

7200.6 

ax 

222.268 

3931.4 

^     248.579 

4917.2 

i< 

274.889 

6013.2 

Tj 

301.200 

7219.4 

tx 

222.660 

3945.3 

%     248.971 

4932.7 

*y 

275.282 

6030.4 

96. 

301.593 

7238.2 

71. 

2-23.053 

3959.2 

%     249.364 

4948.3 

X, 

275.675 

6047.6 

i/ 

301.986 

7257.1 

223.446 

3973.1 

y*     249.757 

4963.9 

£ 

276.067 

6064.9 

•^ 

302.378 

7-276  0 

IX 

223.838 

3987.1 

%     250.149 

4979.5 

88. 

276.460 

6082.1 

xl 

302.771 

7294.9 

SU 

224.231 

4001.1 

%\    250.542 

4995.2 

276.853 

6099.4 

•If 

303.164 

7313.8 

IX 

224.624 

4015.2 

%     250.935 

5010.9 

lx 

277.246 

6116.7 

xi 

303.556 

7332.8 

M 

225.017 

4029.2 

80. 

251.327 

5026.5 

y 

277.638 

6134.1 

S 

303.949 

7351.8 

| 

225.409 

4043.3 

251.720 

5042.3 

278.031 

6151.4 

304.342 

7370.8 

225.802 

4057.4 

Hi    252.113 

5058.0 

$ 

278.424 

6168.8 

97.^ 

304.734 

7389.8 

72. 

226.195 

4071.5 

%j    252.506 

5073.8 

278.816 

6186.2 

if 

305.127 

7408.9 

226.587 

085.7 

yz\    252.898 

5089.6 

y 

279.209 

6203.7 

Tj 

305.520 

7428.0 

i/ 

226.980 

099.8 

%\    253.291 

5105.4 

89. 

'279.602 

6221.1 

3x 

305.913 

7447.1 

a/ 

227.373 

114.0 

%     253.684 

5121.2 

279.994 

6238.6 

y 

306.305 

7466.2 

i^ 

227.765 

128.2 

%     254.076 

5137.1 

•y 

280.387 

6256.1 

xi 

306.698 

7485.3 

M 

228.158 

142.5 

81. 

254.469 

5153.0 

% 

280.780 

6273.7 

1 

307.091 

7504.5 

i 

228.551 

156.8 

254.862 

5168.9 

IX 

281.173 

6291.2 

307.483 

7523.7 

228944 

4171.1 

Kl    255.254 

5184.9 

H 

281.565 

6308.8 

98.    * 

307.876 

7543.0 

73. 

229.336 

4185.4 

%     255.647 

5200.8 

* 

281.958 

6326.4 

i/ 

308.269 

7562.2 

229.729 

4199.7 

3^i    256.040 

5216.8 

% 

282.351 

6344.1 

Jx 

308.661 

7581.5 

| 

230.122 

4214  1 

S 

256.433 

5232.8 

90. 

282.743 

6361  .7 

309.054 

7600.8 

230.514 

4228.5 

1 

256.825 

5248.9 

283.136 

6379.4 

Xl 

309.447 

7620.1 

!,< 

230.907 

4242.9 

257.218 

5264.9 

34 

283.529 

6397.1 

309.840 

7639.5 

xi 

2#lv300 

4257.4 

82. 

257.611 

5281.0 

% 

283.921 

6414.9 

a, 

310.232 

7658.9 

a. 

2jf$  .<J9'2 

4271.8 

258.003 

5297.1 

284.314 

6432.6 

Tx 

310.625 

7678.3 

Tx 

232^085 

4286.3 

i/ 

258.396 

5313.3 

% 

284.707 

6450.4 

99. 

311.018 

7697.7 

74. 

232.478 

4300.8 

Xl    258.789 

5329.4 

JX 

285.100 

6468.2 

H 

311.410 

7717.1 

232.871 

4315.4 

H     259.181 

5345.6 

TX 

285.492 

6486.0 

34 

311.803 

7736.6 

lx 

233  263 

4329.9 

% 

259.574 

5361.8 

91. 

•285.885 

6503.9 

N 

312.196 

7756.1 

xl 

233.656 

4344.5 

259.967 

5378.1 

286.278  i  6521.  8 

3^ 

312.588 

7775.6 

£ 

234.049 

4359.2 

yt 

260.359 

5394.3 

•y 

286.670 

6539.7 

7i 

312.981 

7795.2 

xl 

234.441 

4373.8 

83. 

260.752 

5410.6 

287.063 

6557.6 

313.374 

7814.8 

JL 

234.834 

4388.5 

k 

261.145 

5426.9 

y* 

287.456 

6575.5 

y 

313.767 

7834.4 

a 

235.227 

4403.1 

34 

261.538 

5443.3 

N 

287.848 

6593.5 

100. 

314.159 

7854.0 

75. 

235.619 

4417.9 

% 

261.930 

5459.6 

288.241 

6611.5 

236.012 

4432.6 

262.323 

5476.0 

? 

288.634 

6629.6 

-, 


To  find  orcl  mates,  tl  s,  ao,  «&€».,  of  a  circular  arc  nsm. 

First  find  the  one  d  s  at  the  center,  thus.    Square  the  rad.    Square  d  m, 
or  *a^  the  chord  n  m.     Subtract  this  last  square  from  the  first  one. 
/-"•  Take  the  sq  rt  of  the  rem.     Subtract  this  sq  rt  from  the  rad  for  d  s. 

For  any  other  ord  a  o,  subtract  d  s  from  rad.  Call  the  rem  t.  Square 
rad.  Square  d  a.  Take  this  last  square  from  the  first  one.  Take  sq  rt 
of  the  rem.  From  this  sq  rt  take  t.  The  rem  will  be  a  o.  For  R  R 
ords,  see  pp  416,  633. 


MENSURATION. 


21 


of  circnlar  arcs  not  exceeding:  a  semicircle. 

Knowing  its  chord  and  height,  divide  the  height  by  the  chord.  Find  in  the  column  of  heights  the 
aumber  equal  to  this  quotient.  Take  out  the  corresponding  number  from  the  column  of  lengthg. 
Multiply  this  last  number  by  the  length  of  the  given  chord.  See  p  434  for  another  table. 

TABLE    OF    CIRCULAR    ARCS. 

No  errors. 


H'shti. 

Lengths. 

I'ghts. 

Lengths. 

H'ghts. 

Lengths. 

H'ghts. 

Lengths. 

H'ghts. 

Lengths. 

.001 

1.00002 

.076 

1.01533 

.151 

1.05973 

.226 

1.13108 

.301 

1.22636 

.002 

1.00002 

.077 

1.01573 

.152 

1.06051 

.227 

.13219 

.302 

1.22778 

.004 

1.00303 

.078 

1.01614 

.153 

1.06130 

.223 

.13331 

.303 

1.22920 

.004 

1.03004 

.0^9 

1.01656 

.154 

1.06209 

.229 

.13444 

.304 

1.23063 

.005 

1.0001)7 

.0,30 

1.01698 

.155 

1.06288 

.230 

.13557 

.305 

1.23206 

.006 

1.00010 

.03! 

1.01741 

.156 

1.06368 

.231 

.13671 

.306 

1.23349 

.007 

1.00013 

.032 

1.01784 

.157 

1.06449 

.23'2 

.13785 

.307 

1.23492 

.003 

1.03017 

.033 

1.01828 

.158 

1.06530 

.233 

.13900 

.308 

1.23636 

.009 

1.00022 

.034 

1.01872 

.159 

1.06611 

.234 

.14015 

.309 

1.23781 

.010 

1.00027 

.085 

1.01916 

.160 

1.06693 

.235 

.14131 

.310 

1.23926 

.011 

1.00032 

.0-56 

1.01981 

..161 

1.06775 

.236 

.14247 

.311 

1.24070 

.012 

1.00033 

.037 

1.02006 

.162 

1.06358 

.237 

.14363 

.312 

1.24216 

.013 

1.00045 

.033 

1.020*2 

.163 

1.06941 

.238 

.14480 

.313 

1.24361 

.014 

1.00053 

.039 

1.02098 

.164 

1.07025 

.239 

.14597 

.314 

1.24507 

.015 

1.00031 

.030 

1.02116 

.165 

1.07109 

.240 

.14714 

.315 

1.24654 

.016 

1.00U69 

.031 

1.02192 

.166 

1.07194 

.241 

.14832 

.316 

1.24801 

.017 

1.01078 

.092 

1.022  tO 

.167 

1.07279 

.242 

.14951 

.317 

1.24848 

.018 

1.000*7 

.093 

1.02239 

.168 

1.07365 

.243 

.15070 

•318 

1.25095 

.019 

1.00097 

.094 

1.02339 

.169 

1.07451 

.244 

.15189 

.319 

1.25243 

.020 

1.00107 

.095 

1.02339 

.170 

1.07537 

.245 

.15308 

.320 

1.25391 

.021 

1.00117 

.096 

1.02440 

.171 

1.07624 

.246 

.15428. 

.321 

1.25540 

.022 

1.00128 

.097 

1.02191 

.172 

1.07711 

.24^ 

.15549 

.322 

1.25689 

.023 

1.00140 

.098     | 

1.02542 

.173 

1.07799 

.248 

.15670 

.323 

1.25838 

.024 

1.0  J  1  53 

.099 

1.02593 

.174 

1.07888 

.249 

.15791 

•324 

1.25988 

.02', 

1.00167 

.100 

1.02646 

.175 

1.07977 

.250 

.15912 

•325 

1.26138 

.026 

1.00182 

.101 

1.02398 

.176 

1.08066 

.251 

.16034 

•326 

1.26288 

.027 

1.0019R 

.102 

1.02752 

.177 

1.08156 

.252 

.16156 

•327 

1.26<37 

.023 

1.00210 

.103 

1.02306 

.178 

1.08246 

•253 

.16279 

•328 

1.26588 

.029 

1.00225 

.104 

1.02360 

.179 

1.08337 

.254 

.16402 

.329 

1.26740 

.030 

1.00210 

.105 

1.02914 

.180 

1.08428 

.255 

.16526 

.330 

1.26892 

.031 

1.00258 

.106 

1.02970 

.181 

1.08519 

.256 

.16650 

.331 

1.27044 

.032 

1.00272 

.107 

1.03025 

.182 

1.08611 

.257 

.16774 

.332 

1.27196 

.033 

1.00239 

.103 

1.03032 

.183 

1.08704 

.258 

.16899 

.333 

1.27349 

.034 

1.00307 

.109 

1.03139 

.184 

1.08797 

.259 

.17024 

.334 

1.27502 

.035 

1.00327 

.110 

1.03196 

.185 

1.08890 

.260 

.17150 

.335 

1.27656 

.036 

1.00345 

.111 

1.03254 

.186 

1.08984 

.261 

.17276 

.336 

1.27810 

.037 

1,00364 

.112 

1.03312 

.187 

1.09079 

.262 

.17403 

.337 

1.27964 

.033 

1.00334 

.113 

1.03571 

.188 

1.09174 

.2H3 

.17530 

.338 

1.28118 

.039 

1.00405 

.114 

1.03430 

.189 

1.09269 

.264 

.17657 

.£39 

1.28273 

.040 

1.00425 

.115 

1.03490 

.190 

1.09365 

.265 

.17784 

.340 

1.28428 

.Oil 

1.00447 

.116 

1.03551 

.191 

1.09161 

.266 

.17912 

.341 

1.28583 

.042 

1.00469 

.117 

l.OSfill 

.192 

1.09557 

.267 

.18040 

.342 

1.28739 

.043 

1.00492 

.118 

1.03672 

.193 

1.09654 

.263 

.18169 

.343 

1.28895 

.044 

1.00515 

.119 

1.03734 

.194 

1.09752 

.269 

.18299 

.344 

1.2E052 

.045 

1.00539 

.120 

1.03797 

.195 

1.09850 

.270 

.18429 

.345 

1.2£209 

.046 

1.00563 

.121 

1.03860 

.196 

1.09949 

.271 

.18559 

.346 

1.2S366 

.047 

1.00587 

.122 

1.03923 

.197 

1.10048 

.272 

.18689 

.347 

1.29523 

.043 

1.00S12 

.123 

1.03937 

.198 

1.10147 

.273 

.18820 

.348 

1.29681 

.049 

1.00338 

.121 

1.04051 

.199 

1.10247 

.274 

1.18951 

.349 

1.2S839 

.050 

1.00S65 

.125 

1.04116 

.200 

1.  10347 

.275 

1.19082 

.350 

1.29S97 

.051 

1.00692 

.126 

1.04181 

.201 

1.10447 

.276 

1.19214 

.351 

1.30156 

.Oi2 

1.00720 

.127 

1.04247 

.202 

1.10548 

.277 

1.19346 

.352 

1.30315 

.053 

1.00748 

.128 

1.04313 

.203 

1.  10650 

.278 

1.19479 

.353 

1.30474 

.054 

1.00776 

.129 

1.04380 

.204 

1.10752 

.279 

1.19612 

.354 

1.30634 

.055 

1.00805 

.130 

1.04447 

.205 

1.10855 

.280 

1.19746 

.355 

1.30794 

.056 

1.00334 

.131 

1.04515 

.206 

1.10958 

.281 

1.19880 

.356 

1.30954 

.057 

1.00364 

.132 

1.04584 

.207 

1.11062 

.262 

1.20014 

.357 

1.31115 

.058 

1.00395 

.133 

1.04652 

.208 

1.11165 

.283 

1.20149 

.358 

1.31276 

.059 

1.00926 

.134 

1.04722 

.209 

1.11269 

.284 

1.20284 

.359 

1.31437 

.060 

1.00957 

.135 

1.04792 

.210 

1.11374 

.285 

1.20419 

.360 

1.31599 

.061 

1.00989 

.136 

1.04862 

.211 

1.11479 

.286 

1.20555 

.361 

1.31761 

.062 

1.01021 

.137 

1.01932 

.212 

1.11584 

.237 

1.20691 

.362 

1.31923 

.063 

1.01054 

.138 

1.05003 

.213 

1.11690 

.288 

1.20827 

.363 

1.32086 

.064 

1.01088 

.139 

1.05075 

.214 

1.11796 

.289 

1.20964 

.364 

1.32249 

.065 

1.01123 

.140 

1.05147 

.215 

1.11904 

.290 

1.21102 

.365 

1.32413 

.066 

1.01158 

.141 

1.05220 

.216 

1.12011 

.291 

1.21239 

.366 

1.32577 

.067 

1.01193 

.142 

1.05293 

.'217 

1.12118 

.202 

1.21377 

.367 

1-3J741 

.063 

1.01228 

.143 

1.05337 

.218 

1.12225 

.293 

1.21515 

.368 

1.22C05 

.069 

1.01264 

.144 

1.05441 

.219 

1.12334 

.294 

1.21654 

.369 

1.33069 

.070 

1.01302 

.145 

1.05516 

.220 

1.12444 

.295 

1.21794 

.370 

1.3G234 

.071 

1.01338 

.146 

1.05591 

.221 

1.12554 

.298 

1.21933 

.371 

1.  33399" 

.072 

1.01376 

.147 

1.05667 

.222 

1.12664 

.297 

1.22073 

.372 

1.33564 

.073 

1.01414 

.148 

1.05743 

.223 

1.12774 

.298 

1.22213 

.373 

1.33730 

.074 

1.01453 

.149 

1.05819 

.224 

1.1-2885 

.299 

1.22354 

.374 

1.8389S 

.075 

1.01493 

.150 

1.05896 

.225 

1.12997 

.300 

1.22495 

.375 

1.34063 

22 


MENSURATION. 


TABLE    OF   CIRCULAR  ARCS  — (CONTINUED.) 


H'ghts. 

Lengths. 

H'ghts. 

Lengths. 

H'ghts. 

Lengths. 

H'ghts. 

Lengths. 

H'ghts. 

Lengths. 

.376 

1.31229 

.401 

1.38496 

.426 

.42S45 

.451 

1.475G5 

.476 

1.52346 

.377 

1.34395 

.402 

1.38671 

.427 

.43127 

.452 

1.47753 

.477 

1.52541 

.378 

1.34363 

.403 

1.38846 

.428 

.43309 

.453 

1.47142 

.478 

1.52736 

.379 

1.34731 

.404 

1.39021 

.429 

.43491 

.454 

1.46131 

.479 

1.521-31 

.380 

1.31399 

.405 

1.39H.6 

.430 

.43673 

.455 

1.483'JO 

.480 

1.53126 

.381 

1.35068 

.406 

1.3S372 

.431 

1.43S5G 

.456 

1.48509 

.481 

1.53322 

.382 

1.35237 

.407 

1.39548 

.432 

1.44039 

.457 

.48619 

.482 

1.53518 

.383 

1.3J406 

.408 

1.39724 

.433 

1.44222 

.458 

.48889 

.483 

1.53714 

.384 

1.35575 

.409 

1.39900 

.434 

1.44405 

.459 

.4E079 

.484 

1.53910 

.385 

1.35744 

.410 

.40077 

.435 

1.44589 

.460 

.49269 

.485 

1.54106 

.386 

1.35914 

.411 

.40254 

.436 

1.44773 

.461 

.49480 

.486 

1.54802 

.387 

1.36084 

.412 

.40432 

.437 

1.44957 

.462 

1.4S651 

.487 

1.54419 

.388 

1.38254 

.413 

.40610 

.438 

1.45142 

.463 

1.4C842 

.488 

1.546£6 

.389 

1.36425 

.414 

.40788 

.439 

1.45327 

.464 

1.50033 

.489 

1.548S3 

.390 

1.36596 

.415 

.40966 

.440 

1.45512 

.465 

.50224 

.490 

1.55091 

.391 

1.36767 

.416 

1.41145 

.441 

1.45697 

.466 

1.50416 

.491 

1.55289 

.392 

1.36939 

.417 

1.41324 

.442 

1.458S3 

.467 

.50608 

.492 

1.55487 

.393 

1.37111 

.418 

1.41503 

.443 

1.46069 

.468 

.50800 

.493 

1.55685 

.394 

1.37283 

.419 

1.416S2 

.444 

1.46255 

.469 

.50952 

.494 

1.  55884 

.395 

1.37455 

.420 

1.41861 

.445 

1.46441 

.470 

.51185 

.495 

1.56083 

.396 

1.37628 

.421 

1.42041 

.446 

1.46628 

.471 

.51378 

.4£6 

1.56282 

.397 

1.37801 

.422 

1.  42221 

.447 

1.46m  5 

.472 

.51571 

.497 

1.56481 

.398 

1.37974 

.423 

1.42402 

.448 

1.47002 

.473 

.51764 

.498 

1.56681 

.399 

1.38148 

.424 

1.425S3 

.449 

1.47196 

.474 

1.51958 

.499 

1.56881 

.400 

1.38322 

.425 

1.42764 

.150 

1.47377 

.475 

1.52152 

.500 

1.57080 

If  the  arc  is  greater  than  a  semicircle,  then  by  the  Rem  at  top  of 

page  -23  find  the  diam  of  the  circle.  Then  find  its  circumf.  From  diam  take  ht  of  arc.  The  rem 
will  be  ht  of  the  smaller  arc  of  the  circle.  By  rule  at  top  of  p  21  find  the  length  of  this  smaller  arc. 
Subtract  it  from  circumf. 

The  length  of  1  degree  of  a  circular  arc  is  equal  to  .017453  292  520  X  its  radius. 
"  «      "         "  1  minute        "         "         "         "         "  .000290  8KH  209  X    "         " 
"        "   1  second        "        "        "        "        "  .000004  848  137  X    "        " 

An  arc  of  1°  of  the  earth's  great  circle  is  but  4.6356  feet  longer  than  its 

chord.  Its  lencth  is  69. 16  land  or  statute  miles.     Earth's  equatorial  rad  — 3962.5705  miles.  Polar  3949.67. 
An  arc  of  1°,  rad  1  mile,  is  92.1534  feet:  a  minute  is  1.5359  feet;  a  second  is  .0256  of  a  foot; 
or  very  nearly  5-sixteeuths  of  an  inch.     Arc  of  1°,  rad  1OO  ft  =  1.74533  feet. 

To  find  the  area  of  a  circular  sector  a  d  b  c. 

RULE  1.  Mult  the  length  of  the  arc  a  d  b,  by  the  rad  c  a  or  c  6 ;  and 
divide  prod  by  2. 

REM.  Knowing  the  chord  a  ft,  and  the  height  nd;  or  the  rad.  and 
the  number  of  degs  in  the  arc.  the  length  of  the  arc  itself  can  be  found 
from  either  the  preceding  or  the  following  table. 

RULE  2.  As  360°  is  to  the  number  of  degs  in  the  angle  a  c  b,  so  is 
the  area  of  the  entire  circle  to  the  area  of  the  sector. 


To  find  the  area  of  a  circular  ring:. 

Find  the  areas  of  the  t 
mult  the  sum  of  the  two  d 
Or  mult  together  the  thick 


circles,  and  take  the  least  from  the  greatest.  Or 
b,  cd,  by  their  diff ;  mult  the  prod  by  .7854. 
;  the  mean  diam  II;  and  3.1416. 


To  find  the  rad  of  a  circle  which  shall  have  the 
same  area  of  section  as  a  given  circular  ring 

c  s  dab. 

Draw  any  rad  nr  of  the  outer  circle  ;  and  from  where  said  rad  cuts  the  inner  circle  at  «,  draw  t  s  at 
right  angles  to  it.     Then  will  t  s  be  the  reqd  rad. 

To  find  the  area  of  a  circular  segment,  abed,  not  exceeding  180°. 

By  the  table  on  pages  24  and  25  having  given  both  the  height  d  6; 
and  the  diam  of  the  circle.  Div  the  height  d  b  by  the  diam  of  the  circle  ; 
find  in  the  first  column,  the  number  nearest  to  the  quot;  take  out  from 
the  col  of  areas,  the  corresponding  area;  mult  this  area  by  the  square 
of  the  diam.  If  it  exceeds  ISO  \  See  p  630. 


Then  Plam='qa™gi^feh°rt 


Rem.  Knowing  the  chord  a  c,  and  height  d  b. 
the  rise  is  greater  or  less  than  a  semicircle. 

Having  the  area  of  a  sea-men  t  reqd  to  be  cut  off  from  a  given 
circle,  to  fiaad  its  chord  and  height. 

Div  the  area  by  the  square  of  the  diam  of  the  circle ;  look  for  the  quot  in  the  col  of  areas  in  the 
table  which  follows  the  next  one;  take  out  from  the  table  the  corresponding  height;  and  mult  it  by 
the  diam.  The  prod  will  be  the  reqd  height.  Then  from  the  diam,  take  the  actual  height  thus  found ; 
mult  the  rem  by  the  actual  height;  take  the  sq  rt  of  the  prod  ;  mult  it  by  2,  for  the  reqd  chord. 


MENSURATION. 


23 


To  find  ttie  length  of  a  circular  arc  by  the  following?  table. 

See  also  page  434. 

Knowing  the  rad  of  the  circle,  and  the  measure  of  the  arc  in  deg,  min,  &c. 

RULK.  Add  together  the  lengths  in  the  table  found  respectively  opposite  to  the  deg,  min,  &c,  of 
the  arc.     Mult  the  sum  by  the  rad  of  the  circle. 

Ex.  In  a  circle  of  12.43  feet  rad,  is  an  arc  of  13  deg,  27  min,  8  sec.     How  long  is  the  arc? 
Here,  opposite  13  deg  in  the  table,  we  find,  .2268928 
27  min  "  "        "      .0078540 

8  sec    "  "        "     .0000388 

Sum  =  .2347856 
And  .2347856  X  12.43  or  rad  =  2.918385  feet,  the  reqd  length  of  arc. 


JLEXGTHS  OF  CIRCULAR  ARCS  TO  RAD  1. 


No  errors. 


Deg. 

Length. 

Deg. 

Length. 

Deg. 

Length. 

Min. 

Length. 

Sec. 

Length. 

1 

.0174533 

61 

1.0646508 

121 

2.1118484 

1 

.0002909 

1 

.OOOC048 

2 

.0349086 

62 

1.0821041 

122 

2.1293017 

2 

.0005818 

2 

.0000097 

3 

.0523593 

63 

1.0995574 

123 

2.1467550 

3 

.0008727 

3 

.0000145 

4 

.0698132 

64 

1.1170107 

124 

2.1642083 

4 

.0011636 

4 

.0000194 

5 

.0872665 

65 

1.1344640 

125 

2.1816616 

5 

.0014544 

5 

.0000242 

6 

.1047198 

66 

1.1519173 

126 

2.1991149 

6 

.0017453 

6 

.0000291 

7 

.1221730 

67 

1.1093706 

127 

2.2165682 

7 

.0020362 

7 

.0000339 

8 

.1396263 

68 

1.  1883239 

128 

2.2340214 

8 

.0023271 

8 

.0000388 

9 

.1570796 

69 

1.2042772 

129 

2.2514747 

9 

.0026180 

9 

.0000436 

10 

.1745329 

70 

1.2217305 

130 

2.2689280 

10 

.0029089 

10 

.0000485 

11 

.1919882 

71 

1.2391838 

131 

2.2863813 

11 

.0031998 

11 

.0000533 

12 

.  2094:595 

72 

1.2566371 

132 

2.3U3M346 

12 

.0034907 

12 

.0000582 

13 

.2268928 

73 

1.2740904 

133 

2.3212879 

13 

.0037815 

13 

.0000630 

14 

.24434(51 

74 

1.2915436 

134 

2.9387*12 

4 

.0040724 

14 

.0000679 

15 

.2617994 

75 

1.3089989 

135 

2.35(51945 

5 

.0043633 

15 

.0000727 

16 

.2792527 

76 

1.3264502 

136 

2.373G478 

6 

.0046542 

16 

.0000776 

17 

.2967060 

77 

1.3439035 

137 

2.o911011 

7 

.0049451 

17 

.0000824 

IS 

.3141593 

78 

1.3513568 

138 

2.4085544 

8 

.0052360 

18 

.0000873 

19 

.3316126 

79 

1.3788101 

139 

2.4260077 

9 

.0055269 

19 

.0000921 

20 

.3490659 

80 

1.3982634 

140 

2.4434610 

20 

.0058178 

20 

.0000970 

21 

.3665191 

81 

1.4137167 

141 

2.4609142 

21 

.0061087 

21 

.0001018 

22 

.3839724 

82 

1.4311700 

142 

2.4783675 

22 

.0063995 

22 

.0001067 

23 

.4014257 

83 

1.4486233 

143 

2.4958208 

23 

.0066904 

23 

.0001115 

24 

.4188790 

84 

1.4660766 

144 

2.5132741 

24 

.0069813 

24 

.0001164 

25 

.4:5(53323 

85 

1.4835299 

145 

2.5307274 

25 

.0072722 

25 

.0001212 

26 

.4537856 

86 

1.5009832 

146 

2.5481807 

26 

.0075631 

26 

.0001261 

27 

.4712389 

87 

1.5184364 

147 

2.5656340 

27 

.0078540 

27 

.0001309 

28 

.4886922 

88 

1.5358897 

148 

2.5830873 

28 

.0081449 

28 

.0001357 

29 

.5081455 

89 

1.5533430 

149 

2.6005406 

29 

.0084358 

29 

.0001406 

30 

.5235988 

90 

1.  57079(53 

150 

2.6179939 

30 

.0087266 

30 

.0001454 

31 

.5410321 

91 

1.5882496 

151 

2.6354472 

31 

.0090175 

31 

.0001503 

32 

.5M50.il 

92 

1.6057029 

152 

2.6529005 

32 

.0093084 

32 

.0001551 

33 

.5759587 

93 

1.6231562 

153 

2.6703538 

33 

.0095993 

33 

.0001  BOO 

34 

.5934119 

94 

1.6406095 

154' 

2.6878070 

34 

.0098902 

34 

.0001648 

35 

.6108?!52 

95 

1.65S0628 

155 

2.7052603 

35 

.0101811 

35 

.0001697 

36 

.6283185 

96 

1.6755161 

156 

2.7227136 

36 

.0104720 

36 

.0001745 

37 

.6457718 

97 

1.6929694 

157 

2.7401669 

37 

.0107629 

37 

.0001794 

38 

.6632251 

98 

1.7104227 

158 

2.7576202 

38 

.0110538 

38 

.0001842 

39 

.6806784 

99 

1.7278760 

159 

2.7750735 

39 

.0113446 

39 

.0001891 

40 

.6981317 

100 

1.7453293 

160 

2.7925268 

40 

.0116355 

40 

.0001939 

41 

.7155850 

101 

1.7627825 

161 

2.8099801 

41 

.0119264 

41 

.0001988 

42 

.7330383 

102 

1.7802358 

1(52 

2.8274334 

42 

.0122173 

42 

.0002036 

43 

.7504916 

103 

1.7976891 

163 

2.8448867 

43 

.0125082 

43 

.0002085 

44 

.7679449 

104 

1.8151424 

164 

2.8023400 

44 

.0127991 

44 

.0002133 

45 

.785:5982 

105 

1.8325957 

165 

2.8797933 

45 

.0130900 

45 

.000-2182 

46 

.8028515 

106 

1.8500490 

166 

2.8972466 

46 

.0133809 

46 

.0002230 

47 

.8203017 

107 

1.8(575023 

167 

2.914(5999 

47 

.0136717 

47 

.0002279 

48 

.8377580 

108 

1.8849556 

168 

2.9321531 

48 

.0139626 

48 

.0002327 

49 

.8552113 

109 

1.9024089 

169 

2.9496064 

49 

.0142535 

49  . 

.0002376 

50 

.8726H46 

110 

1.919*622 

170 

2.9670597 

50 

.0145444 

50 

.0002424 

51 

.8901179 

111 

1.93731  "5 

171 

2.9845130 

nl 

.0148353 

51 

.0002473 

52 

.9075712 

112 

1.9547688 

172 

3.0019663 

52 

.0151262 

52 

.0002521 

53 

.9-250215 

113 

1.9722221 

173 

3.0194196 

53 

.0154171 

53 

.0002570 

54 

.9124778 

114 

1.9890753 

174 

8.03(587'-'9 

54 

.0157080 

54 

.0002618 

55 

.9599311 

115 

2.0071286 

175 

3.0543?6" 

55 

.01599*9 

55 

.0002G66 

56 
57 

.9773144 

116 

2.0245819 

176 

3.0717795 

06 

.0162897 

56 

.OCC2715 

58 

1.  0127910 

118 

2.05948S5 

177 

178 

3.0S9?328 
3.10fifiW,l 

57 
58 

.Olfi5*06  . 

.01(58715 

57 
58 

.OOf"/7(53 
.000281  2 

59 

1.0297443 

119 

2.0769418 

179 

3.1241394 

59 

.0171624 

59 

.0002860 

60 

1.0471976 

120 

2.0943951 

180 

3.1415927 

60 

.0174533 

60 

.0002909 

24 


MENSURATION. 


TABLE  OF  AREAS  OF  CIRCULAR  SEGMENTS. 


The  heights  are  in  parts  of  the  diam  of  the  circle. 
For  the  use  of  this  table  see  top  of  page  23. 

If  the  segment  exceed  180°,  see  page  630. 


No  errors. 


a 

m 

3 

Areas. 

ja 

s* 

K 

Areas. 

*i 

So 
M 

Areas. 

• 

Areas. 

«i 

bfl 

Areas. 

.001 

.000042 

.064 

.021168 

.127 

.057991 

.IPO 

.103900 

.253 

.156149 

.002 

.000119 

.065 

.021660 

.128 

.058658 

.191 

.104686 

.254 

.157019 

.003 

.000219 

.066 

.022155 

.129 

.059328 

.192 

.105472 

.255 

.157891 

.004 

.000337 

.067 

.022653 

.130 

.059999 

.193 

.106261 

.256 

.158763 

.005 

.000471 

.068 

.023155 

.131 

.060673 

.194 

.107051 

.257 

.159636 

.006 

.000619 

.069 

.023660 

.132 

.061349 

.195 

.107843 

.258 

.160511 

.007 

.000779 

.070 

.024168 

.133 

.062027 

.196 

.108636 

.259 

.161386 

.008 

.000952 

.071 

.024680 

.134 

.062707 

.197 

.109431 

.260 

.162263 

.009 

.001135 

.072 

.025196 

.135 

.063389 

.198 

.110227 

.261 

.163141 

.010 

.001329 

.073 

.025714 

.136 

.064074 

.199 

.111025 

.262 

.164020 

.011 

.001533 

.074 

.026236 

.137 

.064761 

.200 

.111  824 

.263 

.164900 

.012 

.001746 

.075 

.026761 

.138 

.065449 

.201 

.112625 

.264 

.165781 

.013 

.001969 

.076 

.027290 

.139 

.066140 

.202 

.113427 

.265 

.166663 

.014 

.002199 

.077 

.027821 

.140 

.066833 

.203 

.114231 

.266 

.167546 

.015 

.002438 

.078 

.028356 

.141 

.067528 

.204 

.115036 

.267 

.168431 

.016 

.002685 

.079 

.028894 

.142 

.068225 

.205 

.115842 

.268 

.169316 

.017 

.002940 

.080 

.029435 

.143 

.068924 

.206 

.116651 

.269 

.170202 

.018 

.003202 

.081 

.029979 

.144 

.069626 

.207 

.117460 

.270 

.171090 

.019 

.003472 

.082 

.030526 

.145 

.070329 

.208 

.118271 

.271 

.171978 

.020 

.003749 

.083 

.031077 

.146 

.071034 

.209 

.119084 

.272 

.172868 

.021 

.004032 

.084 

.031630 

.147 

.071741 

.210 

.119898 

.273 

.173758 

.022 

.004322 

.085 

.032186 

.148 

.072450 

.211 

.120713 

.274 

.174650 

.023 

.004619 

.086 

.032746 

.149 

.073162 

.212 

.121530 

.275 

.175542 

.024 

.004922 

.087 

.033308 

.150 

.073875 

.213 

.122348 

.276 

.176436 

.025 

.005231 

.088 

.033873 

.151 

.074590 

.214 

.123167 

.277 

.177330 

.026 

.005546 

.089 

.034441 

.152 

.075307 

.215 

.123988 

.278 

.178226 

.027 

.005867 

.090 

.035012 

.153 

.076026 

.216 

.124811 

.279 

.179122 

.028 

.006194 

.091 

.035586 

.154 

.076747 

.217 

.125634 

.280 

.180020 

.029 

.006527 

.092 

.036162 

.155 

.077470 

.218  !  .126459 

.281 

.180918 

.030 

.006866 

.093 

.036742 

.156 

.078194 

.219  i  .127286 

.282 

.181818 

.031 

.007209 

.094 

.037324 

.157 

.078921 

.220 

.1-28114 

.283 

.182718 

.032 

.007559 

.095 

.037909 

.158 

.079650 

.221 

.128943 

.284 

.183619 

.033 

.007913 

.096 

.038497 

.159 

.080380 

.222 

.129773 

.285 

.184522 

.034 

.008273 

.097 

.039087 

.160 

.081112 

.223 

.130605 

.286" 

.185425 

.035 

.008638 

.098 

.039681 

.161 

.081847 

.224 

.131438 

.287 

.186329 

.036 

.009008 

.099 

.040277 

.162 

.082582 

.225 

.132273 

.288 

.187235 

.037 

.009383 

.100 

.040875 

.163 

.083320 

.226 

.133109 

.289 

.188141 

.038 

.009764 

.101 

.041477 

.164 

.084060 

.227 

.133946 

.290 

.189048 

.039 

.010148 

.102 

.042081 

.165 

.084801 

.228 

.134784 

.291 

.189956 

.040 

.010538 

.103 

.042687 

.166 

.085545 

.229 

.135624 

.292 

.190865 

041 

.010932 

.104 

.043296 

.167 

.086290 

.230 

.136465 

.'293 

.191774 

.042 

.011331 

.105 

.043908 

.1H8 

.087037 

".231 

.137307 

.294 

.192685 

.043 

.011734 

.106 

.044523 

.169 

.087785 

.232 

.138151 

.295 

.193597 

.044 

.012142 

.107 

.045140 

.170 

.088536 

.233 

.138996 

.296 

.1.94509 

.045 

.012555 

.108 

.045759 

.171 

.0892*8 

.234 

.139842 

.297 

.195423 

.046 

.012971 

.109 

.046381 

.172 

.09004-2 

.'235 

.140689 

.298 

.196337 

.047 

.013393 

.110 

.047006 

.173 

.090797 

.236 

.141538 

.299 

.197252 

.048 

.013818 

.111 

.047633 

.174 

.091555 

.237 

.142388 

.300 

.198168 

.049 

.014248 

.112 

.048262 

.175 

.092314 

.238 

.143239 

.301 

.199085 

.050 

.014681 

.113 

.048894 

.176 

.093074 

.239 

.144091 

.302 

.200003 

.051 

.015119 

.114 

.049529 

.177 

.093837 

.240 

.144945 

.303 

.200922 

.052 

.015561 

.115 

.050165 

.178 

.094601 

.241 

.145800 

.304 

.201841 

.053 

.016008 

.116 

.050805 

.179 

.095367 

.242 

.146656 

.305 

.202762 

.054 

.016458 

.117 

.051446 

.180 

.096135 

.243 

.147513 

.306 

.203683 

.055 

.016912 

.118 

.052090 

.181 

.096904 

.244 

.148371 

.307 

.204605 

.056 

.017369 

.119 

.052737 

.182 

.097675 

.245 

.149231 

.308 

.205528 

.057 

.017831 

.120 

.053385 

.183 

.098447 

.246 

.150091 

.309 

.206452 

.058 

.018297 

.121 

.0540:7 

.184 

.099221 

.247 

.150953 

.310 

.207376 

.059 

.018766 

.122 

.054600 

.185 

.099997 

.248 

.151816 

.311 

.208302 

.060 

.019239 

.123 

.055346. 

.186 

.100774 

.249 

.152681 

.312 

.209228 

.061 

.019716 

.124 

.056004 

.187 

.101553 

.230 

.153546 

.313 

.210155 

.062 

.020197 

.125 

.056664 

.188 

.102334 

.251 

.154413 

.314 

.211083 

.063 

.020681 

.126 

.057327 

.189 

.103116 

.252 

.155281 

.315   .212011 

MENSURATION. 


25 


TABUS   OF   AREAS   OF  CIRCULAR   SEGMENTS  —  (CONTINUED.) 


1 

Areas. 

H 

Areas. 

60 
•5 

H 

Areas. 

I 

a 

Areas. 

£ 

P9 

Areas. 

.316 

.212941 

.353 

.247845 

•390 

.283593 

.427 

.319959 

.464 

.356730 

.317 

.213871 

.354 

.248801 

•391 

.284569 

.428 

.320949 

.465 

.357728 

.318 

.214802 

.355 

.249758 

•392 

.285545 

.429 

.321938 

.466 

.358725 

.319 

.215734 

.356 

.250715 

•393 

.286521 

.430 

.32292S 

.467 

.359723 

.320 

.216666 

.357 

.251673 

•394 

.287499 

.431 

.323919 

.46S 

.360721 

.321 

.217600 

.358 

.252632 

•395 

.288476 

.432 

.324909 

.469 

.361719 

.322 

.218534 

.359 

.253591 

•396 

.289454 

.433 

.325900 

.470 

.362717 

.323 

.219469 

.360 

.254551 

•397 

.290432 

.434 

.326891 

.471 

.363715 

.324 

.220404 

.361 

.255511 

•398 

.291411 

.435 

.3-27883 

.472 

.364714 

.325 

.221341 

.362 

.256472 

•399 

.292390 

.436 

.328874 

.473 

.365712 

.326 

.222278 

.363 

.257433 

'400 

.293370 

.437 

.32986e> 

.474 

.366711 

.327 

.223216 

.364 

.258395 

•401 

.294350 

.438 

.330858 

.475 

.367710 

.328 

.224154 

.365 

.259358 

•402 

.295330 

.439 

.331851 

.476 

.368708 

.329 

.225094 

.366 

.260321 

•403 

.296311 

.440 

.332843 

.477 

.369707 

.330 

.226034 

.367 

.261285 

•404 

.297292 

441 

.333836 

.478 

.370706 

.331 

.226974 

.368 

.262249 

•405 

.298274 

.442 

.334829 

.479 

.371705 

.332 

.227916 

.369 

.263214 

•406 

.299256 

.443 

.335823 

.480 

.372704 

.333 

.228858 

.370 

.264179 

•407 

.300238 

.444 

.336816 

.481 

.373704 

.334 

.229801 

.371 

.265145 

•408 

.301221 

.445 

.337810 

.482 

.374703 

.335 

.230745 

.372 

.266111 

•409 

.302204 

.446 

.338804 

.483 

.375702 

.336 

.231689 

.373 

.267078 

•410 

.303187 

.447 

.339799 

.484 

.376702 

.337 

.232634 

.374 

.26^046 

•411 

.304171 

.448 

.340793 

.485 

.377701 

.338 

.233580 

.375 

.269014 

•412 

.305156 

.449 

.341788 

.486 

.378701 

.339 

.234526 

.376 

.269982 

•413 

.306140 

.450 

.342783 

.487 

.379701 

.340 

.235473 

.377 

.270951 

•414 

.307125 

.451 

.343778 

.488 

.380700 

.341 

.236421 

.378 

.271921 

•415 

.308110 

.452 

.344773 

.489  1  .381700 

.342 

.237369 

.379 

.272891 

•416 

.309096 

.453 

.345768 

.490 

.382700 

.343 

.238319 

.380 

.273861 

•417 

.310082 

.454 

.346764 

.491 

.383700 

.344 

.239268 

.381 

.274832 

•418 

.311068 

.455 

.347760 

.492 

.384699 

.345 

.240219 

.382 

.275804 

•419 

.312055 

.456 

.348756 

.493 

.385699 

.346 

.241170 

.383 

.276776 

.420 

.313042 

.457 

.349752 

.494 

.386699 

.347 

.242122 

.384 

.277748 

.421 

.314029 

.458 

.350749 

.495 

.387699 

.348 

.243074 

.385 

.278721 

.422 

.315017 

.459 

.351745 

.496 

.388699 

.349 

.244027 

.386 

.279695 

.423 

.316005 

.460 

.352742 

.497 

.389699 

.350 

.244980 

.387 

.280669 

.424 

.316993 

.461 

.353739 

.498 

.390699 

.351 

.245935 

.388 

.281643 

.425 

.317981 

.462 

.354736 

.499 

.391699 

.352 

.246890 

.389 

.282618 

.426 

.318970 

.463 

.355733 

.500 

.392699 

To  find  the  area  of  a  circular  zone  abed. 

Knowing  the  diam  of  the  circle ;  the  chords  a  b,  and  c  d  ;  and  the  heights 
o  m,  and  «  n,  of  the  segments  a  m  b,  and  end.     First  find  the  area  of  the     • 
entire  circle;  then  by  means  of  the  preceding  table  of  circular  segments, 
find  the  areas  of  the  two  segments  amb,  and  end;  and  subtract  them  from 
the  area  of  the  circle. 

To  find  the  area  of  a  circular  lune  abco. 

A  lune  is  a  crescent- shaped  fig,  comprised  between  two  arcs  of  circles  of 
diff  diams.  Having  the  chord  a  c,  and  the  heights  of  the  two  segments  a  o  c, 
and  a  b  c,  find  the  areas  of  those  segments  ;  take  the  least  of  these  areas  from 
the  greatest ;  the  rem  is  evidently  the  area  of  the  lune. 

THE 


26 


MENSURATION. 


An  ellipse  is  a  curve,  eeee,  Pig  1,  formed  by  an  oblique  section  of  either  a  cone  or  a  cylinder,  pass- 
ing  through  its  curved  surface,  without  touching  the  base.     Its  nature  is  such  that  if  two  line*,  a.* 


,  , 

.  ,  from  any  point  n  in  its  periphery  or  circumf,  to  two  certain  points/ 

and  g,  situated  in  its  long  diani  c  w,  (and  called  the  foci  of  the  ellipse,)  they  will  be  equal  to  any  other 


nf  and  n  q.  Fig  2,  be  drawn 

,  d  in  its  long  diani  c     ,  , 

two  lines,  as  b  /,  and  b  g,  drawn  from  any  other  point,  as  6,  in  the  circumf,  to  the  foci  /  and  g  •  also  any 


Rule.    As  ?/  m2  :  s  a 
Example,    y  m 


ich  lines  will  together  be  equal  to  the  long  diam  c  w.     The  line  c  w  dividing  the  ellipse  into  tw 

equal  parts  lengthwise,  is  called  its  transverse,  or  major  axis,  or  long  diam ;  and  a  b,  which  divides  it 
equally  at  right-angles  to  cw,  is  called  the  conjugate,  or  minor  axis,  or  short  diam.  To  find  the  posi- 
tion of  the  foci  of  au  ellipse,  from  either  end,  as  b,  of  the  short  diam,  measure  off  the  dists  bf  and 
6  g,  Fig  2,  each  equal  to  o  c,  or  one-half  the  long  diain. 
The  parameter  of  an  ellipse  is  a  certain  length  obtained 
thus ;  as  the  long  diam  :  short  diam  :  :  short  diam  :  para- 
meter; or  the 

short  diam  2 

parameter  is  equal  to . 

long  diam 

Any  line  b  a,  or  c  d.  Fig  3,  drawn  from  the  circumf,  to, 
and  at  right  angles  to,  either  diam.  is  called  an  ordirmte  ; 
and  the  part  a  y,  or  c  x  of  that  diam,  between  the  ord  and 
the  circumf,  is  called  an  abscissa,  or  absciss. 

To  find  the  length  of  any  ordinal  Q 
a  b,  drawn  to  the  long  diam .ym. 

Knowing  the  absciss  y  a,  and  the  two  diams  y  m,  ex. 
:  y  a  X  «  m  '•  abz. 
s  x  —  3 :  y  a  =  2 ;  am  =  6. 
Then  as  64  :  9  : :  12_:_  1.6875. 
Hence  a  b  =  1/1.6875  =  1.299. 

Or,  mult  together  the  two  parts  y  a  and  a  m,  Fig  3,  into  which  the  ord  divides  the  long  diam ;  take 
the  sq  rt  of  the  prod;  mult  this  sq  rt  by  the  short  diam ;  div  the  prod  by  the  long  diam. 

REM.  Neither  of  these  rules  (nor  any  other)  applies  also  to  ordinates  like  c  d,  drawn  to  the  short 
diam  x  s. 

To  find  the  circumf  of  an  ellipse. 

Mathematicians  have  furnished  practical  men  with  no  simple  working  rule  for  this  purpose.  The 
so-called  approximate  rules  do  not  deserve  the  name.  They  are  as  follows,  D  being  the  long  diam  : 
and  d  the  short  one.* 

RULE  1.  Circumf  =  3.1416  D  +  <*  •    RULE  ».  3.1416  /  ^P!+^!:\  •   RULE  8.  2.2 

\/  2 

this  is  the  same  as  Rule  2,  but  in  a  diff  shape.     RULE  4.  2j/  D2-f- 1.4674  d2.     Now,  in  an  ellipse 

whose  long  and  short  diams  are  10  and  2,  the  circumf  is  actually  21,  very  approximately;  but  rule  1 
gives  it  =  18.85  ;  rule  2,  or  3,  =  22.65 ;  and  rule  4,  n  20.51.  Again,  if  the  diams-be  10  and  6.  the  cir- 
cumf actually  =  25.59;  but  rule  4  gives  24.72.  These  examples  show  that  none  of  the  rules  usually 
given  are  reliable.  The  following  one  by  the  writer,  is  sufficiently  exact  for  ordinary  purposes;  not 
being  in  error  probably  more  than  1  part 'in  1000.  When  D  is  not  more  than  5  times  as  long  as  d,  then, 
calling  the  diff  between  them,  Diff,  the 


Circumf  =  3.  U16  - 


^EEi   -  ?if!- 
V       2       /  8.8 


If  B  is  more  than  5  times  d,  then  instead  of  dividing  Diff 2  by  8.8,  div  it  by  the  number  in  the 
following  table: 


D  =  6cZ 

9 

D  =  14d 

9.6 

D  =  40  d 

9.98 

7 

9.2 

16 

9.68 

50 

10.04 

8 

9.3 

18 

9.75 

60 

10.10 

9 

9.35 

20 

9.8 

70 

10.17 

10 

9.4 

25 

9.87 

80 

10.23 

12 

9.5 

30 

9.92 

100 

10.35 

In  words,  this  rule  is  as  follows  :  Square  both  D  and  d ;  add  these  squares  together  ;  div  the  sum  by 
2;  call  the  quot  A.  Next  subtract  d  from  D;  square  the  Diff:  div  this  square  by  8.8  (or  by  the 
proper  number  from  the  table) ;  subtract  the  quot  from  A  ;  take  the  sq  rt  of  the  reni;  mult  this  sq  rt 
by  3. 1416. 

The  following  table  of  semi-elliptic  arcs,  has  been  prepared  by  this  rule. 

b 


*  The  full  table  in  Mr.  Haswell's  book,  Adcock's,  and  others,  is  incorrect,  esp*. 
oially  OQ  the  first  p*ge.     For  a  more  recent  rule,  see  p  680. 


MENSURATION. 


27 


TABLE  OF  &EXGTHS   OF   SEm-EEMPTIC  ARCS. 

(Original.) 


Heights. 

Lengths. 

Heights. 

Lengths. 

Heights. 

Lengths. 

Heights. 

Lengths. 

.005 

1.000 

.130 

1.079 

.255 

1.219 

.380 

1.390 

.01 

1.001 

.135 

1.084 

.260 

1.226 

.385 

1.397 

.015 

1.002 

.140 

1.089 

.265 

1.233 

.390 

1.404 

.02 

1.003 

.145 

1.094 

.270 

1.239 

.395 

1.412 

.025 

1.004 

.150 

1.099 

.275 

1.245 

.400 

1.419 

.03 

1.006 

.155 

1.104 

.280 

1.252 

.405 

1.426 

.035 

1.008 

.160 

1.109 

.285 

1.259 

.410 

1.434 

.04 

1.011 

.165 

1.115 

.290 

1.265 

.415 

1.441 

.045 

1.014 

.170 

1.120 

.295 

1.272 

.420 

1.449 

.05 

1.017 

.175 

1.125 

.300 

1.279 

.425 

1.456 

.055 

1.020 

.180 

1.131 

.305 

1.285 

.430 

1.464 

.06 

1.028 

.185 

1.137 

.310 

1.292 

.435 

1.471 

.065 

1.026 

.190 

1.142 

.315 

1.298 

.440 

1.479 

.07 

1.029 

.195 

1147 

.320 

1.305 

.445 

1.486 

.075 

1.032 

.200 

1.153 

.325 

1.312 

.450 

1.494 

.08 

1.036 

.205 

1.159 

.330 

1.319 

.455 

1.501 

.085 

1.039 

.210 

1.165 

.335 

1.325 

.460 

1.509 

.09 

1.043 

.215 

1.171 

.340 

1.332 

.465 

1.517 

.095 

1.046 

.220 

1.177 

•345 

1.339 

.470 

1.524 

.100 

1.051 

.225 

1.183 

.350 

1.346 

.475 

1.532 

.105 

1.055 

.230 

1.189 

.355 

1.353 

,480 

1.540 

.110 

1.059 

.235 

1.196 

.360 

1.361 

.485 

1.547 

.115 

1.064 

.240 

1.202 

.365 

1.368 

.490 

1.555 

.120 

1.069 

.245 

1.207 

.370 

1375 

.495 

1.563 

.125 

1.074 

.250 

1.213 

.375 

1.382 

.500 

1.571 

To  find  the  area  of  an  ellipse. 

Mult  the  two  diams  together;  mult  the  prod  by  .7854.  Ex.  D  =  10;  d  =  6.  Then  10  X  6  X  .7854 
—  47. 124  area.  The  area  of  an  ellipse  is  a  mean  proportional  between  the  areas  of  two  circles,  described 
on  its  two  diams ;  therefore  it  may  be  found  by  mult  together  the  areas  of  those  two  circles  ;  and  taking 
the  sq  rt  of  the  prod.  The  area  of  an  ellipse  is  therefore  always  greater  than  that  of  the  circular  sec- 
tion of  the  cylinder  from  which  it  may  be  supposed  to  be  derived. 

To  find  the  diam  of  a  circle  of  the  same  area  as  a  given  ellipse. 

Mult  together  the  2  diams ;  take  sq  rt  of  prod. 

To  find  the  area  of  an  elliptic  segment  whoso  base  is  parallel 
to  either  diam. 

Div  the  height  of  the  segment,  by  that  diam  of  which  said  height  is  a  part.  From  the  table  of  cir. 
cnlar  segments  take  out  the  tabufar  area  opposite  the  quot.  Mult  together  this  area,  the  long  diam, 
and  the  short  diam. 

To  draw  an  ellipse. 

Having  its  long  and  short  diams  a  I  and  c  d,  Fig  4. 

RULE  1.  From  either  end  of  the  short 
diam,  as  c,  lay  off  thedists  cf,  c/',each  equal 
to  e  a,  or  to  one-half  of  the  long  diam.  The  - 
points  /,  /'are  the  foci  .of  the  ellipse.  Pre- 
pare a  string,/'  n  f,  or  /'  g  f,  with  a  loop  at 
each  end  ;  the  total  length  of  string  from  end 
to  end  of  loop,  being  equal  to  the  long  diam. 
Place  pins  at/ and  /' ;  and  placing  the  loops 
ever  them,  trace  the  curve  by  a  pencil,  which 
in  every  position,  asatn.  or  g,  keeps  the  string 
/'  nf,  or/'  gf,  equally  stretched  all  the  time. 

Note.  Owing  to  the  difficulty  of  keeping 
the  string  equally  stretched,  this  method  is 
not  as  satisfactory  as  the  following. 

RULE  2.  On  the  edge  of  a  strip  of  paper 
w  s,  mark  w  I  equal  to  half  the  short  diam  ; 
and  ws  equal  half  the  long  diam.  Then  in 
whatever  position  this  strip  be  placed,  keep- 
ing I  on  the  long  diam,  ami  s  on  the  short 
diam,  to  will  mark  a  point  in  the  circumf  of 
the  ellipse.  We  may  thus  obtain  as  many  such  points  as  we  please ;  and  then  draw  the  curve  through 
them  by  hand. 

RULE  3. 
de 


28 


MENSURATION. 


To  draw  a  tangent  1 1,  at  any  point  n  of  an  ellipse. 

Draw  nf  and  n/',  to  the  foci ;  bisect  the  angle  fnf  by  the  line  xp ;  draw  t  n  t  at  right  angle* 
to  xp. 

To  draw  a  joint  -n,p,  of  an  elliptic  arch,  from  any  point  •«,  in 
the  arch. 

Proceed  as  in  the  foregoing  rule  for  a  tangent,  only  omitting  tt;  np  will  be  the  reqd  joint. 
THE  PARABOLA. 

•b 


r 


F,'4  1 
The  common  or  conic  parabola, 

o  &  c,  Fig  1,  is  a  curve  formed  by  cutting  a  cone  in  a  direction  b  a,  parallel  to  its  side.  The 
curved  line  o  b  c  itself  is  called  the  perimeter  of  the  parabola  ;  the  line  o  c  is  called  its  base ;  b  a  it* 
height  or  axis  ;  b  its  apex  or  vertex;  any  line  e  «,  or  o  a,  Fig  2,  drawn  from  the  curve,  to,  and  at  right 
angles  to,  the  axis,  is  an  ordinate  ;  and  the  part  s  o,  or  a  b,  of  the  axis,  between  the  ordiuate  and  the 
apex  b,  is  an  abscissa.  The  focus  of  a  parabola  is  that  point  in  the  axis,  where  the  abscissa  b  s,  is 
equal  to  one-half  of  the  ord  e  a.  The  dist  from  the  apex  to  the  focus,  is  called  the  focal  dist.  The 
focus  may  be  entirely  beyond  or  outside  of  the  curve  itself.  Its  dist  from  the  apex  is  found  thus  : 
square  any  ord,  as  o  a ;  div  this  square  by  the  abscissa  6  a  of  that  ord ;  div  the  quot  by  4.  The 
nature  of  the  parabola  is  such  that  its  abscissas,  as  b  s,  b  a,  &c,  are  to  each  other  as,  or  in  proportion 
to,  the  squares  of  their  respective  ords  e  s,  o  a,  &c ;  that  is,  as  6  s  :  b  a  :  :  e  «2  ;  o  a2 ;  or  b  s  :  e  «2  : ;  5  a  : 
o  a*  .  If  the  square  of  any  ord  be  divided  by  its  abscissa,  the  quot  will  be  a  constant  quantity  ;  that 
is,  it  will  be  equal  to  the  square  of  any  other  ord  divided  by  its  abscissa.  This  quot  or  constant  quan- 
tity is  also  equal  to  a  certain  quantity  called  the  parameter  of  the  ellipse.  Therefore  the  parameter 
may  be  found  by  squaring  e  s,  or  o  a,  (one-half  of  the  base,)  and  dividing  said  square  by  the  height 
6  «,  or  b  a,  as  the  case  may  be.  If  the  square  of  any  ord  be  divided  by  the  parameter,  the  quot  will 
be  the  abscissa  of  that  ord. 

To  find  the  length  of  a  parabolic  curve. 

The  so-called  approximate   rule  given  by  various  pocket-books,  is,  like  those  for  the  ellipse, 

entirely  unreliable.    It  is  

2  X  V (1A  base)2  +  1^  times  the  (Heights) 

This  rule  is,  however,  close  (about  1  per  ct  in  excess)  for  parabolas,  whose 
height  is  not  more  than  1-lOth  of  the  base;  and  still  more  so  for  flatter  ones. 
But  in  a  parabola  whose  height  is  2,  and  base  1,  it  gives  a  curve  of  4.73 ;  whereas 
it  is  actually  but  about  4.2  ;  being  an  error  of  nearly  13  per  cent,  or  1  in  1%. 
The  following  by  the  writer  is  correct 
within  perhaps  1  part  in  «00,  in  all  cases  ;  aud  will 
therefore  answer  for  many  purposes. 

Let  adb,  Fig  3,  or  n,  a'd.  Fig  4,  be  the  parabola, 
in  which  are  given  the  base  a  b  or  n  d;  and  the 
height  c  d  or  c  a.  Imagine  the  complete  fig  a  d  b  s, 
or  n  a  d  b,  to  be  drawn  ;  aud  in  either  case,  assume 
its  long  diarn  a  b  to  be  the  chord  or  base ;  aud  one- 
half  the  short  diam,  or  c  d,  to  be  the  height,  of  a 
circular  arc.  Find  the  length  of  this  circular  arc, 
by  means  of  the  rule  and  table  given  for  that  pur- 
pose. Then  div  the  chord  or  base  a  b,  or  n  d  of 
the  parabola,  by  its  height  c  d  or  c  a.  Look  for 
the  quot  in  the  column  of  bases  in  the  following 
table,  and  take  from  the  table  the  corresponding 
multiplier.  Mult  the  length  of  the  circular  arc  by 
this;  the  prod  will  be  the  length  of  arc  adb,  or 
n  a  d,  M  the  case  may  be.  For  bases  of  parabolas 
less  than  .05  of  the  height,  or  greater  than  10  times 
the  height,  the  multiplier  is  1,  and  is  very  approx- 
imate ;  or  in  other  words,  the  parabola  will  be 
of  almost  exactly  the  same  length  as  the  circular 
arc. 

To  find  the  area  of  a  parabola  m  a  n  &. 

Mult  its  base  m  n,  Fig  5,  by  its  height  a  b ;  and  take  %ds  of  the  prod. 
The  area  of  any  segment,  as  u  b  v,  whose  base  M  v  is  parallel  to  mn,  is 
found  in  the  same  way,  using  u  v  and  a  b,  instead  of  mn  and  a  b. 

To  find  the  area  of  a  parabolic  zone,  or  frus- 
tum, as  vi  n  u  v. 

RULE  1.  First  find  by  the  preceding  rule  the  area  of  the  whole  parabola 
m  b  n ;  then  that  of  the  segment  u  b  v ;  and  subtract  the  last  from  the 
first. 

RULE  2.    From  the  cube  of  m  n,  take  the  cube  of  u  v:  call  the  diff  c. 


ULE    .       rom      e  cue  o    m    ,    a  o  . 

From  the  square  of  m  n,  take  the  square  of  tt  v ;  call  the  diff  s.    Div  c  by 
i.    Mult  the  quot  by  %ds  of  the  height  a  «. 


MENSURATION. 


29 


Original. 


Base. 

Mult. 

Base. 

Mult. 

Base. 

Mult. 

Base. 

Mult. 

.05 

1.000 

1.10 

.999 

215 

.949 

3.20 

.983 

.10 

1.001 

1.15 

.997 

2.20 

.951 

3.30 

.984 

.15 

1.002 

1.20 

.995 

2.25 

.954 

3.40 

.985 

.20 

1.004 

1.25 

.993 

2.30 

.956 

3.50 

.986 

.25 

1.006 

1.30 

.990 

2.35 

.958 

3.60 

.987 

.30 

1.007 

1.35 

.987 

2.40 

.960 

3.70 

.988 

.35 

1.007 

1.40 

.984 

2.45 

.962 

3.80 

.989 

.40 

1.008 

1.45 

.980 

2.50 

.963 

3.90 

.990 

.45 

1.009 

1.50 

.977 

2.55 

.965 

4.00 

.991 

.50 

1.010 

1.55 

.974 

2.60 

.967 

4.25 

.992 

.55 

1.010 

1.60 

.970 

2.65 

.969 

4.50 

.993 

.60 

1.010 

1.65 

.966 

2.70 

.970 

4.75 

.994 

.65 

1.011 

1.70 

.963 

2.75 

.972 

5.00 

.995 

.70 

1.011 

1.75 

.960 

2.80 

.973 

5.25 

.996 

.75 

1.010 

1.80 

.957 

2.85 

.975 

5.50 

.997 

.80 

1.009 

1.85 

.953 

2.90 

.976 

5.75 

.998 

.85 

1.008 

1.90 

.950 

2.95 

.978 

6.00 

.998 

.90 

1.006 

1.95 

.946 

3.00 

.979 

7.00 

.999 

.95 

1.004 

2.00 

.942 

305 

.980 

8.00 

1.000 

1.00 

1.002 

2.05 

.944 

3.10 

.981 

10.00 

1.000 

1.05 

1.001 

2.10 

.946 

3.15 

.982 

To  draw  a  parabola, 

cos,  Fig  6.  Make  of  equal  to  the  height  eo.  Draw  ct  and 
»t;  and  divide  each  of  thum  into  any  number  of  equal  parts  ; 
numbering  them  as  in  the  Fig.  Join  1,  1 ;  2,  2  ;  3,  3.  &c  ; 
then  draw  the  curve  bv  hand.  It  will  be  observed  that  the 
intersections  of  the  lines  1,1;  2,  2,  &c,  do  not  give  points  in 
,  the  curve  ;  but  a  portion  of  each  of  those  lines  forms  a  tan- 
gent to  the  curve.  By  increasing  the  number  of  divisions 
on  c  t  and  s  t,  an  almost  perfect  curve  is  formed,  scarcely 
requiring  to  be  touched  up  by  hand.  In  practice  it  is  best 
first  to  draw  only  the  center  portions  of  the  two  lines  which 
cross  each  other  just  above  o  ;  and  from  them  to  work  down- 
ward; actually  drawing  only  that  small  portion  of  each 
successive  lower  line,  which  is  necessary  to  indicate  the 
curve. 

Or  the  parabola  may  be  drawn 
thus : 

Let  6  c.  Fig  7,  be  the  base  ;  and  a  d  the  height.   Draw  the 
rectangle  I  nm  c  ;  div  each  half  of  the  base  into  any  num- 
ber  of  equal  parts,  and  number  them  from  the  center  each 
way.   Div  n  b,  and  m  c  into  the  same  number  of  equal  parts  ; 
and  number  them  from  the  top,  downward.     From  the  points 
on  6  c  draw  vert  lines  ;  and  from  those  at  the  sides  draw  lines 
to  rf.  Then  the  intersections  of  lines  1,  1 ;  2.  2,  &c, 
will  form  points  in  the  parabola.     As  in  the  pre-       ^ 
ceding  case,  it  is  not  necessary  to  draw  the  entire 
lines  ;  but  merely  portions  of'them,  as  shown  be- 
tween d  and  c. 

Or  a  parabola  may  be  drawn  by  first  div  the 
height  a  I,  Fig  5,  into  any  number  of  parts,  either 
equal  or  unequal;  and  then  calculating  the  ordi- 
nates  us,  &c  ;  thus,  as  the  height  a  b  :  square  of 
half  base  am::  any  absciss  6  s  :  square  of  its 
ord  u  a.  Take  the  sq  rt  for  us. 

REM. — When  the  height  of  a  parabola  is  not 
greater  than  l-10th  part  its  base,  the  curve  coin- 
cides so  very  closely  with  that  of  a  circular  arc, 
that  in  the  preparation  of  drawings  for  suspen- 
sion bridges.  &c.,  the  circular  arc  may  be  em- 
ployed; or  if  no  great  accuracy  is  reqd,"the  circle 
may  be  used  even  when  the  height  is  as  great  as 
^ith  base. 

To  draw  a  tangent  w  r,  Fig.  5,  to  a  parabola,  from  any  point  tt. 

Draw  t;  s  perp  to  axis  a  b  ;  prolong  a  I  until  6  w  equals  s  6.    Join  w  v. 

The  Cycloid, 

a  cT>  is  the  curve  described  by  a  point  a  in  the  circumf  of  a  circle,  a  n,  during  one  complete  revo- 
liKion  of  the  circle,  rolled  along  a  straight  line  a  6;  which  is  called  the  base  of  the  cycloid.    Th« 


; 

x 

% 

4-32         101         23^ 
)                                    c*.     ¥i37 

30 


MENSURATION. 


base  a  ft  is  evidently  equal  to  the  circumf  of  th« 
b  generating  circle  an;  and  the  axis,  or  height 
—  d  s  c,  is  equal  to  its  diam.  The  length  a  c  b  =  4- 
times  diam  of  a  n.  The  area  a  c  l>  d  z=  3  times 
area  of  circle  a  n  ;  and  it.-t  ceu  of  gray  is  at  %ths 
of  c  d,  measured  from  the  vertex  c.  To  draw  a 
tang  co,  from  any  poiut  e;  draw  e  s  at  right 
angles  to  the  axis  d  c ;  on  d  c  describe  the  gener- 
ating circle  d  t  c ;  join  t  c  ;  from  e  draw  e  o  parallel 
to  t  c.  The  cycloid  is  the  curve  of  quickest  descent  ; 
so  that  a  body  would  full  from  b  to  c  along  the  curve 
6  m  c,  in  less  time  than  along  the  inclined  plane 
6  t  c,  or  any  other  line. 


PARALLELOPIPEDS. 


A  parallelepiped  is  any  solid  contained  within  six  sides,  all  of  which  are  parallelograms ;  and  those 
of  each  opposite  pair,  parallel  to  each  other.  We  show  but  four  of  them  ;  corresponding  to  the  four 
parallelograms;  namely,  the  cube,  Fig  1,  which  has  all  its  sides  equal  squares;  and  all  its  angles 
right  angles  ;  the  right  rectangular  prism.  Fig  2,  has  all  its  angles  right  angles  ;  each  pair  of  oppo- 
site faces  equal ;  but  not  all  of  its  faces  equal ;  the  Rhomb,  Fig  3,  which  has  a.11  its  sides  equal  rhom- 
buses ;  the  Rhombic  prism,  Fig  4;  its  faces,  rhombuses,  or  rhomboids;  each  pair  of  opposite  faces 
equal ;  but  not  all  its  faces  equal.  All  parallelepipeds  are  prisms. 

To  find  the  solidity  of  any  parallelopiped. 

Mult  the  area  of  any  face,  as  a,  b?  the  perp  dist,  p,  to  the  opposite  face.  A  cube  18  =  1.90985,  its 
inscribed  sphere;  or,  1.27324,  its  inscribed  cylinder;  or,  3.81972,  its  inscribed  c<m<<. 

Ttie  diag  of  a  cube,  or  the  diam  of  its  circumscribing  sphere,  is  equal  to  one  of  its  edges  mult  by 
1.73'J030S.  The  diag  of  a  rhomb,  or  of  a  rhombic  prism,  cannot  be  calculated  by  means  of  its  sides, 
and  their  angles. 

PRISMS. 


9      10 


Whether  right  or  oblique,  regular  or  irregular  ;  m 
other  end ;  or  mult  the  area  measured  perp  to  the  sid 


A  prism  Is  any  solid  whose  two  ends  are  parallel,  similar,  and  equal;  and  whose  sides  are  paral- 
lelograms, as  Figs  5  to  10.  Consequently  the  foregoing  parallelepipeds  are  prisms.  A  right  prism 
is  one  whose  sides  are  perp  to  its  ends,  as  5,  6,  7 ;  when  not  so,  the  prism  is  oblique,  as  8, 9, 10.  When 
all  the  sides  of  the  figs  which  form  the  ends  are  equal,  the  angles  included  between  those  sides  are 
also  equal;  and  the  prism  is  then  said  to  be  regular:  otherwise,  irregular. 

To  find  the  solidity  of  any  prism, 

mult  the  area  of  one  end  by  the  perp  dist,  p,  to  the 
ides  by  the  actual  length  a  b ;  Figs.  5  to  10. 

To  find  the  solidity  of  any  frustum* 
of  any  prism, 

Whose  cross  section,  perp  to  its  sides,  is  either  any  triangle  ; 
any  parallelogram ;  a  square,  (as  Fig  10J4  is  supposed  to  be ;) 
or  a  regular  polygon  of  any  number  of  sides  ;  no  matter  how 
the  two  ends  of  the  frustum  may  be  inclined  with  regard  to 
each  other;  or  whether  one,  or  neither  of  them,  is  parallel 
to  the  base  of  the  original  prism.  Measure  and  arid  together 
the  lengths  of  all  the  parallel  edges  11,  22.  33.  44.  Div  the 
sum  by  the  number  of  them,  for  a  mean  length.  Mult  the 
area  of  cross  section  perp  to  the  sides,  by  this  mean  length, 
for  the  solHity. 

This  rule  may  be  used  for  ascertaining  beforehand,  the 
quantity  of  earth  to  be  removed  from  a  "  borrow  pit."  The 
irregular  surface  of  the  ground  is  first  staked  out  in  squares  ; 
(the  tape-line  being  stretched  horizontally,  when  measuring 
off  their  sides).  These  squares  should  be  of  such  a  size  that 
without  material  error  each  of  them  may  be  considered  to  be 


A 

1 

4      i 

N~- 

\ 

V 

<K 

A 

j- 

7 

C 

*  Generally  misspelt  "frustrum." 


MENSURATION. 


31 


a  plane  surface,  either  hor,  or  inclined.  The  depth  of  the  hor  bottom  of  the  pit  being  determined 
cm;  and  the  levels  being  taken  at  every  corner  of  the  squares,  we  are  thereby  furnished  with  the 
lengths  of  the  tour  parallel  vertical  edges  of  each  of  the  resulting  frustums  of  earth,  in  Figs  10>4  y 
may  be  supposed  to  represent  one  of  these  frustums.* 

If  ttie  frustum  is  that  of  an  irreyular  4-sided,  or  polygonal  prism,  first 

div  its  cross  section  perp  to  its  sides,  into  triangles,  by  lines  drawn  from  ol 

any  one  of  its  angles,  as  a,  1<  ig  \Q%.  Calculate  the  area  of  each  of  these 
triangles  separately  ;  then  consider  the  entire  frustum  to  be  made  up  of 
so  many  triangular  ones;  calculate  the  solidity  of  each  of  these  by  the 
preceding  rule  for  triangular  frustums;  and  add  them  together,  for  the 
solidity  of  the  entire  frustum. 

Tbe  solidity  of  any  frustum  whatever,  of 
any  prism  whatever, 


Or  of  a  cylinder,  may  be  found  thus, 
find  its  area.    Also  find  the  cen  of  grai 
from   the  base   to   said  ceu  of  gray. 
Fig.  10%. 


Consider  either  end  to  be  the  base;  and 
c  of  the  other  end ,  and  the  perp  dist  n  c, 
Mult  this  dist  by  the  area  of  the  base. 


To  find  the  surface  of  any  prism ;  Figs.  5  to  1O. 

Whether  right  or  oblique  ;  regular,  or  irregular ;  mult  the  circumf  of  one  end, 
by  the  perp  dist  p  to  the  other  end ;  this  gives  the  surf  of  the  sides  ;  to  which  add 
the  surf  or  area  of  the  two  ends,  when  reqd.  Or  mult  the  circumf  measd  perp  to 
the  sides  by  the  actual  length  a  b ;  then  add  the  ends. 

CYL.IICDERS. 

A  cylinder  is  any  solid  whose  ends  are  parallel, 
similar,  and  equal  curved  figs  ;  and  whose  sections 
parallel  to  the  ends  arc  everywhere  the  same  as  the 
ends.  Hence  there  are  circular,  elliptic,  (or  cylin- 
droids),  and  other  cylinders;  but  when  not  other- 
wise expressed,  the  circular  one  is  understood.  A 
right  cylinder  is  one  whose  ends  are  perp  to  its  sides, 
as  Fig"  11 ;  when  otherwise,  it  is  oblique,  as  Fig  12. 
If  the  ends  of  a  right  circular  cylinder  be  cut  so  as  to 
make  it  oblique,  it  becomes  an  elliptic  one ;  because 
then  both  its  ends,  and  all  sections  parallel  to  them, 
are  ellipses.  An  oblique  circular  cylinder  seldom 
occurs;  it  n^ay  be  conceived  of  by  imagining  the 
two  ends  of  Fig  12  to  be  c'  ' 


lines  forming  its  curved  sides. 


united  by  straight 


To  find  the  solidity  of  any  cylinder. 

"Whether  circular,  elliptic,  &c  ;  right  or  oblique ;  as  in  prisms,  mult  the  area  of  one  end  by  the  perp 
dist,  p,  to  the  other  end.  Or  mult  the  area  measd  perp  to  the  sides,  by  the  actual  length  a  6,  Figs  11, 
12.  The  solidity  of  a  cylinder  is  3  times  that  of  a  cone  of  the  same  base  and  height. 

To  find  the  surface  of  any  cylinder, 

As  in  prisms,  mult  the  circumf  of  one  end  by  the  perp  dist  p  to  the  other  end  ;  this  gives  the  surf  of 
the  sides;  to  which  add  that  of  the  ends  when  reqd.  Or  mult  the  circumf  measd  perp  to  the  sides  as 
at  c  o,  Fig  i'2,  by  the  actual  length  a  b,  Figs  11  and  12,  and  add  the  ends. 

REM.  —  The  solidity  of  a  right  cvlinder  whose  height  equals  its  diam,  is  to  the  solidity  of  its 
inscribed  sphere,  as  3  is  to  2;  the  curved  surf  of  the  cylinder  (that  is,  its  sides,)  is  equal  to  the  surf 
of  the  sphere :  and  the  entire  surf  of  the  cylinder,  including  its  ends,  is  to  the  surf  of  the  sphere 
.as  3  to  2;  consequently  the  surf  of  the  two  ends  of  such  a  cylinder,  is  equal  to  half  the  surf  of  its 
sides.  Any  slant  end,  c,  fig  10%,  is  au  ellipse,  of  greater  area  than  the  circular  end. 

To  find  the  solidity  of  a  cyl- 
iiidric  ungiila,  when  the 
cutting'  plane  does  not  pass 
through  the  base. 

Figs  13  and  14 :  mult  the  areaof  its  base,  by  half 
the  sum  of  its  greatest  and  least  perp  lengths  on, 
and  c  TO.  Or  mult  the  area  measd  perp  to  the 

est  and  least  actual  lengths  o  t  and  g  m. 

To  find  the  surface  of  a  cyl- 
indric  ungiila,  when  the 
cutting  plane  does  not  pass 
through  the  base. 

Mult  the  circumf  of  its  base,  by  the  half  sum  of  its  greatest  and  least  perp  lengths  o  n,  nnd 
c  TO.  The  prod  will  be  the  curved  surf  of  the  sides;  to  which  add  the  ends  if  reqd.  Or  mult  its  cir- 
cumf measured  perp  to  its  sides,  as  at  x,  Fig  14,  by  the  half  sum  of  the  greatest  and  least  actual 
lengths  o  t,  gm.  Add  the  ends  if  reqd. 

*  Our  text- hooks  on  mpnsuration  strangely  do  not  give  rules  for  frustums  of  prisms,  although  such 
solids  are  of  frequent  occurrence. 


32 


MENSURATION. 


n      Cylindric  uii-n la  when  the  cutting  plane 
passes  through  the  base, 

making  m  a,  less  than  m  c,  or  half  the  diam  of,  the  circle.  For  the 
solidity,  cube  a  6,  and  take  %ds  of  it ;  which  call  p.  Mult  the  area  of 
the  base  a  d  m  6,  by  a  c.  Take  the  prod  from  p :  mult  the  rem  by  the 
height  mn;  div  the  prod  by  a  m.  See  App.  p.  630. 

For  the  convex  surface. 

Mult  the  diam  m  y  of  the  circle,  by  a  b  ;  call  the  prod  ».     Mult  the 
IT)         length  of  the  arc  d  m  b,  by  a  c ;  take  the  prod  from  p.    Mult  the  rem 
by  the  height  mn;  div  the  prod  by  a  m.    See  App.  p.  630  . 

CIRCULAR    RINGS. 


For  the  solidity, 

Mult  the  area  of  transverse  section  of  the  bar  of  which  the  ring  is  made,  by  half  the  sum  of  the 
inner  and  outer  diams,  a  a,  and  b  6,  of  the  ring;  mult  the  prod  by  3.1416. 

For  the  surface, 

Mult  together  the  girt  of  the  bar ;  the  half  sum  of  the  two  diams  ;  and  3.1416. 
PYRAMIDS    AND    CONES. 

d      d 


A  pyramid, 


Figs  1, 2, 3,  is  any  solid  which  has  for  its  base,  a  plane  fig  of  any  number  of  sides ;  and  for  its  sides, 
plane  triangles,  all  terminating  at  one  point  d,  called  its  apex,  or  top.  When  the  base  is  a  regular  fig, 
the  pyramid  is  regular;  otherwise  irregular.  For  regular  figs  see  Polygons,  p.  15. 

A  cone, 

Figs  4  and  5,  is  a  solid,  of  which  the  base  is  a  curved  fig ;  and  which  may  be  considered  as  made  or 
generated  by  a  line,  of  which  one  end  is  stationary  at  a  certain  point  d,  called  the  apex  or  top,  while 
the  line  is  being  carried  around  the  circumf  of  the  base,  which  may  be  a  circle,  ellipse,  or  other  curve. 

The  axis  of  a  pyramid,  or  cone,  is  a  straight  line  d  o  in  Figs  1,  2, 4 ;  and  d  i  in  Figs  3  and  5,  from  the 
apex  d,  to  the  centre  of  the  base.  When  the  axis  is  perp  to  the  base,  as  in  Figs  1,  2,  4,  the  solid  is  said 
to  be  a  right  one;  when  otherwise,  as  Figs  3.  5,  an  oblique  one.  When  the  word  cone  is  used  alone, 
the  right  circular  cone,  Fig  4,  is  understood.  If  snch  a  cone  be  cut,  as  at  t.  t,  obliquely  to  its  base,  the 
new  base  1 1  will  be  an  ellipse  ;  and  the  cone  d  1 1  becomes  an  oblique  elliptic  one.  Fig  5  will  represent 
either  an  oblique  circular  cone,  or  an  oblique  elliptic  one,  according  as  its  base  is  a  circle,  or  an  ellipse. 

To  find  the  solidity  of  any  pyramid,  or  cone, 

Whether  regular  or  irregular;  right  or  oblique;  mult  the  area  of  its  base,  by  one-third  of  its  perp 
height  d  o,  Figs  1  to  5.  Every  pyramid,  or  cone,  has  one-third  of  the  solidity  of  either  a  prism  or  a 
cylinder  having  the  same  area  of  base,  and  the  same  perp  height;  and  one-half  that  of  a  hemisphere 
of  the  same  base  and  height ;  in  other  words,  a  cone,  hemisphere,  and  cylinder  of  the  same  base  and 
height,  have  solidities  as  1,  2.  3. 

To  find  the  surface  of  any  regular  right  pyramid,  or  right 

cone. 

Mult  the  circumf  or  outline  of  its  base,  by  the  slant  height ;  take  half  the  prod.  This  will  give  the 
surf  of  the  sides  ;  to  which  add  that  of  the  base  if  reqd.  In  the  pyramid,  this  slant  height  must  be 
measd  from  d  to  the  middle  of  one  of  the  equal  sides,  and  not  along  one  of  the  edges  of  the  pyramid. 
Mathematicians  have  been  unable  to  devise  any  measurement  of  the  surf  of  an  oblique  cone. 

To  find  the  surface  of  an  irregular  pyramid. 

Whether  right  or  oblique,  each  side  must  be  calculated  as  a  separate  triaugle ;  and  the  several  areas 
added  together.  Add  the  area  of  base  if  reqd. 


MENSURATION. 


33 


To  find  the  solid- 
ity of  any  frus- 
tum of  any  pyr- 
amid, or  cone, 
when  the  base 
and  top  are  par- 
allel. 

RULE  1.  Whether  regular 
or  irregular,  right  or  oblique, 
add  together  the  areas  of  the 
base,  and  top,  and  the  area  of 
their  mean  proportional ;  mult 
the  sum  by  ^d  of  the  perp 
height  o  o,  Figs  6  and  7,  between  the  base  and  top. 

RKM.  For  the  area  of  the  mean  proportional,  (which  is  not  the  area  halfway  between,  and  parallel 
with  the  ends,)  mult  together  the  areas  of  base  and.  top ;  and  take  the  sq  rt  of  the  prod. 

RULE  2.  This  applies  only  to  the  right  circular  conical  frnst.  Add  together  the  squares  of  the  two 
diams;  and  the  prod  of  the  two  diams  ;  mult  the  sum  by  J>£d  the  perp  height  o  o ;  mult  the  prod  by. 7864. 

To  find  the  surface  of  any  frustum  of  a  regular  right  pyramid, 
or  cone,  Figs  6  and  7,  when  the  base  and  top  are  parallel. 

Add  together  the  circumfs  of  the  two  ends ;  mult  the  sum  by  the  slant  height  « t ;  take  half  the  prod. 
This  gives  the  surf  of  the  sides ;  to  which  add  that  of  the  ends  when  reqd.  In  the  frustum  of  the 
pyramid,  the  slant  height  must  be  measd  between  the  middles  s  and  t  of  two  corresponding  sides  of 
the  base,  and  top,  Fig.  6. 

If  the  pyramid  is  not  regular ;  or  if  it  is  oblique,  then  the  surf  of  the  sides  must  be  obtained  for 
each  aide  separately  as  a  trapezoid. 

WEDGES. 
aba  b       a  b 


Fid  9  "         Fi^ld" 
A  wedge 


Is  usually  defined  to  be  a  solid,  Figs  8  and  9,  generated  by  a  plane  triangle,  a  n  c,  moving  parallel  *o 
Itself,  in  a  straight  line.  This  definition  requires  that  the  two  triangular  ends  of  the  wedge  should  be 
parallel  ;  but  a  wedge  may  be  shaped  as  in  Fig  10  or  11..  We  would  therefore  propose  the  following 
definition,  which  embraces  all  the  figs  ;  besides  various  modifications  of  them.  A  solid  of  five  plane 
faces;  one  of  which  is  a  parallelogram  abed,  two  opposite  sides  of  which,  as  a  c  and  b  d,  are  united 
by  means  of  two  triangular  faces  a  c  n,  and  &  dm,  to  an  edge  or  line  n  m,  parallel  to  the  other  opposite 
sides  a  b  and  c  d.  The  parallelogram  abed  may  be  either  rectangular,  or  not  ;  the  two  triangular 
faces  may  be  similar,  or  not  ;  and  the  same  with  regard  to  the  other  two  faces.  The  following  rule 
applies  equally  to  all. 


To  find  the  solidity  of  any  wedge. 

Add  together  the  length  of  the  edge  m  n,  and  twice  the  length  a  b  or  c  d  of  the  back  ;  mult  the  su 
by  the  perp  height  p,  from  the  edge  to  the  back  ;  mult  this  prod  by  the  breadth  of  the  back,  mei 
perp  to  its  two  sides  a  b  and  c  d  ;  div  this  last  prod  by  6. 

PRISMOIDS. 


m 

eiud 


A  prismoid  is 

Any  solid  bounded  by  six  plane  surfaces,  of  which  but  two,  as  a  6  c  d  and  e  f  g  h,  Figs  1  and  2,  ar« 
necessarily  parallel ;  and  at  least  two  other  opposite  ones  not  parallel. 

To  find  the  solidity  of  any  prismoid. 

Add  together  the  areas  of  the  two  parallel  surfaces  ;  and  four  times  the  area  of  the  section  taken 
halfway  between  them,  and  parallel  to  them  ;  mult  the  sum  by  the  perp  dist  between  the  two  parallel 
Bides ;  and  div  the  prod  br  6.  In  Fig  1,  A  n  is  the  perp  dist ;  and  in  Fig  2.  which  represents  a  railroad 
excavation,  it  is  g  c.  To  find  the  areas  referred  to,  see  Trapezofds  and  Trapeziums*  PP  14<  15. 

The  foregoing  rule  is  the  well-known  "prismoidal  formula;"  the  very  extended  application  of 
which  to  other  solids  than  those  which  fall  strictly  within  the  definition  of  the  prismoid,  was  first 


34 


MENSURATION. 


discovered  and  made  known  by  Ellwood  Morris,  civ  eng  of  Philadelphia,  in  1840.  It  embraces  <# 
parallelepipeds,  prisms,  pyramids,  cones,  wedges,  &c,  whether  regular  or  irregular,  right  or  oblique; 
together  with  their  frustums  when  cut  parallel  to  their  bases;  indeed  all  solids  whatever  having  two 
parallel  faces,  or  sides,  provided  these  two  faces  are  uuited  by  surfaces,  whether  plane  or  curved,  upon 
which,  and  through  every  point  of  which,  a  straight  line  may  be  drawn  from  one  of  the  parallel  faces 
to  the  other.  The  following  six  Figs  represent  a  few  such  solids ;  they  may  be  regarded  as  one-chain 
lengths  of  railroad  cuttings ;  a  o  being  the  perp  dist  between  the  two  parallel  ends. 


o 

The  prismoidal  formula  applies  also  to  the  sphere, 
hemisphere,  and  other  spherical  segments;  also  to 
any  sections  such  as  abed,  and  o  n  i  d  6  c,  of  the 
cone,  in  which  the  sides  a  d,  a  c,  or  od,  fc,  are  straight ; 

through  the  apex,  or  top  a.  Also  to  the  cylinder 
when  a  plane  parallel  to  the  sides  passes  th  rough 
both  ends  ,  but  not  if  the  plane  w  x  is  oblique,  as 
in  the  fig,  though  never  erring  more  than  1  in 
142.  In  this  last  case  we  must  imagine  the 
plane  to  be  extended  until  it  cuts  the  side  of  the 
cylinder  likewise  extended  ;  and  then  by  p  32  or  630 
find  the  solidity  of  the  ungula  thus  formed.  Then 
find  the  solidity  of  the  small  ungula  above  w,  also 
thus  formed,  and  subtract  it  from  the  large  one. 

SPHERES    OR    GLOSSES. 
A  sphere 

Is  a  solid  generated  by  the  revolution  of  a  semicircle  around  its  diam.  Any  line  passing  entirely 
through  a  sphere,  and  through  its  center,  is  called  its  axis,  or  diam.  Any  circle  described  on  the 
surface  of  a  sphere,  from  the  center  of  the  sphere  as  the  center  of  the  circle,  is  called  a  great  circle  of 
that  sphere;  in  other  words,  any  entire  circumf  of  a  sphere  is  a  great  circle.  A  sphere  has  a  greater 
content  or  solidity  than  any  other  solid  with  the  same  amount  of  surface;  so  that  if  the  shape  of  a 
sphere  be  any  way  changed,  its  content  will  be  reduced. 

To  find  I  lie  solidity  of  a  sphere.* 

Cube  its  diam;  mult  said  cube  by  .5236.  Or  cube  its  circumf;  and  mult  by  .01689.  Or  mult  its 
surface  by  its  diam ;  and  div  by  6.  Or  refer  to  the  following  table  of  spheres.  The  solidity  of.  a  sphere 
is  %  that  of  its  circumscribing  cylinder;  or  .5236  (hat  of  its  circumscribing  cube;  or  to  4  great 
circles  X  %  diam;  or  to  cube  of  rad  X  4.1888;  or  to  surface  X  %  diam. 


To  find  the  surface  of  a  sphere. 


To  find  the  solidity  of  a  spherical  segment. 

C  RULE  1.  Square  the  rad  on,  of  its  base;  mult '  his  square  by  3  ;  to 

the  prod  add  the  square  of  its  height  o  s  ;  mult  the  sum  by  the  height 
o  s ;  and  mult  this  last  prod  by  .5236. 

RULE  2.  Mult  the  diam  aft  of  the  sphere  by  3;  from  the  prod 
take  twice  the  height  o  s  of  the  segment;  mult  the  rem  by  the  square 
of  the  height  o  s ;  and  mult  this  prod  by  .5236. 
*In  the  following  table  the  surfaces  and  solidities  will  be  inches, 
feet,  yards,  <fcc,  according  as  the  diams  are  considered  to  be  in 
inches,  feet,  yards.  *c. 

The  solidity  of  a  sphere  being  %ds  that  of  its  circumscribing  cylin- 
der, if  we  add  to  any  solidity  in  the  table,  its  half,  we  obtain  that 
of  a  cylinder  of  the  same  diam  as  the  sphere,  and  whose  height 
equals  its  diam. 

*  If  the  diam  is  measured  in  inches,  divide  the  surfaces  in  the  table  by  144.  if  it  is  required  to  reduce- 
them  to  square  feet;  and  divide  the  solidities  by  1728,  if  required  in  cubic  feet. 


MENSURATION. 
SPHERES.    (ORIGINAL.) 

Some  errors  of  1  in  the  last  figure  only. 


35 


a 

Q 

Surface. 

I 

i 

s 

1 

1 

a 

a 

s 

Surface. 

•3 
1 

| 
Q 

1 
I 

•3 
1 

02 

1-64 

.00077 

13-32 

18.190 

7.2949 

9i 

70.87 

210.03 

» 

921.33 

2629.6 

1-32 

.00307 

.00002 

7-16 

18.666 

7.5829 

H 

76.71 

220.89 

y± 

934.83 

2687.6 

3-64 

.00690 

.00005 

15-32 

19.147 

7.8783 

% 

82.66 

232.13 

% 

948.43 

2746.5 

1  16 

.01227 

.00013 

y% 

19.635 

8.1813 

% 

88.69 

243.73 

y% 

962.12 

2806/2 

3-32 

.02761 

.00043 

17-32 

20.129 

8.4919 

% 

94.83 

255.72 

% 

975.91 

2866.8 

.04909 

.00102 

9-16 

20.629 

8.8103 

8. 

201.06 

268.08 

% 

989.80 

2928.2 

5-32 

.07670 

.00200 

19-32 

21.135 

9.1366 

H 

207.39 

280.85 

% 

1003.8 

2990.5 

3-16 

.11045 

.00345 

y* 

21.648 

9.4708 

M 

213.82 

294.01 

18. 

1017.9 

3053.6 

7-32 

.15033 

.00548 

21-32 

22.166 

9.8131 

220.36 

307.58 

1032.1 

3117.7 

\i 

.19635 

.00818 

11-16 

22.691 

10.164 

y% 

226.98 

321.56 

y\ 

1046.4 

3182.6 

9-32 

.24851 

.01165 

23-32 

23.222 

10.522 

6/j 

233.71 

335.95 

% 

1060.8 

3248.5 

5-16 

.30680 

.01598 

23.758 

10.889 

H 

240.53 

350.77 

y% 

1075.2 

3315.3 

11-32 

.37123 

.02127 

25-32 

24.302 

11.265 

% 

247.45 

366.02 

% 

1089.8 

3382.9 

.44179 

.02761 

13-16 

24.850 

11.649 

9. 

254.47 

381.70 

% 

1104.5 

3451.5 

13-32 

.51848 

.03511 

27-32 

25.405 

12.041 

% 

261.59 

397.83 

y» 

1119.3 

3521.0 

7-16 

.60132 

.04385 

% 

25.967 

12.443 

y± 

268.81 

414.41 

19. 

1134.1 

3591.4 

15-32 

.69028 

.05393 

2932 

26.535 

12.853 

% 

276.12 

431.44 

% 

1149.1 

3662.8 

.78540 

•06545 

15-16 

27.109 

13.272 

% 

283.53 

448.92 

y± 

1164.2 

3735.0 

17-32 

.88664 

.07850 

31-32 

27.688 

13.700 

291.04 

466.87 

% 

1179.3 

88U8.2 

9-16 

.99403 

.09319 

3. 

28.274 

14.137 

% 

298.65 

485.31 

y<t 

1194.6 

3882.5 

19-32 

1.1075 

.10960 

1-16 

29.465 

15.039 

% 

306.36 

504.21 

% 

1210.0 

3957.6 

bA 

1.2272 

.12783 

M 

30.680 

15.979 

10. 

314.16 

523.60 

% 

1225.4 

4033.7 

21-32 

1.3530 

.14798 

3-16 

31.919 

16.957 

H 

322.06 

543.48 

V 

1241.0 

4110.8 

11-16 

1.4849 

.17014 

M 

33.183 

17.974 

M 

330.06 

563.86 

20. 

1256.7 

4188.8 

23-32 

1.6230 

.19442 

5-16 

34.472 

19.031 

338.16 

584.74 

^6 

1272.4 

4267.8 

H 

1.7671 

.22089 

% 

35.784 

20.129 

/^ 

346.36 

606.13 

H 

1288.3 

4347.8 

25-32 

1.9175 

.24967 

7-16 

37.122 

21.268 

% 

354.66 

628.04 

1304.2 

4428.8 

13-16 

2.0739 

.28084 

^ 

38.484 

22.449 

363.05 

650.46 

y% 

1320.3 

4510.9 

27-32 

2.2365 

.31451 

9-16 

39.872 

23.674 

y 

371.54 

673.42 

Y* 

1336.4 

4593.9 

% 

2.4053 

.35077 

H 

41.283 

24.942 

11. 

380.13 

696.91 

% 

1352.7 

4677.9 

29-32 

2.5802 

.38971 

11-16 

42.719 

26.254 

388.83 

720.95 

K 

1369.0 

4763.0 

15-16 

2.7611 

.43143 

H 

44.179 

27.611 

K 

397.61 

745.51 

21. 

1385.5 

4849.1 

31-32 

2.9483 

.47603 

13-16 

45.664 

29.016 

406.49 

770.64 

H 

1402.0 

4*36.2 

1. 

3.1416 

.52360 

H 

47.173 

30.466 

y% 

415.48 

796.33 

1418.6 

5024.3 

1-32 

3.3410 

.57424 

15-16 

48.708 

31.965 

% 

424.56 

822.58 

% 

1435.4 

5113.5 

1-16 

3.5466 

.62804 

4. 

50.265 

33.510 

% 

433.73 

849.40 

y% 

1452.2 

5203.7 

3-32 

3.7583 

.68511 

1-16 

51.848 

35.106 

i/ 

443.01 

876.79 

% 

1469.2 

5295.1 

3.9761 

.74551 

M 

53.456 

36.751 

12. 

452.39 

904.78 

% 

1486.2 

5387.4 

5-32 

4.2000 

.80939 

3  16 

55.089 

38.448 

M 

461.87 

933.34 

% 

1503.3 

5480.8 

3-16 

4.4301 

.87681 

56.745 

40.195 

y* 

471.44 

962.52 

22. 

15'20.5 

5575.3 

7-32 

4.6664 

.94786 

5-16 

58.427 

41.994 

H 

481.11 

992/28 

H 

15379 

5670.8 

4.9088 

1.0227 

H 

60.133 

43.847 

y* 

490.87 

1022.7 

X 

1555.3 

5767.6 

9-32 

5.1573 

1.1013 

7-16 

61.863 

45.752 

500.73 

1053.6 

1572.8 

5865.2 

5-16 

5.4119 

1.1839 

63.617 

47.713 

% 

510.71 

1085.3 

y* 

1590.4 

5964.1 

11-32 

5.6728 

1.2704 

9-16 

65.397 

49.729 

TX 

520.77 

1117.5 

% 

1608.2 

6064.1 

5.9396 

1.3611 

67.201 

51.801 

13. 

530.93 

1150.3 

% 

1626.0 

6165.2 

13-32 

6.2126 

1.4561 

11-16 

69.030 

53.929 

541.19 

1183.8 

1643.9 

6267.3 

7-16 

6.4919 

1.5553 

H 

70.88S 

56.116 

M 

551.55 

1218.0 

23.    * 

1661.9 

6370.6 

15-32 

67771 

1.6590 

13-16 

72.759 

58.359 

56200 

1252.7 

y* 

1680.0 

6475.0 

7.0686 

1.7671 

H 

74.663 

60.663 

^ 

572.55 

1288.3 

y± 

1698.2 

6580.6 

17-32 

7.3663 

1.8799 

15-16 

76.589 

63.026 

583.20 

1324.4 

1716.5 

6687.3 

9-16 

7.6699 

1.9974 

5. 

78.540 

65.450 

% 

593.95 

1361.2 

jl 

1735.0 

6795.2 

19-32 

7.9798 

2.1196 

1-16 

80.516 

67.935 

K 

604.80 

1398.6 

1753.5 

6904  2 

% 

8.2957 

2.2468 

x^j 

82.516 

70.482 

14. 

615.75 

1436.8 

% 

1772.1 

7014.3 

21-32 

8.6180 

2.3789 

3-16 

84.541 

73.092 

y* 

626.80 

1475.6 

K 

1790.8 

7125.6 

11-16 

8.9461 

2.5161 

H 

86.591 

75.767 

y± 

637.95 

1515.1 

24. 

1809.6 

7238.2 

23-32 

9.2805 

2.6586 

5-16 

88.664 

78.505 

% 

649.17 

15S5.3 

y* 

1828.5 

7351.9 

H 

9.6211 

2.8062 

H 

90.763 

81.308 

M 

660.52 

1596.3 

y* 

1847.5 

7466.7 

25-32 

9.9678 

2.9592 

7-16 

92.887 

84.178 

% 

671.95 

1637.9 

% 

1866.6 

7583.0 

13-16 

10.321 

3.1177 

y% 

95.033 

87.113 

683.49 

1680.3 

y% 

1885.8 

7700.1 

27-32 

0.680 

3.2818 

9-16 

97.205 

90.118 

% 

695.13 

1723.3 

H 

1905.1 

7818.6 

% 

1.044 

3.4514 

M 

99.401 

93.189 

15. 

706.85 

1767.2 

1924.4 

7938.3 

29-32 

1.416 

3.6270 

11-16 

101.62 

96.331 

/^ 

718.69 

1811.7 

K 

1943.9 

8059.2 

15-16 

1.793 

3.8083 

% 

103.87 

99.541 

M 

730.63 

1857.0 

25. 

1963.5 

8181.3 

31-32 

2.177 

3.9956 

13-16 

106.14 

102.82 

742.65 

1903.0 

H 

1983.2 

8304.7 

2. 

2.566 

4.1888 

% 

108.44 

106.18 

y% 

754.77 

1949.8 

2002.9 

8429.2 

1-32 

2.962 

4.3882 

15-16 

110.75 

09.60 

767.00 

1997.4 

2022.9 

8554.9 

1-16 

3.364 

4.5939 

6. 

113,10 

13.10 

H 

779.32 

2045.7 

¥ 

2042.8 

8682.0 

3-32 

3.772 

4.8060 

117.87 

20.31 

791.73 

2094.8 

2062.9 

8810.3 

4.186 

5.0243 

H 

122.72 

27.83 

16. 

804.25 

2144.7 

H 

2083.0 

8939.9 

5-32 

4.607 

5.2493 

127.68 

35.66 

H 

816.85 

2195.3 

% 

2103.4 

9070.6 

3-16 

5.033 

5.4809 

132.73 

43.79 

829.57 

2246.8 

26. 

21-23.7 

9202.8 

7-32 

5.466 

5.7190 

Y* 

137.89 

152.25 

y 

842.40 

2299.1 

2144.2 

9336.2 

y± 

15.904 

5.9641 

% 

143.14 

161.03 

y% 

855.29 

2352.1 

\s 

2164.7 

9470.9 

9-32 

16.349 

6.2161 

r/ 

148.49 

170.14 

% 

868.S1 

2406.0 

ax 

2185.5 

9606.? 

5-16 

16.800 

6.4751 

7. 

153.94 

179.59 

% 

881.42 

2460.6 

y% 

2206.2 

9744.6 

11-32 

17.258 

6.7412 

15949 

189.39 

y 

894.  6S 

2516.1 

bz 

2227.1 

9882.4 

H 

17.721 

7.0144 

y\ 

165.13 

199.53 

17. 

907.93 

2572.4 

% 

2248.0 

10022 

36 


MENSURATION. 


SPHERES  —  (CONTINUED.) 


I 

1 

I 

1 

1 

1 

a 

.2 

i 

I 

a 

a 

I 

1 

5 

I 

1 

Q 

3 
0! 

1 

5 

1 

'•3 

02 

5 

s 
to 

1 

M 

2269.1 

10164 

~% 

4214.1 

25724 

% 

6756.5 

52222 

H 

98G6.0 

92570 

27. 

22S0.2 

10306 

H 

4243.0 

25988 

rz 

6792.9 

52645 

H 

9940.2 

S31SO 

2311.5 

104.00 

4271.8 

26254 

fix 

6829.5 

53071 

% 

9984.4 

93812 

x4 

2332.8 

10595 

37. 

4300.9 

26522 

8£ 

6866.1 

53499 

10029 

94438 

2354.3 

10741 

4330.0 

267S2 

% 

6902.9 

53929 

% 

10073 

95066 

JX 

2375.8 

108b9 

IX 

4359.2 

27063 

47. 

6939.9 

54362 

K 

10118 

95697 

fi/ 

2397.5 

11038 

a/ 

4388.5 

27337 

6976.8 

54797 

10163 

S6330 

ax 

2419.2 

11189 

IX 

4417.9 

27612 

IX 

7013.9 

55234 

57. 

10207 

S6967 

y 

2441.  1 

11341 

xi 

4447.5 

27889 

% 

7050.9 

55674 

1C252 

97606 

28. 

2463.0 

11494 

% 

4477.1 

28168 

% 

7088.3 

56115 

J4 

10297 

98248 

2485.1 

11649 

% 

4506.8 

28449 

6X 

7125.6 

56559 

n 

10342 

98893 

IX 

2507.2 

11805 

38. 

4536.5 

28731 

ax 

7163.1 

570C6 

% 

1C387 

99541 

az 

2529.5 

11962 

4566.5 

2S016 

TX 

7200.7 

57455 

y% 

10432 

100191 

TX 

2551.8 

12121 

x4 

4596.4 

29302 

48. 

7238.3 

579C6 

H 

10478 

100845 

RX 

2574.3 

12281 

4626.5 

29590 

7276.0 

5£3€0 

10523 

101501 

ax 

2596.7 

12443 

IX 

4656.7 

29880 

JX 

7313.9 

5S815 

58. 

10568 

102161 

y 

2619.4 

126C6 

H 

4C86.9 

30173 

xt 

7351.9 

59274 

10614 

102823 

29. 

2642.  1 

12770 

H 

4717.3 

30-:  66 

7389.9 

59734 

x4 

10660 

103488 

2665.0 

12S36 

4747.9 

30762 

N 

7428.0 

60197 

% 

10706 

104155 

rx 

2687.8 

13103 

39. 

4778.4 

31059 

7466.3 

606C3 

% 

10751 

104826 

% 

2710.9 

13272 

4809.0 

31359 

TX 

7504.5 

61131 

y* 

10798 

105499 

IX 

2734.0 

13442 

•y 

4839.9 

31661 

49. 

7543.1 

61601 

% 

10844 

106175 

RX 

2757.3 

13614 

a/. 

4870.8 

31964 

H 

7581.6 

62074 

TX 

10890 

106854 

% 

2780.5 

13787 

x^ 

4S01.7 

32270 

7620.1 

62549 

59. 

10936 

107536 

% 

2804.0 

13961 

N 

4932.7 

32577 

% 

7658.9 

63026 

K 

10983 

108221 

80. 

2827.4 

14137 

H 

4964.0 

32886 

rx 

7697.7 

63506 

M 

11029 

108909 

2851.1 

14315 

4995.3 

33197 

RX 

7736.7 

63989 

n 

11076 

109600 

u 

2874.8 

14494 

40. 

5026.5 

33510 

ax 

7775.7 

64474 

7& 

11122 

1  10294 

2898.7 

14674 

M 

5058.1 

33826 

% 

7814.8 

64861 

y* 

11169 

110990 

J^X, 

2922.5 

14856 

M 

5089.6 

34143 

50. 

7854.0 

65450 

% 

11216 

111690 

2946.6 

15039 

5121.3 

34462 

7893.3 

65941 

11263 

112392 

»/ 

2970.6 

15224 

IX 

5153.  1 

347S3 

TX 

7932.8 

66436 

60. 

11310 

113098 

If 

2994.9 

15411 

% 

5184.9 

35106 

% 

7972.2 

66934 

Ji 

11357 

113806 

81. 

3019.1 

15599 

ax 

5216.8 

35431 

IX 

8011.8 

67433 

Ji 

11404 

114518 

3043.6 

15788 

TX 

5248.9 

35758 

RX 

8051.6 

67935 

•2 

11452 

115232 

i/ 

3068.0 

15979 

41. 

5281.1 

36087 

H 

8091.4 

68439 

J6 

11499 

115949 

a/ 

3092.7 

16172 

5313.3 

36418 

y* 

8131.3 

68946 

S 

11547 

116669 

?x 

3117.3 

16366 

TX 

5345.6 

36751 

51. 

8171.2 

694£6 

% 

11595 

117392 

% 

3142.1 

16561 

% 

5378.1 

37086 

xi 

8211.4 

69967 

% 

11642 

118118 

P 

3166.9 

16758 

1Z 

5410.7 

37423 

M 

8251.6 

7C482 

61. 

11690 

118847 

3192.0 

16957 

!N? 

5443.3 

37763 

N 

8292.0 

709S9 

/^ 

11738 

119579 

82.    * 

3217.0 

17157 

H 

5476.0 

38104 

8332.3 

71519 

J4 

11786 

120315 

3242.2 

17359 

5508.9 

38448 

RX 

8372.8 

72040 

% 

11  £-34 

121053 

s 

3267.4 

17563 

42. 

5541.9 

38792 

ax- 

8413.4 

72565 

% 

11882 

121794 

% 

3292.9 

17768 

5574.9 

39140 

\/ 

8454.1 

73092 

% 

11931 

122538 

JX 

3318.3 

17974 

IX 

5608.0 

39490 

52. 

8494.8 

73622 

% 

11  980 

123286 

% 

3343.9 

18182 

a/ 

5641.3 

39841 

xi 

8535.8 

74154 

% 

12028 

124036 

3369.6 

18392 

IX 

5674.5 

40194 

x4 

8576.8 

74689 

62. 

12076 

124789 

«x 

3395.4 

18604 

KX 

5708.0 

40551 

8617.8 

75226 

1^5 

12126 

125545 

83.    * 

3421.2 

18817 

ax 

5741.5 

40908 

A 

8658.9 

75767 

x4 

12174 

126305 

3447.3 

19032 

TX 

5775.2 

41268 

8700.4 

76309 

^8 

12223 

127067 

IX 

3473.3 

19248 

43.    8 

5808.8 

41630 

% 

8741.7 

76854 

M 

12272 

'127^32 

ax 

3499.5 

19466 

IX 

5842.7 

41994 

8783.2 

77401 

N 

12322 

128601 

JX 

3525.7 

19685 

y* 

5876.5 

42360 

53. 

8824.8 

77952 

12371 

129373 

RX 

3552.  1 

19907 

5910.7 

42729 

iff 

8866.4 

78505 

K 

12420 

130147 

«x 

3578.5 

20129 

TX 

5944.7 

43099 

i/ 

8S08.2 

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9076.4 

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3712.3 

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9118.5 

81876 

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3739.3 

21501 

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6151.5 

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12768 

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6186.3 

45753 

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9203.3 

83021 

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9331.2 

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9460.2 

86521 

xi     13121 

141320 

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3959.2 

23425 

IX 

6432.7 

48513 

55. 

9.503.2 

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%     1H172 

142142 

RX 

3987.2 

23674 

% 

6)68.3 

48916 

u 

9546.5 

87709 

%  :    132^2 

142966 

x4 

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6503.9 

49321 

9590.0 

88307 

65.          13273 

143794 

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4043.3 

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6T39.7 

49729 

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9633.3 

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%     13324 

144625 

86. 

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24429 

ax 

6575.5 

50139 

9676.8 

89511 

%'    13376 

145460 

4099.9 

24685 

% 

6611.6 

50551 

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90117 

%:    13427 

146297 

I/ 

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J^!    1347-8 

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ax 

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4185.5 

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6720.0  1    51801 

56. 

9852.0 

91953          H>  13582 

148828 

MENSURATION. 
SPHERES— (CONTINUED.) 


37 


I 

s 

1 

3 
O3 

1 

5 

Surface. 

1 

I 

Q 

<£ 
1 

1 

S 

s 

Ul 

1 

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150533 

% 

17496 

217597 

34 

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26518 

406060 

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151390 

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17554 

218693 

21839 

303463 

92. 

26590 

407721 

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152251 

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219792 

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75. 

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21970 

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412726 

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23(534 

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23575 

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H 

28577 

454259 

% 

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176723 

34 

19237 

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23711 

343307 

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N 

15284 

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% 

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252077 

87.  * 

23779 

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% 

19422 

254496 

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% 

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* 

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255713 

23984 

349269 

96. 

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24191 

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29180 

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•% 

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19919 

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97. 

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%   16005 

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80. 

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y% 

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481579 

34   16061 

191389 

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% 

24745 

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% 

29788 

483438 

% 

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192395 

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29865 

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% 

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34   25589 

384894 

v 

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% 

20996 

286061 

%   25660 

386496 

99. 

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508047 

34 

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205789 

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287378 

34  :  25730 

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.  Y%   25802 

389711 

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21  IS!) 

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21318 

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% 

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71. 

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% 

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34   26159 

% 

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% 

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%  j  26230 

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100. 

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402756 

To  find  the  curved  surface  of  a  spherical  segment. 

See  last  Fig.  KUT.E  1.  Mult  the  dinm  a  b  of  the  sphere  from  which  the  segment  is  cut,  by  3.1416; 
mult  the  prod  by  the  height  o  s  of  the  seg.  Add  the  area  of  base  reqd.  REM.  Having  the  diam  n  r 
of  the  seg,  and  its  height  o  s,  the  diam  a  b  of  the  sphere  may  be  found  thus :  Div  the  square  of  half 
the  diam  n  r,  by  its  height  o  s ;  to  the  quot  add  the  height  o  s.  RULE  2.  The  curved  surf  of  either 
ft  segment,  (last  Fig,)  or  of  a  zouc,  (next  Fig,)  bears  the  same  proportion  to  the  surf  of  the  whole 


38 


MENSURATION. 


sphere,  that  the  height  of  the  seg  or  zone  bears  to  the  diam  of  the  sphere.  Therefore,  first  find  tke 
surf  of  the  whole  sphere,  either  by  rule  or  from  the  preceding  table  ;  mult  it  by  the  height  of  the  sez 
or  zone;  div  the  prod  by  diam  of  sphere.  RULE  3.  Mult  the  circumf  of  the  sphere  by  the  height  o  t 
of  the  seg. 

To  find  the  solidity  of  a  spherical  zone* 

Add  together  the  square  of  the  rad  e  d,  the  square  of  rad  o  b 
and  %d  of  the  square  of  the  perp  height  eo;  mult  the  sum  by 
1.5708;  and  mult  this  prod  by  the  height  eo. 

To  find  the  curved  surface  of  a  spher- 
ical zone. 

RULE  1.  Mult  together  the  diarn  m  n  of  the  sphere  ;  the  height 
e  o  of  the  zone,  and  the  number  3.1416.  Or  see  preceding  Rule  2 
for  snrf  of  segments.  Rule  2.  Mult  the  circumf  of  the  sphere,  by 
the  height  of  the  zone. 

To  find  the  solidity  of  a  hollow  spher- 
ical shell. 

Take  from  the  foregoing  table  the  solidities  of  two  spheres  having 

bthe  diams  a  6,  and  c  d.    Subtract  the  least  from  the  greatest.    Here 
a  cor  b  d  is  the  thickness  of  the  shell. 

THE    REGULAR    BODIES. 
A  regular  body,  or  regular  polyhedron. 

Is  one  which  has  all  its  sides,  and  its  solid  angles,  respectively 
similar  and  equal  to  each  other.  There  are  but  five  such  bodies, 
namely,  the  Tetraedron,  bounded  by  four  equilateral  triangles  ;  the  Hexaedron,  or  cube,  bounded  by 
six  squares ;  the  Octaedron,  bounded  by  eight  equilateral  triangles ;  the  Dodecaedron,  bounded  by 
twelve  equilateral  pentagons;  and  the  Icosaedron,  bounded  by  twenty  equilateral  triangles. 

To  find  the  solidity  or  the  surface  of 
a  regular  body. 

For  the  solidity,  cube  the  length  of  one  of  its  edges; 
mult  said  cube  by  the  correspouding  number  in  the  fol- 
lowing column  of  solidities.  For  the  surface,  square  the 
length  of  one  of  its  edges  ;  mult  said  square  by  the  cor- 
responding number  in  the  following  column  of  surfs. 

The  solidity  of  any  body  as  a  &  cm, 

generated  by  a  complete  revolution  of  any  fig  as  a  b  c  a, 
around  one  of  its  sides  as  a  c  for  an  axis,  may  be  found 
thus.  Find  the  area  of  the  generating  fig  a  b  c  a;  also 

find  its  cen  of  grav  G,  (see  top  of  p  443.)  Measure  o  G  perp  to  a  c,  and  calling 
o  G  the  rad  of  a  circle  find  the  corresponding  circumf.  Mult  this  circumf  by  the 

area  of  the  fig  a  b  c  a  before  found.  In  other  words,  mult  the 
area  of  the  generating  fig  by  the  circumf  of  the 
circle  described  by  its  cen  of  grav  while  revolv- 
ing. 

THE  ELLIPSOID,  OR  SPHEROID, 

Is  a  solid  generated  by  the  revolution  of  an  ellipse  around  either  its  long  or  its  short  diam.     When 

around  the  long  (or  trans  verse)  diara,  as  at  a,  Fig  1,  it  is  an  oblong  or  pro- 
late spheroid;  when  around  the  short  (or  conjugate)  one,  as  at  in,  in  Fig  2, 
it  is  oblate. 


STames. 

Surf. 

Solid. 

Tetraedron  
Hexaedi-on  
Octaedron  
Dodecaedron  .. 
Icosaedron  

1.7320 
6. 
3.4641 
20.6458 
8.6602 

.1178 
1. 
.4714 
7.6631 
2.1817 

For  the  solidity  in  either  case,  mult  the  fixed  diam  or  axis  by  the  square 

of  the  revolving  one  ;  and  mult  the  prod  by  .5236. 

THE  PARABOLOID,  OR  PARABOLIC  COXOID, 

next  Fig,  is  a  solid  generated  by  the  revolution  of  a  parabola  a  c  b,  around  its  axis,  c  r. 

For  its  solidity  mult  the  area  of  its  base,  by  half  its  height,  r  c.    Or  inult 

together  the  square  of  the  rad  a  r  of  the  base;  the  height  r  c;  and  the  number  1.5708. 


PLANE   TRIGONOMETRY. 


39 


For  the  solidity  of  a  frustum, 

a  b  g  7i,  the  ends  of  which  are  perp  to  the  axis  r  c  ;  add  together  th» 
squares  of  the  two  diams  a  It  and  g  h  ;  mult  the  sum  b 
mult  the  prod  by  the  decimal  .3927. 


by  the  height  r  I; 


To  find  the  surface  of  a  paraboloid, 

Mult  the  rad  a  r  of  its  base,  by  6.2832;  div  the  prod  by  12  times  the 
square  of  the  height  re:  call  the  quot  p.  Then  add  together  the  square 
of  the  rad  a  r,  and  4  times  the  square  of  the  height  r  c.  Cube  the 
sum ;  take  the  sq  rt  of  this  cube ;  from  the  sq  rt  subtract  the  cube  of 
the  rad  a  r.  Mult  the  rent  by  p. 

Either  the  solidity,  or  the  surface  of  a  frustum,  a  b  g  h,  when  gh  is 
parallel  to  a  6,  may  be  found  by  calculating  for  the  whole  paraboloid, 
and  for  the  upper  portion  c  g  h,  as  two  separate  paraboloids,  and  taking  their  diff. 

THE    CIRCULAR   S  P I X  » I,  E, 

Is  a  solid  abny  generated  by  the  revolution  of  a  circular  segment 
a  b  n  e  a,  around  its  chord  a  n  as  an  axis. 

To  find  its  solidity. 

RULE  1.   First  find  the  area  of  a  b  e,  or  half  the  generating  circular 
segment.    Then  to  the  square  of  a  e,  add  the  square  of  b  e;  div  the  sum 
by  6  y ;  from  the  quot  take  b  e :  mult  the  rein  by  the  area  of  a  e  b ;  call 
theprodp.   Cubeae;  div  the  cube  by  3 ;  from  the  quot  take  p.   Multthe/ 
rem  by  12.5664. 

RULE  2.   When  the  dist  o  c  is  known,  from  the  center  of  the  circle  to  I 
the  cen  of  the  spindle,  then  mult  that  dist  o  e,  by  the  area  of  a  b  e ;  call  ' 
the  prod  p ;  cube  ae;  div  the  cube  by  3  ;  from  the  quot  take  p ;  mult  the 
rem  by  12.5664. 

To  find  its  surface. 

RULK  1.   First  find  the  length  of  the  circular  arc  a  b  n ;  and  mult  it  hy 
the  dist  o  e  from  the  center  of  the  circle  to  the  center  of  the  spindle.  Call 
the  prod  p.    Next  mult  the  length  an  of  the  spindle,  by  the  rad  o  b  of  the  circle.    From  the  prod 
takep;  mult  the  rem  by  6.2832. 

RULE  2.  First  find  the  length  of  the  arc  a  b  n.  Square  a  e ;  also  square  b  e ;  add  these  squares 
together;  div  their  sum  by  b  y  ;  call  the  quot  s;  and  mult  it  by  a  n;  call  the  prod  p.  Next  from  stake 
b  e  ;  mult  the  rem  by  the  length  of  the  arc  a  b  n.  Subtract  the  prod  from  p  ;  mult  the  rem  by  6.2832. 

To  find  the  solidity  of  a  middle  zone  of  a  circular  spindle, 

As  h  a  Jcp 

I  (a  e3  —  -^jp)  X  g  «)  —  (o  e  X  area  of  g  h  I  k\  1  X  6.2832. 


PLANE  TRIGONOMETRY. 


;  given  side  :  reqd  side. 


PLANS  trigonometry  teaches  how  to  find  certain  unknown  parts  of  plane,  or  straight  -  sided  tri- 
angles, by  means  of  other  parts  which  are  known;  and  thus  enables  us  to  measure  inaccessible  dis- 
tances, &c.  A  triangle  consists  of  six  parts,  namely,  three  sides,  and  three  angles;  and  if  we  know 
any  three  of  these,  (except  the  three  angles,  and  in  the  ambiguous  case  under  "  Case  2,")  we  can  find 
the  other  three.  The  following  fcur  cases  include  the  whole  subject;  the  student  should  commit  them 
to  memory. 

Case  1.    Having?  any  two  angles,  and  one  side,  to  find  the 
other  sides  and  angle. 

Add  the  two  angles  together;  and  subtract  their  sum  from  180°;  the  rem 
will  be  the  third  angle.     And  for  the  sides,  as 

Sine  of  the  angle      .  Sine  of  the  angle 

opp  the  given  side    •  opp  the  reqd  side 

Use  the  side  thus  found,  as  the  given  one;  and  in  the  same  manner  find 
the  third  side.     As  a  practical  example  of  the  use  of  Case  1,  if  we  measure 

the  line  or  side  a  b   along  one  shore  of  a  river;  and  also  the  two  angles  ^  _ 

a  b  c,  b  a  c,  to  an  object  c  on  the  opposite  shore,  we  can  calculate  the  dist  ac,          /  \ 

or  6  c,  to  that  object;  or  by  drawing  the  triangle  on  paper,  to  a  scale,  we  can        -J *» 

afterward  measure  a  c,  or  6  c  by  the  scale.  D 

Case  2.  Having  two  sides,  and  the  angle  opposite  to  one  of 

them, 

To  find  the  other  side,  and  angles.     This  is  of  use  only  when  that  gives  side  which  is  opp  the  given 
angle,  is  as  long,  or  longer  than  the  other  given  side. 

Side  opp  the  .  Side  opp  the  .  .  Sine  of  angle      .  Sine  of  angle 
given  angle    •  reqd  angle      •  •  opp  the  former    •  opp  the  latter. 
Having  found  the  sine,  take  out  the  corresponding  angle  from  the  table  of  nat  sine". 
REM.  When  the  given  side  a  c.  next  Fig,  opposite  the  given  angle  a  b  c,  is  shorter  than  the  other  given 
side  a  b,  then  the  above  proportion  alone  does  not  enable  us  to  determine  the  other  parts  ;  for  it  is 
plain  that  after  having;  drawn  the  side  a  6.  and  laid  down  the  given  angle  a  b  c,  and  extended  it  indefi- 
^itely  toward  n,  we  cannot  tell  whether  the  length  of  a  b  is  to  be  drawn  from  a  to  c,  or  from  a  to  d. 


40 


PLANE   TRIGONOMETRY. 


It  a  c  la  as  long  or  longer  than  a  6,  there  can  be  no  douot  ;  for  in  that  case  It 
cannot  be  drawn  toward  6,  but  only  toward  n,  as  in  the  Fig.  Therefore,  in  taking 
field-notes,  the  two  sides  aud  the  angle  opp  one  of  them,  must  not  be  depended 
upon  as  sufficient  data. 

Case  3.    Having*  two  sides,  and  the  angle  in- 
cluded between  them. 

Take  the  angle  from  180°  ;  the  rem  will  be  the  sum  of  the  two  unknown  angles. 
Div  this  sum  by  2  ;  and  find  the  uat  tang  of  the  quot.  Then  as 

The  sum  of  the    .    rrhpjr,i:ff  .  .  Tang  of  half  the  sum  .  Tang  of  half 

two  given  sides    •  •  •  of  the  two  unknown  angles    •  their  diff. 

Take  from  the  table  of  nat  tai>g,  the  angle  opposite  this  last  tang.  Add  thir 
angle  to  the  half  sum  of  the  two  unknown  angles,  aud  it  will  give  the  angle  opp  th( 
longest  given  side;  and  subtract  it  from  the  same  half  sum,  for  the  angle  opp  tho 
shortest  given  side.  Having  thus  found  the  angles,  find  the  third  side  by  Case  1- 


As  a  practical  exam  pie  of  theuse^of  Ci 

n 


- 

,  can  ascertain  the  dist  n  m  across 
a  deep  pond,  by  measuring  two 
lines  n  o  and  m  o  ;  and  the  an- 
gle 71  o  m.  From  this  data  we 
may  calculate  nm;  or  by  draw- 
ing  the  two  sides,  and  the  angle 
on  paper,  by  a  scale,  we  can 
afterward  measure  n  m  on  the 
drawing. 

Case  4.  Having  the 
three  sides, 

To  find  the  three  angles  :  upon  one  side  a  6  as  a  base,  draw  (or  suppose  to  be  drawn)  a  perp  c  g  from 
the  opposite  angle  c.   Find  the  ditf  between  the  other  two  sides,  a  c  and  c  6  :  also  their  sum.   Then,  aa 


Th    ,         .  Sum  of  the  .  .  Diff  of  other  .   Diff  of  the  tw 

s    •  other  two  sides    •  •  two  sides        •  parts  ag  and 


half  that  angle  will  be  equal  to 


ag  and  bg,  of  the  base. 

Add  half  this  diff  of  the  parts,  to  half  the  base  a  b  ;  the  sum  will  be  the  longest  part  a  g  ;  which 
taken  from  the  whole  base,  gives  the  shortest  part  g  b.  By  this  means  we  get  in  each  of  the  small  tri- 
angles a  c  ff  and  cgb,  two  sides,  (namely,  a  c  and  <i  g;  and  c  b  aud  g  b  ;)  and  an  angle  (namely,  the 
right  angle  cya,  or  c  g  b)  opposite  to  one  of  the  given  sides.  Therefore,  use  Case  2  for  finding  the 
angles  a  and  6.  When  that  is  done,  take  their  sum  from  180°,  for  the  angle  a  c  6. 

Or,  «d  mode:  call  half  the  sum  of  the  three  sides,  s;  and  call  the 
two  sides   which  form  either  angle,  m  and  n.    Then  the  uat  sine  of 

(S  —  TO)  X    (fj^«) 

wt  X  n 

Ex.  1.    To  find  the  dist  from  ft  to  an  inac- 
cessible object  c. 

Measure  a  line  a  6  ;  and  from  its   ends  measure  the  angles  c  a  6  and 
cb  a.    Thus  having  found  one  side  and  two  angles  of  the  triangle  a  b  c, 
calculate  a  c  by  means  of  Case  1.     Or  if  extreme  accuracy  is  not  reqd, 
L       draw  the  line  a  6  on  paper  to  any  convenient  scale  ;  then  by  means  of  a 
"       protractor  lay  off  the  angles  c  a  b,  c  b  a;  and  draw  a  c  and  c  b  ;  then 
measure  a  c  by  the  same  scale. 

Ex.  2.    To  find  the  height  of  a  vertical 
object,  n  a. 

Place  the  instrument  for  measuring  angles,  at  any  conve- 
nient spot  o  ;  also  meas  the  dist  o  a  ;  or  if  o  a  cannot  be'aotually 
measd  in  consequence  of  some  obstacle,  calculate  it  by  the 
same  process  as  a  c  in  Fi(?  1.  Then,  first  directing  the  instru 
ment  horiron  tally,*  as  o  9.  measure  the  angle  of  depression. 
s  o  a,  say  12°:  also  the  angle  s  o  n,  say  30°.  These  two  angles 
added  together,  give  the  angle  a  o  n,  42°.  Now.  in  the  small 
triangle  o  s  a  we  have  the  angle  osa  equal  to  90°.  because  n  n 
is  vert,  anr*  o  .s  nor  :  and  since  the  three  angles  of  any  triangle 
are  equal  to  180°,  if  we  subtract  the  angles  osa  (90°),  and  so  a 
H2°)  from  180°.  the  rem  (78-)  will  be  the  angle  o  «  s  or  o  «  n. 
Therefore,  in  the  triangle  on  a.  we  have  one  side  o  a;  and  two 
angles  a  o  n,  and  o  a  n,  to  calculate  the  side  a  n  by  Case  1. 


*  Angles  aiiddists  on  sloping  ground  must  be  measured  hor- 
izontally.   The  graduated  hor 

circle  of  the  instrument  evidently  meas- 

••-"•'•*  n  \  ures  the  an#le  between  two  objects  hori- 

/|\w    \  zontally,  no  matter  how  much  hipher one 

/I  \  ^    \  of  them  may  be  than  the  other:  one  per- 

haps requiring  the  telescope  of  the  instru- 
ment to  be  directed  upward  toward  it; 
and  the  other  downward.  If,  therefore, 
the  sides  of  triangles  lyinc?  upon  sloping 

•-  be  no  accordance  between  the  two.    Tbu.-\ 


PLANE   TRIGONOMETRY. 


41 


RKM.  If,  as  In  Fig  3,  it  should  be  necessary  to  ascertain  the  vert  height  an  from  a  point  o,  entirely 
above  it,  then  both  the  angles  measd  at  o,  namely,  son,  and  s  o  a,  will  be  angles  of  depression,  or 
below  the  hor  line  o  s  assumed  to  measure  them  from.  In  this  case  we  have  the  side  o  a  as  before ; 
the  angle  noa  =  so  a  —  son;  and  the  angle  o  a  n  =  180°  —  (o  s  a  (90°),  and  «  o  a  i)  to  calculate 
an  by  Case  1. 


Or  If,  as  in  Fig  4,  the  observations  are  to  be  taken  from  a  point  o,  entirely  below  the  object  c 
both  the  angles  s  o  a,  a  o  n,  will  be  angles  of  elevation,  or  above  the  assumed  hor  line  e 
have  in  the  triangle  o  n  a,  the  given  side  o  o  as 
before ;  the  angle  a  o  n  =  8  on  —  so  a;  and  the 
angle  on 0  =  180° —  (os  n  (90°),  and  no  «,  ^  to 

calculate  a  n  by  Case  1. 

If  the  object  an,  as  in  Fig  5,  instead  of  being 
vert,  is  inclined;  and  instead  of  its  vert  height, 
we  wish  to  find  its  length  a  n,  we  must  first  as- 
certain its  angle  y  t  i  of  inclination  to  the  hori- 
zon ;  to  which  angle  each  of  the  angles  osn  will 
be  equal.  To  find  this-angle  y  t  i,  suspend  a  plumb- 
line  i  y,  of  any  convenient  kuown  length,  from  the 
object  a  n ;  and  measure  also  y  t  horizontally. 
Then  say  as 

y  t  :  i  y  :  :  1  :  nat  tang  of  angle  yt  i. 

From  the  table  of  nat  tangs  take  out  the  angle 
y  t  i  found  opposite  this  nat  tang;  and  use  it  for 
the  angles  osnor  osa;  instead  of  the  90°  of  Figs 
8  and  4.  Also  when  the  object  inclines,  the  side 
o  o  of  the  triangle  must  be  measd  in  line,  or  in 
range  with  the  inclination.  If  the  object,  as  the 
rock  a  n,  Fig  6,  is  curved  or  irregular,  a  pole  a  a 
may  be  planted  sloping  in  the  direction  a  n  ;  and 


in  the  triangle  a  b  c,  upon  sloping  ground,  the  in 
refore,  the  side  which  c 


t  at  o.  measures  the  hor  angle  ion;  and  not 

the  angle  bac.  Therefore,  the  side  which  corresponds  with  this  hor  angle  i  o  n,  is  the  hor  dist  i  n  ; 
and  not  the  sloping  dist  b  c.  In  other  words,  when  sides  and  angles  are  on  sloping  ground,  we  do 
not  seek  their  actual  measures  ;  but  their  hor  ones.  This  remark  applies  to  all  surveying  for  farms, 
railroads,  triangulations  of  countries,  &c,  &c  ;  and  the  want  of  a  strict  atteutiou  to  it,  is  one  cause 
of  the  small  errors,  almost  unavoidable,  (and  fortunately,  of  but  trifling  consequence  in  practice), 
which  occur  in  all  ordinary  field  operations.  See  p  98. 

When  a  sextant  is  used,  angles  between  objects  at  diff  altitudes,  asp  and 

q,  may  be  measd  hor,  by  first  planting  two  vert  rods 
o  and  s,  in  range  with  the  objects;  and  then  taking 
the  hor  angle  o  n  s,  subtended  by  the  rods. 

Angles  may  be  meascl  without 
any  iiist,  thus:  Measure  100  ft  toward 

each  object,  and  drive  stakes;  measure  the  dist  across  . 

from  one  stake  to  the  other.     Half  this  dist  wjll  be 

the  sine  of  half  the  angle  to  a  rad  of  100  :  and  if  we  move 

the  decimal  point  two  places  to  the  left,  we  get  the  nat 

sine  of  this  one  half  of  the  angle  to  a  rad  of  1,  as  in  the 

inhles.     Thus,  suppose  the  dist  to  be  80.64  feet  ;  then 

40.32  is  the  sine  of  half  the  angle  ;  an  .'  .4032  will  be 

ihe  nat  sine,  opposite  to  which  in  the  table  of  hat 

sines  we  find  thp  angle  23°  47'  ;  which  mult  by  2  gives 

47°  34',  the  reqd  angle.     If  obstacles  prevent  measuring  toward  the  objects,  we  may  measure  directly 

from  them  ;  because,  when  two  lines  intersect,   the  opposite  angles  are  equal.    A  rough  measurement 

'may  be  made  by  sticking  three  pins  vert,  and  a  few  ins  apart,  into  a  small  piece  of  board,  nailed  bar 

to  the  top  of  a  post.      The  pins  would  occupy  the  positions  n  o  s,  of  the  last  figure.     Pencil-lines  may 

then  be  drawn,  connecting  the  pin-holes  ;  and  the  angle  be  measd  with  a  protractor.     By  nailing  a 

piece  of  board  vert  to  a  tree,  and  then  drawing  upon  it  a  short  hor  line,  by  means  of  a  pocket  car- 

penters' spirit-level,  vert  ancles  of  elevation  and  depression  may  be  taken  roughly  in  the  same  way. 

In  this  w.T^  r>ie  writer  has  at  times  availed  himself  of  the  outer  door  of  a  house,  by  opening  it  until  it 

pointed  toward  some  mountain-peak,  the  dist  of  which  he  knew  approximately  ;  but  of  the  height  of 


42 


PLANE   TKIGONOMETRY. 


Fig.  6. 


its  angle  yti  of  inclination  with  the  horizon  found  as  before; 
in  which  case  the  dist  an  is  calculated.  Or  if  the  vert  height  en 
is  sought,  the  point  c  may  first  be  found  by  sighting  upward 
along  a  plumb-line  held  above  the  head. 

Ex.  3.    To  find  the  approximate  height, 
s  x,  of  a  mountain, 

Of  which,  perfeaps,  only  the  very  summit,  z,  is  visible  above 
interposing  forests,  or  other  obstacles ;  but  the  dist,  wit,  of  which 
is  known.  In  this  case,  first  direct  the  instrument  nor,  as  m  h; 
and  then  measure  the  angle  i  m  x. 
Then  in  the  triangle  i  m  x  we  have 
one  side  mi;  the  measd  angle  imx, 
and  the  angle  mix  (SO0),  to  find  ix 
by  Case  1.  But  to  this  i  x  we  must 
add  i  o,  equal  to  the  height  y  m  of  the 
instrument  above  the  ground:  and 
also  o  s.  Now,  o  s  is  apparently  due 
entirely  to  the  curvature  of  the  earth, 
which  is  equal  to  very  nearly  8  ins,  or 
.667  ft  in  one  mile  ;  and  increases  as 
the  squares  of  the  dists;  being  4 
times  8  ins  in  2  miles  ;  9  times  8  ins 
in  3  miles,  &c.  But  this  is  somewhat  diminished  by  the  refraction  of  the  atmosphere  ;  which  varies 
with  temperature,  moisture,  &c ;  but  alwavs  tends  to  make  the  object  x  appear  higher  than  it 

actually  is.  At  an  average,  this  deceptive  elevation  amounts  to  about  -  th  part  of  the  curvature  of 
the  earth ;  and  like  the  latter,  it  varies  with  the  squares  of  the  dists.  Consequently  if  we  subtract  — 

part  from  8  ins,  or  .667  ft,  we  have  at  once  the  combined  effect  of  curvature  and  refraction  for  one 
mile,  equal  to  6.857  ins,  or  .5714  ft :  and  for  other  dists,  as  shown  in  the  following  table,  by  the  use 
of  which  we  avoid  the  necessity  of  making  separate  allowances  for  curvature  and  refraction. 

Table  of  allowances  to  be  added  for  curvature  of  the  earth; 
and  for  refraction;  combined. 


Dist. 
in  yards. 

Allow, 
feet. 

Dist. 

in  miles. 

Allow, 
feet. 

Dist. 

in  miles. 

Allow, 
feet. 

Dist. 
in  miles. 

Allow. 
feet. 

100 

.002 

% 

.036 

6 

20.6 

20 

229 

150 

.004 

1% 

.143 

7 

*       28.0 

22 

277 

200 

.007 

3/f 

.321 

8 

36.6 

25 

357 

300 

.017 

i 

.572 

9 

46.3 

30 

514 

400 

.030 

H£ 

.893 

10 

57.2 

35 

700 

500 

.046 

li| 

1.29 

11 

69.2 

40 

915 

600 

.066 

]M 

1.75 

12 

82.3 

45 

1158 

700 

.090 

2 

2.29 

13 

96.6 

50 

1429 

800 

.118 

2^ 

3.57 

14 

112 

55 

1729 

900 

.149 

3 

5.14 

15 

129 

60 

2058 

1000 

.185 

3% 

7.00 

16 

146 

70 

2SOI 

1200 

.266 

4 

-9.15 

17 

165 

80 

3659 

1500 

.415 

&A 

11.6 

18 

185 

90 

4631 

3000 

.738 

5 

14.3 

19 

206 

100 

5717 

Hence,  if  a  person  whose  eye  is  5.14  ft,  or  112  ft  above  the  sea,  sees  an  object  just  at  the  sea's 
horizon,  that  object  will  be  about  3  miles,  or  14  miles  distant  from  him. 

A  horizontal  line  is  not  a  level  one,  for  a  straight  line  cannot  be  a 

level  one.  The  curve  of  the  earth,  as  exemplified  in  an  expanse  of  quiet  water,  is  level.  In  Fig  7, 
if  we  suppose  the  curved  line  t  y  s  g  to  represent  the  surface  of  the  sea,  then  the  points  t  y  s  and  g  are 
on  a  level  with  each  other.  They  need  not  be  equidistant  from  the  center  of  the  earth,  for  the  sea  at 
the  poles  is  about  13  miles  nearer  it  than  at  the  equator;  yet  its  surface  is  everywhere  on  a  level. 

Up,  and  down,  rei'er  to  sea  level.  I^evel  means  parallel  to  the  curvature 
of  the  sea;  and  horizontal  means  tangential  to  a  level. 


Ex.  4.    If  the  inaccessible  vert  height  c  d,  Fig  8, 


w"hich  he  had  no  idea.  For  allowance  for  curvature  and  refraction  see  above  Table. 
A  triangle  whose  sides  are  as  3,  4,  and  5,  is  right  angled;  and  one 

whose  sides  are  as  7 ;  7;  and  9.  9;  contains  1  right  angle;  and  2  angles  of  45°  each.  As  it  is  fre- 
quently necessary  to  lay  down  angles  of  45°  and  90°  on  the  ground,  these  proportions  may  be  used  for 
the  purpose,  by  shaping  a  portion  of  a  tape-line  or  chain  into  such  a  triangle,  and  driving  a  stake  at 
each  angle.  See  p.  G70. 


PLANE   TRIGONOMETRY. 


occupied  by  the  top  o  of  the  staff;  and  from  o  measure  the  angles  iod  and  doc.  This  being  done,  sub, 

tract  the  angle  toe  from 

180° ;  the  rcm  will  be  the 

angle  c  o  n.*  Consequi 

ly  in  the  triangle  noc, 

have  one  side  no,  and  two 

angles,  c  n  o  and  con,  to 
&  find  by  Case  1  the  side  o  c. 

Again,  take  the  angle  too"5 

from  180° ;  the  remainder 

will  be  the  angle  n  o  d,  so 

that  in  the  triangle  d  n  o 

we  have  one  side  n  o,  and 

the  two  angles  d  n  o  and 

nod,  to  find  by  Case  1 

the  side  od.     Finally,  in 

the  triangle  cod,  we  have 

two  sides  c  o  and  o  d,  and 

their  included  angle  c  o  d, 

to  find  c  d,  the  reqd  vert 

height. 

REM.  If  c  d  were  in  a  valley,  or  on  a  hill,  and  the  observations  reqd  to  be  made  from  either  higher 
or  lower  ground,  the  operation  would  be  precisely  the  same. 

Ex.  5.     See  Ex  10. 

To  find  the  dist  ao,  Fig  9,  between  two  entirely  inaccessible 
objects, 

Measure  a  side  n  TO  ;  at  n  measure  the  angles  anm  and  o  n  m ;  also  at  TO  measure  the  angles  o  m  n,  and 
o  TO  n.  This  being  done,  we  have  in  the  triangle  a  n  TO,  one  side  n  m,  Fig  9,  and  the  angles  anm,  aud 
nma;  hence,  by  Case  1,  we  can  calculate  the  side  an. 
Again,  in  the  triangle  o  TO  n  we  have  one  side  n  TO,  and 
the  two  angles  o  TOW,  and  TO  no;  hence,  by  Case  1,  we  can 
calculate  the  side  n  o.  This  being  done,  we  have  in  the 
triangle  ano,  two  sides  on,  and  no;  and  their  included 
angle  ano;  hence,  by  Case  3,  we  can  calculate  the  side 
oo,  which  is  the  reqd  dist.  It  is  plain  that  in  this  manner 
we  may  obtain  also  the  position  or  direction  of  the  inacces- 
sible line  ao ;  for  we  can  calculate  the  angle  n  ao  ;  and  can 
therefrom  deduce  that  of  ao;  and  thus  be  enabled  to  run 
a  line  parallel  to  it,  if  required*  By  drawing  n  m  on  pa- 
per by  a  scale,  and  laying  down  the  four  measd  angles, 
the  dist  oo  may  be  measd  upon  the  drawing  by  the  same  scale. 

If  the  position  of  the  inaccessible  dist  c  n.  Fig  10,  be  such  that 
we  can  place  a  stake  p  in  line  with  it.we  may  proceed  thus  :  Place 
the  instrument  at  any  suitable  point  s,  and  take  the  angles  psc 
andean.  Also  find  the  angle  cpa,  and  measure  the  distj»*.  Then 
in  the  triangle  p  a  c  find  8  c  by  Case  1 ;  again,  the  exterior  angle 
n  c  s,  being  equal  to  the  two  interior  and  opposite  angles  c  p  a, 
and  p  a  c,  we  have  in  the  triangle  can,  one  side  and  two  angles 
to  find  c  n  by  Case  1. 

Ex.  6.  To  find  a  dist  ab,  Fig  11.  of  which 
the  ends  only  are  accessible. 

From  a  and  ft,  measure  any  two  lines  o  e,  6  c,  meeting  at  c ;  also 
measure  the  angle  a  c  ft.  Then  in  the  triangle  aft  e  we  have  two 
sides,  and  the  included  angle,  to  find  the  third  side  o  ft  by  Case  3. 

Ex.  7.  To  find  the  vert  height  o  -m,  of  a 
hill,  above  a  given  point  i. 

Place  the  instrument  atf;  measure  a  TO.  Directing 
the  instrument  hor,  as  on,  take  the  angle  nam.  Then, 
since  a  n  TO  is  90°  Fig  12,  we  have  one  side  a  TO,  and 
two  angles,  nam  and  an  TO,  to  find  nm  by  Case  I. 
Add  no,  equal  to  ai,  the  height  of  the  instrument. 
Also,  if  the  hill  is  a  long  one,  add  for  curvature  of  the 
earth,  and  for  refraction,  as  explained  in  Example  3,  *»  ..-•' 
Fig  7.  Or  the  instrument  may  be  placed  at  the  top  of 
the  hill ;  and  an  angle  of  depression  measured  ;  instead 
of  the  angle  of  elevation  n  am. 

REM.  1.  It  is  plain,  that  if  the  height  OTO  be  previously 
known,  and  we  wish  to  ascertain  the  dist  from  its  sum- 
mit TO  to  any  point  i,  the  same  measurement  as  before, 
of  the  angle  n  a  TO,  will  enable  us  to  calculate  o  TO  by 
Case  1.  So  in  Ex.  2,  if  the  height  na  be  known,  the  angles  measd  in  that  example,  will  enable  us 
to  compute  the  dist  a  o ;  so  also  in  Figs  3,  4,  5,  and  7 ;  in  all  of  which  the  process  is  so  plain  as  to 
require  no  further  explanation. 

REM.  2.  The  height  of  a  vert  object  by  means  Of  its  shadow.  Plant  one  end  of 
a  straight  stick  vert  in  the  ground  ;  and  measure  its  shadow  ;  also  measure  the  length  of  the  shadow 
of  the  object.  Then,  as  the  length  of  the  shadow  of  the  stick  is  to  the  length  of  the  stick  above 

*  Because,  when  two  straight  lines,  as  o  o,  and  ni,  intersect  each  other  at  any  inclination  whatever, 
the  two  adjacent  angles  con  and  toe  amount  to  180°.  Therefore,  if  one  of  them  is  given,  we  can  find 
*he  other  by  subtractiag  ih9  sriven  one  from  180°. 


44 


PLANE   TRIGONOMETRY. 


ground,  so  is  the  length  of  the  shadow  of  the  object,  to  its  height.    If  the  object  is  inclined,  the  stick 
must  be  equally  inclined. 

Rein.  3.  Or  the  height  of  a  vert  object  raw, 

Fig  12^,  whose  distance  r  m  is  known,  may  be  found  by 
its  reflection  in  a  vessel  of  water,  or  in  a  piece  of 


looking  glass  placed  perfectly  hori: 
"  ;  reflectoi 


tal  at  r;  for  as  r  a  is  to  the  height 
~isr~  x- 


made 

i  c  is  to  c  o,  so  is  i  m  to  m  n. 


The  following  examples  may  be  regarded  as  substitutes  for  strict  trigonome- 
try :  and  will  at  times  be  useful,  iu  case  a  table  of  sines,  &c,  is  not  at  hand  for 
making  trigonometrical  calculations. 

Ex.  8.  To  find  the  dist  alt,  of  which  one  end  only 
is  accessible. 

Drive  a  stake  at  any  convenient  point  a ;  from  a  lay  off  any  angle  ft  a  c.  In 
the  line  a  c,  at  any  convenient  point  c,  drive  a  stake ;  and  from  c  lay  off  an  angle 
acd,  equal  to  the  angle  ft  a  c.  In  the  line  c  d,  at  avy  convenient  point,  as  d, 
drive  a  stake.  Then,  standing  at  d,  and  looking  at  ft,  place  a  stake  o  in  range 
with  d  ft  ;  and  at  the  same  time  in  the  line  ac.  Measure  ao,  o  c,  and  erf;  then, 
from  the  principle  of  similar  triangles,  as 

o  c  :  c  d  :  :  a  o  :  ab. 


Or  thus: 

Fig  14,  n  h  being  the  dist,  place  a  stake  at  n ;  and  lay  off  the  angle  h  n  m  90°. 
At  any  convenient  dist  n  m,  place  a  stake  m.  Make  the  angle  h  m  y  —  90°;  a  id 
place  a  stake  at  y,  in  range  with  h  n.  Measure,  n  y  and  n  m ;  then,  from  the 
principle  of  similar  triangles,  as 

it  y  :  n  m  :  :  n  in  :  n  h. 

Or  thus,  Fig  14.    Lay  off  the  angle  hnm  =  90°,  placing  a  stake 

m,  at  any  convenient  dist  n  m.  Measure  n  m.  Also  measure  the  angle  n  m  h. 
IT-  Ji  1  /  Find  nat  tanS  of  n  m  A  by  Table  p  103.  Mult  this  nat  tang  by  n  m.  The  prod 
1  10  I T-  will  be  n  h, 

Or  thus.    Lay  off  angle  h  n  m  =  90°.    From  m  measure  the 

augle  n  TO  A,  and  lay  off  angle  nmy  equal  to  it,  placing  a  stake  at  y  in  range 
with  h  n.    Then  is  n  y  =  n  h. 

Or  thus,  without' measuring 
any  angle ; 

t  u  being  the  dist.  Make  u  v  of  any  convenient 
length,  in  range  with  t  u.  Measure  any  v  o ;  and 
o  x  equal  to  it.  in  range.  Measure  u  o ;  and  oy 
equal  to  it  in  range.  Place  a  stake  z  in  range  with 
both  x  11,  and  t  o.  Then  will  y  z  be  both  equal  to 
t  u,  anil  parallel  to  it. 


Or  thus,  without  measuring  any  angle. 

Drive  two  stakes  t  and  u,  in  range  with  the  object  8.  From  t  lay  off  any 
convenient  dist  t  x,  in  any  direction.  From  u  lay  off  u  w  parallel  to  t  x, 
placing  win  range  with  x  s.  Make  u  v  equal  to  t  x.  Measure  w  v,  v  x,  and 
x  t.  Then,  as 

tv  v  :  v  x  :  \xt\ts. 

Or  thus.    At  a  lay  off  angle  o  a  c  =  5°  43'.    Meas- 
U   ure  o  c,  and  mult  it  by  10  for  a  o,  too  long  only  1  part  iu  935.6. 


PLANE    TRIGONOMETRY. 


45 


Fi6  17 


Ex.  9.    To  find  the  dist  a  ft,  of  which  the 
ends  only  are  accessible. 

From  a  lay  off  the  angle  bnc;  and  from  6,  the  angle  a  b  d.  cnch 
90°.  Make  a  c  and  h  d  equal  to  each  other.  Then,  cd~a  1>.  Or 
a  f)  may  he  considered  as  the  dist  across  the  river  in  Pigs  15,  13.  or 
14:  and  be  ascertained  in  the  same  way.  Or  measure  any  dist.  Fig 
17,  a  o;  and  make  a  n  in  line  and  equal  to  it,.  Also  measure  ho; 
and  mRke  om  in  Sine  and  equal  to  it.  Then  will  ran  be  both  paral- 
lel to  a  6,  and  equal  to  it. 


Ex.  1O.    See  Ex.  4.    To  find  the  entirely 

inaccessible  clist  //  z,  and  also 

its  direction. 

At  any  two  convenient  points  a  and  b,  from  each  of  which 
y  and  z  can  be  seen,  drive  stakes.  Then  we  have  the  four 
corners  of  a  four-sided  figure,  in  which  are  given  the  directions 
of  three  of  its  sides,  and  of  its  two  diags.  This  data  enables  us 
to  lay  out  on  the  ground,  the  small  four-sided  fig  a  c  o  t,  exactly 
similar  to  the  large  one.  Thus,  in  the  Hue  a  b  place  a  stake 
c;  and  make  co  parallel  to  fez;  o  being  at  the  same  time  in 
range  of  the  diag  a  z.  Also,  from  c  make  c  i  parallel  to  6  y ; 
i  being  at  the  same  time  in  range  of  a  y.  Then  will  i  o  be  in 
the  same  direction  as  y  z,  or  parallel  to  it.  Measure  a  c,  a  b, 
and  to ;  then  evidently,  from  the  principle  of  similar  figures,  as 
a  c  :  a  b  :  :  i  o  :  i/  z. 

If  y  z  were  a  visible  line,  such  as  a  fence  or  road,  we  could 
from  a  divide  it  into  any  required  portions.  Thus,  if  we  wish 
to  place  a  stake  halfway  between  y  and  z,  first  place  one  half- 
way between  i  and  o;  then  standing  at  a,  by  means  of  signals, 
place  a  person  in  range  on  y  z.  Or,  to  find  along  n  b,  a  point  t 
perp  to  y  z  at  y,  first  make  ois  =  90° ;  and  measure  a  «.  Then, 

o  i  :  a  s  :  :  y  z  :  a  t. 
Ex.  11.  To  find  the  position  of  a  point,  »,  Fig?  19, 

By  means  of  two  angles  a  n  b  and  b  n  n.  taken  from  it  to  the  three  objects  a  b  c,  whose  positions 
and  dists  apart  are  known. 

The  use  of  this  problem  is  more  frequent  in  marine  than  in  land  surveying.  It  is  chiefly  employed 
for  determining  the  position  n  of  a 
boat  from  which  soundings  are  being 
taken  along  a  coast.  As  the  boat 
moves  from  point  to  point  to  take 
fresh  soundings,  it  becomes  necessary  A^ 
to  make  a  fresh  observation  at  each 
point,  in  order  to  define  its  position 
on  the  chart.  An  observation  consists 
in  the  measurement  by  a  sextant  of 
the  two  angles  an  b,  b  n  c.  to  the  sig- 
nals a  6  c,  previously  arranged  on  the 
shore.  When  practicable,  this  method 
should  be  rejected;  and  the  observa- 
tions taken  to  the  boat  at  the  same 
instant,  by  two  observers  on  shore,  at 
two  of  the  stations.  The  boat  to  show 
a  signal  at  the  proper  moment.  The 
most  expeditious  mode  of  fixing  the 
point  n  upon  the  map,  is  to  draw  three 
lines,  forming  the  two  angles,  and  ex- 
tended indefinitely,  on  a  piece  of  trans- 
parent paper.  Place  the  paper  upon  the  map,  and  move  it  about  until  the  three  lines  pass  through 
the  three  stations  ;  then  prick  through  the  point  n  wherever  it  happens  to  come. 

Instead  of  the  transparent  paper,  an  instrument  called  a  station  pointer  may  be  used  when  there 
are  many  points  to  be  fixed. 

But  the  position  of  the  point  n  can  he  found  more  correctly  by  describing  two  circles,  as  in  Fig  19, 
each  of  which  shall  pass  through  n  and  two  of  the  station  points'.  The  question  is  to  find  the  centers 
oand  x  of  two  such  circles.  This  is  very  simple.  We- know  that  the  angle  a  ob  at  the  center  of  a  circle  is 
twice  as  great  as  ary  amgle  a  n  b  at  the  circumf  of  the  same  circle,  when  both  are  subtended  by  the 
same  chord  a  h.  Consequently ,  if  the  angle  a  n  b,  observed  from  the  boat,  is  say.  50°,  the  angle  no  b 
must  be  100°.  And,  since  the  three  angles  of  every  plane  triangle  are  equal  to  180°,  the  two  anglf  s 
o  ab  and  o  b  a  are  together  equal  to  180°  —  100°'=  80°.  And,  since  the  two  sides  a  o  and  b  o  are 
equal  (being  radii  of  the  same  circle),  therefore,  the  angles  oab  and  o  ba  are  equal;  and  each  equal  to 

_-  rr  40P.     Consequently,  on  the  map  we  have  only  to  lay  down  at  a  and  b,  two  angles  of  40°;  the 

point  o  of  intersection  will  be  the  center  of  the  circle  a  b  n.   Proceed  in  the  same  way  with  the  angle 
6  n  c,  to  find  the  center  x.    Then  the  intersection  of  the  two  circles  at  n  will  be  the  point  sought. 


46 


CONTENTS  OF  CYLINDERS,  OR  PIPES. 


Contents  for  one  foot  in  length,  in  Cub  Ft,  and  in  U.  S.  Gallons  of 

231  cub  ins,  or  7.4805  Galls  to  a  Cub  Ft.    A  cub  ft  of  water  weighs  about  62^  Ibs  ;  and  a  gallon 
about  8%  Ibs.     Dium*  2,  8,  or  1O  times  as  great,  give  4,  9,  or  100  times  the  content. 

For  the  weight  of  water  in  pipes,  see  Table  2^,  page  540. 

No  errors. 


For  1  ft.  in 

For  1  ft  in 

For  1  ft.  in 

length. 

length. 

length. 

Diam. 

Diam. 

Diam. 

Diam. 

Diam. 

in 

in  deci- 

.  a 

fc,    05 

in 

in  deci- 

. a 

fc.    00 

Diam. 

in  deci- 

3 

. 

Ins. 

mals  of 

0)  °g 

o  a 

Ins. 

mals  of 

"£  "^ 

o  a 

in 

mals  of 

W  °rf 

o  a 

a  foot. 

£££ 

|i 

a  foot. 

fc  ££ 

|i 

Ins. 

a  foot. 

£  ££ 

§-°' 

.O  a   3* 

If* 

i|* 

1| 

5-16 

.0208 
.0260 

.0003 
.0005 

.0025 
.0040 

7.  4 

.5625 
.5833 

.2485 
.2673 

1.859 
1.999 

19. 

1.583 

1.625 

1.969 
2.074 

14.73 
15.51 

% 

.0313 

.0008 

.0057 

i/ 

.6042 

.2867 

2.145 

20.  2 

1.667 

2.182 

16.32 

7-16 

.0365 

.0010 

.0078 

1/5 

.6250 

.3068 

2.295 

17 

1.708 

2.292 

17.15 

17J 

0417 

.0014 

.0102 

74 

.6458 

.3276 

2.450 

21. 

1.750 

2.405 

17.99 

9-16 

.0469 

.0017 

.0129 

8. 

.6667 

.3491 

2.611 

17 

1.792 

2.521 

18.86 

57 

.0521 

.0021 

.0159 

i/ 

.6875 

.3712 

2.777 

22. 

1.833 

2.640 

1975 

11-16 

.0573 

.0026 

.0193 

% 

.7083 

.3941 

2.948 

17 

1.875 

2.761 

20.66 

3// 

.0625 

.0031 

.0230 

3% 

.7292 

.4176 

3.125 

23.. 

1.917 

2.885 

21.58 

13-16 

.0677 

.0036 

.0269 

9. 

.7500 

.4418 

3.305 

\/ 

1.958 

3.012 

22.53 

Ys 

.0729 

.0042 

.0312 

\/ 

.7708 

.4667 

3.491 

24. 

2.000 

3.142 

23.50 

15-16 

.0781 

.0048 

.0359 

\/ 

.7917 

.4922 

3.682 

25. 

2.083 

3.409 

25.50 

1. 

.0833 

.0055 

.0408 

74 

.8125 

.5185 

3.879 

26. 

2.167 

3.687 

27.58 

i// 

.1042 

.0085 

.0638 

10. 

.8333 

.5454 

4.080 

27. 

2.250 

3.976 

29.74 

17 

.1250 

.0123 

.0918 

I/ 

.8542 

.5730 

4.286 

28. 

2.333 

4.276 

31.99 

74 

.1458 

.0167 

.1249 

P    .8750 

.6013 

4.498 

29. 

2.417 

4.587 

3431 

2. 

.1667 

.0218 

.1632 

.8958 

.6303 

4.715 

30. 

2.500 

4.909 

36.72 

i/^ 

.1875 

.0276 

.2066 

11.  4 

.9167 

.6600 

4.937 

31. 

2.583 

5.241 

39.21 

i^J 

.2083 

.0341 

.2550 

l// 

.9375 

.6903 

5.164 

32. 

2.667 

5.585 

41.78 

74 

.2292 

.0412 

.3085 

17 

.9583 

.7213 

5.396 

33. 

2.750 

5.940 

44.43 

3. 

.2500 

.0491 

.3672 

74 

.9792 

.7530 

5.633 

34. 

2.833 

6.305 

47.13 

\/ 

.2708 

.0576 

.4309 

12.      1  Foot. 

.7854 

5.875 

35. 

2.917 

6.681 

49.98 

X/ 

.2917 

.0668 

.4998 

Ml.  042 

.8522 

6.375 

36. 

3000 

7.069 

52.88 

74 

.3125 

.0767 

.5738 

13.     !  1.083 

.9218 

6.895 

37. 

3.083 

7.467 

55.86 

4. 

.3333 

.0873 

.6528 

17  1.125 

.9940 

7.43c 

38. 

3.16T 

7.876 

58.92 

i/^ 

.3542 

.0985 

.7369 

14.     11.167 

1.069 

7.997 

39. 

3.250 

8.296 

62.06 

ITJ 

.3750 

.1104 

.8263 

Y^  1.208 

1.147 

8.578 

40. 

3.333 

8.727 

65.28 

74 

.3958 

.1231 

.9206 

15. 

1.250 

1.227 

9.180 

41. 

3.417 

9.168 

68.58 

5. 

.4167 

.1364 

1.020 

if,  1.292 

1.310 

9.801 

42. 

3.500 

9.621 

71.97 

i/; 

.4375 

.1503 

1.125 

16.     (1.333 

1.396 

10.44 

43. 

3.583 

10.085 

75.44 

ITJ 

.4583 

.1650 

1.234 

/fj  1.375 

1.485 

11.11 

44. 

3.667 

10.559 

78.99 

74 

.4792 

.1803 

1.349 

17.     il.417 

1.576 

11.79 

45. 

3.750 

11.045 

82.62 

6. 

.5000 

.1963 

1.469 

H!  L458 

1.670 

12.49 

46. 

3.833  111.  541 

86.33 

i// 

.5208 

.2131 

1.594 

18.     |1.500 

1.767 

13.22 

47. 

3.917   12.048 

90.13 

/I 

.5417 

.2304 

1.724 

%  1.542 

1.867 

13.96 

48. 

4.000 

12.566 

94.00 

1 

Table  continued,  but  with  the  dianis  in  feet. 


Diam. 

Cub. 

U.  s. 

Diam. 

Cub. 

U.  S. 

Dia. 

Cub. 

U.S. 

Dia. 

Cub. 

U.  S. 

Feet. 

Feet. 

Galls. 

Feet. 

Feet. 

Galls. 

Feet. 

Feet. 

Galls. 

Feet. 

Feet. 

Galls. 

4 

12.57 

94.0 

7 

38.49 

287.9 

12 

113.1 

846.1 

24 

452.4 

3384 

\S 

14.19 

106.1 

X 

41.28 

308.8 

13 

132.7 

992.8 

25 

490.9 

3672 

7% 

15.90 

119.0 

g 

44.18 

330.5 

14 

153.9 

1152. 

26 

530.9 

3971 

% 

17.72 

132.5 

X 

47.17 

352.9 

15 

176.7 

1322. 

27 

572.6 

4283 

5 

19.64 

146.9 

8 

50.27 

376.0 

16 

201.1 

1504. 

28 

615.8 

4606 

74" 

21.65 

161.9 

M 

56.75 

424.5 

17 

227.0 

1698. 

29 

660.5 

4941 

/^ 

23.76 

177.7 

9 

63.62 

475.9 

18 

254.5 

1904. 

30 

706.9 

5288 

% 

25.97 

194.3 

1A 

70.88 

530.2 

19 

283.5 

2121. 

31 

754.8 

5646 

6 

28.27 

211.5 

10 

78.54 

587.6 

20 

314.2 

2350. 

32 

804.3 

6017 

74" 

30.68 

229.5 

\i 

86.59 

647.7 

21 

346.4 

2591. 

33 

855.3 

6398 

X£ 

33.18 

248.2 

11 

95.03 

710.9 

22 

380.1 

2844. 

34 

907.9 

6792 

% 

35.79 

267.7 

X 

103.90 

777.0 

23 

415.5 

3108. 

35 

962.1 

7197 

DIGGING,  ETC.,  OF   WELLS. 


47 


DIGGING,  &C,  OF  WELLS. 


s  twice  as  great  as  those  in  the  table,  for  the  cub  yds  of  digging,  take  out  those  opposite 
the  greater  diani  ;  and  mult  them  by  4.  Thus,  for  the  cub  yds  in  each  foot  of  depth  of  a 
in  diam,  first  take  out  from  the  table  those  opposite  the  diani  of  153*  feet  ;  namely  6  989 
X  4  ~  27.956  cub  yds  reqd  for  the  31  ft  diam.  But  for  the  stone  lining  or  walling,  bricks 


For  diams  twice  as  great  as  those  in  the  table,  for  the  cub  yds  of  digging,  take  out  those  opposite 
one  half  of  the  gre 
well  31  feet  in  dia 

Then  6.989  X  4  ~  2.  .  , 

or  plastering,  mult  the  tabular  quantity  opposite  half  the  greater  diam,  by  2.  Thus,  the  perches  of 
stone  walling  for  each  foot  of  depth  of  a  well  of  31  ft  diam,  will  be  2.073  X  2  ~  4.146.  If  the  wall  is 
more  or  less  than  one  foot  thick,  within  usual  moderate  limits,  it  will  generally  be  near  enough  for 
practice  to  assume  that  the  number  of  perches,  or  of  bricks,  will  increase  or  decrease  in  the  same  pro- 
portion. 

The  size  of  the  bricks  is  taken  at  8J4  X  4  X  2  inches  ;  and  to  be  laid  dry,  or  without  mortar.  In 
practice  an  addition  of  about  5  per  cent  should  be  made  for  waste.  The  brick  lining  is  supposed  to 
be  1  brick  thick,  or  8%  ins. 

CAUTIOX.  —  Be  careful  to  observe  that  the  diams  to  he  used  for  the  digging, 
are  greater  than  those  for  the  walling,  bricks,  or  plastering.  No  errors. 


For  each  foot  of  depth. 

For  each  foot  of  depth. 

For  this 

For  these  three  cols  use  the 

For  this 

For  these  three  cols  use  the 

col  use  the 

diam  in  clear  of  the  "lining. 

col  use  the 

diam  in  clear  of  the  lining. 

Diam. 

Diameter 

Diam. 

Diameter 

in 

of  the 

in 

of  the 

Feet. 

Digging. 

Cub  Yds. 
of 
Digging. 

Stone 
Lining 
I  ft  thick. 
Perches  of 
25  Cub  Ft. 

No.  of 
Bricks  in 
a  Lining 
1  Brick 
thick. 

Square 
Yards  of 
Plaster- 
ing. 

Feet. 

Digging. 

Stone 
Lining 
1  ft  thick. 
Perches  of 
25  Cub  Ft. 

No.  of 
Bricks  in 
a  Lining 
1  Brick 
thick. 

Square 
Yards 
of  Plas- 
tering. 

Cub  Yds. 
Digging. 

1. 

.0291 

.2513 

57 

.3491 

X 

5.107 

1.791 

750 

4.625 

34 

.0455 

.2827 

71 

.4364 

X 

5.301 

1.822 

764 

4.713 

^£ 

.0654 

.3142 

85 

.5236 

K 

5.500 

1.854 

778 

4.800 

M 

.0891 

.3456 

99 

.6109 

14. 

5.701 

1.885 

792 

4.887 

2. 

.1164 

.3770 

114 

.6982 

34 

5.907 

1.916 

806 

4.974 

34 

.1473 

.4084 

J28 

.7855 

3l2 

6.116 

1.948 

820 

5.062 

X. 

.1818 

.4398 

142 

.8727 

% 

6.329 

1.979 

834 

5.149 

K 

.2200 

.4712 

156 

.9600 

15. 

6.545 

2.011 

849 

5.236 

3. 

.2618 

.5027 

170 

1.047 

34 

6.765 

2.042 

863 

5.323 

34 

.3073 

.5341 

184 

1.135 

% 

6.989 

2.073 

877 

5.411 

^ 

.3563 

.5655 

198 

1.222 

% 

7.216 

2.105 

891 

5.498 

H 

.4091 

.5969 

212 

1.309 

16. 

7.447 

2.136 

905 

5.585 

4. 

.4654 

.6283 

227 

1.396 

H 

7.681 

2.168 

919 

5.673 

.5254 

.6597 

241 

1.484 

y* 

7.919 

2.199 

933 

5.760 

i^ 

.5890 

.6912 

255 

1.571 

% 

8.161 

2.231 

948 

5.847 

H 

.6563 

.7226 

269 

1.658 

17. 

8.407 

2.262 

962 

5.934 

5. 

.7272 

.7540 

283 

1.745 

34 

8.656 

2.293 

976 

6.022 

.8018 

.7854 

297 

1.833 

J4 

8.908 

2.325 

990 

6.109 

£ 

.8799 

.8168 

311 

1.920 

H 

9.165 

2.356 

1004 

6.196 

% 

.9617 

.8482 

326 

2.007 

18. 

9.425 

2.388 

1018 

6.283 

6. 

1.047 

.8796 

340 

2.095 

34 

9.688 

2.419 

1032 

6.371 

34 

1.136 

.9111 

354 

2.182 

X 

9.956 

2.450 

1046 

6.458 

H 

1.229 

.9425 

368 

2.269 

H 

10.23 

2.482 

1061 

6.545 

« 

1.325 

.9739 

382 

2.356 

19. 

10.50 

2.513 

1075 

6.633 

7. 

1.425 

1.005 

396 

2.444 

34 

10.78 

2.545 

1089 

6.720 

34 

1.529 

1.037 

410 

2.531 

X 

11.06 

2.576 

1103 

6.807 

M 

1.636 

1.068 

425 

2.618 

% 

11.35 

2.608 

1117 

6.894 

N 

1.747 

1.100 

439 

2.705 

20. 

11.64 

2.639 

1131 

6.982 

8. 

1.862 

1.131 

453 

2.793 

34 

11.93 

2.670 

1145 

7-069 

34 

1.980 

1.162 

467 

2.880 

34 

12.22 

2.702 

1160 

7.156 

k 

2.102 

1.194 

481 

2.967 

H 

12.52 

2.733 

1174 

7.243 

% 

2.227 

1.225 

495 

3.054 

21. 

12.83 

2.765 

1188 

7.331 

9. 

2.356 

1.257 

509 

3.142 

34 

13.14 

2.796 

1202 

7.418 

34 

2.489 

1.288 

523 

3.229 

34 

13.45 

2.827 

1216 

7.505 

% 

2.625 

1.319 

538 

3.316 

H 

13.76 

2.859 

1230 

7.593 

% 

2.765 

1.351 

552 

3.404 

14.08 

2.890 

1244 

7.680 

10. 

2.909 

1.382 

566 

3.491 

34 

14.40 

2.922 

1259 

7.767 

34 

3.056 

1.414 

580 

3.578 

34 

14.73 

2.953 

1273 

7.854 

3^ 

3.207 

1.445 

594 

3.665 

H 

15.06      ' 

2.985 

1287 

7.942 

X 

3.362 

1.477 

608 

3.753 

23. 

15.39 

3.016 

1301 

8.029 

11. 

3.520 

1.508 

622 

3.840 

34 

15.72 

3.047 

1315» 

8.116 

34 

3.682 

1  539 

637 

3.927 

34 

16.06 

3.079 

1329 

8.203 

g 

3.847 

1.571 

651 

4.014 

H 

16.41 

3.110 

1343 

8.  ''91 

H 

4.016 

1.602 

665 

4.102 

24. 

16.76 

3.142 

1357 

8.378 

12. 

4.189 

1.634 

679 

4.189 

34 

17.11 

3.173 

1372 

8.465 

34 

4.365 

1.665 

693 

4.276 

34 

17.46 

3  204 

1386 

8.552 

J6 

4.545 

1.696 

707 

4.364 

H 

17.82 

3.236 

1400 

8.6*0 

« 

4.729 

1.728 

721 

4.451 

25. 

18.18 

3.267 

1414 

8.727 

13. 

4.916 

1.759 

736 

4.538 

A  i 

sub  yd 

=  2021 

r.  s.  s:a 

Is. 

If  perches  are  named  in  a  contract,  it  is  necessary,  in  order  to  prevent  fraud* 

to  specify  the  number  of  cub  feet  contained  in  the  perch  ;  for  stone-quarriers  have  one  perch,  stone- 
masons another,  Ac.  Engineers,  on  this  account,  contract  by  the  cubic  yard.  The  perch  should  be 
done  away  with  entirely  ;  Perches  of  25  cub  ft  X  .926  =  cub  yds  ;  and  cub  yds-j-  .926-  pers  of  25  cub  ft. 


48 


SQUARE  AND  CUBE   HOOTS. 


Square  Roots  and  Cube  Roots  of  Numbers  from  .1  to  2$. 

No  errors. 


No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Rt. 

No. 

Sq.  Rt. 

C.  Rt. 

No. 

Sq.  Rt. 

C.  Rt. 

.1 

.01 

.001 

.316 

.464 

7 

2.387 

1.786 

.4 

3.661 

2.375 

.15 

.0225 

.0034 

.387 

.531 

8 

2.408 

1.797 

.6 

3.688 

2.387 

.2 

.04 

.008 

.447 

.585 

9 

2.429 

1.807 

.8 

3.715 

2.399 

.25 

.0625 

.0156 

.500 

.630 

6 

2.449 

1.817 

14. 

3.742 

2.410 

.3 

.09 

.027 

.548 

.669 

1 

2.470 

1.827 

.2 

3.768 

2.422 

.35 

.1225 

.0429 

.592 

.705 

2 

2.490 

1.837 

.4 

8.795 

2.433 

.4 

.16 

.064 

.633 

.737 

3 

2.510 

1.847 

.6 

3.821 

2.444 

.45 

.2025 

.0911 

.671 

.766 

4 

2.530 

1.857 

.8 

8.847 

2.455 

.5 

.25 

.125 

.707 

.794 

.5 

2.550 

1.866 

15. 

3.873 

2.466 

.55 

.3025 

.1664 

.742 

.819 

.6 

2.569 

1.876 

.2 

3.899 

2.477 

.6 

.36 

.216 

.775 

.843 

2.588 

1.885 

.4 

3.924 

2.488 

.65 

.4225 

.2746 

.806 

.866 

.8 

2.608 

1.895 

.6 

3.950 

2.499 

.7 

.49 

.343 

837 

.888 

.9 

2.627 

1.904 

.8 

3.975 

2.509 

.75 

.5625 

.4219 

.866 

.909 

7. 

2.646 

1.913 

16. 

4. 

2.520 

.8. 

.64 

.512 

.894 

.928 

.1 

2.665 

1.922 

.2 

4.025 

2.530 

.85 

.7225 

.6141 

.922 

.947 

.2 

2.683 

1.931 

.4 

4.050 

2.541 

,9 

.81 

.729 

.949 

.965 

.3 

2.702 

1.940 

.6 

4.074 

2.551 

.95 

.9025 

.8571 

.975 

.983 

.4 

2.720 

1.949 

.8 

4.099 

2.561 

1. 

1.000 

1.000 

1.000 

1.000 

.5 

2.739 

1.957 

17. 

4.123 

2.571 

.05 

1.103 

1.158 

1.025 

1.016 

.6 

2.757 

1.966 

.2 

4.147 

2.581 

1.1 

1.210 

1.331 

1.049 

1.032 

.7 

2.775 

1975 

.4 

4.171 

2.591 

.15 

1.323 

1.521 

1.072 

1.018 

.8 

2.793 

1.983 

.6 

4.195 

2.601 

1.1 

1.440 

1.728 

1.095 

1.063 

.9 

2.811 

1.992 

.8 

4.219 

2.611 

.'25 

1.563 

1.953 

.118 

1.077 

8. 

2.828 

2.000 

18. 

4.243 

2.621 

1.3 

1.690 

2.197 

.140 

1.091 

.1 

2.846 

2.008 

.2 

4.266 

2.630 

.35 

1.823 

2.460 

.162 

1.105 

.2 

2.864 

2.017 

.4 

4.290 

2,640 

1.4 

1.960 

2.714 

.183 

1.119 

.3 

2.881 

2.025 

.6 

4.313 

2.650 

.45 

2.103 

3.019 

.204 

1.132 

.4 

2.898 

2.033 

.8 

4.336 

2.659 

1.5 

2.250 

3.375 

.225 

1.145 

.5 

2.915 

2.041 

19. 

4.359 

2.668 

.55 

2.403 

3.724 

.215 

1.157 

.6 

2.933 

2.049 

.2 

4.382 

2.678 

1.6 

2.560 

4.096 

.265 

1.170 

.7 

2.950 

2.057 

.4 

4.405 

2.687 

.65 

2.723 

4.492 

.285 

1.182 

.8 

2.966 

2.065 

,6 

4.427 

2.696 

1.7 

2.890 

4.913 

.304 

1.193 

.9 

2.983 

2.072 

.8 

4.450 

2.705 

.75 

3.063 

5.359 

.323 

1.205 

9. 

3. 

2.080 

20. 

4.472 

2.714 

1.8 

3.240 

5.832 

.342 

1.216 

.1 

3.017 

2.088 

.2 

4.494 

2.723 

.85 

3.423 

6.332 

.360 

1.228 

.2 

3.033 

2.095 

.4 

4.517 

2.732 

1.9 

3.610 

6.859 

.378 

1.239 

.3 

3.050 

2.103 

.6 

4.539 

2.741 

.95 

3.803 

7.415 

.396 

1.249 

.4 

3.066 

2.110 

.8 

4.561 

2.750 

2. 

4.000 

8.000 

.414 

1.260 

.5 

3.082 

2.1J8 

21. 

4.583 

2.759 

.1 

4.410 

9.261 

.419 

1.281 

.6 

3.098 

2.125 

.2 

4.604 

2.768 

.2 

4.840 

10.65 

.483 

1.301 

.7 

3.114 

2-133 

.4 

4.626 

2.776 

.3 

5.290 

12.17 

.517 

1.320 

.8 

3.130 

2.140 

.6 

4.648 

2.785 

.4 

5.760 

13.82 

.549 

1.339 

.9 

3.146 

2.147 

.8 

4.669 

2.794 

.5 

6.250 

15.63 

.581 

1.357 

10. 

3.162 

2.154 

22. 

4.690 

2.802 

.6 

6.760 

17.58 

.612 

1.375 

.1 

3.178 

2.162 

.2 

4.712 

2.810 

.7 

7.290 

19.68 

.643 

1.392 

.2 

3.194 

2.169 

A 

4.733 

2.819 

.8 

7.840 

21.95 

.673 

1.409 

.3 

3.209 

2.176 

.6 

4.754 

2.827 

.9 

8.410 

24.39 

.703 

1.426 

.4 

3.225 

2.183 

.8 

4.775 

2.836 

8. 

9. 

27. 

.732 

1.442 

.5 

3.240 

2.190 

23. 

4.796 

2.844 

.1 

9.61 

29.79 

.761 

1.458 

.6 

3.256 

2.197 

.2 

4.817 

2.852 

.2 

10.24 

32.77 

.789 

1.474 

3.271 

2.204 

.4 

4.837 

2.860 

.3 

10.89 

35.94 

.817 

1.489 

.8 

3.286 

2.210 

.6 

4.858 

2.868 

.4 

11.56 

39.30 

1.844 

1.504 

.9 

3.302 

2.217 

.8 

4.879 

2.876 

.5 

12.25 

42.88 

1.871 

1.518 

11. 

3.317 

2.224 

24. 

4.899 

2.884 

.6 

12.96 

46.66 

1.897 

1.533 

.1 

3.332 

2.231 

.2 

4.919 

2.892 

.7 

13.69 

50.65 

1.924 

1.547 

.2 

3.347 

2.237 

.4 

4.940 

2.900 

.8 

14.44 

54.87 

1.949 

1.560 

.3 

3.362 

2.244 

.6 

4.960 

2.908 

.9 

15.21 

59.32 

1.975 

1.574 

.4 

3376 

2.251 

.8 

4.980 

2.916 

4. 

16. 

64. 

2: 

1.587 

.5 

3.391 

2.257 

25. 

5. 

2.924 

16.81 

68.92 

2.025 

1.601 

.6 

3.406 

2.264 

.2 

5.020 

2.932 

.2 

17.64 

74.09 

2.049 

1613 

.7 

3.421 

2.270 

.4 

5.040 

2.940 

.3 

18.49 

79.51 

2.074 

1.626 

.8 

3.435 

2.277 

.6 

5.060 

2.947 

.4 

19.36 

85.18 

2.098 

1.639 

.9 

3.450 

2.283 

.8 

5.079 

2.955 

.5 

20.25 

91.13 

2.121 

1.651 

12. 

3.464 

2289 

26. 

5.099 

2.962 

.6 

21.16 

97.34 

2.145 

1.663 

.1 

3.479 

2.296 

.2 

5.119 

2.970 

J7 

22.09 
23.04 

103.8 
110.6 

2.168 
2.191 

1.675 
1.687 

.2 
.3 

3.493 
3.507 

2.302 
2.308 

.4 
.6 

5.138 
5.158 

2.978 
2.985 

!u 

5. 

^j 

24.01 
25. 
26.01 

117.6 
125. 
132.7 

2.214 
2.236 

2.258 

1.698 
1.710 
1.721 

.4 

.5 
.6 

3.521 
3.536 
3.55« 

2.315 
2.321 
2.327 

.8 
27. 
.2 

5.177 
5.196 
5.215 

2.993 
3.000 
3.007 

!a 

.3 
.4 
.5 
.6 

27.04 
28.09 
29.16 
30.25 
31.36 

140.6 
148.9 
157.5 
166.4 
175.6 

2.280 
2.302 
2.324 
2.345 
2.366 

1.732 
1.744 
1.754 
1.765 
1.776 

.7 
.8 
.9 
13. 
.2 

3.564 
3.578 
3.592 
3.606 
3.633 

2.333 
2339 
2.345 
2.351 
2.363 

.4 

.6 

.8 
28. 
.2 

5.235 
5.254 
5.273 
5.292 
5.310 

3.015 
3.022 
3.029 
3.037 
3.044 

SQUARES,  CUBES,  AND  ROOTS. 


TABLE  of  Squares.  Cubes.  Square  Roots,  and  Cube  Roots, 
of  Numbers  from  1  to  1OOO. 


REMARK  ON  THE  FOLLOWING  TABLE. 
add  1  to  the  fourth  and  final  decimal  it 


Wherever  the  effect  of  a  fifth  decimal  in  the  roots  would  be  to 
the  table,  the  addition  has  boen  made.  No  errors. 


No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Rt. 

No. 

Square 

Cube. 

Sq.  Rt. 

C.  Rt. 

] 

1 

l 

3.0000 

1.0000 

61 

3721 

226981 

7.8102 

3.9365 

2 

4 

8 

1.4142 

1.25J9 

62 

3844 

238828 

7.8740 

3.9579 

3 

9 

27 

1.7321 

1.4422 

63 

3969 

250047 

7.9373 

3.9791 

4 

16 

64 

2.0000 

1.5874 

64 

4096 

262144 

8.0000 

4. 

5 

25 

123 

2.2361 

1.7100 

65 

4225 

274625 

8.0623 

4.020T 

6 

36 

216 

2.4495 

1.8171 

66 

4356 

287496 

8.1240 

4.0412 

7 

49 

343 

2.6458 

1.9129 

67 

4489 

300763 

8.1854 

4.0615 

8 

64 

512 

2.8284 

2.0000 

68 

4624 

311432 

8.2462 

4.0817 

9 

81 

729 

3.0000 

2.0801 

69 

4761 

328509 

8.3066 

4.1016 

10 

100 

1000 

3.1623 

2.1544 

70 

4900 

343000 

8.3666 

4.1213 

11 

121 

1331 

3.3166 

2.2240 

71 

5041 

357911 

8.4261 

4.1408 

12 

114 

1728 

3.4641 

2.2894 

72 

5184 

373248 

8.4853 

4.1602 

13 

169 

2197 

3.6056 

2.3513 

73 

5329 

38S017 

8.5440 

4.1793 

14 

196 

2744 

3.7417 

2.4101 

74 

5*76 

405224 

8.6023 

4.1983 

15 

225 

3375 

3.8730 

2.4062 

75 

5625 

421875 

8.6603 

4.2172 

16 

256- 

4096 

4.0000 

2.5198 

76 

5776 

438976 

8.7178 

4.2358 

17 

289 

4913 

4.1231 

2.5713 

77 

£929 

456533 

8.7750 

4.2543 

18 

324 

5832 

4.2426 

2.6207 

78 

C084 

474552 

8.&318 

4.2727 

19 

361 

6859 

4.3589 

2.6681 

79 

6241 

493039 

8.8882 

4.2908 

20 

400 

8000 

4.4721 

2.7144 

80 

6400 

512000 

8.9443 

4.3089 

21 

441 

9261 

4.5826 

2.7589 

81 

6561 

531441 

9. 

4.3267 

22 

484 

10648 

4.6904 

2.8020 

82 

6724 

551368 

9.0554 

4.3445 

23 

529 

12167 

4.7958 

2.8439 

83 

6889 

571V87 

9.1104 

4.3621 

24 

576 

13824 

4.8990 

2.8845 

84 

7056 

592704 

9.1652 

4.3795 

25 

625 

15625 

5.0000 

2.9240 

85 

7225 

614125 

9.2195 

4.3968 

26 

676 

17576 

5.0990 

2.9625 

86 

7396 

636056 

9.2736 

4.4140 

27 

729 

19683 

5.1C62 

3.0000 

87 

7569 

658503 

9.3274 

4.4310 

28 

784 

21952 

5.2915 

3.0366 

88 

7744 

681472 

9.3808 

4.4480 

29 

841 

24389 

5.3852 

3.0723 

89 

7921 

704969 

9.4340 

4.4647 

30 

900 

27000 

5.4772 

3.1072 

90 

8100 

729000 

9.4868 

4.4814 

31 

961 

29791 

5.5678 

3.1414 

91 

82S1 

753571 

9.5394 

4.4979 

32 

1024 

32768 

5.6569 

3.1748 

92 

8464 

778688 

9.5917 

4.5144 

33 

1039 

35937 

5.7446 

3.2075 

93 

8G49 

804357 

9.6437 

4.5307 

34 

1156 

39304 

5.8310 

3.2396 

94 

8836 

830584 

9.6954 

4.5468 

35 

1225 

42875 

5.9161 

3.2711 

95 

9025 

857375 

9.7468 

4.5629 

36 

1296 

46656 

6.0000 

3.3019 

96 

9216 

884736 

9.7980 

4.5789 

37 

1369 

50653 

6.0828 

3.3322 

97 

9409 

912673 

9.8489 

4.5917 

38 

1444 

54872 

6.  Hi  It 

3.3620 

98 

9H04 

941192 

9.8995 

4.6104 

39 

1521 

59319 

6.2450 

3.3912 

99 

9801 

970299 

9.9499 

4.G2G1 

40 

ItiOO 

64000 

6.3246 

3.4200 

100 

10000 

1000000 

10. 

4.6416 

41 

1681 

68921 

6.4031 

3.4482 

101 

10201 

1030301 

10.0499 

4.6570 

42 

1764 

74088 

6.4807 

3.4760 

102 

10404 

1061208 

10.0995 

4.6723 

43 

1849 

79507 

6.5574 

3.5034 

103 

1060;) 

1092727 

10.1489 

4.6875 

44 

1936 

85184 

6.6332 

3.530:5 

104- 

0816 

1124864 

10.1980 

4.7027 

45 

2025 

91125 

6.7082 

3.5569 

105 

1025 

1157625 

10.2470 

4.7177 

46 

2116 

97336 

6.7823 

3.5830 

106 

1236 

1191016 

10.2956 

4.7326 

47 

2209 

103823 

6.8557 

3.0088 

107 

1449 

1225043 

10.3441 

4.7475 

48 

21304 

110592 

6.9282 

3.6312 

108 

1664 

1259712" 

10.3923 

4.7622 

49 

2401 

117019 

7.0000 

3.6593 

1  >9 

1881 

1295029 

10.4403 

4.7769 

50 

2500 

125000 

7.0711 

3.6840 

1  0 

2100 

1331000 

10.4881 

4.7914 

61 

2601 

132G51 

7.1414 

3.7084 

1  1 

2321 

1367631 

10.5357 

4.8059 

52 

2704 

140608 

7.2111 

3.7325 

1  2 

2544 

1404928 

10.5830 

4.8203 

53 

2809 

148877 

7.2801 

3.7563 

1  3 

2769 

1442897 

10.6301 

4.a346 

54 

2<>16 

157464 

7.34*5 

8.T798 

1  4 

2996 

1481544 

10.6771 

4.8488 

55 

3025 

166375 

7.4162 

3.8030 

115 

3225 

1520875 

10.7238 

4.8629 

56 

3136 

175616 

7.4833 

3.8259 

116 

3456 

1560896 

10.7703 

4.8770 

57 

3249 

185193 

7.5498 

3.51485 

117 

3689 

1601613 

10.8167 

4.8910 

58 

3364 

195112 

7.6158 

3.870-) 

118 

3924 

1643032 

10.8628 

4.9049 

59 

3181 

205379 

7.6H11 

3.  8930 

119 

14161 

1685159 

10.9087 

4.9181 

60 

3600 

216000 

7.7460 

3.9149 

120 

14400 

1728000 

10.9545 

4.9324 

50 


SQUARES,  CUBES,  AND   ROOTS. 


TABLE  of  Squares,  Cubes,  Square  Roots,  and  Cube  Roots, 
of  Numbers  from  1  to  1OOO  —  (CONTINUED.) 


Square. 

Cube. 

Sq.  Rt. 

C.  lit. 

No. 

Square. 

Cube. 

Sq.  Rt. 

C.I 

14641 

1771561 

11. 

4.9461 

186 

34596 

6434856 

13.6382 

5.7( 

14884 

1815848 

11.0454 

4.9597 

187 

34969 

6539203 

13.6748 

5.7 

15129 

1860867 

11.0905 

4.9732 

188 

35344 

6644672 

13.7113 

5.7 

15376 

1906624 

11.1355 

4.9866 

189 

35721 

6751269 

13.7477 

5.7, 

15625 

1953125 

11.1803 

5. 

190 

36100 

6859000 

13.7840 

5.7< 

15876 

2000376 

11.2250 

5.0133 

191 

36481 

6967871 

13.8203 

5.7, 

16129 

2048383 

11.2694 

5.0265 

192 

36864  . 

7077888 

13.8564 

5.7< 

16384 

2097152 

11.3137 

5.0397 

193 

37249 

7189057 

13.8924 

5.7 

16641 

2146689 

11.3578 

5.0528 

194 

37636 

7301384 

13.9284 

5.7* 

16900 

2197000 

11.4018 

5.0658 

195 

38025 

7414875 

13.9642 

5.71 

17161 

2248091 

11.4455 

5.0788 

196 

88416 

7529536 

14. 

5.« 

17424 

2299968 

11.4891 

5.0916 

197 

38809 

7645373 

14.0357 

5.8 

17689 

2352637 

11.5326 

5.1045 

198 

39204 

7762392 

14.0712 

5.8 

17956 

2406104 

11.5758 

5.1172 

199 

39601 

7880599 

14.1067 

5.8. 

18225 

2460375 

11.6190 

5.1299 

200 

40000 

8000000 

14.1421 

6.* 

18496 

2515456 

11.6619 

5.1426 

201 

40401 

8120601 

14.1774 

5.8 

18769 

2571353 

11.7047 

5.1551 

202 

40804 

8242408 

14.2127 

5.8 

19044 

2628072 

11.7473 

5.1676 

203 

41209 

8365427 

14.2478 

5.8 

19321 

2685619 

11.7898 

5.1801 

204 

41616 

8489664 

14.2829 

5.8 

19600 

2744000 

11.8322 

5.1925 

205 

42025 

8615125 

14.3178 

5.8S 

19881 

2803221 

11.8743 

5.2048 

206 

42436 

8741816 

14.3527 

5.9( 

20164 

2863288 

11.9164 

5.2171 

207 

42849 

8869743 

14.3875 

5.9 

20449 

2924207 

11.9583 

5.2293 

208 

43264 

8998912 

14.4222 

5.9 

20736 

2985984 

12. 

5.2415 

209 

43681 

9129329 

14.4568 

5.9 

21025 

3048625 

12.0416 

5.2536 

210 

44100 

9261000 

14.4914 

5.9 

21316 

3112136 

12.0830 

5.2656 

211 

44521 

9393931 

14.5258 

5.9 

21609 

3176523 

12.1244 

5.2776 

212 

44944 

9528128 

14.5602 

5.9 

21904 

3241792 

12.1055 

5.2896 

213 

45369 

9663597 

14.5945 

5.9 

22201 

3307949 

12.2066 

5.3015 

214 

45796 

9800344 

14.6287 

5.9* 

22500 

3375000 

12.2474 

5.3133 

215 

46225 

9938375 

14.6629 

5.9< 

22801 

3442951 

12.2882 

5.3251 

216 

46656 

10077696 

14.6969 

6. 

23104 

3511808 

12.3288 

5.3368 

217 

47089 

10218313 

14.7309 

6.0 

23409 

3581577 

12.3693 

5.3485 

218 

47524 

10360232 

14.7648 

6.0 

23716 

3652264 

12.4097 

5.3601 

219 

47961 

10503459 

14.7986 

6.0 

24025 

3723875 

12.4499 

5.3717 

220 

48400 

10648000 

14.8324 

6.0. 

24336 

3796416 

12.4900 

5.3832 

221 

48841 

10793861 

14.8661 

6.0< 

24619 

3869893 

12.5300 

5.3947 

222 

49284 

10941048 

14.8997 

6.0, 

24964 

3944312 

12.5698 

5.4061 

223 

49729 

11089567 

14.9332 

6.0( 

25281 

4019679 

12.6095 

5.4175 

224 

50176 

11239424 

14.9666 

6.0 

25600 

4096000 

12.6491 

5.4288 

225 

50625 

11390625 

15. 

6.0. 

25921 

4173281 

12.6886 

5.4401 

226 

51076 

11543176 

15.0333 

6.0J 

26244 

4251528 

12.7279 

5.4514 

227 

'51529 

11697083 

15.0665 

6.K 

26569 

43IS0747 

12.7671 

5.4626 

228 

51984 

11852352 

15.0997 

6.H 

26896 

4410944 

12.8062 

5.4737 

229 

52441 

12008989 

15.1327 

6.11 

27225 

4492125 

12.8452 

5.4848 

230 

52900 

12167000 

15.1658 

6.1' 

27556 

4574296 

12.8841 

5.4959 

231 

53361 

12326391 

15.1987 

6.K 

27889 

4657463 

12.9228 

5.5069 

232 

53824 

12487168 

15.2315 

6.14 

28224 

4741632 

12.9615 

5.5178 

233 

54289 

12649337 

15.2643 

6.  If 

28561 

4826809 

13. 

5.5288 

234 

54756 

12812904 

15.2971 

6.1f 

28900 

4913000 

13.0384 

5.5397 

235 

55225 

12977875 

15.3297 

6.11 

29241 

5000211 

13.0767 

5.5505 

236 

55696 

13144256 

15.3623 

6.11 

29584 

5088448 

13.1149 

5.5613 

237 

56169 

13312053 

15.3948 

6.18 

29929 

5177717 

13.1529 

5.5721 

238 

56644 

13481272 

15.4272 

6.1f 

30276 

5268024 

13.1909 

5.5828 

239 

57121 

13651919 

15.4596 

6.2C 

30625 

5359375 

13.2288 

5.5934 

240 

57600 

13824000 

15.4919 

6.21 

30976 

5451776 

13.2665 

5.6041 

241 

58081 

13997521 

15.5242 

6.25 

31329 

5545233 

13.3041 

5.6147 

242 

58564 

14172488 

15.5563 

6.23 

31684 

5639752 

13.3417 

5.6252 

243 

59049 

14348907 

15.5885 

6.24 

32041 

5735339 

13.3791 

5.6357 

244 

59536 

14526784 

15.6205 

6.24 

32400 

5832000 

13.4164 

5.6462 

245 

60025 

14706125 

15.6525 

6.25 

32761 

5929741 

13.4536 

5.6567 

246 

60516 

14886936 

15.6844 

6.26 

33124 

6028568 

13.4907 

5.6671 

247 

61009 

15069223 

15.7162 

6.27 

33489 

6128487 

13.5277 

5.6774 

248 

61504 

15252992 

15.7480 

6.28 

33856 

6229504 

13.5647 

5.6877 

249 

62001 

15438249 

15.7797 

3.28 

34225 

6331625 

13.6015 

5.6980 

250 

62500 

15625000 

15.8114 

6.28 

SQUARES,  CUBES,  AND  ROOTS. 


51 


TABLE  of  Squares,  Cubes,  Square  Roots,  and  Cube  Roots, 
of  Numbers  from  1  to  1OOO  —  (CONTINUED.) 


No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Rt 

No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Rt. 

251 

63001 

15813251 

15.8430 

6.3080 

316 

99856 

31554496 

17.7764 

6.8113 

252 

63504 

16003008 

118745 

6.3164 

317 

100489 

31855013 

17.8045 

6.8185 

253 

64009 

16194277 

15.9060 

6.3247 

318 

101124 

32157432 

17.8326 

6.8256 

254 

64516 

16387064 

15.9374 

6.3330 

319 

101761 

32461759 

17.8606 

6.8328 

255 

65025 

16581375 

15.9687 

6.3413 

320 

102400 

32768000 

17.8885 

6.8399 

256 

65536 

16777216 

16. 

6.3496 

321 

103041 

33076161 

17.9165 

6.8470 

257 

66049 

16974593 

16.0312 

6.3579 

322 

103684 

33386248 

17.9444 

6.8541 

258 

66564 

17173512 

16.0624 

6.3661 

323 

104329 

33698267 

17.9722 

6.8612 

259 

67081 

17373979 

16.0935 

6.3743 

324 

104976 

34012224 

18. 

6.8683 

280 

67600 

17576000 

16.1245 

6.3825 

325 

105625 

34328125 

18.0278 

6.8753 

261 

68121 

17779581 

16.1555 

6.3907 

326 

106276 

34645976 

18.0555 

6.8824 

262 

68644 

17984728 

16.1864 

6.3988 

327 

106929 

34965783 

18.0831 

6.8894 

263 
264 

69696 

18399744 

16.2481 

6.4070 
6.4151 

328 
329 

107584 
108241 

35287552 
35611289 

18.1108 

18.1384 

6.8964 
6.9034 

265 

70225 

18609625 

16.2788 

6.4232 

330 

108900 

35937000 

18.1659 

6.9104 

266 

70756 

18821096 

16.3095 

6.4312 

331 

109561 

36264691 

18.1934 

6.9174 

267 

71289 

19034163 

16.3401 

6.4393 

332 

110224 

36594368 

18.2209 

6.9244 

268 

71824 

19248832 

16.3707 

6.4473 

333 

110889 

36926037 

18.2483 

6.9313 

269 

72361 

19465109 

16.4012 

6.4553 

334 

111556 

37259704 

18.2757 

6.9382 

270 

72900 

19683000 

16.4317  . 

6.4633 

335 

112225 

37595375 

18.3030 

6.9451 

271 

73441 

19902511 

16.4621 

6.4713 

336 

112896 

37933056 

18.3303 

6.9521 

272 

73984 

20123648 

16.4924 

6.4792 

337 

113569 

38272753 

18.3576 

6.9589 

273 

74529 

20346417 

16.5227 

6.4872 

338 

114244 

38614472 

18.3848 

6.9658 

274 

75076 

20570824 

16.5529 

6.4951 

339 

114921 

38958219 

18.4120 

6.9727 

275 

75625 

20796875 

16.5831 

6.5030 

340 

115600 

39304000 

18.4391 

6.9795 

276 

76176 

21024576 

16.6132 

6.5108 

341 

116281 

39651821 

18.4662 

6.9864 

277 

76729 

21253933 

16.6433 

6.5187 

342 

116964 

40001688 

18.4932 

6.9932 

278 

77284 

21484952 

16.6733 

6.5265 

343 

117649 

40353607 

18.5203 

7. 

279 

77841 

21717639 

16.7033 

6.5343 

344 

118336 

40707584 

18.5472 

7.0068 

280 

78400 

21952000 

16.7332 

6.5421 

345 

119025 

41063625 

18.5742 

7.0136 

281 

78961 

22188041 

16.7631 

6.5499 

346 

119716 

41421736 

18.6011 

7.0203 

282 

79524 

22425768 

16.7929 

6.5577 

347 

120409 

41781923 

18.6279 

7.0271 

283 

80089 

22665187 

16.8226 

6.5654 

348 

121104 

42144192 

18.6548 

7.0338 

284 

80656 

22906304 

16.8523 

6.5731 

349 

121801 

42508549 

18.6815 

7.0406 

285 

81225 

23149125 

16.8819 

6.5808 

350 

122500 

42875000 

18.7083 

7.0473 

286 

81796 

23393656 

16.9115 

6.5885 

351 

123201 

43243551 

18.7350 

7.0540 

287 

82369 

2363.9903 

16.9411 

6.5!lo2 

352 

123904 

43614208 

18.7617 

7.0607 

288 

82944 

23887872 

16.9706 

6.6039 

353 

124609 

43986977 

18.7883 

7.0674 

289 

83521 

24137569 

17. 

6.6115 

354 

125316 

44361864 

18.8149 

7.0740 

290 

84100 

24389000 

17.0294 

6.6191 

355 

126025 

44738875 

18.8414 

7.0807 

291 

84681 

24642171 

17.0587 

6.6267 

356 

126736 

45118016 

18.8680 

7.0873 

292 

85264 

24897088 

17.0880 

6.6343 

357 

1274-49 

45499293 

18.8944 

7.0940 

293 

85849 

25153757 

17.1172 

6.6419 

358 

128164 

45882712 

18.9209 

7.1006 

294 

86436 

25412184 

17.1464 

6.6494 

359 

128881 

46268279 

18.9473 

7.1072 

295 

87025 

25672375 

17.1756 

6.6569 

360 

129600 

46656000 

18.9737 

7.1138 

2% 

87616 

25934336 

17.2047 

6.6644 

361 

130321 

47045881 

19. 

7.1204 

297 

88209 

26198073 

17.2337 

6.G719 

362 

131044 

47437928 

19.0263 

7.1269 

298 

88804 

26463592 

17.2627 

6.6794 

363 

131769 

47832147 

19.0526 

7.1335 

299 

89401 

26730899 

17.2916 

6.6869 

364 

132496 

48228544 

19.078« 

7.1400 

300 

90000 

27000000 

17.3205 

6.6943 

365 

133225 

48627125 

19.1050 

7.1466 

301 

90601 

27270901 

17.3494 

6.7018 

366 

133956 

49027896 

19.1311 

7.1531 

302 

91204 

27543608 

17.3781 

6.7092 

367 

134689 

49430863 

19.1572 

7.1596 

303 

91809 

27818127 

17.4069 

6.7166 

368 

135424 

49836032 

19.1833 

7.1661 

304 

92H6 

28094464 

17.4356 

6.7240 

369 

136161 

50243409 

19.2094 

7.1726 

305 

93025 

28372625 

17.4642 

6.7313 

370 

136900 

50653000 

19.2354 

7.1791 

306 

93636 

28652616 

17.4929 

6.7387 

371 

137641 

51064811 

19.2614 

7.1855 

307 

94249 

28934443 

17.5214 

6.7460 

372 

138384 

51478848 

19.2873 

7.1920 

308 

94864 

29218112 

17.5499 

6.7533 

373 

139129 

51895117 

19.3132 

7.1984 

309 

95481 

29503629 

17.5784 

6.7606 

374 

139876 

52313624 

19.3391 

7.2048 

310 

96100 

29791000 

17.6068 

6.7679 

375 

140625 

52734375 

19.3649 

7.2112 

311 

96721 

30080231 

17.6352 

6.7752 

376 

141376 

53157376 

19.3907 

7.2177 

312 

97344 

30371328 

17.6635 

6.7H24 

377 

142129 

53582633 

19.4165 

7.2240 

313 

97969 

30664297 

17.6918 

6.7897 

378 

142884 

54010152 

19.4422 

7.2304 

314 

98596 

30959144 

17.7200 

6.7969 

379 

143641 

54439939 

19.4679 

7.2368 

315 

992/0 

31255875 

17.7482 

6.8041 

380 

144400 

54872000 

19.4936 

7.2432 

52 


SQUARES,  CUBES,  AND  ROOTS. 


TABLE  of  Squares,  Cubes,  Square  Roots,  and  Cube  Roots, 
of  Numbers  from  1  to  100O  — (CONTINUED.) 


No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Ht. 

No. 

Square. 

Cube. 

Sq.  Bt. 

C.Rt. 

381 

145161 

55306341 

19.5192 

7.2485 

446 

198916 

88716536 

21.1187 

7.6403 

382 

145324 

55742968 

19.5448 

7.2558 

447 

199809 

89314623 

21.1424 

7.6460 

383 

146689 

56181887 

19.5704 

7.2622 

448 

200704 

89915392 

21.1660 

7.6517 

384 

147456 

56623104 

19.5959 

7.2685 

449 

201601 

90518849 

21.1896 

7.6574 

385 

148225 

57066625 

19.6214 

7.2748 

450 

202500 

91125000 

21.2132 

7.6631 

886 

148996 

57512456 

19.6469 

7.2811 

451 

203401 

91733851 

21.2368 

7.6688 

387 

149769 

57960603 

19.6723 

7.2874 

452 

204304 

92345408 

21.2603 

7.6744 

388 

150544 

58411072 

19.6977 

7.2936 

453 

205209 

92959677 

21.2838 

7.6801 

389 

151321 

58863869 

19.7231 

7.2999 

454 

206116 

93576G64 

21.3073 

7.6857 

390 

152100 

59319000 

19.7484 

7.3061 

455 

207025 

94196375 

21.3307 

7.6914 

391 

152881 

59776471 

19.7737 

7.3124 

456 

207936 

94818816 

21.3542 

7.6970 

392 

153664 

60236288 

19.7990 

7.3186 

457 

208849 

95443993 

21.3776 

7.7026 

393 

154449 

60698457 

19.8242 

7.3248 

458 

209764 

96071912 

21.4009 

7.7082 

394 

155236 

61162984 

19.8494 

7.3310 

459 

210681 

96702579 

21.4243 

7.7138 

395 

156025 

61629875 

19.8746 

7.3372 

460 

211600 

97336000 

21.4476 

7.7194 

3% 

156816 

62099136 

19.8997 

7.3434 

461 

212521 

97972181 

21.4709 

7.7250 

397 

157609 

62570773 

19.9249 

7.3196 

462 

213444 

98611128 

21.4942 

7.7306 

398 

158404 

63044792 

19.9499 

7.3558 

4fi3 

214369 

99252847 

21.5174 

7.7362 

399 

159201 

63521199 

19.9750 

7.3019 

464 

215296 

99897344 

21.5407 

7.7418 

400 

160000 

64000000 

20. 

7.3681 

465 

216225 

100544625 

21.5639 

7.7473 

401 

160801 

6*481201 

20.0250 

7.3742 

466 

217156 

101194696 

21.5870 

7.7529 

402 

161604 

64964808 

20.0499 

7.3803 

467 

218089 

101847563 

21.6102 

7.7584 

403 

162409 

65450827 

20.0749 

7.3861 

4(58 

219024 

102503232 

21.6333 

7.7639 

404 

163216 

65939264 

20.0998 

7.3925 

469 

219961 

103161709 

21.6564 

7.7695 

405 

164025 

66430125 

20.1246 

7.3986 

470 

220900 

103823000 

21.6795 

.  7.7750 

406 

164836 

66923416 

20.1494 

7.4047 

471 

221841 

104487111 

21.7025 

7.7805 

407 

165849 

67419143 

20.1742 

7.4108 

72 

222784 

105154048 

21.7256 

7.7860 

408 

166464 

67917312 

20.1990 

7.4169 

73 

223729 

105823817 

21.7486 

7.7915 

409 

167281 

68417929 

20.2237 

7.4229 

74 

224676 

106496424 

21.7715 

7.7970 

410 

168100 

68921000 

20.2485 

7.4290 

75 

225625 

107171875 

21.7945 

7.8025 

411 

168921 

69426531 

20.2731 

7.4350 

76 

226576 

107850176 

21.8174 

7.8079 

412 

169744 

69934528 

20.2978 

7.4410 

77 

227529 

108531333 

21.8403 

7.8134 

413 

170569 

70444997 

20.3224 

7.4470 

78 

228484 

109215352 

21.8632 

7.8188 

414 

171396 

70957944 

20.3470 

7.4530 

479 

229441 

109902239 

21.8861 

7.8243 

415 

172225 

71473375 

20.3715 

7.4590 

480 

230400 

110592000 

21.9089 

7.8297 

416 

173056 

71991296 

20.3961 

7.4650 

481 

231361 

111234641 

21.9317 

7.8352 

417 

173889 

72511713 

20.4206 

7.4710 

482 

232324 

111980168 

21.9545 

7.8406 

418 

174724 

73034632 

20.4450 

7.4770 

483 

233289 

112678587 

21.9773 

7.8460 

419 

175561 

73560059 

20.4695 

7.4829 

4S4 

234256 

113379904 

22. 

7.8514 

420 

176400 

74088000 

20.4939 

7.4889 

485 

235225 

114084125 

22.0227 

7.8568 

421 

177241 

74618461 

20.51&? 

7.4948 

486 

.  236196 

114791256 

22.0454 

7.8622 

422 

178084 

75151448 

20.5426 

7.5007 

487 

237169 

115501303 

22.0681 

7.8676 

423 

178929 

756869<>7 

20.5670 

7.5067 

488 

23814  t 

116214272 

22.0907 

7.8730 

424 

179776 

76225024 

20.5913 

7.5126 

48:> 

233121 

11G9301C9 

22.1133 

7.8784 

425 

180625 

76765625 

20.6155 

7.5185 

490 

240100 

117649000 

22.1359 

7.8837 

426 

181476 

77308776 

20.6398 

7.5244 

431 

241081 

118370771 

22.1585 

7.8891 

427 

182329 

77354483 

20.6640 

7.5302 

492 

242064 

119095488 

22.1811 

7.8944 

428 

183184 

78402752 

20.6882 

7.5361 

493 

243049 

11982315 

22.2036 

7.8998 

429 

184041 

78953589 

20.7123 

7.5420 

494 

244036 

120553784 

22.2261 

7.9051 

430 

184900 

79507000 

20.7364 

7.5478 

495 

245025 

121287375 

22.2486 

7.9105 

431 

185761 

80062991 

20.7605 

7.5537 

4ns 

246016 

122023936 

22.2711 

7.9158 

432 

186624 

80621568 

20.7346 

7.5595 

497 

247009 

122763473 

22.2935 

7.9211 

433 

187489 

81182737 

20.8087 

7.5654 

498 

24S004 

123505992 

22.3159 

7.9264 

434 

188356 

81746504 

20.8327 

7.5712 

499 

249001 

124251499 

22.3383 

7.9317 

435 

189225 

82312875 

20.8567 

7.5770 

500 

250000 

125000000 

22.3607 

7.9370 

436 

190096 

82881856 

20.8806 

7.5828 

501 

251001 

125751501 

22.3830 

7.9423 

437 

190969 

83453453 

20.9045 

7.5S86 

502 

252004 

12850300fc 

22.4054 

7.9476 

438 

191844 

84027672 

20.9284 

7.5944 

503 

253009 

127263527 

22.4277 

7.9523 

439 

192721 

84604519 

20.9523 

7.6001 

504 

254016 

12802KK51 

22.4499 

7.9581 

440 

193600 

85184000 

20.9762 

7.6059 

505 

255025 

128787623 

22.4722 

7.9634 

441 

194481 

85766121 

21. 

7.6117 

506 

256036 

129554216 

22.4944 

7.9686 

442 

1  95364 

86350888 

21.0238 

7.6174 

507 

257019 

1303233*1 

22.5167 

7.9739 

443 

196249 

86938307 

21.0476 

7.6232 

508 

258064 

131096515 

22.5389 

7.9791 

444 

197136 

87528384 

21.0713 

7.6289 

509 

259081 

1318722X 

22.5610 

7.9843 

445 

198025 

88121125 

21.0950 

7.6346 

510 

260100 

13265100C 

22.5832 

7.9896 

SQUARES,  CUBES,  AND  ROOTS. 


53 


TABLE  of  Squares,  Cubes,  Square  Roots,  and  Cube  Roots, 
of  Numbers  from  1  to  1OOO  — (CONTINUED.) 


No. 

Square. 

Cube. 

Sq.  Ht. 

C.  lit. 

No. 

Square. 

Cube. 

Sq.  Kt. 

C.  Kt. 

511 

261121 

133432831 

22.6053 

7.9948 

576 

331776 

191102976 

24. 

8.3203 

512 

262144 

13A217728 

22.6274- 

8. 

577 

332929 

192100033 

24.0208 

8.3251 

513 

263169 

135005697 

22.6495 

8.0052 

578 

334084 

193100552 

24.0416 

8.3300 

514 

264196 

135796744 

22.6716 

8.0104 

579 

335241 

194104539 

24.0624 

8.3348 

5-15 

265225 

136590875 

22.6936 

8.0156 

580 

336400 

195112000 

24.0832 

8.3396 

516 

266256 

137388096 

22.7156 

8.0208 

581 

337561 

196122941 

24.1039 

8.3443 

517 

267289 

138188413 

22.7376 

8.0260 

582 

338724 

197137368 

24.1247 

8.3491 

518 

268324 

138991832 

22.7596 

8.0311 

583 

339889 

198155287 

24.1454 

8.3539 

519 

269361 

139798359 

22.7816 

8.0363 

584 

341056 

199176704 

24.1661 

8.3587 

520 

270400 

140608000 

22.8035 

8.0415 

585 

342225 

200201625 

24.1868 

8.3634 

521 

271441 

141420761 

22.8254 

8.0466 

586 

343396 

2012*056 

24.2074 

8.3682 

522 

272484 

142236648 

22.8473 

8.0517 

587 

344569 

202262003 

24.2281 

8.3730 

523 

273529 

143055667 

22.8692 

8.0569 

588 

345744 

203297472 

24.2487 

8.3777 

524 

274576 

143877824 

22.8910 

8.0620 

589 

346921 

204336469 

24.2693 

8.3825 

525 

275625 

144703125 

22.9129 

8.0671 

590 

348100 

205379000 

24.2899 

8.3872 

526 

276676 

145531576 

22.9347 

8.0723 

591 

349281 

206425071 

24.3105 

8.3919 

527 

277729 

146363183 

22.9565 

8.0774 

592 

350464 

207474688 

24.3311 

8.3967 

528 

278784 

147197952 

22.9783 

8.0825 

593 

351649 

208527857 

24.3516 

8.4014 

529 

279841 

148035889 

23. 

8.0876 

594 

352836 

209584584 

24.3721 

8.4061 

530 

280900 

148877000 

23.0217 

8.0927 

595 

354025 

210644875 

24.392S 

8.4108 

531 

281961 

149721291 

23.0434 

8.0978 

596 

355216 

211708736 

24.4131 

8.4155 

532 

283024 

150568768 

23.0651 

8.1028 

597 

356409 

212776173 

24.4336 

8.4202 

533 

284089 

151419437 

23.0868 

8.1079 

598 

357604 

213847192 

24.4540 

8.4249 

534 

285156 

152273304 

23.1084 

8.1130 

599 

358801 

214921799 

24.4745 

8.4296 

535 

286225 

153130375 

23.1301 

8.1180 

600 

360000 

216000000 

24.4949 

8.4343 

536 

287296 

153990656 

23.1517 

8.1231 

601 

361201 

217081801 

24.5153 

8.4390 

537 

288369 

154854153 

23.1733 

8.1281 

602 

362404 

218167208 

24.5357 

8.4437 

538 

289444 

155720872 

23.1948 

8.1332 

603 

363609 

219256227 

24.5561 

8.4484 

539 

290521 

156590819 

23.2164 

8.1382 

604 

364816 

220348864 

24.5764 

8.4530 

540 

291600 

157464000 

23.2379 

8.1433 

605 

366025 

221445125 

24.5967 

8.4577 

541 

292681 

158340421 

23.2594 

8.1483 

606 

367236 

222545016 

24.6171 

8.4623 

542 

293764 

159220088 

23.2809 

8.1533 

607 

368449 

223648543 

24.6374 

8.4670 

543 

294849 

160103007 

23.3024 

8.1583 

608 

369664 

224755712 

24.6577 

8.4716 

544 

295936 

160989184 

23.3238 

8.1633 

609 

370881 

225866529 

24.6779 

8.4763 

545 

297025 

161878625 

23.3452 

8.1683 

610 

372100 

226981000 

24.6982 

8.4809 

546 

298116 

162771336 

23.3666 

8.1733 

611 

373321 

228099131 

24.7184 

8.4856 

547 

299209 

163667323 

23.3880 

8.1783 

612 

374544 

229220928 

24.7386 

8.4902 

648 

300304 

164566592 

23.4094 

8.1833 

613 

375769 

230346397 

24.T588 

8.4948 

549 

301401 

165469149 

23.4307 

8.1882 

614 

376996 

231475544 

24.7790 

8.4994 

550 

302500 

166375000 

23.4521 

8.1932 

615 

378225 

232608375 

24.7992 

8.5040 

551 

303601 

167284151 

23.4734 

8.1982 

616 

379456 

233744896 

24.8193 

8.5086 

552 

304704 

168196608 

23.4947 

8.2031 

617 

380689 

234885113 

24.8395 

8.5132 

553 

305809 

169112377 

23.5160 

8.2081 

618 

381924 

236029032 

24.8596 

8.5178 

554 

306916 

170031464 

23.5372 

8.2130 

619 

383161 

237176659 

24.8797 

8.5224 

555 

308025 

170953875 

23.5584 

8.2180 

620 

384400 

238328000 

24.8998 

8.5270 

556 

309136 

171879616 

23.5797 

8.2229 

621 

385641 

239483061 

24.9199 

8.5316 

557 

310249 

172808693 

23.6008 

8.2278 

622 

386884 

240641848 

24.9399 

8.5362 

558 

311364 

173741112 

23.6220 

8.2327 

623 

388129 

241804367 

24.9600 

8.5408 

559 

312481 

174676879 

23.6432 

8.2377 

624 

389376 

242970624 

24.9800 

8.5453 

560 

313600 

175616000 

23.6643 

8.2426 

625 

390625 

244140625 

25. 

8.5499 

561 

314721 

176558481 

23.6854 

8.2475 

626 

391876 

245314376 

25.0200 

8.5544 

562 

315844 

177504328 

23.7065 

8.2524 

627 

393129 

246491883 

25.0400 

8.5590 

563 

316969 

178453547 

23.7276 

8.2573 

628 

394384 

247673152 

25.0599 

8.5635 

564 

318096 

179406144 

23.7487 

8  2621 

629 

395641 

248858189 

25.0799 

8.5681 

565 

319225 

180362125 

23.7697 

8.2670 

630 

396900 

250047000 

25.0998 

8.5726 

566 

320356 

181321496 

23.7908 

8.2719 

631 

398161 

251239591 

25.1197 

8.5772 

567 

321489 

182284263 

23.8118 

8.2768 

632 

399424 

252435968 

25.1396 

8.5817 

568 

322624 

183250432 

23.8328 

8.2816 

633 

400689 

253636137 

25.1595 

8.5862 

569 

323761 

184220009 

23.8537 

8.2865 

634 

401956 

254840104 

25.1794 

8.5907 

570 

324900 

185193000 

23.8747 

8.2913 

635 

403225 

256047875 

25.1992 

8.5952 

571 

326041 

186169411 

23.8956 

8.2962 

636 

404496 

257259456 

25.2190 

8.5997 

572 

327184 

187149248 

23.9165 

8.3010 

637 

405769 

258474853 

25.2389 

8.6043 

573 

328329 

188132517 

23.9374 

8.3059 

638 

407044 

259694072 

25.2587 

8.6088 

574 

329476 

189119224 

23.9583 

8.3107 

639 

408321 

260917119 

25.2784 

8.613'Jt 

§75 

330625 

190109375 

23.9792 

8.3155 

640 

409600 

262144000 

25.2982 

8.6177 

54 


SQUARES,  CUBES,  AND   ROOTS. 


TABLE  of  Squares,  Cnbes,  Square  Roots,  and  Cube  Roots, 
of  X aim  her*  from  1  to  10OO  —  (CONTINUED.) 


I 

Square. 

Cube. 

Sq.  Rt. 

C.  Rt. 

No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Rt. 

410881 

263374721 

25.3180 

8.6222 

706 

498436 

351895816 

26.5707 

8.9043 

412164 

264609288 

25.3377 

8.6267 

707 

499849 

353393243 

26.5895 

8.9085 

413449 

265847707 

25.3574 

8.6312 

708 

501264 

354894912 

26.6083 

8.9127 

414736 

267089984 

25.3772 

8.6357 

709 

502681 

356400829 

26.6271 

8.9169 

416025 

268336125 

25.3969 

8.6401 

710 

504100 

357911000 

26.6458 

8.9211 

417316 

269586136 

25.4165 

8.6446 

711 

505521 

359425431 

26.6646 

8.9253 

418609 

270840023 

25.4362 

8.6490 

712 

506944 

360944128 

26.6833 

8.9295 

419904 

272097792 

25.4558 

8.6535 

713 

508369 

362467097 

26.7021 

8.9337 

421201 

273359449 

25.4755 

8.6579 

714 

509796 

363994344 

26.7208 

8,9378 

422500 

274625000 

25.4951 

8.6624 

715 

511225 

365525875 

26.7395 

8.9420 

423801 

275894451 

25.5147 

8.6668 

716 

512656 

367061696 

26.7582 

8.9462 

425104 

277167808 

25.5343 

8.6713 

717 

514089 

368601813 

26.7769 

8.9503 

426409 

278445077 

25.5539 

8.6757 

718 

515524 

370146232 

26.7955 

8.9545 

427716 

279720264 

25.5734 

8.6801 

719 

516961 

371694959 

26.8142 

8.9587 

429025 

281011375 

25.5930 

8.6845 

720 

518400 

373248000 

26.8328 

8.9628 

430336 

282300416 

25.6125 

8.6890 

721 

519841 

374805361 

26.8514 

8.9670 

431649 

283593393 

25.6320 

8.6934 

722 

521284 

376367048 

26.8701 

8.9711 

432964 

284890312 

25.6515 

8.6978 

723 

522729 

377933067 

26.8887 

8.9752 

434281 

286191179 

25.6710 

8.7022 

724 

524176 

379503424 

26.9072 

8.9794 

435600 

287496000 

25.6905 

8.70G6 

725 

525625 

381078125 

26.9258 

8.9835 

436921 

288804781 

25.7099 

8.7110 

726 

527076 

382657176 

26.9444 

8.9876 

438244 

290117528 

25,7294 

8.7154 

727 

528529 

384240583 

26.9629 

8.9918 

439569 

291434247 

25.7488 

8.7198 

728 

529984 

385828352 

26.9815 

8.9959 

440896 

292754944 

25.7682 

8.7241 

729 

531441 

387420489 

27. 

9. 

442225 

294079625 

25.7876 

8.7285 

730 

532900 

389017000 

27.0185 

9.0041 

443556 

295408296 

25.8070 

8.7329 

731 

534361 

390617891 

27.0370 

9.0082 

444889 

296740963 

25.8263 

8.7373 

732 

535824 

392223168 

27.0555 

9.0123 

446224 

298077632 

25.8457 

8.7416 

733 

537289 

393832837 

27.0740 

9.0164 

447161 

299418309 

25.8650 

8.7460 

734 

538756 

395446904 

27.0924 

9.0205 

448900 

300763000 

25.8844 

8.7503 

735 

540225 

397065375 

27.1109 

9.0246 

450241 

302111711 

25.9037 

8.7547 

736 

541696 

398688256 

27.1293 

9.02H7 

451584 

303464448 

25.9230 

8.7590 

737 

543169 

400315553 

27.1477 

9.0328 

452929 

304821217 

25.9422 

8.7634 

738 

544644 

401947272 

27.1662 

9.0369 

454276 

306182024 

25.9615 

8.7677 

739 

546121 

403583419 

27.1846 

9.0410 

455625 

307546875 

25.9808 

8.7721 

740 

547600 

405224000 

27.2029 

9.0450 

456976 

308915776 

26. 

8.7764 

741 

549081 

406869021 

27.2213 

9.0491 

458329 

310288733 

26.0192 

8.7807 

742 

550564 

408518488 

27.2397 

9.0532 

459684 

311665752 

26.0384 

8.7850 

743 

552049 

410172407 

27.2580 

9.0572 

461041 

313046839 

26.0576 

8.7893 

744 

553536 

411830784 

27.2764 

9.0613 

462400 

314432000 

26.0768 

8.7937 

745 

555025 

413493625 

27.2947 

9.0654 

463761 

315821241 

26.0960 

8.7980 

746 

556516 

415160936 

27.3130 

9.0694 

465124 

317214568 

26.1151 

8.8023 

747 

558009 

416832723 

27.3313 

9.0735 

466489 

318611987 

26.1343 

8.8066 

748 

559504 

418508992 

27.3496 

9.0775 

467856 

320013504 

26.1534 

8.8109 

749 

561001 

420189749 

27.3679 

9.0816 

469225 

321419125 

26.1725 

8.8152 

750 

562500 

421875000 

27.3861 

9.0856 

470596 

322828856 

26.1916 

8.8194 

751 

564001 

423564751 

27.4044 

9.0896 

471969 

324242703 

26.2107 

8.8237 

752 

565504 

425259008 

27.4226 

9.0937 

473344 

325660672 

26.2298 

8.8280 

753 

567009 

426957777 

27.4408 

9.0977 

474721 

327082769 

26.2488 

8.8323 

754 

568516 

428661064 

27.4591 

9.1017 

476100 

328509000 

26.2679 

8.83S6 

755 

570025 

430368875 

27.4773 

9.1057 

477481 

329939371 

26.2869 

8.8408 

756 

571536 

432081216 

27.4955 

9.1098 

478864 

331373888 

26.3059 

8.8451 

757 

573049 

433798093 

27.5136 

9.1138 

480249 

33'/812557 

26.3249 

8.8493 

758 

574564 

435519512 

27.5318 

9.1178 

481636 

334255384 

26.3439 

8.8536 

759 

576081 

437245479 

27.5500 

9.1218 

483025 

335702375 

26.3629 

8.8578 

760 

577600 

438976000 

27.5681 

9.1258 

484416 

337153536 

26.3818 

8.8621 

761 

579121 

440711081 

27.5862 

9.1298 

485809 

338608873 

26.4008 

8.8663 

762 

580644 

442450728 

27.6043 

9.1338 

487204 

340068392 

26.4197 

8.8706 

763 

582169 

444194947 

27.6225 

9.1378 

488601 

341532099 

26.4386 

8.8748 

764 

583696 

445943744 

27.6405 

9.1418 

490000 

343000000 

26.4575 

8.8790 

765 

585225 

447697125 

27.6586 

9.1458 

491401 

344472101 

26.4764 

8.8833 

766 

586756 

449455096 

27.6767 

9.1498 

492804 

345948408 

26.4953 

8.8875 

767 

588289 

451217663 

27.6948 

9.1537 

494209 

347428927 

26.5141 

8.8917 

768 

589824 

452984832 

27.7128 

9.1577 

495616 

348913664 

26.5330 

8.8959 

769 

591361 

454756609 

27.7308 

9.1617 

487026 

360402625 

26.6518 

8.9001 

770 

592900 

466533006 

27.7488 

9.1657 

SQUARES,  CUBES,  AND  ROOTS. 


55 


TABLE  of  Squares,  Cubes,  Square  Roots,  and  Cube  Roots, 
of  Numbers  from  1  to  1OOO  —  (CONTINUED.; 


No. 

Square. 

Cube. 

Sq.  St. 

C.  Rt. 

No. 

Square 

Cube. 

Sq.  Rt. 

C.  R*. 

771 

594441 

45831401 

27.7669 

9.1696 

836 

698896 

58427705 

28.9137 

9.4204 

772 

595984 

46009964 

27.7849 

9.1736 

837 

700569 

58637625d 

28.9310 

9.424! 

773 

597529 

46188991 

27.8029 

9.1775 

838 

702244 

58848047 

28.9482 

9.4279 

774 

599076 

46368482 

27.8209 

9.1815 

839 

703921 

5905897111 

28.9655 

9.4316 

T75 

600625 

46548437 

27.8388 

9.1855 

840 

705600 

59270400 

28.9828 

9.4354 

776 

602176 

467288576 

27.8568 

9.1894 

841 

707281 

59482332 

29. 

9.4391 

777 

603729 

469097433 

27.8747 

9.1933 

842 

708964 

596947688 

29.0172 

9.4429 

778 

605284 

47091095 

27.8927 

9.1973 

843 

710649 

599077107 

29.0345 

9.4466 

779 

606841 

472729139 

27.9106 

9.2012 

844 

712336 

601211584 

29.0517 

9.4503 

780 

608400 

474552000 

27.9285 

9.2052 

845 

714025 

603351120 

29.0689 

9.4541 

781 

609961 

47637954 

27.9464 

9.2091 

846 

715716 

605495736 

29.0861 

9.4578 

782 

611524 

478211768 

27.9643 

9.2130 

817 

717409 

607645423 

29.1033 

9.4615 

783 

613089 

480048687 

27.9821 

9.2170 

848 

719104 

609800192 

29.1204 

9.4652 

784 

614656 

481890304 

28. 

9.2209 

849 

720801 

611960049 

29.1376 

9.4690 

785 

616225 

483736625 

'  28.0179 

9.2248 

850 

722500 

614125000 

29.1548 

9.4727 

786 

617796 

485587656 

28.0357 

9.2287 

851 

724201 

616295051 

29.1719 

9.4764 

787 

619369 

487443403 

28.0535 

9.2326 

852 

725904 

618470208 

291890 

9.4801 

788 

620944 

489303872 

28.0713 

9.2365 

853 

727609 

620650477 

29.2062 

94838 

789 

622521 

491169069 

28.0891 

9.2404 

854 

729316 

622835864 

29.2233 

9.4875 

790 

624100 

493039000 

28.1069 

9.2443 

855 

731025 

625026375 

29.2404 

9.4912 

791 

625681 

494913671 

28.1247 

9.2482 

856 

732736 

627222016 

29.2575 

9.4949 

792 

627264 

496793088 

28.1425 

9.2521 

857 

734449 

62942279: 

29.2746 

9.4986 

793 

628849 

498677257 

28.1603 

9.2560 

858 

736164 

631628712 

29.2916 

9.5023 

794 

630136 

500566184 

28.1780 

9.2599 

859 

737881 

633839779 

29.3087 

9.5060 

795 

632025 

502459875 

28.1957 

9.2638 

860 

739600 

636056000 

29.3258 

9.5097 

796 

633616 

504358336 

28.2135 

9.2677 

861 

741321 

638277381 

29.3428 

9.5134 

797 

635209 

506261573 

28.2312 

9.2716 

862 

743044 

640503928 

29.3598 

9.5171 

798 

636804 

508169592 

28.2489 

9.2754 

863 

744769 

642735647 

29.3769 

9.5207 

799 

638401 

510082399 

28.2666 

9.2793 

864 

746496 

644972544 

29.3939 

9.5244 

800 

640000 

512000000 

28.2843 

9.2832 

865 

748225 

647214625 

29.4109 

9.5281 

801 

641601 

513922401 

28.3019 

9.2870 

866 

749956 

649461896 

29.4279 

9.5317 

802 

643204 

515849608 

28.3196 

9.2909 

867 

751689 

651714363 

29.4449 

9.5354 

803 

644809 

517781627 

28.3373 

9.2948 

808 

753424 

653972032 

29.4618 

9.5391 

804 

646416 

519718464 

28.3549 

9.2986 

869 

755161 

<;.->62::i9i>9 

29.4788 

9.5427 

805 

648025 

521660125 

28.3725 

9.3025 

870 

756900 

658503000 

29.4958 

9.5464 

806 

649636 

523606616 

28.3901 

9.3063 

871 

758641 

660776311 

29.5127 

9.5501 

807 

651249 

525557943 

28.4077 

9.3102 

872 

760384 

663054848 

29.5296 

9.5537 

808 

652864 

527514112 

28.4253 

9.3140 

873 

762129 

665338617 

29.5466 

9.5574 

809 

654481 

529475129 

28.4429 

9.3179 

874 

763876 

667627624 

29.5635 

9.5610 

810 

656100 

531441000 

28.4605 

9.3217 

875 

765625 

669921875 

29.5804 

9.5647 

811 

657721 

533411731 

28.4781 

9.3255 

876 

767376 

672221376 

29.5973 

9.5683 

812 

659344 

5353*732* 

28.4956 

9.3294 

877 

769129 

674526133 

29.6142 

9.5719 

813 

660969 

537367797 

28.5132 

9.3332 

878 

770884 

676836152 

29.6311 

9.5756 

814 

662596 

539353144 

28.5307 

9.3370 

879 

772641 

679151439 

29.6479 

9.5792 

815 

664225 

541343375 

28.5482 

9.3408 

880 

774400 

681472000 

29.6648 

9.5828 

816 

665856 

543338496 

28.5657 

9.3447 

881 

776161 

683797841 

29.6816 

9.5865 

817 

667489 

645338513 

28.5832 

9.3485 

882 

777924 

686128968 

29.6985 

9.5901 

818 

669124 

547343432 

28.6007 

9.3523 

883 

779689 

688465387 

29.7153 

9.5937 

819 

670761 

5W:!.Vi2.V» 

28.6182 

9.3561 

884 

781456 

690807104 

29.7321 

9.5973 

820 

672400 

551368000 

28.6356 

9.3599 

885 

783225 

693154125 

29.7489 

9.6010 

821 

674041 

553387661 

28.6531 

9.3637 

886 

784996 

695506456 

29.7658 

9.6046 

822 

675684 

555412248 

28.6705 

9.3675 

887 

786769 

697864103 

29.7825 

9.6082 

823 

677329 

557441767 

28.6880 

9.3713 

888 

788544 

700227072 

29.7993 

9.6118 

824 

678976 

55947(5224 

2S.70.i4 

9.3751 

889 

790321 

702595369 

29.8161 

9.6154 

825 

680625 

561515625 

28.7228 

9.3789 

890 

792100 

704969000 

29.8329 

9.6190 

826 

682276 

563559976 

28.7402 

9.3827 

891 

793881 

707347971 

29.8496 

9.6226 

827 

683929 

565«092«3 

28.7576 

9.3865 

892 

795664 

709732288 

29.8664 

9.6262 

828 

685584 

567M3552 

28.7750 

9.3902 

893 

797449 

712121957 

29.8831 

9.6298 

829 

687241 

569722789 

28.7924 

9.3940 

894 

799236 

714516984 

29.8998 

9.6334 

830 

688900 

571787000 

28.8097 

9.3978 

895 

801025 

716917375 

29.9166 

9.6370 

831 

690561 

573856191 

28.8271 

9.4016 

896 

802816 

719323136 

29.9333 

9.6406 

832 

692224 

575930368 

28.8444 

9.4053 

897 

804609 

721734273 

29.9500 

9.6442 

833 

693889 

578009537 

28.8617 

9.4091 

898 

806404 

724150792 

29.9666 

9.6477 

834 

695556 

580093704 

28.8791 

9.4129 

899 

808201 

726572699 

29.9833 

9.6513 

835 

697225 

682182S75 

28.8964 

9.4166  J 

900 

810000 

729000000 

30. 

9.6540 

56 


SQUARES,  CUBES,  AND  ROOTS. 


TABLE  of  Square**,  Cubes,  Square  Roots,  and  Cube  Roots, 
of  Numbers  from  1  to  1OOO  —  (CONTINUED.) 


No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Rt. 

No. 

Square. 

Cube. 

Sq.  Rt, 

C.  Rt. 

901 
902 

811801 
813604 

731432701 

733870808 

30.0167 
30.0333 

9.6585 
9.6620 

951 
952 

904101 
906304 

b60085351 
862501408 

30.8383 
30.8545 

9.8339 
0.8374 

903 

815409 

736314327 

30.0500 

9.6656 

953 

908209 

8G5523177 

30.8707 

9.8408 

904 

817216 

738763264 

30.0666 

9.6692 

954 

910116 

868250664 

30.8S69 

9.8443 

905 

819025 

741217625 

30.0832 

9.6727 

955 

912025 

870983875 

30.9031 

9.8477 

906 

820836 

743677416 

30.0998 

9.6763 

956 

913936 

873722816 

30.9192 

9.8511 

907 

822649 

746142643 

30.1164 

9.6799 

957 

915849 

876467-193 

30.9354 

9.8546 

908 

824464 

748613312 

30.1330 

9.6834 

958 

917764 

879217912 

30.9516 

9.8580 

909 

826281 

75  1081)429 

30.1496 

9.6870 

959 

919681 

881974079 

30.9677 

9.8614 

910 

828100 

753571000 

30.1662 

9.6905 

960 

921600 

884736000 

30.9839 

9.8648 

911 

829921 

756058031 

30.1828 

9.6941 

961 

923521 

887503681 

31. 

9.8683 

912 

831744 

758550528 

30.1993 

9.6976 

962 

925444 

890277128 

31.0161 

9.8717 

913 

833569 

761048497 

30.2159 

9.7012 

963 

927369 

893056347 

31.0322 

9.8751 

914 

835396 

76:5551944 

30.2324 

9.7047 

964 

929296 

895841344 

31  .0483 

9.8785 

915 

837225 

766060875 

30.2490 

9.7082 

965 

931225 

898632125 

31.0644 

9.8819 

916 

839056 

768575296 

30.2655 

9.7118 

966 

933156 

901428696 

31.0805 

9.8854 

917 

840889 

771095213 

30.2820 

9.7153 

967 

935089 

904231063 

31.0966 

9.8888 

918 

842724 

773620632 

30.2985 

9.7188 

968 

937024 

907039232 

31.1127 

9.8922 

919 

844561 

776151559 

30.3150 

9.7224 

969 

938961 

909853209 

31.1288 

9.8956 

920 

846400 

778688000 

30.3315 

9.7259 

970 

940900 

912673000 

31.1448 

9.8990 

921 

848241 

781229961 

30.3480 

9.7294 

971 

942841 

915498611 

31.1609 

9.9024 

922 

850084 

783777448 

30.3645 

9.7329 

972 

914784 

918330048 

31.1769 

9.i;058 

923 

851929 

786330467 

30.3809 

9.7364 

973 

946729 

921167317 

31.1929 

9.9092 

924 

853776 

788889024 

30.3974 

9.7400 

974 

948676 

924010424 

31.2090 

9.9126 

925 

855625 

791453125 

30.4138 

9.7435 

975 

950625 

926859375 

31.2250 

9.9160 

926 

857476 

791022776 

30.4302 

9.7470 

976 

952576 

929714176 

31.2410 

9.9194 

927 

859329 

796597983 

30.4467 

9.7505 

977 

954529 

932574833 

31.2570 

9.9227 

928 

861  184 

799178752 

30.4631 

9,7540 

978 

956484 

935441352 

31.2730 

9.9261 

929 

863041 

801765089 

30.4795 

9.7575 

979 

958441 

938313739 

31.2890 

9.9295 

930 

864900 

804357000 

30.4959 

9.7610 

980 

960400 

941192000 

31.3050 

9.9329 

931 

866761 

806954491 

30.5123 

9.76-15 

981 

962361 

944076141 

31.3209 

9.9363 

932 

868624 

809557568 

30.5287 

9.7680 

982 

964324 

946966168 

31.3369 

9.9396 

933 

870489 

8l2166'j:J7 

30.5450 

9.7715 

983 

966289 

949862087 

31.3528 

9.9430 

934 

872356 

814780504 

30.5614 

9.7750 

9S4 

968256 

952763904 

31.3688 

9.9464 

935 

874225 

817400375 

30.5778 

9.7785 

985 

970225 

955671625 

31.3847 

9.9497 

936 

876096 

820025856 

30.5941 

9.7819 

986 

972196 

958585256 

31.4006 

9.9531 

937 

877969 

H22656953 

30.6105 

9.7854 

987 

974169 

961504803 

31.4166 

9.9565 

938 

879844 

S2.V293672 

30.6268 

9.7889 

988 

976144 

964430272 

31.4325 

9.9598 

939 

881721 

827936019 

30.6431 

9.7924 

989 

978121 

967361669 

31.4484 

9.9632 

940 

883600 

830584000 

30.6594 

9.7959 

990 

980100 

970299000 

31.4643 

9.9666 

941 

885481 

833237621 

30.6757 

9.7993 

991 

982081 

973242271 

31.4802 

9.9699 

942 

687364 

835896888 

30.6920 

9.8028 

992 

984064 

976191488 

SI.  4960 

9.9733 

943 

889249 

838561807 

30.7083 

9.8063 

993 

986049 

979146657 

31.5119 

9.9766 

944 

891136 

841232384 

30.7246 

9.8097 

994 

988036 

982107784 

81.5278 

9.9800 

945 

893025 

843908625 

30.7409 

9.8132 

995 

990025 

985074875 

31.5436 

9.9833 

946 

894916 

846590536 

30.7571 

9.8167 

996 

992016 

988047936 

31.5595 

9.9866 

947 

896809 

849278123 

30.7734 

9.8201 

997 

994009 

991026973 

31.5753 

9.95)00 

948 

898704 

851971392 

30.7896 

9.8236 

998 

996004 

994011992 

81.5911 

9.9933 

949 

900601 

854670349 

30.8058 

9.8270 

999 

998001 

997002999 

81.6070 

9.9967 

950 

902500 

857375000 

30.8221 

9.8305 

1000 

1000000 

1000000000 

31.6228 

10. 

SQUARE  AND  CUBE  ROOTS. 


Square  Roots  and  Cube  Roots  of  X ambers  from  1000  to  1OOOO. 


Num. 

Sq.  Rt. 

Cu.Rt 

Num. 

Sq.  Rt. 

Cu.Rt. 

Num. 

Sq.  Rt. 

Cu.  Rt. 

Num. 

Sq.  Rt. 

Cu.  Rt. 

1005 

31.70 

10.02 

1405 

3  .48 

11.20 

1805 

42.49 

12.18 

2205 

46.06 

13.02 

1010 

31.78 

10.03 

1410 

3  .55 

11.21 

1810 

42.54 

12.19 

2210 

47.01 

13.03 

1015 

31.86 

10.05 

1415 

3  .62 

11.23 

1815 

42.60 

12.20 

2215 

47.06 

13.04 

1020 

31.94 

10.07 

1420 

3  .68 

11.24 

1820 

42.66 

12.21 

2220 

47.12 

13.  05 

1025 

32.02 

10.08 

1425 

3  .75 

11.25 

1825 

42.72 

12.22 

2225 

47.17 

13.05 

1030 

32.09 

10.10 

1430 

3  .82 

11.27 

1830 

42.78 

12.23 

2230 

47.22 

13.06 

1035 

32.17 

10.12 

1435 

37.88 

-11.28 

1835 

42.84 

12.24 

2235 

47.28 

13.07 

1040 

32.25 

10.13 

1440 

37.95 

11.29 

1840 

42.90 

12.25 

2240 

47.33 

13.08 

1045 

32.33 

10.15 

1445 

38.01 

11.31 

1845 

42.95 

12.26 

2245 

47.38 

13.09 

1050 

32.40 

10.16 

1450 

38.08 

11..-52 

1850 

43.01 

12.28 

2250 

47.43 

13.10 

1055 

32.48 

10.18 

1455 

38.14 

11.33 

1855 

43.07 

12.29 

2255 

47.49 

13.11 

1060 

32.56 

10.20 

1460 

38.21 

11.34 

1860 

43.13 

12.30 

2260 

47.54 

13.12 

1065 

32.63 

10.21 

1465 

38.28 

11.36 

1865 

43.19 

12.31 

2265 

47.59 

13.13 

1070 

32.71 

10.2:5 

1470 

38.34 

11.37 

1870 

43.24 

12.32 

2270 

47.64 

13.14 

1075 

32.79 

10.24 

1475 

38.41 

11.38 

1875 

43.30 

12.33 

2275 

47.70 

13.15 

1080 

32.86 

10.26 

1480 

38.47 

11.40 

1880 

43.36 

12.34 

2280 

47.75 

13.16 

1085 

32.94 

10.28 

1485 

38.54 

11.41 

1885 

43.42 

12.35 

2285 

47.80 

13.17 

1090 

33.02 

10.29 

1490 

38.60 

11.12 

1890 

4:5.17 

12.36 

2290 

47.85 

13.18 

1095 

33.09 

10.31 

1495 

38.67 

11.4:5 

1895 

43.53 

12.37 

2295 

47.91 

13.19 

1100 

33.17 

10.32 

1500 

38.73 

11.15 

1900 

43.59 

12.39 

2300 

47.96 

13.20 

1105 

33.24 

10.34 

1505 

38.79 

11.46 

1905 

43.65 

12.40 

2305 

48.01 

13.21 

1110 

33.32 

10.85 

1510 

38.86 

11.47 

1910 

43.70 

12.41 

2310 

48.06 

13.22 

1115 

33.39 

10.37 

1515 

38.92 

11.49 

1915 

43.76 

12.42 

2315 

48.11 

13.23 

1120 

33.47 

10.38 

1520 

38.99 

11.50 

1920 

43.82 

12.43 

2320 

48.17 

13.24 

1125 

33.54 

10.40 

1525 

39.05 

11.51 

1925 

43.87 

12.44 

2325 

48.22 

13.25 

1130 

33.62 

10.42 

1530 

39.12 

11.52 

1930 

43.93 

12.45 

2330 

48.27 

13.26 

1135 

33.69 

10.43 

1535 

39.18 

11.51 

1935 

43.99 

12.46 

2335 

48.32 

13.27 

1140 

33.76 

10.45 

1540 

39.24 

11.55 

1940 

41.05 

12.47 

2340 

48-37 

13.28 

1145 

33.84 

10.46 

1545 

39.31 

11.56 

1915 

44.10 

12,48 

2345 

48.43 

13.29 

1150 

33.91 

10.48 

1550 

39.37 

11.57 

1950 

44.16 

12.49 

2350 

48.48 

13.30 

1155 

33.99 

10.49 

1555 

39.43 

11.  o9 

1955 

44.22 

12.50 

2355 

48.53 

13.30 

1160 

34.06 

10.51 

1560 

39.50 

11.60 

1960 

44.27 

12.51 

2360 

48.58 

13.31 

1165 

34.13 

10.52 

1565 

39.56 

11.61 

1965 

44.33 

12.5:5 

2365 

48.63 

13.32 

1170 

34.21 

10.54 

1570 

39.62 

11.62 

1970 

44.38 

12.54 

2370 

48.68 

13.33 

1175 

34.28 

10.55 

1575 

39.69 

11.63 

1975 

44.44 

12.55 

2375 

48.73 

13.34 

1180 

34.35 

10.57 

1580 

39.75 

11.65 

1980 

44.50 

12.56 

2380 

48.79 

13.35 

1185 

34.42 

10.58 

1585 

39.81 

11.66 

1985 

44.55 

12.57 

2385 

48.84 

13.36 

1190 

34.50 

10.60 

1590 

39.87 

11.67 

1990 

44.61 

12.58 

2390 

48.89 

13.37 

1195 

34.57 

10.61 

1595 

39.94 

11.68 

1995 

44.67 

12.59 

2395 

48.94 

13.38 

1200 

34.64 

10.63 

1600 

40.00 

11.70 

2000 

44.72 

12.60 

2400 

48.99 

13.39 

1205 

34.71 

10.61 

1605 

40.06 

11.71 

2005 

44.78 

12.61 

2405 

49.04 

13.10 

1210 

34.79 

10.66 

1610 

40.12 

11.72 

2010 

44.83 

12.62 

2410 

49.09 

13.41 

1215 

34.86 

10.67 

1615 

40.19 

11.73 

2015 

44.89 

12.63 

2415 

49.14 

13.42 

1220 

84.93 

10.69 

1620 

40.25 

11.74 

2020 

44.94 

12.64 

2420 

49.19 

13.43 

1225 

35.00 

10.70 

1625 

40.31 

11.76 

2025 

45.00 

12.65 

2425 

49.24 

13.43 

1230 

35.07 

10.71 

1630 

40.37 

11.77 

2030 

45.06 

12.66 

2430 

49.30 

13.44 

1235 

35.14 

10.73 

1635 

40.44 

11.78 

2035 

45.11 

12.67 

2435 

49.35 

13.45 

1240 

35.21 

10.71 

1640 

40.50 

11.79 

2040 

45.17 

12.68 

2440 

49.40 

13.46 

1245 

35.28 

10.76 

1645 

40.56 

11.  SO 

2045 

45.22 

12.69 

2445 

49.45 

13.47 

1250 

35.36 

10.77 

1650 

40.62 

11.82 

2050 

45.28 

12.70 

2450 

49.50 

13.48 

1255 

35.43 

10.79 

1655 

40.68 

11.83 

2055 

45.33 

12.71 

2460 

49.60 

13.50 

1260 

35.50 

10.80 

1660 

40.74, 

11.84 

2060 

45.39 

12.72 

2470 

49.70 

13.52 

1265 

35.57 

10.82 

1665 

40.80 

11.85 

2065 

45.44 

12.73 

2480 

49.80 

13.54 

1270 

35.64 

10.83 

1670 

40.87 

11.86 

•2070 

45.50 

12.74 

2490 

49.90 

13.55 

1275 

35.71 

10.84 

1675 

40.93 

11.88 

2075 

45.55 

12.75 

2500 

50.00 

13.57 

1280 

35.78 

10.86 

1680 

40.99 

11.89 

2080 

4561 

12.77 

2510 

50.10 

13.59 

1285 

35.85 

10.87 

1685 

41.05 

11.90 

2085 

45.66 

12.78 

2520 

50.20 

13.61 

1290 

35.92 

10.89 

1690 

41.11 

11.91 

2090 

45.72 

12.79 

2530 

50.30 

13.63 

1295 

35.99 

10.90 

1695 

41.17 

11.92 

2095 

45.77 

12.80 

2540 

50.40 

13.64 

1300 

36.06 

10.91 

1700 

41.23 

1  1  .9:5 

2100 

45.83 

12.81 

2550 

50.50 

13.66 

1305 

36.12 

10.93 

1705 

41.29 

11.95 

2105 

45.88 

12.82 

2560 

50.60 

13.68 

1310 

36.19 

10.94 

1710 

41.35 

11.96 

2110 

45.93 

12.83 

2570 

50.70 

13.70 

1315 

86.26 

10.96 

1715 

41.41 

11.97 

2115 

45.99 

12.81 

2580 

50.79 

13.72 

1320 

86.33 

10.97 

1720 

41.47 

11.98 

2120 

46.04 

12.85 

2590 

50.89 

13.73 

1325 

36.40 

10.98 

1725 

41.53 

11.99 

2125 

46.10 

12.86 

2600 

50.99 

13.75 

1330 

36.47 

11.00 

1730 

41.59 

12.00 

2130 

46.15 

12.87 

2610 

51.09 

1:5.77 

1335 

36.54 

11.01 

1735 

41.65 

12.02 

2135 

46.21 

12.88 

2620 

51.19 

13.79 

1340 

36.61 

11.02 

1740 

41.71 

12.03 

2140 

46.26 

12.89 

2630 

51.28 

13.80 

1345 

36.67 

11.01 

1745 

41.77 

12.04 

2145 

46.31 

12.SO 

2640 

51.38 

13.82 

1350 

36.74 

11.05 

1750 

41.83 

12.05 

2150 

46.37 

12.91 

2650 

51.48 

13.84 

1355 

36.81 

11.07 

1755 

41.89 

12.06 

2155 

46.42 

12.92 

2660 

51.58 

13.86 

1360 

36.88 

11.08 

1760 

41.95 

12.07 

2160 

46.48 

12.9:5 

2670 

51.67 

13.87 

1365 

36.95 

11.09 

1765 

42.01 

12.09 

2165 

46.53 

12.94 

2680 

51.77 

13.89 

1370 

37.01 

11.11 

1770 

42.07 

12.10 

2170 

46.58 

12.95 

2690 

51.87 

13.91 

1375 

37.08 

11.12 

1775 

42.13 

llll 

2175 

46.64 

12.96 

2700 

51.96 

13.92 

1380 

37.15 

11.13 

1780 

42.19 

12.12 

2180 

46.69 

12.97 

2710 

52.06 

13.94 

1385 

37.22 

11.15 

1785 

42.25 

12.13 

2185 

46.74 

12.98 

2720 

52.15 

13.96 

1390 

37.28 

11.16 

1790 

42.31 

12.14 

2190 

46.80 

12.99 

2730 

52.25 

13.98 

1395 

37.155 

11.17 

1795 

42.37 

12.15 

2195 

46.85 

13.00 

2740 

52.35 

13.99 

1400 

37.42 

11.19 

1800 

42.43 

12.16 

2200 

46.1W 

13.01 

2750 

62.44 

14.01 

68 


SQUARE  AND  CUBE  ROOTS. 


Square  Roots  and  Cube  Roots  of  Numbers  from  100O  tolOOOO 

— (CONTINUED.) 


Num. 

Sq.  Rt. 

Cu.  Rt. 

Num. 

Sq.  Rt. 

Cu.  Rt. 

Num. 

Sq.  Rt. 

Cu.Rt. 

Num. 

Sq.  Rt. 

Cu.  Rt. 

2760 

52.54 

14.03 

3550 

59.58 

15.25 

4340 

65.88 

16.31 

5130 

71.62 

17.25 

2770 

52.63 

14.04 

3560 

59.67 

15.27 

4350 

65.95 

16.32 

5140 

71.69 

17.26 

2780 

52.73 

14.06 

3570 

59.75 

15.28 

4360 

66.03 

16.34 

5150 

71.76 

17.27 

2790 

52.82 

14.08 

3580 

59.83 

15.30 

4370 

66.11 

16.35 

5160 

71.83 

17.28 

2800 

52.92 

14.09 

3590 

59.92 

15.31 

4380 

66.18 

16.36 

5170 

71.90 

17.29 

2810 

53.01 

14.11 

3600 

60.00 

15.33 

4390 

66.26 

16.37 

5180 

71.97 

17.30 

2820 

53.10 

14.13 

3610 

60.08 

15.34 

4400 

66.33 

16.39 

5190 

72.04 

17.31 

2830 

53.20 

14.14 

3620 

60.17 

15.35 

4410 

66.41 

16.40 

5200 

72.11 

17.32 

2840 

53.29 

14.16 

3630 

60.25 

15.37 

4420 

66.48 

16.41 

5210 

72.18 

17.34 

2850 

53.39 

14.18 

3640 

60.33 

15.38 

4430 

66.56 

16.42 

5220 

72.25 

17.35 

2860 

53.48 

14.19 

3650 

60.42 

15.40 

4440 

66.63 

16.44 

5230 

72.32 

17.36 

2870 

53.57 

14.21 

3660 

60.50 

15.41 

4450 

66.71 

16.45 

5240 

72.39 

17.37 

2880 

53.67 

14.23 

3670 

60.58 

15.42 

4460 

66.78 

16.46 

5250 

72.46 

17.38 

2890 

53.76 

14.24 

3680 

60.66 

15.44 

4470 

66-86 

16.47 

5260 

72.53 

17.39 

2900 

53.85 

14.26 

3690 

60.75 

15.45 

4480 

66.93 

16.49 

5270 

72.59 

17.40 

2910 

53.94 

14.28 

3700 

GO.  83 

15.47 

4490 

67.01 

16.50 

5280 

72.66 

17.41 

2920 

54.04 

14.29 

3710 

60.91 

15.48 

4500 

67.08 

16.51 

5290 

72.73 

17.42 

2930 

54.13 

14.31 

3720 

60.99 

15.49 

4510 

67.16 

16.52 

5300 

72.80 

17.44 

2910 

54.22 

14.33 

3730 

61.07 

15.51 

4520 

67.23 

16.53 

5310 

72.87 

17.45 

2950 

54.31 

14.34 

3740 

61.16 

15.52 

4530 

67.31 

16.55 

5320 

72.94 

17.46 

2960 

54.41 

14.36 

3750 

61.24 

15.54 

4540 

67.38 

16.56 

5330 

73.01 

17.47 

2970 

54.50 

14.37 

3760 

61.32 

15.55 

4550 

67.45 

16.57 

5340 

73.08 

17.48 

2980 

54.59 

14.39 

3770 

61.40 

15.56 

4560 

67.53 

16.58 

5350 

73.14 

17.49 

2990 

54.68 

14.41 

3780 

61.48 

15.58 

4570 

67.60 

16.59 

5360 

73.21 

17.50 

3000 

54.77 

14.42 

3790 

61.56 

15.59 

4580 

67.68 

16.61 

5370 

73.28 

17.51 

3010 

54.86 

14.44 

3800 

61  64 

15.60 

4590 

67.75 

16.62 

5380 

73.35 

17.52 

3020 

54.95 

14.45 

3810 

61.73 

15.62 

4600 

67.82 

16.63 

5390 

73.42 

17.53 

3030 

55.05 

14.47 

3820 

61.81 

15.63 

4610 

67.90 

16.64 

5400 

73.48 

17.54 

3010 

55.14 

14.49 

3830 

61.89 

15.65 

4620 

67.97 

16.66 

5410 

73.55 

17.55 

3050 

55.23 

14.50 

3840 

61.97 

15.66 

4630 

68.04 

16.67 

5420 

73.62 

17.57 

3060 

55.32 

14.52 

3850 

62.05 

15.67 

4640 

68.12 

16.68 

5430 

73.69 

17.58 

3070 

55.41 

14.53 

3860 

62.13 

15.69 

4650 

68.19 

16.69 

5440 

73.76 

17.59 

3080 

55.50 

14.55 

3870 

62.21 

15.70 

4660 

68.26 

16.70 

5450 

73.82 

17.60 

3090 

55.59 

14.57 

3880 

62.29 

15.71 

4670 

68.34 

16.71 

5460 

73.89 

17.61 

3100 

55.68 

14.58 

3890 

62.37 

15.73 

4680 

68.41 

16.73 

5470 

73.96 

17.62 

3110 

55.77 

14.60 

3900 

62.45 

15.74 

4690 

68.48 

16.74 

5480 

74.03 

17.63 

3120 

55.86 

14.61 

3910 

62.53 

15.75 

4700 

68.56 

16.75 

5490 

74.09 

17.64 

3130 

55.95 

14.63 

3920 

62.61 

15.77 

4710 

68.63 

16.76 

5500 

74.16 

K.65 

3140 

56.04 

14.64 

3930 

62.69 

15.78 

4720 

68.70 

16.77 

5510 

74.23 

17.66 

3150 

56.12 

14.66 

3940 

62.77 

15.79 

4730 

68.77 

.  16.79 

5520 

74.30 

17.67 

3160 

56.21 

14.67 

3950 

62.85 

15.81 

4740 

68.85 

16.80 

5530 

74.36 

17.68 

3170 

56.30 

14.69 

3960 

62.93 

15.82 

4750 

68.92 

16.81 

5540 

74.43 

17.69 

3180 

56.39 

14.71 

3970 

63.01 

15.83 

4760 

68.99 

16.82 

5550 

74.50 

17.71 

:*3190 

56.48 

14.72 

3980 

63.09 

15.85 

4770 

69.07 

16.83 

5560 

74.57 

17.72 

3200 

56.57 

14.74 

3990 

63.17 

15.86 

4780 

69.14 

16.85 

5570 

74.63 

17.73 

S210 

56.66 

14.75 

4000 

63.25 

15.87 

4790 

69.21 

16.86 

5580 

74.70 

17.74 

3220 

56.75 

14.77 

4010 

63.32 

15.89 

4800 

69.28 

16.87 

5590 

74.77 

17.75 

3230 

56.83 

14.78 

4020 

63.40 

15.90 

4810 

69.35 

16.88 

5600 

74.83 

17.76 

3240 

56.92 

14.80 

4030 

63.48 

15.91 

4820 

69.43 

16  89 

5610 

74.90 

17.77 

3250 

57.01 

14.81 

4040 

63.56 

15.93 

4830 

69.50 

16.90 

5620 

74.97 

17.78 

3260 

57.10 

14.83 

4050 

63.64 

15.94 

4840 

69.57 

16.92 

5630 

75.03 

17.79 

3270 

57.18 

14.84 

4060 

63.72 

15.95 

4850 

VJ.64 

16.93 

5640 

75.10 

17.80 

3280 

57.27 

14.86 

4070 

63.80 

15.97 

4860 

69.71 

16.94 

5650 

75.17 

17.81 

3290 

57.36 

14.87 

4080 

63.87 

15.98 

4870 

69.79 

16.95 

5660 

75.23 

17.82 

3300 

57.45 

14.89 

4090 

63.95 

15.99 

4880 

69.86 

16.96 

5670 

75.30 

17.83 

3310 

57.53 

14.90 

4100 

64.03 

16.01 

4890 

69.93 

16.97 

5680 

75.37 

17.84 

3320 

57.62 

14.92 

4110 

64.11 

16.02 

4900 

70.00 

16.98 

5690 

75.43 

17.85 

3330 

57.71 

14.93 

4120 

64.19 

16.03 

4910 

70.07 

17.00 

5700 

75.50 

17.86 

3340 

57.79 

14.95 

4130 

64.27 

16.04 

4920 

70.14 

17.01 

5710 

75.56 

17.87 

3350 

57.88 

14.96 

4140 

64.34 

16.06 

4930 

70.21 

17.02 

5720 

75.63 

17.88 

3360 

57.97 

14.98 

4150 

64.42 

16.07 

4940 

70.29 

17.03 

5730 

75.70 

17.89 

3370 

58.05 

14.99 

4160 

64.50 

16.08 

4950 

70.36 

17.04 

5740 

75.76 

17.90 

3380 

58.14 

15,01 

4170 

64.58 

16.10 

4960 

70.43 

17.05 

5750 

75.83 

17.92 

3390 

58.22 

15.02 

4180 

64.65 

16.11 

4970 

70.50 

17.07 

5760 

75.89 

17.93 

3400 

58.31 

15.04 

4190 

64.73 

16.12 

4980 

70.57 

17.08 

5770 

75.96 

17.94 

3410 

58.40 

15.05 

4200 

64.81 

16.13 

4990 

70.64 

17.09 

5780 

76.03 

17.95 

3420 

58.48 

5.07 

4210 

64.88 

16.15 

5000 

70.71 

17.10 

5790 

76.09 

17.96 

3430 

58.57 

5.08 

4220 

64.96 

16.16 

5010 

70.78 

17.11 

5800 

76.16 

17.97 

3440 

58.65 

5.10 

4230 

65.04 

16.17 

5020 

70.85 

17.12 

5810 

70.  '22 

17.98 

3450 

58.74 

5.11 

4240 

65.12 

16.19 

5030 

70.92 

17.13 

5820 

76.29 

17.99 

3460 

58.82 

5.12 

4250 

65.19 

16.20 

5040 

70.99 

17.15 

5830 

76.35 

18.00 

3470 

58.91 

15.14 

4260 

65.27 

16.21 

5050 

71.06 

17.16 

5840 

76.42 

18.01 

3480 

58.99 

15.15 

4270 

65.35 

16.22 

5060 

71.13 

17.17 

5850 

76.49 

18.02 

3490 

59.08 

15.17 

4280 

65.42 

16.24 

5070 

71.20 

17.18 

5860 

76.55 

8.03 

3500 

59.16 

15.18 

4290 

65.50 

16.25 

5080 

71.27 

17.19 

5870 

76.62 

8.04 

3510 

59.25 

15.20 

4300 

6557 

16.26 

5090 

71.34 

17.20 

5880 

76.68 

8.05 

3520 

59.33 

15.21 

4310 

65.65 

16.27 

5100 

71.41 

17.21 

5890 

76.75 

8.06 

3530 

59.41 

15.23 

4320 

65.73 

16.29 

5110 

71.48 

17.22 

5900 

76.81 

8.07 

3549 

58.50 

15.24 

4330 

65.80 

16.30 

6120 

71.55 

17.24 

5910 

76.88 

18.08 

SQUARE  AND  CUBE  ROOTS.  Otf 

Square  Roots  and  Cube  Roots  of  tf  ambers  from  1OOO  to  1OOOO 

—  (CONTINUED.) 


Num. 

Sq.  Rt. 

Cu.  Rt. 

Num. 

Sq.  Rt.JGu.Rt. 

Num. 

Sq.  Rt. 

Cu.  Rt. 

Num. 

Sq.  Rt.j  Cu.Rt, 

5920 

76.94 

18.09 

6710 

81.91 

18.86 

7500 

86.60 

19.57 

8290 

91.05 

20.24 

5930 

77.01 

18.10 

6720 

81.98 

18.87 

7510 

8(>.(i6 

19.58 

8300 

91.10 

20.25 

5940 

77.07 

18.11 

6730 

82.04 

18.88 

7520 

86.72 

19.59 

8310 

91.16 

20.26 

5950 

77.14 

18.12 

6740 

82.10 

18.89 

7530 

86.78 

19.60 

8320 

91.21 

20.26 

5960 

77.20 

18.13 

6750 

82.16 

18.90 

7540 

86.83 

19.61 

8330 

91.27 

20.27 

5970 

77.27 

18.14 

6760 

82.22 

18.91 

7550 

86.89 

19.62 

8340 

91.32 

20.28 

5980 

77.33 

18.15 

6770 

82.28 

18.92 

7560 

86.95 

19.63 

8350 

91.38 

20.29 

5990 

77.40 

18.16 

6780 

82.34 

18.93 

7570 

87.01 

19.64 

8360 

91.43 

20.30 

6000 

77.46 

18.17 

6790 

82.40 

18.94 

7580 

87.06 

19.64 

8370 

91.49 

20.30 

6010 

77.52 

18.18 

6800 

82.46 

18.95 

7590 

87.12 

19.65 

8380 

91.54 

20.31 

6020 

77.59 

18.19 

6810 

82.52 

18.95 

7600 

87.18 

19.66 

8390 

91.60 

20.32 

6030 

77.65 

18.20 

6820 

82.58 

18.96 

7610 

87.24 

19.67 

8400 

91.65 

20.33 

6040 

77.72 

18.21 

6830 

82.64 

18.97 

7620 

87.29 

19.68 

8410 

91.71 

20.34 

6050 

77.78 

18.22 

6840 

82.70 

18.98 

7630 

87.35 

19.69 

8420 

91.76 

20.34 

6060 

77.85 

18.23 

6850 

82.76 

18.99 

7640 

87.41 

19.70 

8430 

91.82 

20.35 

6070 

77.91 

18.24 

6860 

82.83 

19.00 

7650 

87.46 

19.70 

8440 

91.87 

20.36 

6080 

77.97 

18.25 

6870 

82.89 

19.01 

7660 

87.52 

19.71 

8450 

91.92 

20.37 

6090 

78.04 

18.26 

6880 

82.95 

19.02 

7670 

87.58 

19.72 

8460 

91.98 

20.38 

6100 

78.10 

18.27 

6890 

83.01 

19.03 

7680 

87.64 

19.73 

8470 

92.03 

20.38 

6110 

78.17 

18.28 

6900 

83.07 

19.04 

7690 

87.69 

19.74 

8480 

92.09 

20.39 

6120 

73.23 

18.29 

6910 

83.13 

19.05 

7700 

87.75 

19.75 

8490 

92.14 

20.40 

6130 

78.29 

18.30 

6920 

83.19 

19.06 

7710 

87.81 

19.76 

8500 

92.20 

20.41 

6140 

78.36 

18.31 

6930 

83.25 

19.07 

7720 

87.8b 

19.76 

8510 

92.25 

20.42 

6150 

78.42 

18.32 

6940 

83.31 

19.07 

7730 

87.92 

19.77 

8520 

92.30 

20.42 

6160 

78.49 

18.33 

6950 

83.37 

19.08 

7740 

87.98 

19.78 

8530 

92.36 

20.43 

6170 

78.55 

18.34 

6960 

83.43 

19.09 

7750 

88.03 

19.79 

8540 

92.41 

20.44 

6180 

78.61 

18.35 

6970 

83.49 

19.10 

7760 

88.09 

19.80 

8550 

92.47 

20.45 

6190 

78.68 

18.36 

6980 

83.55 

19.11 

7770 

88.15 

19.81 

8560 

92.52 

20.46 

6200 

78.74 

18.37 

6990 

83.61 

19.12 

7780 

88.20 

19.81 

8570 

92.57 

20.46 

6210 

78.80 

18.38 

7000 

83.67 

19.13 

7790 

88.26 

19.82 

8580 

92.63 

20.47 

6220 

78.87 

18.39 

7010 

83.73 

19.14 

7800 

88.32 

19.83 

8590 

92.68 

20.48 

6230 

78.93 

18.40 

7020 

83.79 

19.15 

7810 

88.37 

19.84 

8600 

92.74 

20.49 

6240 

78.99 

18.41 

7030 

83.85 

19.16 

7820 

88.43 

19.85 

8610 

92.79 

20.50 

6250 

79.06 

18.42 

7040 

83.90 

19.17 

7830 

88.49 

19.86 

8620 

92.84 

20.50 

6260 

79.12 

18.43 

7050 

83.»o 

19.17 

7840 

88.54 

19.87 

8630 

92.90 

20.51 

6270 

79.18 

18.44 

7060 

84.02 

19.18 

7850 

88.60 

19.87 

8640 

92.95 

20.52 

6280. 

79.25 

18.45 

7070 

84.08 

19.19 

7860 

88.66 

19.88 

8650 

93.01 

20.53 

6290 

79.31 

18.46 

7080 

84.14 

19.20 

7870 

88.71 

19.89 

8660 

93.06 

20.54 

6300 

79.37 

18.47 

7090 

84.20 

19.21 

7880 

88.77 

19.90 

8670 

93.11 

20.54 

6310 

79.44 

18.48 

7100 

84.26 

19.22 

7890 

88.83 

19.91 

8680 

93.17 

20.55 

6320 

79.50 

18.49 

7110 

84.32 

19.23 

7900 

88.88 

19.92 

8690 

93.22 

20.56 

6330 

79.56 

18.50 

7120 

84.38 

19.24 

7910 

88.94 

19.92 

8700 

93.27 

20.57 

6340 

79.62 

18.51 

7130 

84.44 

19.25 

7920 

88.99 

19.93 

8710 

93.33 

20.57 

6350 

79.69 

18.52 

7140 

84.50 

19.26 

7930 

89.05 

19.94 

8720 

93.38 

20.58 

6360 

79.75 

18.53 

7150 

84.56 

19.26 

7940 

89.11 

19.95 

8730 

93.43 

20.59 

6370 

79.81 

18.54 

7160 

84.62 

19.27 

7950 

89.16 

19.96 

8740 

93.49 

20.60 

6380 

79.87 

18.55 

7170 

84.68 

19.28 

7960 

89.22 

19.97 

8750 

93.54 

20.61 

6390 

79.94 

18.56 

7180 

84.73 

19.29 

7970 

89.27 

19.97 

8760 

93.59 

20.61 

6400 

80.00 

18.57 

7190 

84.79 

19.30 

7980 

89.33 

19.98 

8770 

93.65 

20.62 

6410 

80.06 

18.58 

7200 

84.85 

19.31 

7990 

89.39 

19.99 

8780 

93.70 

20.63 

6420 

80.12 

18.59 

7210 

84.91 

19.32 

8000 

89.44 

20.00 

8790 

93.75 

20.64 

6430 

80.19 

18.60 

7220 

84.97 

19.33 

8010 

89.50 

20.01 

8800 

93.81 

20.65 

6440 

80.25 

18.60 

7230 

85.03 

19.34 

8020 

89.55 

20.02 

8810 

93.86 

20.65 

6450 

80.31 

18.61 

7240 

85.09 

19.35 

8030 

89.61 

20.02 

8820 

93.91 

20.66 

6460 

80.37 

18.62 

7250 

85.15 

19.35 

8040 

89.67 

20.03 

8830 

93.97 

20.67 

6470 

80.44 

18.63 

7260 

85.21 

19.36 

8050 

89.72 

20.04 

8840 

94.02 

20.68 

6480 

80.50 

18.64 

7270 

85.26 

19.37 

8060 

89.78 

20.05 

8850 

94.07 

20.68 

6490 

80.56 

18.65 

7280 

85.32 

19.38 

8070 

89.83 

20.06 

8860 

94.13 

20.69 

6500 

80.62 

18.66 

7290 

85.38 

19.39 

8080 

89.89 

20.07 

8870 

94.18 

20.70 

6510 

80.68 

18.67 

7300 

85.44 

19.40 

8090 

89.94 

20.07 

8880 

94.23 

20.71 

6520 

80.75 

18.68 

7310 

85.50 

19.41 

8100 

90.00 

20.08 

8890 

94.29 

20.72 

6530 

80.81 

18.69 

7320 

85.56 

19.42 

8110 

90.06 

20.09 

8900 

94.34 

20.72 

6540 

80.87 

18.70 

7330 

85.62 

19.43 

8120 

90.11 

20.10 

8910 

94.39 

20.73 

6550 

80.93 

18.71 

7340 

85.67 

19.43 

8130 

90.17 

20.11 

8920 

94.45 

20.74 

6560 

80.99 

18.72 

7350 

85.73 

19.44 

8140 

90.22 

20.12 

8930 

94.50 

20.75 

6570 

81.06 

18.73 

7360 

85.79 

19.45 

8150 

90.28 

20.12 

8940 

94.55 

20.75 

6580 

81.12 

18.74 

7370 

85.85 

19.46 

8160 

90.33 

20.13 

8950 

94.60 

20.76 

6590 

81.18 

18.75 

7380 

85.91 

19.47 

8170 

90.39 

20.14 

8960 

94.66 

20.77 

6600 

81.24 

18.76 

7390 

85.97 

19.48 

8180 

90.44 

20.15 

8970 

94.71 

20.78 

6610 

81.30 

18.77 

7400 

86.02 

19.49 

8190 

90.50 

20.16 

8980 

94.76 

20.79 

6620 

81.36 

18.78 

7410 

86.08 

19.50 

8200 

90.55 

20.17 

8990 

94.82 

20.79 

6630 

81.42 

18.79 

7420 

86.14 

19.50 

8210 

90.61 

20.17 

9000 

94.87 

20.80 

6640 

81.49 

18.80 

7430 

86.20 

19.51 

8220 

90.66 

20.18 

9010 

94.92 

20.81 

6650 

81.55 

18.81 

7440 

86.26 

19.52 

8230 

90.72 

20.19 

9020 

94.97 

20.82 

6660 

81.61 

18.81 

7450 

86.31 

19.53 

8240 

90.77 

20.20 

9030 

95.03 

20.82 

6670 

81.67 

18.82 

7460 

86.37 

19.54 

8250 

90.83 

20.21 

9040 

95.08 

20.83 

6680 

81.73 

18.  83 

7470 

86.43 

19.55 

8260 

90.88 

20.21 

9050 

95.13 

20.84 

6690 

81.79 

18.84 

7480 

86.49 

19.56 

8270 

90.94 

20.22 

9060 

95.18 

20.85 

«700 

81.83 

18.85 

7490 

86.54 

19.57 

8-280 

90.99 

20.23 

9070 

95.24 

20.86 

60 


SQUARE  AND  CUBE  ROOTS. 


Square  Roots  and  Cube  Roots  of  Numbers  from  1000  to  10000 

—  (CONTINUED.) 


Num. 

Sq.  Rt. 

Cu.  Rt. 

Num. 

Sq.  Rt. 

Cu.  Rt. 

Num. 

Sq.  Rt. 

Cu.  Rt. 

Num. 

Sq.  Rt. 

Cu.  Rt. 

9080 

95.29 

20.86 

9320 

96.54 

21.04 

9550 

97.72 

2  .22 

9780 

9W.S9 

1.39 

9090 

95.34 

20.87 

9330 

96.59 

21.05 

9560 

97.78 

2  .22 

9790 

98.94 

1.39 

9100 

95.39 

20.88 

9340 

96.64 

21.06 

9570 

97.83 

2  .23 

9800 

98.99 

.40 

9110 

95.45 

20.89 

9350 

96.70 

21.07 

9580 

97.88 

2  .24 

9810 

99.05 

1.41 

9120 

95.50 

20.89 

9360 

96.75 

21.07 

9590 

97.93 

2  .25 

9820 

99.10 

.41 

9130 

95.55 

20.90 

9370 

96.80 

21.08 

9600 

97.98 

2  .25 

9830 

99.15 

.42 

9HO 

95.60 

20.91 

9380 

96.85 

21.09 

9610 

98.03 

21.26 

9840 

99.20 

1.43 

9150 

95.66 

20.92 

9390 

96.90 

•21.10 

9620 

98.08 

21.27 

9850 

99.25 

1.44 

9160 

95.71 

20.92 

9400 

96.95 

21.10 

9630 

98.13 

21.28 

9860 

99.30 

1.44 

9170 

95.76 

20.93 

9410 

97.01 

21.11 

9640 

98.18 

21.28 

9870 

99.35 

1.45 

9180 

95.81 

20.94 

9420 

97.06 

21.12 

9650 

98.23 

21.29 

9880 

99.40 

1.46 

9190 

95.86 

20.95 

9430 

97.11 

21.1  a 

9660 

98.29 

21.30 

9890 

99.45 

1.47 

9200 

95.92 

20.95 

9440 

97.16 

21.13 

9670 

98.34 

21.30 

9900 

99.50 

21.47 

9210 

95.97 

20.96 

9450 

97.21 

21.14 

9680 

98.39 

21.31 

9910 

99.55 

21.48 

9220 

96.02 

20.97 

9460 

97.26 

21.15 

9690 

98.44 

21.32 

9920 

99.60 

21.49 

9230 

96.07 

20.98 

9470 

97.31 

21.16 

9700 

98.49 

21.33 

9930 

99.65 

21.49 

9240 

96.12 

20.98 

9480 

9Y.37 

21.16 

9710 

98.54 

21.33 

9940 

99.70 

21.50 

9250 

96.18 

20.99 

9490 

97.42 

21.17 

9720 

98.59 

21.34 

9950 

99.75 

21.51 

9260 

96.23 

21.00 

9500 

97.47 

21.18 

9730 

98.64 

21.35 

9960 

99.80 

21.52 

9270 

96.28 

21.01 

9510 

97.52 

21.19 

9740 

98.69 

21.36 

9970 

99.85 

21.52 

9280 

96.33 

21.01 

9520 

97.57 

21.19 

9750 

98.74 

21.36 

9980 

99.90 

21.53 

9290 

96.38 

21.02 

9530 

97.62 

21.20 

9760 

98.79 

21.37 

9990 

99.95 

21.54 

9300 

96.44 

21.03 

9540 

97.67 

21.21 

9770 

98.84 

21.38 

10000 

100.00 

21.54 

9310 

96.49 

21.04 

To  find  Square,  or  Cube  Roots  of  largre  numbers  not  con* 
tained  in  the  column  of  numbers  of  the  table. 

Such  roots  may  sometimes  be  taken  at  once  from  the  table,  bj  merely  regarding  the  columns  of 
powers  as  beiug  columns  of  numbers;  and  those  of  numbers  as  being  those  of  roots.  Thus,  if  the 
sq  rt  of  25281  is  reqd,  first  find  that  number  in  the  column  of  squares;  and  opposite  to  it,  in  the 
column  of  numbers,  is  its  sq  rt  159.  For  the  cube  rt  of  857375,  tind  that  number  in  the  column  of 
cubes ;  and  opposite  to  it,  in  the  col  of  numbers,  is  its  cube  rt  95.  When  the  exact  number  is  not  con- 
tained in  the  column  of  squares,  or  cubes,  as  the  case  may  be,  we  may  use  instead  the  number  nearest 
to  it,  if  no  great  accuracy  is  reqd.  But  when  a  considerable  degree  of  accuracy  is  necessary,  the 
following  very  correct  methods  may  be  used. 

For  the  square  root. 

This  rule  applies  both  to  whole  numbers,  and  to  those  which  are  partly  (not  wholly)  decimal.  First, 
in  the  foregoing  manner,  take  out  the  tabular  number,  which  is  nearest  to  the  given  one  ;  and  also  its 
tabular  sq  rt.  Mult  this  tabular  number  by  3  ;  to  the  prod  add  the  given  number.  Call  the  sum  A. 
Then  mult  the  given  number  by  3  ;  to  the  prod  add  the  tabular  number.  Call  the  sum  B.  Then 

A  :  B  :  :  Tabular  root  :  Reqd  root. 

Ex.  Let  the  given  number  be  946.53.  Here  we  find  the  nearest  tabular  number  to  be  947;  and  its 
tabular  sq  rt  30.7734.  Hence, 

947  =  tab  num  1  f  946.53  =  given  num. 

3  3 


2841 
946.53  —  given  num. 

3787.53  —  A. 


and      •{  2839.59 

947     =  tab  num. 


1.3786.59  -  B. 


A.  B.  Tab  root.     Reqd  root. 

Then  3787.53    :    3786.59    :  :    30.7734    :    30.7657  +. 

The  root  as  found  by  actual  mathematical  process  is  also  30.7657  -{-. 

For  the  cube  root. 

This  rule  applies  both  to  whole  numbers,  and  to  those  which  are  partly  decimal.  First  take  out  tTie 
tabular  number  which  is  nearest  to  the  given  one ;  and  also  its  tabular  cube  rt.  Mult  this  tabular 
number  by  2 ;  and  to  the  prod  add  the  given  number.  Call  the  sum  A.  Then  mult  the  given  number 
by  2;  and  to  the  prod  add  the  tabular  number.  Call  the  sum  B.  Then 

A  :  B  :  :  Tabular  root  :  Reqd  root. 

Ex.  Let  the  given  number  be  7368.  Here  we  find  the  nearest  tabular  number  (in  the  column  of 
•ttfte*)  to  be  6859 ;  and  its  tabular  cube  rt  19.  Hence, 

6859  —  tab  num.       ^  f     7368  =  given  num. 


6859  =  tab  num.       ">  r     7368 

!  _j 

t  and       J    14736 

b±9 

I  21595 


13718 
7368  =  given  num. 


—  tab  num. 
21595  =  B. 


21086  =  A. 
A.  B. 

Then,  as  21086      :      21595 
The  root  as  found  by  correct  mathematical  process  is  19.4588.    The  engineer  rarely  requires  even 


Tab  Root.    Reqd  Rt. 
:      19      :      19.4585 


GEOMETRY.  61 

this  degree  of  accuracy ;  for  his  purposes,  therefore,  this  process  is  greatly  preferable  to  the  ordinary 
laborious  one. 

To  find  the  square  root  of  a  number  which  is  wholly 
decimal. 

Very  simple,  and  correct  to  the  third  numeral  figure  inclusive.  If  the  number  does  not  contain  at 
least  five  figures,  counting  from  the  first  numeral,  and  including  it,  add  one  or  more  ciphers  to  make 
five.  If,  after  that,  the  whole  number  is  not  separable  into  twos,  add  another  cipher  to  make  it  so. 
Then  beginning  at  the  first  numeral  figure,  and  including  it,  assume  the  number  to  be  a  whole  one. 
In  the  table  find  the  number  nearest  to  this  assumed  one  ;  take  out  its  tabular  sq  rt ;  move  the  deci- 
mal point  of  this  tabular  root  to  the  left,  half  MS,  many  places  as  the  finally  modified  decimal  number 
has  figures. 

Ex.  What  is  the  sq  rt  of  the  decimal  .002?  Here,  in  order  to  have  at  least  five  decimal  figures, 
counting  from  the  fipst  numeral  (2),  and  including  it,  add  ciphers  thus,  .00,20,00,0.  But,  as  it  is  not 
now  separable  into  twos,  add  another  cipher,  thus,  .00,20,00,00.  Then  beginning  at  the  first  numeral 
(2),  assume  this  decimal  to  be  the  whole  number  200000.  The  nearest  to  this  in  the  table  is  199809; 
and  the  sq  rt  of  this  is  447.  Now,  the  decimal  number  as  finally  modified,  namely,  .00.20,00,00,  has 
eight  figures  ;  one-half  of  which  is  4;  therefore,  move  the  decimal  point  of  the  root  447,  four  places  to 
the  left;  making  it  .0447.  This  is  the  reqd  sq  rt  of  .002,  correct  to  the  third  numeral  7  included. 

To  find  the  cube  root  of  a  number  which  is  wholly  decimal. 

Very  simple,  and  correct  to  the  third  numeral  inclusive. 

If  the  number  does  not  contain  at  least  five  figures,  counting  from  the  first  numeral,  and  including 
it,  add  one  or  more  ciphers  to  make  five.  If,  after  that,  the  number  is  not  separable  into  threes,  add 
one  or  more  ciphers  to  make  it  so.  Then  beginning  at  the  first  numeral,  and  including  it,  assume 
the  number  to  be  a  whole  one.  In  the  table  find  the  number  nearest  to  this  assumed  one,  and  take 
out  its  tabular  cub  rt.  Move  the  decimal  point  of  this  rt  to  the  left,  one-third  as  many  places  as  the 
fiually  modified  decimal  number  has  figures. 

Ex.  What  is  the  cube  rt  of  the  decimal  .002  ?  Here,  in  order  to  have  at  least  five  figures,  counting 
from  the  first  numeral  (2),  and  including  it,  add  ciphers  thus,  .002.000,0.  But  as  it  is  not  now  separ- 
able into  threes,  add  two  more  ciphers  to  make  it  so ;  thus,  .002,000,000.  Then  beginning  with  the 
first  numeral  (2),  assume  the  decimal  to  be  the  whole  number  2000000.  The  nearest  cube  to  this  in 
the  table  in  the  column  of  cubes,  is  2000376;  and  its  tabular  cube  rt  as  found  in  the  col  of  numbers, 
is  126.  Now,  the  decimal  number  as  flnnlly  modified,  namely,  .002  000  000,  has  nine  figures ;  one-third 
of  which  is  3;  therefore,  move  the  decimal  point  of  the  root  126,  three  places  to  the  left,  making  it 
.126.  This  la  the  reqd  cube  rt  of  the  decimal  .002,  correct  to  the  third  numeral  6  included. 

To  find  roots  by  logarithms,  see  p.  614. 


GEOMETKY. 

Lines,  Figures,  Solids,  defined.    Strictly  speaking  a  geometrical  line 

is  simply  length,  or  distance.  The  lines  we  draw  on  paper  have  not  only  length,  but  breadth  and 
thickness  ;  still  they  are  the  most  convenient  symbol  we  can  employ  for  denoting  a  geometrical  line. 

Straight  lines  are  also  called  right  lines.  A  vertical  line  is  one  that  points 
toward  the  center  of  the  earth;  and  a  horizontal  one  is  at  right  angles  to  a 
vert  one.  A  plane  figure  is  merely  any  flat  surface  or  area  entirely  enclosed 

by  lines  either  straight  or  curved  ;  which  are  called  its  outline,  boundary,  circumf,  or  periphery.  "We 
often  confound  the  outline  with  the  fig  itself  as  when  we  speak  of  drawing  circles,  squares,  &c ;  for 
we  actually  draw  only  their  outlines.  Geometrically  speaking,  a  fig  has  length  and  breadth  only ;  no 

thickness.    A  solid  is  any  body ;  it  has  length,  breadth,  and  thickness. 

Geometrically  similar  figs  or  solids,  are  not  necessarily  of  the  same 
size ;  but  only  of  precisely  the  same  shape.  Thus,  any  two  squares  are,  scien- 
tifically speaking,  similar  to  each  other  ;  so  also  any  two  circles,  cubes,  &c,  no  matter  how  different 
they  may  be  in  size.  When  they  are  not  only  of  the  same  shape,  but  of  the  same  size,  they  are  said 

to  be  similar,  and  equal. 

The  quantities  01  lines  are  to  each  other  simply  as  their  lengths;  but 

the  quantities,  or  areas,  or  surfaces  of  similar  figures,  are  as,  or  in  proportion 
to,  the  squares  of  any  one  of  the  corresponding  lines  or  sides  which  enclose  the 
figures,  or  which  may  be  drawn  upon  them  ;  and  the  quantities,  or  solidities  of 
similar  solids,  are  as  the  cubes  of  any  of  the  corresponding  lines  which  form 

their  edges,  or  the  figures  by  which  they  are  enclosed. 

Angles.  When  two  straight,  or  right  lines  meet  each  other  at  any  inclina- 
tion, the  inclination  is  called  an  angle;  and  is  measured  by  the  degrees  con- 
tained in  the  arc  of  a  circle  described  from  the  point  of  meeting  as  a  center.  Since  all  circles,  whether 
large  or  small,  are  supposed  to  be  divided  into  360  degrees,  it  follows  that  any  number  of  degrees  of  a 
small  circle  will  measure  the  same  degree  of  inclination  as  will  the  same  number  of  a  large  one. 

When  two  straight  lines,  as  o  n  and  a  6,  meet  in  such  a  manner  that  the  inclination  o  n  a  is  equal 

to  the  inclination  o  n  &,  then  the  two  lines  are  said  to  be 
perpendicular  to  each  other;  and  the  angles  o  n  a  and 
o  n  b,  are  called  right  angles ;  and  are  each  measd  by,  or  ri /_J__Vu 

are  equal  to,  90°,  or  one-fourth  part  of  the  circumf  of  a  circle.     Any  angle, 

&sced,  smaller  than  a  right  angle,  is  called  acute  or  sharp ; 
and  one  c  ef,  larger  than  a  right  angle,  is  called  obtuse,  or 
blunt.  When  one  line  meets  another  without  crossing  it,  as  in  the  two  foregoing  figures,  the  two 

angles  o  n  n,  and  o  n  6,  are  called  contiguous,  or  adjacent;  so  also  c  e  d, 

and  c  ef.  Two  adjacent  angles  are  together  always  equal  to  two  right  angles;  or  to  180°.  There- 
fore,  if  we  know  the  number  of  degrees  contained  in  one  of  them,  and  subtract  it  from  180°,  we  obtain 
the  other. 


62 


GEOMETRY. 


When  two  straight  lines  cross  each  other,  forming  four  angles, 
either  pair  of  those  angles  which  point  in  exactly  opposite  direc- 
tions are  called  opposite,  or  improperly,  verti- 
cal angles ;  thus,  the  pair  s  u  t,  and  v  u  w  are  op- 
posite angles  ;  also  the  pair  a  u  v  and  t  u  w.  The  opposite  angles 
of  any  pair  are  always  equal  to  each  other. 

"When  a  straight  line  a  6  crosses  two  parallel  lines  c  d,  e  f,  the 

alternate  angles  which  form  a  kind  of  Z,  are  equal 

to  each  other.  Thus,  the  angles  don,  and  o  n  f,  are  equal:  as  are 
also  con,  and  one.  Also  the  two  internal  angles  on  the  same  side 
of  o  b,  are  equal  to  tw>  right  angles,  or  180°;  thus,  the  two  angles  c 
o  n,  and  onf=  18(P ;  as  do  also  don  and  one. 

An  interior  angle, 

In  any  fig,  is  any  angle  formed  inside  of  that  fig,  by  the  meet- 
ing of  two  of  its  sides,  as  the  angles  c  a  b,  a  b  c,  b  c  a,  of  this 
triangle.  All  the  interior  angles  of  any  straight-lined  figure  of 
any  uumber  of  sides  whatever,  are  together  equal  to  twice  as 
many  right  angles  minus  four,  as  the  figure  has  sides.  Thus,  a 
triangle  has  3  sides ;  twice  that  number  is  6 ;  and  6  right  angles, 
or  6  X  90°  =  540° ;  from  which  take  4  right  angles,  or  360° ;  and 
there  remain  180°,  which  is  the  number  of  degrees  in  every 
plane,  or  straight  lined  triangle.  This  principle  furnishes  an 
easy  means  of  testing  our  measurements  of  the  angles  of  any 
fig ;  for  if  the  sum  of  all  our  measurements  does  not  agree  with 
*he  sum  given  by  the  rule,  it  is  a  proof  that  we  have  committed  some  error. 

An  exterior  angle 

Of  any  straight-lined  figure,  is  any  angle,  as  a  b  d,  formed  OUTSIDE  of  that  fig, 
by  the  meeting  of  any  side,  as  a  b,  with  the  prolongation  of  an  adjacent  side, 
as  c  b  ;  so  likewise  the  angles  c  a  s,  and  b  cw.  All  the  exterior  angles  of  any 
straight-lined  fig,  no  matter  how  many  sides  it  may  have,  amount  to  four 
right  angles,  or  360°.  Thus,  in  »he  foregoing  fig,  the  three  exterior  angles 
a  b  d,  c  as,  and  b  cw,  amount  to  360°.  In  any  fig,  those  angles,  as  g.  h, 

.    and  /,  which  point  outward,  are  called  salient ;  those  whicfc 
1J    point  inward,  as  t,  are  called  re-entering*. 


All  angles,  as  n  a  m,  n  o  m,  at  the  circumf  of  a  semicircle,  and  stand- 
ing on  its  diam  n  m,  are  right  angles ;  or,  as  it  is  usually  expressed, 

all  angles  in  a  semicircle  are  right  angles. 

An  angle  n  s  x  at  the  centre  of  a  circle,  is  twice  as  great  as  an  angle  n 
m  x  at  the  circumf,  when  both  stand  upon  the  same  arc  n  x. 


All  angles,  as  y  dp,  y  e  p,  y  g  p.  at  the  circumf  of  a  circle,  and  standing 
upon  the  same  arc.  as  y  p,  are  equal  to  each  other ;  or,  as  usually  expressed, 

all  angles  in  the  same  segment  of  a  circle  are 
equal. 

r  The  complement  of  an  angle  is  usually  said  to 
S  be  what  it  wants  of  being  90°;  and  its  supplement, 
wrhat  it  wants  of  being  180°.  But  in  mathematical  strictness 
its  complement  is  its  difference  from  9O°,  whether 
greater  or  less;  and  its  supplement  in  like  manner  js  its 
diff  from  180°. 

If  any  two  chords,  as  a  b,  o  c,  cross  each  other, 

then  as  o  n  :  n  b  : :  a  n  :  n  c.  Hence,  nb  X  an  =  on'Xnc.  That 
is,  the  product  of  the  two  parts  of  one  of  the  lines,  is  =  the  pro- 
duct of  the  two  parts  of  the  other  line. 


Sines,  Tangents,  Ac. 

Sine,  a  s,  of  any  angle,  a  c  6,  or  which  is  the  same  thing,  the  sine  of  any  circular  are,  a  &, 
which  subtends  or  measures  the  angle,  is  a  straight  line  drawn  from  one  end,  as  a,  of  the  arc,  at  right 
angles  to,  and  terminating  at,  the  rad  c  b,  drawn  to  the  other  end  b  of  the  arc.  It  is,  therefore,  equal 
to  half  the  chord  an,  of  the  arc  abn,  which  is  equal  to  twice  the  arc  ab  ;  or,  the  sine  of  an  angle  is 


GEOMETRY. 


63 


^tways  equal  to  half  the  chord  of  twice  that  angle;  and  vice  versa,  the  chord  of  an  angle  is  always 

iqual  to  twice  the  sine  of  half  the  angle. 

The  sine  t  c  of  an  angle  t  c  b,  or  of  an  arc  yy 

t  a  b,  of  90°,  is  equal  to  the  rad  of  the  arc 

or  of  the  circle  ;  and  this  sine  of  90°  is 

greater  than  that  of  any  other  angle. 

Cosine  c  5  of  an  angle  a  c  &, 

is  that  part  of  the  rad  which  lies  between 
the  sine  and  the  center  of  the  circle.  It 
is  always  equal  to  the  sine  y  a  of  the 

c  b  wants  of  being  90°.  The  prefix  co  be- 
fore sines,  &c,  means  complement ;  thus, 
cosine  means  sine  of  the  complement. 

Versed  sine  s  b  of  any  angle 

a  c  b,  is  that  part  of  the  rad  which  lies 
between  the  sine,  and  the  outer  end  6. 
It  is  very  common,  but  erroneous,  when 
speaking  of  bridges,  &c,  to  call  the  rise 
or  height  s  b  of  a.  circular  arch  a  b  n,  its 
rersed  sine;  while  it  is  actually  the  versed 
sine  of  only  half  the  arch.  This  absurdity 
should  cease ;  for  the  word  rise  or  height 
is  not  only  more  expressive.but  is  correct. 


a  c  6,  is  a  line  drawn  from,  and  at  right 
angles  to,  the  end  6  or  a  of  either  rad  c  6, 
or  c  a,  which  forms  one  of  the  legs  of  the 
angle  ;  and  terminating  as  at  w,  or  c?,  in 
the  prolongation  of  the  rad  which  forms 
the  other  leg.*  This  last  rad  thus  pro- 
longed, that  is,  c  ic,  or  c  d,as  the  case  may 

be,  is  the  secant  of  the  angle 

a  c  ft.  The  two  tangents  are  equal;  so 
also  are  the  two  secants.  The  angle  t  c  I  being  supposed  to  be  90°,  the  angle  t  c  a  becomes  the  com- 
plement  of  the  angle  a  c  b,  or  what  a  c  6  wants  of  being  90° ;  and  the  sine  y  a  of  tbis  complement ;  its 
versed  sine  t  y,  its  tangent  t  o  ;  and  its  secant  c  o,  are  respectively  the  co-sine,  co-versed  sine  ;  co- 
tangent ;  and  co- secant,  of  the  angle  acb.  Or,  vice  versa,  the  sine,  &c.  of  a  c  b,  are  the  cosine.  &c. 
of  t  c  a;  because  the  angle  a  c  6  is  the  complement  of  the  angle  tea.  When  the  rad  c  b,  c  a,  or  c  t, 
is  assumed  to  be  equal  to  unity,  or  1,  the  corresponding  sines,  tangents,  &c,  are  called  natural  ones; 
and  th<nr  several  lengths  for  diff  angles,  for  said  rad  of  unity,  have  been  calculated  ;  constituting  the 
•well-known  tables  of  nat  sines,  &c.  In  any  circle  whose  rad  is  either  larger  or  smaller  than  1,  the 
sines,  <fec,  of  the  angles  will  be  in  the  same  proportion  larger  or  smaller  than  those  in  the  tables,  and 
are  consequently  found  by  mult  the  sine,  &c,  of  the  table,  by  said  larger  or  smaller  rad. 
#  This  is  the  trigonometrical  tang.  For  others  see  Remark  p  101. 


Plane,  or  Straight-sided,  Triangles. 

See  also  Mensuration,  pages  13,  14 ;  and  Trigonometry,  page  39,  etc. 

The  following  apply  to  any  plane  triangle,  whether  oblique  or  right-angled  : 
1.  The  three  angles  amount  to  18<P,  or  two  right  angles. 

3.  Any  exterior  angle,  as  A  C  n,  is  equal  to  the  two  interior  and  opposite 
ones,  A  and  B. 

8.  The  greater  side  is  opposite  the  greater  angle. 

4.  The  sides  are  as  the  sines  of  the.opposite  angles.    Thus,  the  side  a  is  to 
the  side  b  as  the  sine  of  A  is  to  the  sine  of  B. 

5.  If  any  angle  as  s  be  bisected  by  a  line  s  o,  the  two  parts  mo,  on  of 
the  opposite  side  m  n  will  be  to  each  other  as  the  other  two  sides  s  m,  sn; 


6.  If  lines  be  drawn  from  each  angle  r  s  t  to  the 
center  of  the  opposite  side,  they  will  cross  each 

other  at  one  point,  a,  and  the  short  part  of  each  Tn      g" 

of  the  lines  will  be  the  third  part  of  the  whole  line. 
Also,  a  is  the  cen  of  grav  of  the  triangle. 

7.  If  lines  be  drawn  bisecting  the  three  angles,  they  will  meet  at  a  point 
perpendicularly  equidistant  from  each  side,  and  consequently  the  center 

(•  of  the  greatest  circle  that  can  be  drawn  in  the  triangle. 
"• w      8.  If  a  line  s  n  be  drawn  parallel  to  any  side  c  a,      ^ 
the  two  triangles  r  t  n,  r  c  a,  will  be  similar. 

9.  To  divide  any  triangle  a  c  r  into  two  equal  parts  by  a  line  «  n  parallel  to 

any  one  of  its  sides  c  a.     On  either  one  of  the  other  sides,  as  a  r,  as  a  diara, 

describe  a  semicircle  a  o  r;  and  find  its  middle  o.    From  r  (opposite  c  a),  with 

radius  r  o,  describe  the  arc  o  n.     From  n  draw  n  s.  par- 

0  allel  to  c  o. 

>\  1O.   To  find  the  greatest  parallelogram  that  can  be 

_    Jr   \  drawn  in  any  given  triangle  o  n  b.     Bisect  the  three  aides  at  a  c  e,  and  join 

Q(  \C          a  c,  a  e,  e  c.     Then   either  a  e  b  c,  a  e  c  o,  or  a  c  e  n,  each   equal  to  half  the 

/\      /\          triangle,  will   be  the  reqd   parallelogram.     Any  of   these   parallelograms  can 
/       \S       \«       plainly  be  converted  into  a  rectangle  of  equal  area,  and  the  greatest  that  can  be 
i  in  the  triangle. 
!•£.  If  a  line  a  c  bisects  any  two  sides  o  &,  o  n,  of  a  triangle,  it  will  be  par- 


pla 
dr 


. 

allel  to  the  third  side  n  b,  and  half  as  long  as  it. 


64 


GEOMETRY. 


11.  To  find  the  greatest  square  that  can  be  drawn  in  any  triangle  axr.    From 
an  uugle  as  a  draw  a  perp  a  n  to  the  opposite  side  xr,  and  find  its  length.   Then 

o  n,  or  a  side  v  t  of  the  square  will  =  — 

Hem.— If  the  triangle  is  such  that  two  or  three  such  perps  can  be  drawn,  then 
two  or  tnree  equal  squares  may  be  found. 

Right-angled  Triangles. 

All  the  foregoing  apply  also  to  right-angled  triangles;  but  what  follow  apply  to  them  only. 

Call  the  right  angle  A,  and  the  others  B  and  C;  and  call  the  sides  respectively 
JJ  opposite  to  them  a,  b,  and  c.     Then  is 


U 


6  =  a  X  N.  Sine  B  =  a  X  N.  Cos  C  =  c  X  N.  Cot  C  =  c  X  N.  Tang  B. 
c  =  a  X  N.  Sine  C  =  o  X  N.  Cos  B  =  b  X  N.  Tang  C. 

Also  N.  Sine  of  C  =£;  N.  Cos  C  —  ^/  N.  Tang  C  =  ~. 

And  N.  Sine  of  B  -  6;  N.  Cos  B  ~  -;  N.  Tang  B  -  -. 
a  a  c 

AndN.  Sine  of  A  or  9(P  =  1.     N.  Cos  A  =  0.     N.  Tang  A  =  infinity. 

1.  If  from  the  right  angle  o  a  line  o  w  be  drawn  perp  to  the  hypothenuse  or  long  side'  h  g,  then  the 
two  small  triangles  owh,  o  w  g,  and  the  large  one  otig,  will  be  similar. 
Also  g  w :  w  o ::  w  o :  w  h ;  and  g  w  X  w  h  —  w  o%. 

%.  A  line  drawn  from  the  right  angle  to  the  center  of  the  long  side  will 
be  half  as  long  as  said  side. 

8.  If  on  the  three  sides  o  h,  o  g,  g  h  we  draw  three  squares  t,  u  v,  or 
three  circles,  or  triangles,  or  any  other  three  figs  that  are  similar,  then  the 
area  of  the  largest  one  will  be  equal  to  the  areas  of  the  two  others. 

4.  In  a  right-angled  triangle  whose  sides  are  as  3,  4,  and  5  (as  is  the  tri- 
angle AB  C),  the  angles  are  very  approximately  90°:  53°  7'  48.38";  and 

/^ •    36°  52'  11.62".     Their  N.  Sines,  1. ;  .8  ;  and  .6.     Their  N.  Tangs,  infinity ; 

/ij  1.3333;  and  .75. 

\/  &.  One  whose  sides  are  as  7,  7,  and  9.9,  has  very  approx  one  angle  of  90° 

and  two  of  45°  each,  near  enough  for  all  practical  purposes. 

PARALLELOGRAMS. 

A  parallelogram  is  any  four-sided  straight-lined  figure  whose 
opposite  sides  are  equal,  as  a  b  c  d ;  or  a  square,  &c.  (See  Men- 
suration.) Any  line  drawn  across  a  parallelogram  from  two 
opposite  angles,  is  called  a  diagonal,  as  a  c,  or  b  d.  A  diag  divides 
a  parallelogram  into  two  equal  parts  ;  as  does  also  any  line  mn 
drawn  through  the  center  of  either  diag ;  and  moreover,  the  line 
TO  n  itself  is  div  into  two  equal  parts,  or  is  bisected.  Twodiags 
bisect  each  other;  they  also  divide  the  parallelogram  into  four 
triangles  of  equal  areas.  Any  two  adjacent  angles  of  any  paral- 
lelogram are  equal  to  two  right  angles,  or  180°;  as  the  angles 
dab  and  a  ft  c;  or  a  b  c  and  bed;  and  the  four  angles  are 
always  equal  to  four  right  angles,  or  360°. 
The  sum  of  the  squares  of  the  four  sides,  is  equal  to  the  sum  of  the  squares  of  the  two  diags. 

Rem.— Simple  a§  the  following  operations  appear,  it  is  only  by  care,  and  good  instruments,  that 
they  are  made  to  give  accurate  results.  Several  of  them  can  be  much  better  performed  by  means  of  a 
metallic  triangle  having  one  perfectly  accurate  right  angle.  In  the  field,  the  tape-line,  chain,  or  a 
ineasuring-rod  will  take  the  place  of  the  dividers  and  ruler  used  indoors. 


To  divide  a  given  line,  a  b,  into  two  equal  parts. 

>From  its  ends  a  and  b  as  centers,  and  with  any  rad  greater  than  one-half  of  a  b, 
describe  the  arcs  c  and  d,  and  join  «/.  If  the  line  a  b  is  very  long,  first  lay  off 
equal  dists  a  o  and  bg,  each  way  from  thP  ends,  so  as  to  approach  conveniently 
near  to  each  other  ;  and  then  proceed  as  if  o  g  were  the  line  to  be  divided.  Or 
measure  a  6  by  a  scale,  and  thus  ascertain  its  center. 


To  divide  a  given  line,  m  n,  into  any 
given  number  of  equal  parts. 

Prom  TO  and  n  draw  any  two  parallel  lines  TO  o  and  n  a, 
to  an  indefinite  dist;  and  o'n  them,  from  TO  and  n  step  off  the 
reqd  number  of  equal  parts  of  any  convenient  length  :  final- 
ly, join  the  corresponding  points  thus  stepped  off.  Or  only 
one  line,  as  TOO.  may  be  drawn  and  stepped  off.  as  to  s; 
then  join  sn;  and  draw  the  other  short  Hues  parallel  to  it. 

To  divide  a  given  line, tn  n,  into  two  parts  which  shall  have 
a  given  proportion  to  each  other. 

This  is  done  on  the  same  principle  as  the  last ;  thus,  let  the  proportion  be  as  1  to  3.    First  draw 
any  Hue  mo;  and  with  any  convenient  opening  of  the  dividers,  make  mx  equal  to  one  step  ;  and  xs 


GEOMETRY. 


65 


From  any  s  i veil  point,  p,  on  a  line  *  t, 
to  draw  a  perp,  j>  a. 

From  p,  with  any  convenient  opening  of  the  dividers,  step  off  the 

equals  po,pa.    Prom  o  and  g  as  centers,  with  any  opening  greater 

than  half  o  g,  describe  the  two  short  arcs  b  and  c ;  and  join  a  p. 

Or  still  better,  describe  four  arcs,  and  join  a  y. 

Or  from  p  with  any  convenient  scale  describe  two 

short  arcs  g  and  c  either  one  of  them  with  a  radius  3,  and  the  other 
with  a  rad  4.    Then  from  g  with  rad  5  describe  the  arc  6.    Join  p  a. 


If  the  point  p  is  at  one  end  of  the  line, 
or  very  near  it, 

Extend  the  line,  if  possible,  and  proceed  as  above.  But  if  this 
cannot  be  done,  then  from  any  convenient  point,  w,  open  the  divid- 
ers to  p,  and  describe  the  semicircle,  *  p  o ;  through  o  w  draw  o  w 
«;  join  j>  t. 

Or  use  the  last  foregoing  process  with 

rads  3,  4,  and  5. 


From  a  given  point,  o,  to  let  fall  a 
perp  o  s,  to  a  given  line,  ni  n. 

.  From  o,  measure  to  the  line  m  n,  any  two  equal  dists,  o  c, 
o  e ;  and  from  c  and  e  as  centers,  with  any  opening  greater 
than  half  of  c  e,  describe  the  two  arcs  a  and  6  ;  join  o  t.  Or 
from  any  point,  as  d  on  the  line,  open  the  dividers  to  o,  and 
describe  the  arc  o  g  ;  make  i  x  equal  to  i  o ;  and  join  o  x. 


If  the  line,  a  b,  is  on  the  gronnd, 

And  a  perp  is  reqd  to  be  drawn  from  c,  first  measure  off  any  twc 
equal  dists,  c  m,  c  n.  At  m  and  n,  hold  the  ends  of  a  piece  of  string, 
tape-line,  or  chain,  man;  then  tighten  out  the  string,  &c,  as  shown 
by  m  g  n  \  «  being  its  center.  Then  will  «  c  be  the  reqd  perp.  Or  if 
the  perp  x  z  is  to  be  drawn  from  the  end  of  the  line  w  x,  first  measure  xy 
upon  the  line,  and  equal  to  three  feet;  then  holding  the  end  of  a  tape- 
line  at  x,  and  its  nine  feet  mark  at  y,  hold  the  four  feet  mark  at  z,  keep- 
ing zx  and  z  y  equally  stretched.  Then  zx  will  be  the  reqd  perp,  because 
8,4,  and  5,  make  the  sides  of  a  right-angled  triangle.  Instead  of  3,  4,  and 
6,  any  multiples  of  those  numbers  may  be  used,  such  as  6,  8,  and  10 ;  or 
9,  12,  15,  &c :  also  instead  of  feet,  we  may  use  yards,  chains,  &c. 


y  a 


Through  a  given  point,  ft,  to  draw  a 
line,  a  c,  parallel  to  another  line, 

ef. 

"With  the  perp  dist,  a  e,  from  any  point,  n,  In  ef,  describe 
an  arc,  t;  draw  a  c  just  touching  the  arc. 


At  any  point,  a,  in  a  line  a  b, 
to  make  an  angle  ca  b, equal 
to  a  given  angle,  m  n  o. 

From  n  and  a,  with  any  convenient  rad,  describe 
Hie  arcs  nt.de;  measure  s  t,  and  make  e  d  equal      ' 
to  it;  through  a  d  draw  a  c. 


66 


GEOMETRY. 


To  bisect,  or  divide  any  angle,  w  x  y,  into 
two  equal  parts. 

From  x  set  off  any  two  equal  dists,  x  r,  x  a.  From  r  and  «  with  any  rad 
describe  two  arcs  intersecting,  as  at  o ;  and  join  o  x.  If  the  two  sides  of 
the  angle  do  not  meet,  as  c  /  and  g  h,  either  first  extend  them  until  the/ 
do  meet;  or  else  draw  lines  x  w,  and  xy,  parallel  to  them,  and  at  equal 
dists  from  them,  so  as  to  meet;  then  proceed  as  before. 


To  describe  a  circle  through  any  three 
points,  a  be,  not  in  a  straight  line. 

Join  the  points  by  the  lines  a  b,  be;  from  the  centers  of  these  lines  draw 
the  dotted  perps  meeting,  as  at  o,  which  will  be  the  center  of  the  circle. 
Or  from  b,  with  any  convenient  rad,  draw  the  arc  m  n;  and  from  a  and  c, 
with  the  same  rad,  draw  arcs  y  and  z;  then  two  lines  drawn  through  the 
Intersections  of  these  arcs,  will  meet  at  the  center  o. 

To   describe  a   circle   to    touch    the   three 
angles  of  a  triangle  is  plainly  the  same  as  this. 
To  inscribe  a  circle  in  a  triangle  draw  two  lines 

bisecting  any  two  of  the  angles.    Where  these  lines  meet  is  the  center  of 
the  circle. 


\m 


On  a  given  line  »/-a%to  draw  a  square, 

w  x  n,  m. 

From  w  and  x,  with  rad  10  x,  describe  the  arcs  xry  and  w  r  e. 
From  their  intersection  r,  and  with  rad -equal  to  %  of  w  x,  describe 
s  s  s.  From  to  and  z  draw  w  n  and  xm  tangential  to  a  «  *,  and 
ending  at  the  other  arcs ;  join  n  m. 


To  find  the  center  e,  of  a  given  circle. 

Draw  any  chord  a  b ;  and  from  the  middle  of  it  o,  draw  at  right  angles  to 
it,  a  diam  d  g ;  find  the  center  c  of  this  diam. 


To  draw  a  tangent,  I  e  i,  to  a  circle,  from  any 
given  point,  f,  in  its  circumf. 

Through  the  center  n,  and  the  given  point  e,  draw  n  o ;  make  e  o  equal  to 
e  n ;  from  n  and  o,  with  any  rad  greater  than  half  of  o  n,  describe  the  two 
pairs  of  arc  i  i;  join  their  intersections  i  i. 

.  Here,  and  in  the  following  three  figs,  the  tangents  are  ordinary  or  geo- 
metrical  ones ;  and  may  eud  where  we  please.  But  the  trigonometrical 
tangent  of  a  given  angle,  must  end  in  a  secant,  as  in  the  top  fig  of  p  68. 

Or  from  e  lay  off  two  equal  distances  e  c,  e  t ;  and  draw  i  i 

parallel  to  c  t. 

To  draw  a  tang,  a  *  ft,  to  a  circle,  from  a  point, 
a,  which  is  outside  of  the  circle. 

Draw  a  c,  and  on  it  describe  a  semicircle ;  through  the  intersection,  a,  draw 
a  sb.  Here  c  is  the  center  of  the  circle. 


To  draw  a  tang,  g  7tt  from  a  circular  arc,  g  a  c9 

Of  which  n  a  is  the  rise.    With  rad  g  a,  describe  an  arc,  «  a  o.     Make  *  a 
equal  to  s  a.    Through  t  draw  g  ft. 


GEOMETRY. 


67 


To  draw  a  tang  to  two  circles. 

First  draw  the  line  m  n,  just  touching  the  two 
sircles ;  this  gives  the  direction  of  the  tang.  Then 
from  the  centers  of  the  circles  draw  the  radii,  o  o,  perp 
to  m  n.  The  points  t  t  are  the  tang  points.  If  the 
tang  is  in  the  position  of  the  dotted  line,  sy,  the  ope- 
ration is  the  same. 

Rein.     This  empirical   method  is  at 

least  a«  accurate  as  the  scientific  ones,  especially  if 
a  correct  triangular  ruler  is  used  for  the  radii. 


To  draw  a  hexagon,  each  side  of  which  shall 
be  equal  to  a  given  line,  a  b. 

From  a  and  ft,  with  rad  a  b,  describe  the  two  arcs;  from  their  intersection, 
t,  with  the  same  rad,  describe  a  circle;  around  the  circumf  of  which,  step  off 
the  same  rad. 


To  draw  an  octagon,  with  each  side 
equal  to  a  given  line,  c  e. 

From  c  and  e  draw  two  perps,  cp,  ep.  Also  prolong  c  e  toward 
/  and  </;  and  from  c  and  e,  with  rad  equal  c  e,  draw  the  two 
quadrants  ;  and  find  their  centers  h  h ;  join  c  h,  and  e  h  ;  draw 
h  a  and  h  t  parallel  to  cp  ;  and  make  each  of  them  equal  to  c  e; 
make  c  o,  and  e  o,  each  equal  to  h  h  ;  join  o  o,  o  «,  aud  o  t. 


To  draw  an  octagon  in  a  given  square. 

From  each  corner  of  the  square,  and  with  a  rad  equal  to  half  its  diag, 
describe  the  four  arcs;  aud  join  the  points  at  which  they  cut  the  sides  of  the 
square. 

To  draw  any  regular  polygon,  with  each  side 
equal  to  m  n. 

Div  360  degrees  by  the  number  of  sides;  take  the  quot  from  180°;  div  the 
rem  by  2.  This  will  give  the  angle  cm  n,  or  cnm.  At  m  and  n  lay  down  these 
angles  by  a  protractor:  the  sides  of  these  angles  will  meet  at  a  point,  c,  from 
which  describe  the  circle  mtiy;  and  around  its  circumf  step  off  dists  equal  to 
mn. 

In  any  circle,  m  n  y,  to  draw  any  regular 
polygon. 

Div  360°  by  the  number  of  sides  ;  the  quot  will  be  the  angle  men,  at  the  center. 
Lay  off  this  angle  by  a  protractor  ;  aud  its  chord  m  n  will  be  one  side ;  which 
step  off  around  the  circumf. 

On  the  given  line,  a  .<?,  to  draw  a 
cynia  recta,  a  c  «. 

Find  the  center  c,  of  a  s.  From  a,  c.  and  «,  with  one-half 
of  a  s  ae  rad,  draw  the  four  small  arcs  at  o,  o.  The  inter- 
sections o,  o,  are  the  centers  for  drawing  the  cy ma,  with 
the  same  rad.  By  reversing  the  position  of  the  arcs,  we 
obtain  the  cyma  reversa,  or  »gee,  d  e  f. 


To  describe  the  arc  of  a  circle  too  large  for  the  dividers. 


Let  a  c  be  the  chord  ;  and  o  b  the  height  of 
the  reqd  arc,  as  laid  down  on  the  drawing.  On 
a  separate  strip  of  paper,  s  em  n,  draw  a  c.  o  b, 
and  a  b :  also  5  e,  parallel  to  the  chord  ac.  It 
is  well  to  make  6  «,  and  h  e,  each  a  little  longer 
than  a  b.  Then  cut  off  the  paper  carefully 
along  the  lines  s  b  and  b  e,  so  as  to  leave  remain- 
ing only  the  strip  s  a  b  e  m  n.  Now,  if  the 
straight  sides  s  b  and  6  e  be  applied  to  the  draw- 
ing, so  that  any  parts  of  them  shall  touch  at 


68 


GEOMETRY. 


rfhe   same  tii 
•f  the  arc,  at 


ne  the  points  a  and  b,  or  b  and  c,  the  point  b  on  the  strip  will  be  in  the  circum* 
id  mav  be  pricked  off.     Thus,  any  number  of  points  in  the  arc  may  be  found,  and  aftef 

ward  united  to  form  the  curve. 
Or  thus  :  draw  the  span  a  b. 
the  rise  re;  andac,  6  c.  Fron> 
c  with  rad  c  r  describe  a  circle. 
Hake  each  of  the  arcs  o  t  an4 
t 1  equal  to  r  o  or  r  i ;  and  draw 
ct,  cl.  Dive*,  cl,cr, each  into 
half  as  many  equal  parts  as  thft 
curve  is  to  be  divided  into, 
Draw  the  lines  b  1,  b  2,  b  3 ;  and 
a  4,  a  5,  a  6,  extended  to  meet 
the  first  ones  at  e,  s,  h.  Then 
e,  s,  h,  are  points  in  one-half  t'ha 
curve.  Then  for  the  other  half, 
draw  similar  lines  from  a  to  7, 
8, 9 ;  and  others  from  b  to  meet 
•  them,  as  before.  Trace  the 
•O  curve  by  hand. 

T»  Also  see  Remark  in  "Men- 

suration,"  page  17,   after  "To 

6  Hud  the  length  of  a  circular 

arc." 


To  draw  an  oval,  or  false  ellipse. 

When  only  the  long  diam  a  ft  is  given,  the  following 
will  give  agreeable  curves,  of  which  the  span  o  b  witt 
not  exceed  about  three  times  the  rise  c  o.  On  a  ft  de- 
scribe two  intersecting  circles  «f  any  rad;  through 
their  intersections  8,  v,  draw  e  g;  make  s  g  and  v  « 
each  equal  to  the  diam  of  one  of  the  circles.  Through 
the  centers  of  the  circles,  draw  e  y,  eh,  g  d,  g  t.  From 
e  describe  h  i  y ;  and  from  g  describe  dot. 


When  the  span,  m  n,  and  the 
rise,  s  t,  are  both  given. 

Make  any  t  w  and  m  r,  equal  to  each  other, 
bin  each  less  than  t  8.  Drawrw;  and  through 
its  center  o  draw  the  perp  t  o  y.  Draw  y  r  z. 
Make  n  x  equal  m  r,  and  draw  y  x  b.  From  x  and 
r  describe  n  c  and  m  a, ;  and  from  y  descri  be 
a  t  c.  By  making  s  d  equal  to  *  y,  we  obtain 
the  center  for  the  other  side  of  the  oval. 

The  beauty  of  the  curve  will  depend  upon 
what  portion  of  t  s  is  taken  for  m  r  and  t  w. 
When  an  oval  is  very  fiat,  more  than  three  cen- 
ters are  required  for  drawing  a  graceful  curve; 
but  the  finding  of  these  centers  is  quite  as  trou- 
blesome as  to  draw  the  correct  ellipse. 

For  drawing  the  ellipse  and  parabola,  see 
"Mensuration." 


To  reduce  any  polygon,  as  abcdefa,  to  a  triangle  of  the 
same  area. 


If  we  produce  the  side  fa  toward  w;  and  draw  b  g  parallel  to  a,  c,  and  join  g  c,  we  get  equal  trl- 
ingles  acb,  and  a  c  <7,  both  on  the  same  base  a  c;  and  both  of  the  same  perp  heij,>.v.  inasmuch  as 
jhej  are  between  ths  two  parallels  a  c  and  g  b.  But  the  part  act  forms  a  portion  *»f  both  these  trl- 


ARITHMETIC. 


69 


p 
h 


angles,  or  in  other  words,  is  common  to  loth.  Therefore,  if  it  be  taken  away  from  both  triangles, 
the  remaining  parts,  i  c  6  of  one  of  them,  and  i  g  a  of  the  other,  are  also  equal.  Therefore,  if  the 
c  6  be  left  off  from  the  polygon,  and  the  part  i  g  a  be  taken  into  it,  the  polygon  g  /  e'd  c  i  g  will 
the  same  area  as  afe  d  c  b  a  ;  but  it  will  have  but  nve  sides,  while  the  other  has  six.  Again, 
if  e  s  be  drawn  parallel  to  d  f,  and  ds  joined,  we  have  upon  the  same  base  es,  and  between  the  same 
parallels  e  a  and  df,  the  two  equal  triangles  e  s  d,  and  e  «/.  with  the  part  cos  common  to  both  ;  and 
consequently  the  remaining  part  e  o  d  of  one,  and  o  sf  of  the  other,  *ire  equal.  Therefore,  if  oaf  be 
left  off  from  the  polygon,  and  e  o  d  be  taken  into  it,  the  new  polygon  y  «  d  eg,  Fig  2,  will  have  the  same 
area  as  g  f  e  d  c  g;  but  it  has  but  four  sides,  while  the  other  'aas  five.  Finally,  if  g  s,  Fig  2,  be 
extended  toward  n;  and  d  n  drawn  parallel  to  c  a  ;  and  c  n  joined,  we  have  on  the  same  base  c  a,  and 
between  the  same  parallels  c  «  and  d  n,  the  two  equal  triangles  c  s  n,  and  c  «  d,  with  the  part'c  8  t 
common  to  both.  Therefore,  if  we  leave  out  c  d  t,  and  take  in  a  t  »,  we  have  the  triangle  one  equal 
to  the  polygon  g  t  d  c  g,  Fig  2;  or  to  afedcba,  Fig  1. 
Vhis  simple  method  is  applicable  to  polygons  of  any  number  of  sides. 


fo  reduce  a  large  fig:,  abed  e  fg,  to  a  smaller 
similar  one. 

From  any  interior  point  o,  which  had  better  be  near  the  center,  draw  lines     Jt 
to  all  the  angles  a,  b.  c,  &c.    Join  these  lines  by  others  parallel  to  the  sides     5 
of  the  fig.    If  it  should  be  reqd  to  enlarge  a  small  fig,  draw,  from  any  point 
o  within  it,  lines  extending  beyond  its  angles  ;  and  join  these  lines  by  others 
parallel  to  the  sides  of  the  small  fig. 


To  reduce  a  map  to  one  on  a  smaller  scale. 

The  best  method  is  by  dividing  the  large  map  into  squares  by  faint  lines,  with  a  very  soft  lead- 
pencil;  and  then  drawing  the  reduced  map  upon  a  sheet  of 
smaller  ?quares.  A  pair  of  proportional  dividers  will  assist 
much  in  fixing  points  intermediate  of  the  sides  of  the  squares. 
If  the  large  map  would  be  injured  by  drawing,  and  rubbing 
out  the  squares,  threads  may  be  stretched  across  it  to  form  the 
squares. 

Maps,  plans,  and  drawings  of  all  kinds,  are  now  copied, 
reduced,  enlarged,  and  multiplied,  cheaply  and  expeditiously,  by 
photography.  For  this  purpose  they  should  be  prepared  only  in 
plain  black  and  white;  shading  should  be  done  in  lines;  not  in 
washes;*  and  care  be  taken  to  make  the  lettering  and  every  part 
rather  more  strong  and  distinct  than  for  ordinary  drawings. 

Ill  a  rectangular  fig,  fj  h  s  fl, 

Representing  an  open  panel,  to  find  the  points  o  o  o  o  in  its 
sides ;  and  at  equal  dists  from  the  angles  g,  and  s ;  for  inserting 
a  diag  piece  o  o  o  o,  of  a  given  width  I  I,  measured  at  right 
angles  to  its  length.  From  g  and  s  as  centers,  describe  several 
concentric  arcs,  as  in  the  Fig.  Draw  upon  transparent  paper, 
two  parallel  lines  a  o.  c  c.  at  a  distance  apart  equal  to  II;  and 
placing  these  lines  on  top  of  the  panel,  move  them  about  until  it 
is  shown  by  the  arcs  that  the  four  dists  g  o,  y  o,  »  o,  s  o,  are 
equal.  Instead  of  the  transparent  paper,  a  strip  of  common 
paper,  of  the  width  1 1  may  be  used. 

RKM.  Many  problems  which  would  otherwise  be  very  difficult, 
may  be  thus  solved  with  an  accuracy  sufficient  for  practical 

purposes,  by  means  of  transparent  paper. 


AKITHMETIC, 


ON  this  subject  we  shall  merely  give  a  few  examples  for  refreshing  the  memory  of  those  who  for 
want  of  constant  practice  cannot  always  recall  the  processes  at  the  moment. 

Subtraction  of  Vulgar  Fractions. 


Addition  of  Vulgar  Fractions. 


34  +  1f=¥ 


*  Not  absolutely  necessary. 


70  ARITHMETIC, 

Multiplication  of  Vulgar  Fractions. 


Division  of  Vulgar  Fractions. 

l+i=|=>-  i*i=  ¥  =  *=«•***=«-¥  =  ¥  =  *=••  *+*=«=>&• 

«*+'f=V+*=tt=V  =  «f      ^1  =  1-1=  47»=65. 

To  find  the  greatest  common  divisor  of  a  Vulgar  Fraction. 

Ex.l.    OfTW          70)U2(2  Ex.2.    Off  $.          »>JJ(* 

35)70(2  ~4)20(5 

70      Ans35.  20      Ans  4. 

To  reduce  a  Vulgar  Fraction  to  its  lowest  terms. 

First  find  the  greatest  common  divisor  ;  then  divide  both  the  numerator  and  denominator  by  it. 
Thus,  in  the  preceding  example  ^ffg  =  -J  Ans.    And  f  J  =  2^  Ans. 

To  reduce  a  Vulgar  Fraction  to  a  decimal  form* 

Divide  the  numerator  by  the  denominator.    Thus, 

4  =2  )1.  0(0.5  Ans.  V3  =  4)13(3.25  Ans.  14  =  40)32.0(0.8  Ans. 

2  10  .     »    .11  320 


20 

20 

Reduce  3  inches  to  the  decimal  of  a  foot.    There  are  12  ins  in  a  foot;  therefore,  the  question  i> 
to  reduce  -j3^  to  a  decimal.    Therefore,  12)3.0(0.25  of  a  foot.  Ans. 


Reduce  2  ft  3  ins  to  the  decimal  of  a  yard.    There  are  36  ins  in  a  yard;  and  27  ins  in  2  ft  3  ins; 
therefore,  f^  of  a  yard  =  36)27.0(0.75  of  a  yd.  Ans. 

180 
180 

How  many  feet  and  ins  are  there  in  .75  of  a  yard  ?    Here 
.75 
3  ft  in  a  yd. 

Ft  2). 25 

12    ins  in  a  ft. 

Ins  3.00        Ans  2  ft  3  ins. 

How  many  feet  and  ins  are  there  in  .0625  of  a  yard  ? 
.0625 

3  feet  in  a  yd. 

No  feet,  .1875 

12  ina  in  a  ft. 

ft.        ins. 
Ins  2.2500  Ans.   0        2.25. 

How  many  cubic  feet  are  there  in  .314  of  a  cub  yard  ?    And  cub  ins  in  .46  of  a  cub  ft  ? 
.314  .46 

27  cub  ft  in  a  yd.  1728  cub  ins  a  cub  ft. 

2198  ~368 

628  92 

322 

8.478  cub  ft.  Ans.  46 

794.88  cub  ins.    Ans. 


ARITHMETIC.  71 


Decimals. 

ADDITION.  Add  together  .25  and  .75;  also  .006,  1.3472,  and  43. 
25 

1.00    Ans.  

44.3532    An& 

SUBTRACTION.   Subtract  .25  from  .75 ;  also  .0001  from  I ;  also  6.30  from  9.01. 
.75  1.  9-01 

.25  .0001  6.30 

.50  Ans.  .9999    Ans.  2.71    Am. 

MULTIPLICATION.   Mult  3  X  .3 ;  also  .3  X  .3  ;  also  .3  X  .03;  also  4.326  X  .003. 
3                                  .3                                    .3  4.326 

.3  .3  .03  .003 

.9  Aus.  .09  Ans.  .009  Ans.  .012978  Ans. 

DIVISION.  Divide  3.  bj  .3 ;  also  .3  by  .3 ;  also  .3  by  .03 ;  also  4.326  by  .0003. 

.3)3.0(10.  Ans.  .3).3(1.  Ans.       ,03).30(10.  Ans.        .0003)4.3260(14420.  Ana. 

3  33  3 

—5  ~~o  — 

Divide  62  by  87.042.  87.042)62.0000(0.712,  &c.  Ans.  12 

4  _60.y294_  

"lOTWO 

87042 


200180  6 

6 

Divide  .006  by  20.  20.000). 0060000<0.0003  Ans.  

60000  0 

Duodecimals. 

Duodecimals  refer  to  square  feet  of  144  sq  ins ;  to  twelfths  of  a  square  or  duodecimal  foot ;  each 
such  twelfth  being  called  an  inch;  and  being  equal  to  12  square  inches ;  and  to  twelfths,  each  equal 
to  the  12th  of  a  duodecimal  inch,  or  to  one  square  inch.  The  dimensions  of  the  thing  to  be  measd 
are  supposed  to  be  taken  in  common  feet,  ins,  and  12ths  of  an  inch ;  but  as  ordinary  measuring 
rules  are  divided  into  8ths  of  an  inch,  it  is  usually  guess-work  to  some  extent.  Duodecimals  are 
very  properly  going  out  of  use,  in  favor  of  decimals ;  we  shall  therefore  give  no  rule  for  them.  By 
means  of  our  table  of  "Inches  reduced  to  Decimals  of  a  Foot,"  page  75,  all  dimensions  in  feet,  ins, 
and  8ths,  &c,  cau  be  at  once  taken  out  in  ft  and  decimals  of  a  foot. 

Single  Rule  of  Three ;  or,  Simple  Proportion. 

If  3  men  lay  10000  bricks  in  a  certain  time,  how  many  could  6  men  lay  in  the  same  time  ?    Thej 
will  evidently  lay  more ;  therefore,  the  second  term  of  the  proportion  must  be  greater  than  the  first. 
3:6::  10000  :  20000  Ans. 


If  3  men  require  10  hours  to  lay  a  certain  number  of  bricks,  how  many  hours  would  6  men 
require  ?  They  will  evidently  require  less  time ;  therefore,  the  second  term  of  the  proportion  must 
be  less  than  the  first. 

6  :  3  :  :  10  :  5  Ans. 
3 

6)30 
5  Ans 

Double  Rule  of  Three;  or,  Compound  Proportion. 

If  three  men  can  lay  4000  bricks  in  2  days,  how  many  men  can  lay  12000  in  3  days  ?    Here  we  se» 
that  4000  bricks  require.  3  X  2  =  6  days'  work;  therefore  12000  will  require, 
4000  :  12000  : :  6  :    18  days'  work. 

But  there  are  only  3  days  to  do  the  18  days  work  in ;  therefore  the  number  of  men  must  be  ^  =  6 
men.  Ans. 

A  moment's  reflection  will  suffice  to  reduce  any  case  of  double  rule  of  three  to  this  simple  form. 

Arithmetical   Progression, 

In  a  series  of  numbers,  is  a  progressive  increase  or  decrease  in  each  successive  number,  by  the  addi- 
tion or  subtraction  of  the  same  amount  at  each  step;  as  in  1,  2,  3,  4,  5,  Ac.,  in  which  1  is  added  at 
each  step ;  or  10,  8,  6,  4,  &c.,  in  which  2  is  subtracted  at  each  step;  or  y±<  *4,  H,  1,  1^.  &c.  In  any 
such  series  the  numbers  are  called  its  terms ;  and  the  equal  increase  or  decrease  at  each  step  its  com- 
mon difference. 

To  find  the  com  diff,  knowing  the  first  and  last  terms  ;  and  the  number  of  terms.  Find  the  did 
between  the  first  and  last  terms.  From  the  number  of  terms  subtract  1.  Div  tin  diff  just  found,  bj 
ihe  rem. 


72 


ARITHMETIC. 


To  find  the  last  term,  knowing  the  first  term  ;  the  com  diff  ;  and  the  number  of  terms.  Prom  the 
number  of  terms  take  1.  Mult  the  rem  by  the  com  diff.  To  the  prod  add  the  first  term. 

To  find  the  number  of  terms,  having  the  first  and  last  ones  ;  and  the  com  diff.  Take  the  diff 
between  the  first  and  last  terms.  Div  this  diff  by  the  com  diff.  To  the  quot  add  1. 

To  find  the  sum  of  all  the  terms,  having  the  first  and  last  ones;  and  the  number  of  terms.  Add 
together  the  first  and  last  terms.  Div  their  sum  by  2.  Mult  the  quot  by  the  number  of  terms. 

Geometrical  Progression, 

In  a  series  of  numbers,  is  a  progressive  increase  or  decrease  in  each  successive  number,  by  the  same 
multiplier  or  divisor  at  each  step  ;  as  3,  9,  27.  81,  &c,  where  each  succeeding  term  is  increased  by  mult 
the  preceding  one  by  3.  Or  48,  24.  12,  6,  &c,  or  27,  13}^,  6%,  3%,  &c,  where  each  succeeding  term  is 
found  by  dividing  the  preceding  one  by  2.  The  multiplier  or  divisor  is  called  the  common  ratio  of  the 
series,  or  progression. 

To  find  the  last  term,  knowing  the  first  one  ;  the  ratio  ;  and  the  number  of  terms.  Raise  the  ratio 
to  a  power  1  less  than  the  number  of  terms.  Mult  this  power  by  the  first  term. 

Ex.  First  term  10  ;  ratio  8  ;  number  of  terms  8  ;  what  is  the  last  term  ?  Here  the  number  of  terms 
being  8,  the  ratio  3  must  be  raised  to  the  7th  power  ;  thus  : 

3X3X3X3X3X3X3  =  2187,  =  7th  power.    And  2187  X  10  =  21870  last  term.   Ans. 

A  man  agreed  to  buy  8  fine  horses  ;  paying  $10  for  the  first;  $30  for  the  second;  $90  for  the  third, 
&c  :  how  much  will  the  last  one  cost  him?  Ans,  $21870,  as  before. 

To  find  the  sum  of  all  the  terms,  knowing  the  first  one  ;  the  ratio  ;  and  the  number  of  terms.  Raise 
the  ratio  to  a  power  equal  to  the  whole  number  of  terms.  From  this  power  subtract  1.  Div  the  rem 
by  1  less  than  the  ratio.  Mult  the  quot  by  the  first  term. 

Ex.  As  before.  What  is  the  sum  of  all  the  terms?  Here  the  ratio  must  be  raised  to  the  8th 
power  ;  thus,  3X3X3X3X3X3X3X3  —  6561  =  8th  pow.  And  6560  div  by  1  less  than  the  ratio  * 

6560 
3,  =  -  =  3280.    And  3280  X  10  (or  number  of  terms)  —  32800  =  sum.  Ans. 

In  the  foregoing  case,  the  8  horses  would  cost  $32800. 

Permutation 

Shows  in  how  many  positions  any  number  of  things  can  be  arranged  in  a  row.  To  do  this,  mult 
together  all  the  numbers  used  in  counting  the  things.  Thus,  in  how  many  positions  in  a  row  can  9 
things  be  placed  ?  Here, 

1X2X3X4X5X6X7X8X9  =  362880  positions.   Ans. 

Combination 

Shows  how  many  combinations  of  a  few  things  can  be  made  out  of  a  greater  number  of  things.  To 
do  this,  first  set  down  that  number  which  indicates  the  greater  number  of  things  ;  and  after  it  a  series 
of  numbers,  diminishing  by  1,  until  there  are  in  all  as  many  as  the  number  of  the  few  things  that 
are  to  form  each  combination.  Then  beginning  under  the  last  one,  set  down  said  number  of  few 
things  ;  and  going  backward,  set  down  another  series,  also  diminishing  by  1,  until  arriving  under  the 
first  of  the  upper  numbers.  Mult  together  all  the  upper  numbers  to  form  one  prod  ;  and  all  the  lower 
ones  to  form  another.  Div  the  upper  prod  by  the  lower  one. 

Ex.  How  many  combinations  of  4  figures  each,  can  be  made  from  the  9  figs  1,  2,  3,  4,  5,  6,  7,  8,  9; 
or  from  9  any  things  ? 


Alligation 

Shows  the  value  of  a  mixture  of  different  ingredients,  when  the  quantity  and  value  of  each  of  these 
last  is  known. 

Ex.  What  is  the  value  of  a  pound  of  a  mixture  of  20  Ibs  of  sugar  worth  15  cts  per  Ib  ;  with  30  Ibs 
worth  25  cts  per  Ib  ? 

Ihs.    cts.     cts. 

S      «"•">»*  '•£  =»•".*». 

50  Ibs.      1050  cts. 

Equation  of  Payments. 

A  owes  B  $1200;  of  which  $400  are  to  be  paid  in  3  months;  $500  In  4  months  ;  and  $300  in  6 
months  ;  all  bearing  interest  until  paid  ;  but  it  has  been  agreed  to  pay  all  at  once.   Now,  at  what  time 
must  this  payment  be  made  so  that  neither  party  shall  lose  any  interest  ? 
$          months. 

500     X     4     =     2000          Therefore,  —  —  =  4.16,  &c,  months.   Ans. 
800     X     6     =     1800 

1200  5000 

A  owes  B  $1000  to  be  paid  in  12  days  ;  and  $500  to  be  paid  in  3  months.    What  would  be  the  time. 
tor  paying  all  at  once  ? 

$    days. 

1000  X  12  =   12000  57000 

500  X  90  =  45000     Therefore,  -j^  =  88  days.  An*. 

1500  67000 


WEIGHTS  AND  MEASURES.  73 


Simple  Interest. 

What  is  the  simple  interest  on  $865.32  eta  for  one  year,  at  6  per  ct  per  annum  ? 
Principal.       Interest.          Principal.  Interest. 

$100       :       $6       ::       $865.32       :       $51.9192 


Sets. 


100)5191.92(51.9192  Ans.  =  51.91-j 

What  is  the  interest  on  $865.32  cts  for  1  year,  3  months,  and  10  days,  at  7  per  cent  per  annum? 

First  calculate  the  interest  for  1  year  only  ;  thus: 

Prin.  Int.  Prin.  Int. 

$100       :        $7       ::       $865.32       :       $60.5724 

7 

100)6057.24(60.5724 

Then  say,  If  1  year  or  365  days  give  $60.5724  int,  what  will  465  days  give?  or 
Days.  Int.  Days.  Int. 

365       :       $60.5724       :  :       465       :        $77.16,  &c.  Ans. 

At  5  per  ct  simple  interest,  money  doubles  itself  in  20  years;  at  6  per  ct,  in  16%  years;  and  at  7 
per  ct,  in  14^  years.  Simple  Interest  is  Single  Kule  of  Three. 

Compound  Interest. 

When  money  is  borrowed  for  more  than  a  year  at  compound  interest,  find  the  simple  interest  at  the 
end  of  the  first  year,  and  add  it  to  the  principal,  for  a  second  principal.  Find  the  simple  interest  on 
this  second  enlarged  principal  for  the  next  year,  and  add  it  to  the  enlarged  principal  for  a  third  prin- 
cipal ;  and  so  on  for  each  successive  year.  1 

At  5  per  ct  compound  interest,  money  doubles  itself  in  about  14^-  years ;  at  6  per  ct,  in  about  11.9 
years;  and  at  7  per  ct,  in  about  10%  years. 

Discount 

Is  a  deduction  of  a  part  of  the  interest,  when  money  at  interest  is  paid  before  it  is  due.  Or  it  is  a 
deduction. of  the  whole  of  the  interest  in  advance,  at  the  time  the  money  is  lent.  In  the  first  case,  if 
I  borrc  w  $100  for  1  year  at  8  per  ct,  I  must  at  the  end  of  the  year  pay  back  $108 ;  but  if  I  pay  at  the 
end  of  3  months,  I  must  add  only  $2,  or  the  interest  for  those  3  months,  paying  back  $102;  and  the 
diff  of  $6  is  the  discount.  Therefore,  to  find  the  discount  in  such  cases,  first  find  the  interest  for 
the  full  time  ;  then  that  for  the  short  time ;  and  take  the  diff. 

In  the  second  case,  if  I  borrow  $100  from  a  bank  for  one  year,  at  6  per  ct,  I  receive  but  100  — 6  r:  $94; 
but  at  the  end  of  the  year  I  must  pay  back  $100.  By  discounting  in  this  manner,  the  bank  actually 
gains  more  than  6  per  ct ;  for  it  gains  $6  for  the  use  of  $94  for  1  year.  In  the  United  States,  the  banks 
deduct  discount  for  3  days  more  than  the  time  stipulated  in  the  note;  these  are  called  "days  of 
grace." 

Commission,  or  Brokerage, 

Is  a  percentage  tor  so  much  per  each  $100)  paid  to  commission  merchants  for  selling  our  goods;  or 
to  brokers,  or  other  kinds  of  agents,  for  transacting  business  for  us.  It  is  Single  Rule  of  Three. 

Ex.  If  a  broker  makes  purchases  for  me  to  the  amount  of  $9362,  at  2  per  ct,  what  is  his  brokerage? 
Say,  as 

Purchase.     Brokerage.      Purchase.         Brokerage. 
$100        :        $2        :  :        $9362        :        $187.24 

Insurance 

Is  a  percentage  (called  a  premium)  paid  to  a  company  for  insuring  our  property  against  fire,  &c. 
The  company,  or  insurers,  (called  also  underwriters,)  deliver  to  the  person  insured,  a  paper  bearing 
their  seal,  &c,  and  called  the  Policy  of  Insurance  ;  which  contains  the  conditions  of  the  transaction. 
Insurance  is  calculated  like  Commissions,  <fcc. ;  being  merely  Single  Rule  of  Three. 

Fellowship. 

A  puts  $6000  into  a  business  in  partnership  with  B,  who  puts  in  $9000.     At  the  end  of  a  year  they 

have  made  $2400 ;  how  much  is  each  one's  share?  Here,  $6000-f-$9000:=$15000jointcapital;  then  say, 

Joint  cap.        Total  gain.  A's  cap.          A's  share. 

$15000        :        $2400        ::         $6000        :         $960 

B's  cap.  B's  share. 

And  $15000       :       $2400       :  :       $9000       :       $1440 


[GETS  AND  MEASUKES, 


UNITED  STATES  and  British  measures  of  length  and  weight,  of  the  same  denomination,  may,  for  dK 
ordinary  purposes,  be  considered  as  equal  ;  but  the  liquid  and  dry  measures  of  the  same  denomina- 
tion differ  widely  in  the  two  countries.  The  standard  measure  of  length  of  both  countries  is  theo- 
retically that  of  a  pendulum  vibrating  seconds  at  the  level  of  the  sea,  in  the  lat  of  London,  in  a 
vacuum,  with  Fahr's  thermom  at  62°.  The  length  of  such  a  pendulum  is  supposed  to  be  divided  into 
39.1393  equal  parts,  called  inches  ;  and  36  of  these  inches  were  adopted  as  the  standard  yard  of  both 
countries.  But  the  Parliamentary  standard  having  been  destroyed  by  fire,  in  1834  it  was  found  to 
be  impossible  to  restore  it  by  measurement  of  a  pendulum  ;  and  the  present  Brit  standard  yard  is,  in 
consequence,  shorter  than  that  of  the  U.  S.  by  the  latest  comparison,  about  1  part  in  40,000,  or  .03 
inch  in  100  ft;  or  1.584  ins  in  a  mile.  But  at  a  temperature  of  6'2°.25  Fah  for  the  British  standard 
and  59°.62  for  the  U.  S.  one,  the  two  are  of  the  same  length,  and  on  this  basis  the  U.  S.  govt  declares 
the  measures  of  the  two  countries  to  be  the  same ;  as  In  our  tables. 


74  WEIGHTS   AND   MEASURES. 

Troy  Weight. 

U.  a  and  British. 

24  grains 1  pennyweight,  dwt. 

20  pennyweights 1  ounce  —  480  grains. 

12  ounces 1  pound  —  240  dwts  =  5760  grains. 

Troy  weight  Is  used  for  gold  and  silver.* 

A  carat  of  the  jewellers,  for  precious  stones,  is  in  the  U.  S.  zr  3.2  grs ;  in  London.  3.17  grs  •  in 
Paris,  3.18  RTS,  divided  into  4  jewellers'  grs.  In  troy,  apothecaries',  and  avoirdupois,  the 
grain  la  the  same. 

Apothecaries9  Weight* 

U.  S.  and  British. 

20  grains 1  scruple. 

3  scruples ldranir=6.  „--. 

8  drams 1  ounce  =  24  scruples  =  480  grs. 

12  ounces 1  pound  rr  96  drams  =  288  scruples  =  5760  grains. 

In  troy  and  apoth  weights,  the  grain,  ounce,  and  pound  are  the  same. 

Avoirdupois,  or  Commercial  Weight. 

C.  S.  and  British. 

27.34375  grains ]  dram. 

16  drams / 1  ounce  =  437^  grains. 

16  ounces 1  pound  —  256  drams  ~  7000  grains. 

28  pounds 1  quarter  =  448  ounces. 

4  quarters 1  hundredweight  =  112  fts. 

20  hundredweights 1  ton  =  80  quarters  =  2240  Its. 

A  stone  —  14  pounds.    A  quintal  -  100  pounds  avoir. 

The  standard  of  the  avoir  pound*  which  is  the  one  in  common  commercial  use,  is  the 
•weight  of  27.7015  cub  ins  of  pure  distilled  water,  at  its  maximum  density  at  about  39°.2  Fahr,  in 
latitude  of  London,  at  the  level  of  the  sea;  barom  at  30  ins.    But  this  involves  an  error  of 
about  1  part  in  1362,  for  the  1  ft  of  water  —  27.68122  cub  ins. 
A  troy  Ib  r:  .82286  avoir  ft.     An  avoir  ft  =  1.21528  troy  ft,  or  apoth. 
A  troy  oz  ^  1.09714  avoir  oz.    An  avoir  oz  =  .911458  troy  oz.  or  apoth. 

Long  Measure. 

U.  8.  and  British. 
By  law,  the  U.  8.  standards  of  length,  as  well  as  of  weight,  are  made  the  same  as  the  British 

12  inches 1  foot  =  .3047973  metre. 

3  feet 1  yard  =  36  ins  =  .9143919  metre. 

5^  yards 1  rod,  pole,  or  perch  =  16^  feet  =  198  ins. 

40  rods 1  furlong  =  220  yards  =  660  feet. 

8  furlongs 1  statute,  or  land  mile  =  320  rods  =  1760  yds  =  5280  ft  =  63360  ins. 

3  miles 1  league  —  24  furlongs  =  960  rods  —  5280  yds  =  15840  ft. 

A  point  =  y\  inch.  A  line  =  6  points  —  ^  inch.  A  palm  =  3  ins.  A  hand  =  4  ins.  A 
span  =  9  ins.  A  fathom  =  6  feet.  A  cable's  length  =  120  fathoms  =  720  feet.  A  Counter's 
surveying  chain  is  66  feet,  or  4  rods  long.  It  is  divided  into  100  links  of  7.92  ins  long.  80  Gun- 
ter's  chains  —  1  mile, 

A  knot,  or  sea  mile,  is  the  length  of  1  minute  of  longitude  of  the  earth  at  the  equator,  at  the 
level  of  the  sea,  or  the  ^Ylro~77  Part  °^  tne  earth's  equatorial  circumf;  or  about  1.152664  common 
statute  or  land  miles ;  2028.69  yards  ;  or  6086.07  feet ;  consequently  one  degree  of  long  at  the  equator 
=  69.160  land  miles ;  and  a  land  mile  —  .86755  of  a  nautical  mile.  Sometimes  1  minute  of  mean  lati- 
tude is  taken  as  a  nautical  mile.  A  min  of  lat  at  the  equator  is  about  6046  ft;  and  at  the  poles 
about  6107  ;  the  mean  of  which  is  6076^  ft. 

At  the  equator  1°  of  lat  =  68.70  land  miles ;  at  lat  20°  =  68.78 ;  at  40°  = 

69.00 ;  at  60°  —  69.23  ;  at  80°  rr  69.39  ;  at  90°  -  69.41. 

*  The  U.  S.  gold  dollar  weighs  25.8  grs;  and  contains  23.22  grs  of  pure  gold. 
**  "        1O  ••  "        258    grs;     "  "       232.20  grs 

M  «        2O  M  <<        516    grg.     ,,  ,.      464.40  grs  "         " 

Perfectly  pure  gold  is  worth  Si  per  23.22  grs  =  $20.67183  per  troy  oz  —  $18.84151  per  avoir  oz. 
Standard  CW  S.  coin)  is  worth  $18.60465  per  troy  oz  =  $16.95736  per  avoir  oz.  It  consists  of  9 
parts  by  wt  of  pure  gold,  to  1  part  alloy.  Its  value  is  that  of  the  pure  gold  only  ;  the  cost  of  the  alloy 
and  of  the  coinage  being  borne  by  Gov.  A  cub  foot  of  pure  gold  weighs  about  1204  avoir  fts ; 
and  is  worth  $362963,  A  cub  inch  weighs  about  11.148  avoir  oz  ;  aud  is  worth  $210.04. 

Pure  gold  Is  called  fine,  or  24  carat  gold  ;  and  when  alloyed,  the  alloy  is  supposed  to  be  divided 
into  24  parts  by  wt,  and  according  as  10,  15,  or  20,  &c,  of  these  "parts  are  pure  gold,  the  alloy  is  said 
to  be  10,  15,  or  '20,  &c,  carat. 

Pure  silver  fluctuates  in  value;  thus  during  1878  1879,  it  ranged  between  $1.05  and  $1.18  per 
troy  oz,  or  $.957  and  $1.076  per  avoir  oz.  A  cub  inch  weighs  about  5.528  troy,  or  6.065  avoir  ounces. 
The  U.  8.  silver  dollar  weighs  412.5  grs  troy;  but  its  subdivisions  weigh  at  the  rate  of  about  8 
pe:  ct  less.  All  consist  of  9  parts  silver  to  1  part  alloy. 

The  average  fineness  of  California  native  gold,  by  some  thousands  of  assays  at  the  U. 
S.  mint  in  Philada,  is  88.5  parts  gold,  11.5  silver.  Some  from  Georgia,  99  per  ct  gold. 


WEIGHTS  AND   MEASURES. 


75 


[Lengths  of  a  Degree  of  Longitude  in  different  Latitudes,  and 
at  the  level  of  the  Sea.  f 

These  lengths  are  in  common  land  or  statute  miles,  of  5280  ft.    Since  the  figure  of  the  earth  has 
never  been  precisely  ascertained,  these  are  but  close  approximations. 


Degof 
Lat. 

Miles. 

Degof 
Lat. 

Miles. 

Degof 
Lat. 

Miles. 

Degof 
Lat. 

Miles. 

Degof 
Lat. 

Miles. 

Degof 
Lat. 

Miles. 

0 
2 
4 
6 

8 
10 
12 

69.16 
69.12 
6S.99 
68.78 
68.49 
68.12 
67.66 

14 
16 
18 
20 
22 
24 
26 

67.12 
66.50 
65.80 
65.02 
64.15 
63,21 
62.20 

28 
30 
32 
34 
36 
38 
40 

61.11 
59.94 
68.70 
57.39 
56.01 
54.56 
53.05 

42 
44 
46 
48 
50 
52 
54 

51.47 
49.83 
48.12 
46.36 
44.54 
42.67 
40.74 

56 
58 
60 
62 
64 
66 
68 

38.76 
36.74 
34.67 
32.55 
30.40 
28.21 
25.98 

70 
72 
74 
76 

78 
80 
82 

23.72 
21.43 
19.12 
16.78 
14.42 
12.05 
9.66 

Intermediate  ones  may  be  found  correctly  by  simple  proportion. 

Inches  reduced  to  Decimals  of  a  Foot. 


Ins. 

Foot. 

Ins. 

Foot. 

Ins. 

Foot. 

Ins. 

Foot. 

Inn. 

Foot. 

Ins. 

Foot. 

0 

.0000 

2 

.1667 

4 

.3333 

e 

.5000 

8 

.6667 

10 

.8333 

1-32 

.0026 

.1693 

.3359 

.5026 

.6693 

.8359 

1-16 

.0052 

.1719 

.3385 

.5052 

.6719 

.8385 

8-32 

.0078 

.1745 

.3411 

.5078 

.6745 

.8411 

U 

.0104 

X 

.1771 

X 

.3438 

X 

.5104 

X 

.6771 

X 

.8438 

5-32 

.0130 

.1797 

.3464 

.5130 

.6797 

.8464 

3-16 

.0156 

.1823 

.3490 

.5156 

.6823 

.8490 

7-32 

.0182 

.1849 

.3516 

.5182 

.6849 

.8516 

K 

.0208 

X 

.1875 

X 

.3542 

X 

.5208 

X 

.6875 

X 

.8542 

9-32 

.0234 

.1901 

.3568 

.5234 

.6901 

.8568 

5-16 

.0260 

.1927 

.3594 

.5260 

.6927 

.8594 

11-32 

.0286 

.1953 

.3620 

.5286 

.6953 

.8620 

13*2 

.0313 
.0339 

X 

.1979 
.2005 

X 

.3646 
.3672 

X 

.5313 
.5339 

X 

.6979 
.7005 

X 

!8672 

7-16 

.0365 

.2031 

.3698 

.5365 

.7031 

.8698 

15-32 

.0391 

.2057 

.3724 

.5391 

.7057 

.8724 

X 

.0417 

X 

.2083 

X 

.3750 

X 

.5417 

X 

.7083 

X 

.8750 

17-32 

.0443 

.2109 

.3776 

.5443 

.7109 

.8776 

9-16 

.0469 

.2135 

.3802 

.5469 

.7135 

.8802 

19-32 

.0495 

.2161 

.3828 

.5495 

.7161 

.8828 

% 

.0521 

X 

.2188 

X 

.3854 

X 

.5521 

X 

.7188 

X 

.8854 

21-32 

.0547 

.2214 

.3880 

.5547 

.7214 

.8880 

11-16 

.0573 

.2240 

.8906 

.5573 

.7240 

.8906 

23-32 

.0599 

.2266 

.3932 

.5599 

.7266 

.8932 

% 

.0625 

X 

.2292 

% 

.3958 

X 

.5625 

X 

.7292 

X 

.8958 

25-32 

.0651 

.2318 

.3984 

.5651 

.7318 

.8984 

13-16 

.0677 

.2344 

.4010 

.5677 

.7344 

.9010 

27-32 

.0703 

.2370 

.4036 

.5703 

.7370 

.9036 

% 

.0729 

X 

.2396 

X 

.4063 

X 

.5729 

% 

.7396 

% 

.9063 

29-32 

.0755 

.2422 

.4089 

.5755 

.7422 

.9089 

15-16 

.0781 

.2448 

.4115 

.5781 

.7448 

.9115 

31-32 

.0807 

.2474 

.4141 

.5807 

.7474 

.9141 

1 

.0833 

3 

.2300 

5 

.4167 

7 

.5833 

9 

.7500 

11 

.9167 

.0859 

.2526 

.4193 

.5859 

.7526 

.9193 

.0885 

.2552 

.4219 

.5885 

.7552 

.9219 

.0911 

.2578 

.4245 

.5911 

.7578 

.9245 

X 

.0938 

X 

.2604 

X 

.4271 

X 

.5938 

H 

.7604 

x 

.9271 

.0964 

.2630 

.4297 

.5964 

.7630 

.9297 

.0990 

.2656 

.4323 

.5990 

.7656 

.9323 

.1016 

.2682 

.4349 

.6016 

.7682 

.9349 

X 

.1042 

X 

.2708 

X 

.4375 

X 

.6042 

X 

.7708 

X 

.9375 

.1068 

.2734 

.4401 

.6068 

.7734 

.9401 

.1094 

.2760 

.4427 

.6094 

.7760 

.9427 

.1120 

.2786 

.4453 

.6120 

.7786 

.9453 

X 

.1146 

X 

.2813 

% 

.4479 

X 

.6146 

X 

.7813 

X 

.9479 

.1172 

.2839 

.4505 

.6172 

.7839 

.9505 

.1198 

.2865 

.4531 

.6198 

.7865 

.9531 

.1224 

.2891 

.4557 

.6224 

.7891 

.9557 

X 

.1250 

X 

.2917 

X 

.4583 

X 

.6250 

X 

.7917 

X 

.9583 

.1276 

.2943 

.4609 

.6276 

.7943 

.9609 

.1302 

.2969 

.4635 

.6302 

.7969 

.9635 

.1328 

.2995 

.4661 

.6328 

.7995 

.9661 

X 

.1354 

X 

.3021 

% 

.4688 

X 

.6354 

X 

.8021 

X 

.9688 

.1380 

.3047 

.4714 

.6380 

.8047 

.9714 

.1406 

.3073 

.4740 

.6406 

.8073 

.9740 

.1432 

.3099 

.4766 

.6432 

.8099 

.9766 

% 

.1458 

% 

.3125 

X 

.4792 

X 

.6458 

X 

.8125 

X 

.9792 

.1484 

.3151 

.4818 

.6484 

.8151 

.9818 

.1510 

.3177 

.4844 

.6510 

.8177 

.9844 

.1536 

.3203 

.4870 

.6536 

.8203 

.9870 

X 

.1563 

% 

.3229 

X 

.4896 

X 

.6563 

X 

.8229 

X 

.9896 

.1589 

.3255 

.4922 

.6589 

.8255 

.9922 

.1615 

.3281 

.4948 

.6615 

.8281 

.9948 

.1641 

.3307 

.4974 

.6641 

.8307 

.9974 

76 


WEIGHTS  AND   MEASURES. 


Square,  or  Land  Measure. 

C.  8.  and  British. 

144  square  inches 1  sq  foot.     100  sq  ft  ~  1  square. 

9  sq  feet 1  sq  yard  —  129(j  sq  ins. 

30%  sq  yards 1  sq  rod  —  27214  sq  feet. 

40  sq  rods 1  rood  —  1210  sq  yds  —  10890  sq  feet. 

4  roods 1  acre  —  160  rods- 4840  sq  yds- 43560  sq  feet. 

A  section  of  land  is  1  mile  sq,  or  27878400  sq  ft ;  or  3097600  sq  yds ;  or  640  acres.  An  acre 
contains  10  sq  Gunter's  chains.  A  sq  acre  i*  208.710  feet;  a  sq  half  acre,  147.581  ft;  and  a  sq 
quarter  acre,  104.355  ft  on  each  side.  A  circular  acre  is  235.504  feet:  a  circular  half  acre  = 
166.527  ft;  and  a  circular  quarter  acre  —  117.752  ft  diam.  A  circular  Inch  is  a  circle  of  1  inch 
diam;  a  sq  ft  =  183.346  cir  ins.  Also  1  sq  Inch  =  1.27324  cir  ms ;  and  1  cir  inch  =  .7854  of  a  sq 


inch. 


Cubic,  or  Solid  Measure. 

U.  8.  and  British. 


1728  cubic  inches 1  cubic,  or  solid  foot. 

27  cubic  feet 1  cubic,  or  solid  yard. 

A  cord  of  wood  =  128  cub  ft ;  being  4  ft  X  4  ft  X  8  ft.  A  perch  of  masonry  actually  con- 
tains 24%  cub  ft;  being  163^  ft  X  1%  ft  X  1  ft.  It  is  generally  taken  at  25  cub  ft ;  but  by  some  at  22, 
&c;  and  there  is  every  probability  that  a  payer  will  be  cheated  unless  the  number  of  cubic  ft  be  dis- 
tinctly agreed  upon  in  his  contract,  it  is  gradually  falling  into  disuse  among  engineers;  and  the  cub 
yd  is  very  properly  taking  its  place.  To  reduce  cub  yds  to  perches  of  25  cub  ft,  mult  by  1.080; 
and  to  reduce  perches  to  cub  yds,  mult  bv  .926.  The  Brit  rod  of  brickwork,  of  house-builders,  is 
16>£  feet  square,  by  14  inches"(l^  English  bricks)  thick  =  272^  sq  ft  of  14  inch  wall.  It  is  conven- 
tionally taken  at  272  sq  ft;  which  gives  317 X  cub  ft.  In  Brit  engineering  works  the  rod  is  306  cub 
ft,  or  HJi  cub  yds.  The  Montreal,  (Canada,)  toise  =  261^  cub  ft;  or  9.6852  cub  yds,  or  10.46 
perches  of  25  cub  ft.  The  Canadian  chaldron  =  58.64  cub  ft.  A  ton  (2240  fts>  of  Pennsylvania 
anthracite,  when  broken  for  domestic  use,  occupies  from  41  to  43  cub  ft  of  space ;  the  mean  of  which 
is  equal  to  1.556  cub  yds:  or  a  cube  of  3.476  ft  on  each  edge.  Bituminous  coal  44  to .48  cub  ft;  mean 
equal  to  1.704  cub  yd ;  or  a  cube  of  3.583  ft  on  each  edge.  Coke  80  cub  ft. 

A  cubic  foot  is  equal  to 


1728  cub  ins,  or  3300.23  spherical  ins. 

.037037  cub  yard,  or  1.90985  spherical  ft. 

.002832  rayriolitre,  or  decastere. 

.028316  kilolitre,  or  cubic  metre,  or  stere. 

.283161  hectolitre,  or  decistere. 

2.83161  decalitres,  or  centisteres. 

28.3161  litres,  or  cub  decimetres. 

283.161  decilitres. 

2831.61  centilitres. 

28316.1  millilitres,  or  cub  centimetres. 

.803564  U.  S.  struk  bushel  of  2150.42  cub  ins, 

1.24445  cub  ft. 
.779013  Brit  bushel  of  2218.191  cub  ins,  or  1.28368 

cub  ft. 
3.21426  U.  S.  pecks. 

A  cubic  inch  is  equal  to 

16.38663  millilitres;  or  1.638663  centilitres;  or  .1638663  decilitre;  or  .01638663  HUe;  or  to  .0005787 
cub  ft;  or  to  .138528  U.  S.  gill ;  or  1.90985  spherical  ins. 

A  cubic  yard  is  equal  to 


3.11605  Brit  pecks. 

7.48052  U.  S.  liquid  galls  of  231  cub  ins. 

6.42851  U.  S.  dry  galls. 

6.23210  Brit  galls  of  277.274  cub  ins. 

29.92208  U.  S.  liquid  quarts. 

25.71405  U.  S.  dry  quarts. 

24.92H42  Brit  quarts. 

59.84416  U.  S.  liquid  pints. 

51.42809  U.  S.  dry  pints. 

49.85684  Brit  pints. 

239.37662  U.  S.  gills. 

199.42737  Brit  gills. 

.26667  flour  barrel  of  3  struck  bushels. 

.23748  U.  S.  liquid  barrel  of  31}£  galls. 


27  cub  feet,  or  to  201.974  U.  S.  galls. 

46656  cub  ins. 

.0764534  myriolitre. 

.764534  kilolitre,  or  cub  metre. 

7.64534  hecatolitres. 


I      76.4534  decalitres. 

764.534  litres,  or  cub  decimetres. 
7645.34  decilitres. 
21.69623  U.  S.  bushels  (struck). 
21.03336  Brit  bushels. 

Liquid  Measure. 

U.  8.  only. 

The  basis  of  this  measure  in  the  U.  S.  is  the  old  Brit  wine  gallon  of  231  cub  ins ;  or  8.33888  Tbs 
avoir  of  pure  water,  at  its  max  density  of  about  39°. 2  Pahr  ;  the  barom  at  30  ins.  A  cylinder  7  ins 
diam,  aud  6  ins  high,  contains  230.904"  cub  ins,  or  almost  precisely  a  gallon  ;  as  does  also  a  cube  of 
6.1358  ins  on  an  edge.  Also  a  gallon  —  .13368  of  a  cub  ft ;  and  a  cub  ft  contains  7.48052  galls  ;  nearly 
7)4  galls.  This  basis  however  involves  an  error  of  about  1  part  in  1362,  for  the  water  actu- 
ally weighs  8.345008  Ibs. 

cub  ins. 

4  gills 1  pint     =28.875. 

2  pints 1  quart  —  57.750  —  8  gills. 

4  quarts 1  gallon  —  231 .  —  8  pints  =  32  gills. 

In  the  U.  S.  and  Great  Brit.  1  barrel  of  wine  r 


63  gallons 1  hogshead. 

2  hogsheads 1  pipe,  or  butt. 

2  pipes 1  tun. 

brandy  =  31  %  galls  ;  in  Pennsylvania,  a  half 


barrel,  16  galls;  a  double  barrel,  64  gulls;  a  puncheon,  84  galls;  a  tierce,  42  galls.  A  liquid 
measure  barrel  of  31^  galls  contains  4.211  cub  ft  —  a  cube  of  1.615  ft  on  an  edge ;  or  3.384  U.  S.  struck 
bushels.  A  gill  =  7.21875  cub  ias.  The  following  cylinders  contain  some  of  these  measures 
very  approximately. 


Diam.  Height. 

cub  ins.  Ins.  Ins, 

Gill  (7.21875)  ......     1%     ............     3 


............ 

Pint  ..............     3J*     ............     3 

Quart  .............    3%     ............    6 


Diam.  Height. 

Ins.  Ins. 

Gallon 7     6 

2  gallons 7     12 

Sgallons 14     12 

lOgallons 14    15 


WEIGHTS   AND   MEASURES. 


77 


cone  is  not  to  be  less  than  6 
struck  ones:  or  to  1.55556  cub  ft. 


To  reduce  U.  &.  liquid  measures  to  Brit  ones  of  the  same  denomina- 

tion, divide  by  1.20032;  or  near  enough  for  common  use,  by  1.2  ;  or  to  reduce  Brit  to  U.  S.  multiply 
by  1.2.  ^ 

Dry  Measure. 

U.  8.  only. 
The  basis  of  this  is  the  old  British  Winchester  struck  bushel  of  2150.42  cub 

Ins  ;  or  77.627413  pounds  avoir  of  pure  water  at  its  max  density.     Its  dimensions  by  law  are  18J^  ins 
inner  diam  ;  19.^  ins  outer  diam  ;  and  8  ins  deep  ;  and  when  heaped,  the 
ins  high;  which  makes  a  heaped  bushel  equal  to 

Edge  of  a  cube  of 
equal  capacity. 
2  pints     1  quart,  -  67.2006  cub  ins  -  1.16365  liquid  qt  .......................     4.066  ins. 

4  quarts   1  gallon,  =  8  pints,  =  268.8025  cub  ins,  =  1.16365  liq  gal  .............     6.454    "" 

2  gallons  1  peck,     =  16  pints,  =  8  quarts,  -  537.6050  cub  ins  .................     8.131   " 

4  pecks     1  struck  bushel,  —  64  pints,  =  32  quarts,  rrb  gals,  -2150.4200  cub  ins.  12.908   " 

A  struck  bushel  =  1.24445  cub  ft.    A  cub  ft  =  .80356  of  a  struck  bushel. 
The  dry  flour  barrel  =  3.75  cub  ft;  =  3  struck  bushels.    The  dry  barrel  is 

not,  however,  a  legalized  measure;  and  no  great  attention  is  given  to  its  capacity;  consequently, 
barrels  vary  considerably.  A  barrel  of  flour  contains  by  law,  196  fids.  In  ordering  by  the  barrel,  the 
amount  of  its  contents  should  be  specified  in  pounds  or  galls. 

To  reduce  U.  S.  dry  measures  to  Brit  imp  ones  of  the  same  name,  div 

by  1.031516;  and  to  reduce  Brit  ones  to  U.  S.  mult  by  1.031516  ;  or  for  common  purposes  use  1.032. 

British  Imperial  Measure,  both  liquid  and  dry. 

This  system  is  established  throughout  Great  Britain,  to  the  exclusion  of  the  old  ones.    Its  basis  is 
the  imperial  gallon  of  277.274  cub  ins,  or  10  ft>s  avoir  of  pure  water  at  the  temp  of  62°  Fahr,  when 

the  barom  is  at  30  ins.    This  basis  involves  an  error  of  about  1  part  in 

1836,  for  10  as  of  the  water  -  only  277.123  cub  ins. 


Avoir  Ihs. 
of  water. 

Cub.  ins. 

Cub.  ft. 

Edge  of  a  cube  of 
equal  capacity. 
Inches. 

4  gills      1  pint  .... 

1.25 

34.6592 

3  2605 

2  pints     1  quart  

2  50 

69  3185 

4  1079 

2  quarts  1  pottle  

5. 

138  637 

5  1756 

10. 

277  274 

6  5208 

2  gallons  1  peck  . 

20      1 

554  548 

8  2157 

4  p°cks    1  bushel  

80       I     Dry 

2218  192 

1  2837 

13  0417 

320       f  meas 

8872  768 

5  1347 

2  coombs  1  quarter......  

640.     J 

17745.536 

10.2694 

The  imp  gall  =  .  16046  cub  ft;  and  1  cub  ft  =  6. 23210  galls.  The  imp  gal  =  1.20032,  or  very  nearly  \\ 
U.  S.  liquid  galls. 

A  cylinder  1  foot  in  diameter,  and  1  foot  high,  contains 

.02909  cub  yard.  47.0C16  U.  S.  liquid  pints. 

.7854  cub  foot.  188.0064  U.  S.  liquid  gills. 

1357. 1712  cub  inches.  4.8947  Brit  imp  gallons. 

.63112  U.  S.  dry  bushels.  19.5788  Brit  imp  quarts. 

2.5245  U.  S.  dry  pecks.  39.1575  Brit  imp  pints. 

20.1958  U.  S.  dry  quarts.  156.6302  Brit  imp  gills. 

40.3916  U.  S.  dry  pints.  222.395  Decilitres.  • 

5.8752  U.  S.  liquid  gallons.  22.2395  litres. 

23.5008  U.  S.  liquid  quarts.  2.22395  decalitres. 

.222395  hectolitre. 


A  cylinder  1  inch  in  diameter,  and  1  foot  high,  contains 

.2719  Brit  imp  pint. 
1.0877  Brit  imp  gill. 
15.4441  centilitres. 
1.54441  decilitres. 

.154441  litres. 


.005454  cub  foot. 
9.4248  cub  inches. 

.2806  U.  S.  dry  pint. 

.3264  liquid  pint. 
1.3056  U.  S.  gill. 

A  sphere  1  foot  in  diameter,  contains 

.01939  cub  yard.  31 .3344  U.  S.  liquid  pints. 

.5236  cub  foot.  125.3376  U.  S.  liquid  gills. 

904.781  cub  inches.  3.2631  Brit  imp  gallons. 

.42075  U.  S.  bushel.  13.6525  Brit  imp  quarts. 

1.6830  U.  S.  pecks.  26.1050  Brit  imp  pints. 

13.4639  U.  S.  dry  quarts.  104.4201  Brit  imp  gills. 

26.9278  U.  S.  dry  pints.  14.8263  litres. 

8.9168  U.  S.  liquid  gallons.  1.48263  decalitres. 

15.6672  U.  S.  liquid  quarts.  .148263  hectolitres. 

A  sphere  1  inch  in  diameter,  contains 

.000303  cub  foot.  .06043  Brit  gill. 

.5236  cub  inch.  8.580  millilitre. 

.07253  U.  S.  gill.  .8580  centilitre. 

.08580  deciliu*. 


78 


WEIGHTS   AND   MEASURES. 


French  Measures  of  Length. 

By  U.  8.  and  British  Standard. 


Ins. 

Ft. 

Yds. 

Miles. 

Millimetre*  

.039370 

.003281 

Centimetre"!"           .  ...        . 

39370428 

032809 

Decimetre  

3.9370428 

3280869 

.1093623 

MetreJ 

39  370428 

3  280809 

1  093623 

Decametre    1 

393  70428 

32  80869 

10  93623 

Hectometre                    ••         .              I 

Road 

328  0869 

109  3623 

0621375 

measures 

3280  869 

1093.623 

.6213750 

Myriametre  j 

32808.69 

10936.23 

6.213750 

*  Nearly  the  ^V  part  of  an  inch.  t  Full  %  inch. 

I  Very  nearly  3  ft,  3%  ins,  which  is  too  long  by  only  1  part  in  8616. 

French  Square  Measure. 

By  U.  8.  and  British  Standard. 


Sq.  Ins. 

Sq.  Feet. 

Sq.  Yds. 

Acres. 

Sq  Millimetre  

.001550 

.00001076 

.0000012 

Sq  Centimetre 

155003 

00107641 

0001196 

So  Decimetre      .  

15  5003 

10764101 

.0119601 

Sq  Metre,  or  Centiare  
Sq  Decametre  or  Are       .. 

1550.03 
155003 

10.764101 
10764101 

1.19601 
1196011 

.000247 
.024711 

10764.101 

1196.011 

.247110 

107641.01 

11960.11 

2.47110 

Sq  Kilometre      

3861090  sq  miles. 

10764101 

1196011. 

247.110 

Sq  Myriametre  

38.61090           " 

24711.0 

French  Cubic,  or  Solid  Measure. 

According  to  U.  8.  Standard. 

Only  those  marked  "  Brit"  are  British. 


Millilitre.orcub 
Centimetre.... 

Cub  Ins. 

.0610254 

(Liquid.  .0084537  gill. 
\       "        .0070428  Brit  gill. 
(Dry.       .0018162  dry  pint. 

Centilitre  

.610254 

(  Liquid.  .084537  gill. 
\       "        .070428  Brit  gill. 
(Dry.       .018162  dry  pint. 

Decilitre  

6.10254 

(Liquid.  .84537  gill  =  .21134  pint. 
\       "        .70428  Brit  gill  =  .17607  Brit  pint. 

(Dry.       .18162  dry  pint. 

Litre,  or    cubic 
Decimetre  

61.0254 

(Liquid.  1.05671  quart  =  2.1134  pints. 
4       "        .88036  Brit  quart  =  .1.7607  Brit  pints. 
(Dry.       .11351  peck  =  .9081  dry  qt  =  1.8162  dry  pt. 

Decalitre,        or 
Centistere  

610.254 
Cub  Ft. 

.353156 

(Liquid.  2.64179  U.  S.  liquid  gal. 
-j       "        2.20090  Brit  gal. 
(Dry.       .283783  bush  =  1.1351  peck  =  9.081  dry  qts. 

Hectolitre,      or 
Decistere  

3.53156 

(Liquid.  26.4179  U.  S.  liquid  gal. 
\       "        22.0090  Brit  gal. 
(Dry.       2.83783  bush. 

Kilolitre,        or 
Cubic    Metre, 
or  Stere  

35.3156 

(  Liquid.  264.179  U.  S.  liquid  gal.) 
\       «        220.090  Brit  gal.               V  Cub  yds,  1.3080. 
(Dry.       28.3783  bush.                    j 

Myriolitre,      or 
Decastere  

353.156 

{  tit*'  S£iiliquM  gia-}cu"  1*.  "-"BO. 

WEIGHTS  AND  MEASURES. 


79 


The  French  Metre.* 

The  French  metre  was  intended  to  be  the  one  ten-millionth  part  of  the  dist  10m  either  pole  of  the 
earth  to  the  equator :  but  after  it  had  been  introduced  into  use,  errors  were  discovered  in  the  calcu- 
lations employed  for  ascertaining  that  dist ;  so  that  the  French  metre,  like  the  Brit  standard  yard, 
is  not  what  it  was  intended  to  be. 

The  U.  S.  Govt  adopts  for  its  length  1.093623  yds  =  3.280869  ft  =  39.370428 
ins  U.  S.  or  British  measure.  But  in  ordinary  business  transactions 

39.37  ins  are  a  legal  metre.    At  3  ft  3%  ins,  the  length  is  but  1  part  in  8616  too  great. 

French  Weights,  reduced  to  common  Commercial  or  Avoir 
Weight,  of  1  pound  =  16  ounces,  or  70OO  grains. 


Milligramme  

Grains. 
015432 

.15432 

Decigramme  

15432 

Gramme                •• 

15432 

By  law  a  5-cent  nickel  —  5  grammes  

Pounds  av 

.022046 

Hectogramme  

22046 

2.2046 

Myriogramme  

22046 

Quintal         

22046 

22046 

The  gramme  is  the  basis  of  French  weights ;  and  is  the  weight  of  a  cub  centimetre  of  distilled 
water  at  its  max  density,  at  sea  level,  in  lat  of  Paris  ;  barom  29.922  ins. 

French  Measures  of  the  "  Systeine  Usuel." 

This  system  was  in  use  from  about  1812  to  1840,  when  it  was  forbidden  by  law  to  use  even  its  names. 
This  was  done  in  order  to  expedite  the  general  use  of  the  tables  which  we  have  before  given.  But  as 
the  Systeme  Usuel  appears  in  books  published  during  the  above  interval,  we  add  a  table  of  some  of  its 
values, 

Measures  of  Length. 


Ligne  usuel 
x*ouce  usuel 
Pied  usuel, 
Aune  usuel, 
Toise  usuel, 

Yards. 

Feet. 

Inches. 

.09113 
1.09362 
13.12344 
47.245 

78.74172 

or  inch,  =  1 
ar  foot,  =  12 

2  Hgnes 

.09113 
1.09362 
3.93708 
6.56181 

™ 

.36454 
1.31236 

2.18727 

—  6  pieds 

Weights,  Usuel. 

Cubic,  or  Solid,  TJsuel. 

Grain  usuel 
Gros  usuel.. 

.8375  grains. 
60.297         •• 
1.10258  avoir  oz. 
.55129  avoir  Ib. 
1.10258  avoir  Ib. 

Litron  usuel,  or  1  litre 
Boisseau  usuel  

=  1.7608  British  pint. 
2.7512  British  gala. 

Once  usuel  . 

Marc  usuel  . 

Livre  usuel, 
or  pound, 

f 

5 

Before  1812,  or  before  the  "Systeme  usuel,"  the  Old  System,  "  Systeme  Ancien,"  was  in  use. 

French  Measures  of  the  "Systeme  Ancien." 


Lineal. 

Square. 

Cubic. 

Point  ancien     0148  ins 

Sq  ins 

Sq  ft 

Sq.  yds 

C   ins 

C.  ft. 

C.yda. 

Ligne  ancien    0888  ins 

00789 

.0007 

Pouce  ancien,  1.06577  ins  =  .0888  ft  
Pied  ancien   12  7892  ins  ~  1  06577  ft 

1.1359 

1  1359 

1.2106 

1  2106 

Aune  ancien,  46.8939  ins  =  3.  90182  ft  =  1.30261  yds 
Toise  anci°n  ~  6  3946  ft~  2  1315  yds 

40  8908 

4.5434 

261  482 

9  6845 

League  —  2282  toises  ~  2  7637  miles 

There  is,  however,  much  confusion  about  these  old  measures.  Different  measures  had  the  same 
Dame  in  different  provinces. 

•5f  If  the  efforts  now  being  made  to  introduce  the  metre  into  general  use  in  the  United  States 
should  succeed,  they  will  be  a  source  of  extreme  embarrassment  to  many  millions  of  persons  for 
many  years. 


80 


WEIGHTS  AND    MEASURES. 


Russian. 
Verst  =  .6629  U.  S.  or  British  mile.    Pood  =  36.114  ft>s  avoir. 

Spanish. 

The  castellano  of  Spain  and  New  Granada,  for  weighing  gold,  is  variously  estimated,  from  71.07 
to  71.04  grains.  Ai  71.055  grs,  (the  mean  between  the  two,)  an  avoir,  or  common  commercial  ounce 
contains  6.1572  castel ;  and  a  Ib  avoir  contains  98.515.  Also  a  troy  ounce  =  6.7553  castel :  and  a  troy 
ft  -  81.01.4  castel.  Three  U.  S.  gold-dollars  weigh  about  1.1  castel. 

The  Spanish  mark,  or  marco,  for  precious  metals,  in  South  America,  may  be  taken  in  practice, 
as  .5065  of  a  ft  avoir.  In  Spain,  .507(5  ft.  In  other  parts  of  Europe,  it  has  a  great  number  of  val- 
ues ;  most  of  them,  however,  being  between  .5  and  .54  of  a  pound  avoir.  The  .5065  of  a  ft  =  3545>£ 
grs  :  and  .5076  ft  —  3553.2  grs.  1  Marco  r:  50  castellanos  —  400  tomine  =  4800  Spanish  gold-gr*. 

The  arroba  has  various  values  in  different  parts  of  Spain.  That  of  Castile,  or  Madrid,  is  25.4025 
fts  avoir ;  the  tonelnda  of  Castile  =  2032.2  fts  avoir ;  the  quintal  =  101.61  fts  avoir ;  the  libra 
rr  1.0161  fts  avoir;  the  cantara  of  wine,  &c,  of  Castile -4.2ftt  U.  S.  galls;  that  of  Havana=r  4.1  galls. 

The  vara  of  Castile  :=  32.8748  ins,  or  almost  precisely  32%  ins  ;  or  2  ft  8%  ins.  The  fancgadu 
of  land  since  180,1  =  1.5871  acres  —  69134.08  sq  ft.  The  fanega  of  corn,  kc.  ~  1.59914  U.  S.  struck 
bushels.  In  California,  the  vara  by  law  =  33.372  U.  S.  ms ;  and  the  legua  =  5000  varas :  or 
2.6335  U.  S.  miles. 

Civil,  or  Common  Clock  Time. 

60  thirds,  marked  '"  1  second,  marked  ". 

60  seconds  1  minute  '. 

60  minutes  1  hour,  =  3600  sec. 

24  hours  1  civil  day,  —  1440  min.  —  86400  sec. 

7  days  1  week,  =  168  hours  rr  10080  min. 

4  weeks  1  civil  month,  =  28  days  —  672  hours. 

13  civil  months,  (or  52  weeks,)  1  day,  5  hours,  48  min,  49  y^  sec ;  or  365  days,  5  hours,  48  min,  ^y^- 
Bee,  —  1  civil  year.  A  solar  day  is  the  time  between  two  successive  solar  noons,  or  transits  of  the 
BUU  over  the  meridian  of  a  place.  These  intervals  are  not  of  equal  lengths  all  the  year  round.  The 
average  length  of  all  the  solar  days  is  called  the  mean  solar  day;  and  is  the  same  as  the  common 
civil  day  of  24  hours  of  clock  time.  Civil  noon  is  at  12  o'clock  ;  but  solar,  or  apparent  noon,  may  be 
about  14}^  min  before ;  or  16>£  min  after  12  of  correct  clock  time.  A  sidereal  day  is  the  interval 
between  two  passages  of  the  same  star  past  the  range  of  two  fixed  objects;  ami  is  the  precise  time 
reqd  for  one  complete  rev  of  the  earth  on  its  axis^  The  sidereal  day  never  varies  :  but  is  always  equal 
to  23  hours,  56  min,  4.09  sec;*  so  that  a  star  will  on  any  night  appear  to  set,  or  to  pass  the  range  of 
any  two  fixed  objects,  3  min,  55.91  sec  earlier  by  the  clock,  than  it  did  on  the  night  before,!  so  that 
the  number  of  sidereal  days  in  a  civil  year  is  1  greater  than  that  of  the  civil  days. 

An  astronomical  day  begins  at  noon,  and  its  hours  are  counted  from  0  to  24.  In  comparing  it 
•with  the  civil  day,  the  last  is  supposed  to  begin  at  the  midnight  before  the  noon  at  which  the  first  began. 

*  This  gives  a  means  of  regulating1  a  watch  with  much  accuracy  and 

by  a  very  simple  process.  The  writer,  after  having  regulated  his  chronometer  watch  for  a  year  by 
this  method  only,  differed  but  a  few  seconds  from  the  actual  time  as  deduced  from  careful  solar  obser- 
vations. Even  a  person  not  accustomed  to  ranging  objects  very  accurately,  need  scarcely  err  a  min- 
ute in  a  period  of  any  number  of  years.  It  having  occurred  to  him  that  the  motion  of  a  star  in  a 
second  or  two  might  be  visible  to  the  naked  eye.  he  stuck  a  pin  horizontally  into  a  window-jamb  ;  and 
placing  his  eye  close  to  it,  sighted  along  one  side  of  it,  at  a  large  star  setting  behind  the  top  of  a  roof 
about  100  feet  distant,  and  found  that  his  conjecture  was  correct.  Those  stars  which  are  farthest 
from  the  poles  appear  to  move  the  fastest,  and  are  therefore  the  best.  Those  less  than  of  the  second  mag- 
nitude are  not  satisfactory.  If  the  first  observations  of  a  given  star  be  made  as  late  as  midnight,  that 
same  star  will  answer  for  about  three  months,  until  at  last  it  will  begin  to  pass  the  range  in  daylight. 
Before  this  happens,  the  observer  must  transfer  the  time  to  another  star  which  sets  later ;  if  near 
midnight,  the  better,  as  it  will  serve  for  a  longer  time  A  window  looking  west  is  the  best.  The 
longer  the  range,  the  greater  will  be  the  apparent  motion  of  the  star;  and.  consequently,  the  obser- 
vations will  be  more  correct.  If  such  a  range  can  be  secured  as  will  strike  the  heavens  at  an  angle 
of  at  least  40°  above  the  horizon,  the  error  from  refraction  will  not  appreciably  affect  an  observation  ; 
at  a  much  less  angle  it  may  do  so  to  the  extent  of  three  or  four  seconds.  A  caudle  must  be  so  placed 
as  to  render  the  pin  and  the  watch  visible  at  the  same  time.  A  little  practice  will  render  the  process 
very  easy,  and  supersede  the  necessity  for  more  remarks  on  the  subject.  Of  course,  a  memorandum 
must  be  made  and  preserved  of.  the  date,  hour,  minute,  and  (approximately)  second,  at  which  the 
first  passage  of  the  star  took  place.  Subsequent  passages  will  occur  earlier,  as  shown  in  the  follow- 
ing table.  The  watch  must  be  previously  known  to  be  right,  when  taking  the  first  observation,  if  we 
require  afterward  to  keep  the  correct  time.  Any  person  who  will  take  the  trouble  thus  to  observe, 
and  note  down  throughout  a  year,  about  half  a  dozen  stars  following  each  other  at  tolerably  equal 
intervals  of  time,  will  on  almost  any  clear  night  afterward  be  able,  after  a  short  calculation,  to  ascer- 
tain the  correct  clock  time.  The  writer  observed  the  passages  of  two  or  three  stars  behind  different 
ranges,  on  the  same  nights,  in  order  to  obtain  a  mean  of  several  observations ;  his  object  being  to 
Ascertain  how  pocket  chronometers  of  the  best  makers  would  keep  time  under  the  vicissitudes  of  tem- 
perature, railroad  travelling.  Ac,  &c,  to  which  they  are  ordinarily  exposed.  He  used  two  of  the  best 
for  this  purpose,  and  the  result  was  that  their  changes  of  rate  were  at  times  as  great  as  from  three  to 
eight  seconds  per  day.  For  ordinary  purposes,  therefore,  they  are  of  but  little,  if  any,  more  service 
than  a  good  common  watch,  of  one-fourth  the  cost. 
T  More  accurately  3  mitt,  55.90944  sec. 


WEIGHTS   AND   MEASURES. 


81 


TABLE  showing?  bow  much  earlier  a  star  passes  a  given 
range,  on  each  succeeding-  night.  —  (Original.) 


Nights. 
1 

Min.              Sec. 
3               55.91 

Nights. 

H.       Min.         Sec. 
43            15.01 

Nights. 
21 

H.        Mia.          Sec. 
1            22            34.11 

2 

7               51.82 

12 

47            10.92 

22 

1            26            30.02 

3 

11                47.73 

13 

51              6.83 

23 

1            30            25.93 

4 

15                43.64 

14 

55              2.74 

24 

1            34            21.84 

5 

19               39.55 

15 

58            58.65 

25 

1            38            17.75 

6 

23               35.46 

16 

1              2            54.56 

26 

1            42            13.66 

27                31.37 

17 

6           50.47 

27 

1           46             9.57 

31                27.28 

18 

10           46.38 

28 

1            50              5.48 

9 

35                23.19 

19 

14            42.29 

29 

1            54              1.39 

10 

39               19.10 

20 

18            38.20 

30 

1           57           57.30 

31 

2              1            53.21 

Approximate  Tallies  of  foreign  Coins,  in  17.  S.  Currency. 

It  is  difficult  to  obtain  positive  information  respecting  these;  and  it  is  probable  that  many  of  the 
values  in  this  table  are  in  error  from  1  to  5  per  cent.  Since,  however,  many  of  the  coins  appear  in 
statements  of  costs  of  engineering  works  abroad,  it  is  convenient  to  have  even  such  an  approximation. 


Dolls. 

Cts. 

Dolls. 

Cts. 

Augustus.  Saxony  
Carlin.    Sardinia.   ... 

3 

g 

98 
21 

Mohur.  Bombay  
"         Bengal 

7 
8 

20 
15 

Carolin.     Bavaria 

4 

93 

3 

84 

Crown.    Great  Britain  

1 

13 

Ounce    Sicily 

2 

50 

"          Spain  (Half  Pistole). 

1 

95 

Pistola    Rome 

3 

37 

"         Baden  Bavaria  N  Ger- 

Pistole    Spain  . 

3 

90 

1 

6 

10 

««         Sicily            

96 

Peseta    Spain 

20 

27 

Pistareen    Spain 

20 

11           Sweden    Norwa.        **"* 

27 

Piastre    Turkey    Old 

42 

Copeck    Russia. 

% 

"             "           New 

4  2^ 

Dollar.  Bolivia,  new  

96 

"             *'           other  authori- 

TIT 

"      U.  S.  of  Colombia..     .. 

93  5 

ties  19  cts  ;  11  cts,  &c 

"      Chili,  Peru   Ecuador  .. 

93 

1 

4 

"      Liberia  

1 

5 

"      Mexico  

1 

Para  

9 

"      Sandwich  Islds  .... 

1 

Pound.  Great  Britain  

4 

87T7 

Doubloon.  Spain  Mexico. 

15 

65 

"        Canada,  N   Scotia  New 

"         Central  America 

f  14 
<      to 

50 

Bruns,  Newfoundland 
Reale  plate    Spain  

4 

10 

I  15 

65 

5 

"         New  Granada 

15 

34 

"     Central  America    average 

5*f 

Drachm.  Greece 

19 

2  Reales    Ecuador          ' 

18% 

Ducat.  Austria,  Bohemia,  Ham- 
burg, Hanover  

2 

28 

Rix  dollar.  Hamburg.  Hanover. 
"      "        Sweden,  Holland... 

1 
1 

10 
5 

"      Sweden 

2 

l)0 

1 

1 

81 

"      "        Bavaria,       Austria 

1 

32 

Hungary  

97 

Franc.  France,  Belgium,  Ac  

19.4 

"  10  thalers.  Prussia  

8 

5  Franc  piece  

97 

"  thaler.   Prussia,  Poland,  N. 

Florin.  Austria,  Silesia  
"       Holland,  Netherlands,  S. 
Germany  

48 
38 

Germany,  Bremen, 
Saxony,  Hanover, 
"  species  thaler    Saxony  

69 

98 

1 

66 

45 

"         (silver!  H   n 

56 

"PeC'  Lacof^'    . 

45000 

"         P  us  i 

55 

Reis  (1000)    Brazil             

1 

8 

Guilder    Netherland 

40 

Rouble    Russia 

75 

26 

3 

95 

Gulden.  Baden  

40 

Schilling    Hamburg  

2 

Guinea.    Great  Britain,  21  shil- 

Shilling.  Great  Britain  

23 

V 

Groschen    Poland   Prussia. 

tdk 

Scudo.  Piedmont  

1 

3*'4 

5  Groschen      "            " 

12T° 

"         Naples    Sicily  

95 

1 

"         Sardinia    

92 

Imperial    Russia 

7 

92 

1 

Kreutzer.  Bavaria 

K 

,, 

1 

28 

10.  Austria  
"         60.        "      or  florin.. 
Livre.  France.  Sardinia  (Franc) 
"       Tuscany,  Venice 

8 
48 
18J4 
16 

Souverain.  Austria,  Bohemia... 
Sovereign.  Gr  Britain,  or  pound 
Sous.  France  very  nearly 
Star  Pagoda    Madras    

3 
4 

1 

57 

86 
1 
81 

Lira.  Milan  ..     ... 

19 

Stiver    Holland             ...nearly 

I 

Marc.    Germany. 

24 

30 

Maximilian.   Bavaria. 

3 

30 

Testoon    Portueal 

12 

Milrea.  Portugal. 

1 

8 

Zecchin    Turkey  

1 

40 

Moidore.      -          

6 

50 

Zecchino.  Rome  

2 

27 

82 


TKAVERSE   TABLE. 


Traverse  Table  for  a  Distance  =  1. 


Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

^T 

or 

or 

or 

N.  S. 

E.W. 

N.  S. 

E.W. 

N.S. 

E°W. 

0°0' 

2 

1.0000 
1.0000 

.0000 
.0006 

90°0' 
58 

2°0' 
2 

.9994 
.9994 

.0349 
.0355 

88°0' 
58 

4°0' 
2 

.9976 
.9975 

.0698 
.0703 

86°0' 
58 

4 

1.0000 

.0012 

56 

4 

.9993 

.0361 

56 

4 

.9975 

.0709 

56 

6 

1.0000 

.0017 

54 

6 

.9993 

.0366 

54 

6 

.9974 

.0715 

54 

8 

1.0000 

.0023 

.52 

8 

.9993 

.0372 

52 

,  8 

.9974 

.0721 

52 

10 

1.0000 

.0029 

50 

10 

.9993 

.0378 

50 

10 

.9974 

.0727 

50 

12 

1.0000 

.0035 

48 

12 

.9993 

.0384 

48 

12 

.9973 

.0732 

*8 

14 

1.0000 

.0041 

46 

14 

.9992 

.0390 

46 

14 

.9973 

.0738 

46 

16 

1.0000 

.0047 

44 

16 

.9992 

.0396 

44 

16 

.9972 

.0744 

44 

18 

1.0000 

.0052 

42 

18 

.9992 

.0401 

42 

18 

.9972 

.0750 

42 

20 

1.0000 

.0058 

40 

20 

.9992 

.0407 

40 

20 

.9971 

.0756 

40 

22 

1.0000 

.0064 

38 

22 

.9991 

.0413 

38 

22 

.9971 

.0761 

38 

24 

1.0000 

.0070 

36 

24 

.9991 

.0419 

36 

24 

.9971 

.0767 

36 

26 

1.0000 

.0076 

34 

26 

.9991 

.0425 

34 

26 

.9970 

.0773 

34 

28 

1.0000 

.0081 

32 

28 

.9991 

.0430 

32 

28 

.9970 

.0779 

32 

30 

1  .0000 

.0087 

30 

30 

.9990 

.0436 

30 

30 

.9969 

.0785 

30 

32 

1.0000 

.0093 

28 

32 

.9990 

.0442 

23" 

32 

.9969 

.0790 

28 

34 

1.0000 

.0099 

26 

34 

.9990 

.0448 

26 

34 

.9968 

.0796 

36 

36 

.9999 

.0105 

24 

36 

.9990 

.0454 

24 

36 

.9968 

.0802 

24 

38 

.9999 

.0111 

22 

38 

.9989 

.0459 

22 

38 

.9967 

.0808 

22 

40 

.9999 

.0116 

20 

40 

.9989 

.0465 

20 

40 

.9967 

.0814 

20 

42 

.9999 

.0122 

18 

42 

.9989 

.0471 

18 

42 

.9966 

.0819 

18 

44 

.9999 

.0128 

16 

44 

.9989 

.0477 

16 

44 

.9966 

.0825 

16 

46 

.9999 

.0134 

14 

46 

.9988 

.0483 

u 

46 

.9965 

.0831 

14 

48 

.9999 

.0140 

12 

48 

.9988 

.0488 

12 

48 

.9965 

.0837 

12 

50 

.9999 

.0145 

10 

50 

.9988 

.0494 

10 

50 

.9964 

.0843 

10 

52 

.9999 

.0151 

8 

52 

.9987 

.0500 

8 

52 

.9964 

.0848 

8 

54 

.9999 

.0157 

6 

54 

.9987 

.0506 

6 

54 

.9963 

.0854 

6 

56 

.9999 

.0163 

4 

56 

.9987 

.0512 

4 

56 

.9963 

.0860 

4 

58 

.99*9 

.0169 

2 

58 

.9987 

.0518 

2 

58 

.9962 

0866 

2 

1°0' 

.9998 

.0175 

89°0 

3°0' 

.9986 

,0523 

87°0' 

5°0' 

.9962 

.0872 

85°0' 

2 

.9998 

.0180 

58 

2 

.9986 

.0529 

58 

2 

.9961 

.0877 

58 

4 

.9998 

.0186 

56 

4 

.9986 

.0535 

56 

4 

.9961 

.0883 

56 

6 

.9998 

.0192 

54 

6 

.9985 

.0541 

54 

6 

.9960 

.0889 

54 

8 

.9998 

.0198 

52 

8 

.9985 

.0547 

52 

8 

.9960 

.0895 

52 

10 

.9998 

.0204 

50 

10 

.9985 

.0552 

50 

10 

.9959 

.0901 

50 

12 

.9998 

.0209 

48 

12 

.9984 

.0558 

48 

12 

.9959 

.0906 

48 

14 

.9998 

.0215 

46 

14 

.9984 

.0564 

46 

14 

.9958 

.0912 

46 

16 

.9998 

,  .0221 

44 

16 

.9984 

.0570 

44 

16 

.9958 

.0918 

44 

18 

.9997 

.0227 

42 

18 

.9983 

.0576 

42 

18 

'.9957 

.0924 

42 

20 

.9997 

.0233 

40 

20 

.9983 

.0581 

40 

20 

.9957 

.0929 

40 

22 

.9997 

.0239 

38 

22 

.9983 

.0587 

38 

22 

.9956 

.0935 

38 

24 

.9997 

.0244 

36 

24 

.9982 

.0593 

36 

24 

.9956 

.0941 

36 

26 

.9997 

.0250 

34 

26 

.9;)82 

.0599 

34 

26 

.9955 

.0947 

34 

28 

.9997 

.0256 

32 

28 

.9982 

.0605 

32 

28 

.9955 

.0953 

32 

30 

.9997 

.0262 

30 

30 

.9981 

.0610 

30 

30 

.9954 

.0958 

30 

32 

.9996 

.0268 

28 

32 

.9981 

.0616 

28 

32 

.9953 

.0964 

28 

34 

.9996 

.0273 

26 

34 

.9981 

.0622 

26 

34 

.9953 

.0970 

26 

36 

.9996 

.0279 

24 

36 

.9980 

.0628 

24 

36 

.9952 

.0976 

24 

38 

.9996 

.0285 

22 

38 

.9980 

.0634 

22 

38 

.9952 

.0982 

22 

40 

.9996 

.0291 

20 

40 

.9980 

.OS40 

20 

40 

.9951 

.0987 

20 

42 

•9996 

.0297 

18 

42 

.9979 

.0645 

18 

42 

.9951 

.0993 

18 

44 

.9995 

.0302 

16 

44 

.9979 

.0651 

16 

44 

.9950 

.0999 

16 

46 

.9995 

.0308 

14 

46 

.9978 

.0657 

14 

46 

.9949 

.1005 

14 

48 

.9995 

.0314 

12 

48 

.9978 

.0663 

12 

48 

.9949 

.1011 

12 

50 

.9995 

.0320 

10 

50 

.9978 

.0669 

10 

50 

.9948 

.1016 

10 

52 

.9995 

.0326 

8 

52 

.9977 

.0674 

8 

52 

.9948 

.1022 

8 

54 

.9995     .0332 

6 

54 

.9977 

.0680 

6 

54 

.9947 

.1028 

6 

56 

.9994     .0337 

4 

56 

.9976 

.0686 

4 

56 

.9946 

.1034 

4 

58 

.9994 

.0343 

2 

58 

.9976 

.0692 

1 

58 

.9946 

.1039 

2 

2°0' 

.9994 

.0349 

88D0' 

4°0' 

.9976 

.0698 

86°0' 

6°0' 

.9945 

.1045 

84°0* 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

or 

or 

or 

E.W. 

N.  S. 

E.W. 

N.  S. 

E.W. 

N.S. 

TRAVERSE   TABLE. 


83 


Traverse  Table  for  a  Distance  =  1.     (CONTINUED.) 


Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

or 

or 

or 

or 

or 

or 

N.  S. 

E.  W. 

N.  S. 

E.  W. 

N.  S. 

E.W. 

«°0 

.9940 
.9945 

.1045 
.1051 

84°0' 
58 

8°0' 
2 

.9903 
.9902 

.1392 
.1397 

82°0' 
58 

10°0' 

| 

.9848 
.9847 

.1736 
.1742 

80^0' 
58 

.9944 

.1057 

56 

4 

.99U1 

.1403 

56 

4 

.9846 

.1748 

56 

.9943 

.1063 

54 

6 

.9900 

.1409 

54 

6 

.9845 

.1754 

54 

.9943 

.1068 

52 

8 

.9899 

.1415 

52 

8 

.9844 

.1758 

52 

1 

.9942 

.1074 

50 

0 

.9899 

.1421 

50 

10 

.9843 

.  765 

50 

1 

.9942 

.1080 

48 

1 

.9898 

.1426 

48 

12 

.9842 

.  771 

48 

1 

.9941 

.1086 

46 

4 

.9897 

.1432 

46 

14 

.9841 

.  777 

46 

1 

.9940 

.1092 

44 

6 

.9896 

.1438 

44 

16 

.9840 

.  782 

44 

18 

.9940 

.1097 

42 

8 

.9895 

.1444 

42 

18 

.9839 

.  788 

42 

20 

.9939 

.1103 

40 

20 

.9894 

.1449 

40 

20 

.9838 

.1794 

40 

22 

.9938 

.1109 

38 

22 

.9894 

.1455 

38 

22 

.9837 

.1799 

38 

24 

.9938 

.1115 

36 

24 

.9893 

.1461 

36 

24 

.9836 

.1805 

36 

26 

.9937 

.1120 

34 

26 

.9892 

.1467 

34 

26 

.9835 

.1811 

34 

28 

.9936 

.1126 

32 

28 

.9891 

.1472 

32 

28 

.9834 

.1817 

32 

30 

.9936 

.1132 

30 

30 

.9890 

.1478 

30 

30 

.9833 

.1822 

30 

32 

.9935 

.1138 

28 

32 

.9889 

.1484 

28 

32 

.9831 

.1828 

28 

34 

.9934 

.1144 

26 

34 

.9888 

.1490 

26 

34 

.9830 

.1834 

26 

36 

.9934 

.1149 

24 

36 

.9888 

.1495 

24 

36 

.9829 

.1840 

24 

38 

.9933 

.1155 

22 

38 

.9887 

.1501 

22 

38 

.9828 

.1845 

22 

40 

.9932 

.1161 

20 

40 

.9886 

.1507 

2O 

40 

.9827 

.1851 

20 

42 

.9932 

.1167 

18 

42 

.9885 

.1513 

18 

42 

.9826 

.1857 

18 

44 

.9931 

.1172 

16 

44 

.1518 

16 

44 

.9825 

.1862 

16 

46 

.9930 

.1178 

14 

46 

!9883 

.1524 

14 

46 

.9824 

.1868 

14 

48 

.9930 

.1184 

12 

48 

.9882 

.1530 

12 

48 

.9823 

.1874 

12 

60 

.9929 

.1190 

10 

50 

.9881 

.1536 

10 

50 

.9822 

J880 

10 

52 

.9928 

.1196 

8 

52 

.9880 

.1541 

8 

52 

.9821 

.1885 

8 

54 

.9928 

.1201 

6 

54 

.9880 

.1547 

6 

54 

.9820 

.1891 

6 

56 

.9927 

.1207 

4 

56 

.9879 

.1553 

4 

56 

.9818 

.1897 

4 

58 

.9926 

.1213 

2 

58 

.9878 

.1559 

2 

58 

.9817 

.1902 

2 

7°0' 

.9925 

.1219 

83^0' 

9°0 

.9877 

.1564 

81  °0' 

11°0' 

.9816 

.1908 

79°^ 

I 

.9925 

.1224 

58 

2 

.9876 

.1570 

58 

2 

.9815 

.1914 

58 

4 

.9924 

.1230 

56 

4 

.9875 

.1576 

56 

4 

.9814 

.1920 

56 

€ 

.9923 

.1236 

54 

6 

.9874 

.1582 

54 

6 

.9813 

.1925 

54 

8 

.9923 

.1242 

52 

8 

.9873 

.1587 

52 

8 

.9812 

.1931 

52 

10 

.9923 

.1248 

50 

10 

.9872 

.1593 

50 

10 

.9811 

.1937 

50 

12 

.9921 

.1253 

48 

12 

.9871 

.1599 

48 

12 

.9810 

.1942 

48 

14 

.9920 

.1259 

46 

14 

.9870 

.1605 

46 

14 

.9808 

.1948 

46 

16 

.9920 

.1265 

44 

16 

.9869 

.1610 

44 

16 

.9807 

1954 

44 

18 

.9919 

.1271 

42 

18 

.9869 

.1616 

42 

18 

.9806 

.1959 

42 

20 

.9918 

.1276 

40 

20 

.9868 

.1622 

40 

20 

.9805 

.1965 

40 

'22 

.9917 

.1282 

38 

22 

.9867 

.1628 

38 

22 

.9804 

.1971 

38 

•24 

.9917 

.1288 

36 

24 

.9866 

.1633 

36 

24 

.9803 

.1977 

36 

•26 

.9916 

.1294 

34 

26 

.9865 

.1639 

34 

26 

.9802 

.1982 

34 

28 

.9915 

.1299 

32 

28 

.9864 

.1645 

32 

28 

.9800 

.1988 

32 

30 

.9914 

.1305 

30 

30 

.9863 

.1650 

30 

30 

.9799 

.1994 

30 

M 

.9914 

.1311 

28 

32 

.9862 

.1656 

28 

32 

.9798 

.1999 

28 

H 

.9913 

.1317 

26 

34 

.9861 

.1662 

H 

34 

.9797 

.2005 

26 

36 

.9912 

.1323 

24 

36 

.9860 

.16(58 

24 

36 

.9796 

.2011 

24 

58 

.9911 

.1328 

22 

38 

.9859 

.1673 

22 

38 

.9795 

.2016 

22 

40 

.9911 

.1384 

20 

40 

.9858 

.1679 

20 

40 

.9793 

.2022 

20 

42 

.9910 

.1340 

18 

42 

.9857 

.1685 

18 

42 

.9792 

.2028 

18 

44 

.9909 

.1346 

16 

44 

.9856 

.1691 

16 

44 

.9791 

.2034 

16 

46 

.9908 

.1351 

14 

46 

.9855 

.1^96 

14 

46 

.9790 

.2039 

14 

48 

.9907 

.1357 

12 

48 

.9854 

.1702 

12 

48 

.9789 

.2045 

12 

50 

.9907 

.1363 

10 

50 

.9853 

.1708 

10 

50 

.9787 

.2051 

10 

52 

.9906 

.1360 

8 

52 

.9852 

.1714 

8 

52 

.9786 

.2056 

8 

54 

.9905 

.1374 

6 

54 

.9851 

.1719 

6 

54 

.9785 

.2062 

6 

56 

.9904 

.1880 

4 

56 

.9850 

.1725 

4 

56 

jtm 

.2068 

4 

58 

.9903 

,1*86 

2 

58 

.9849 

.1731 

2 

58 

.1)783 

.2073 

2 

8°0' 

.9903 

.1392 

82°0 

1000' 

.9848 

.1736 

80°0' 

12°0' 

.9781 

.2079 

78°0' 

Dep. 

Lat. 

Dep. 

Lnt, 

Dep. 

Lat. 

or 

or 

or 

E.  W. 

N.  8. 

E.  W. 

N.  S. 

E.W. 

N.S. 

84 


TRAVERSE  TABLE. 


Traverse  Table  for  a  Distance  =  1.  (CONTINUED.) 


Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

or 

or 

or 

or 

N.S. 

E.  W. 

1 

N.S. 

E.  W. 

N.S. 

E.  W. 

12°0' 
2 

.9781 
.9780 

.2079 

.2084 

78°0' 

58 

14°0 
2 

.9703 
.9702 

.2419 
.2425 

76°0' 
58 

16°0' 
2 

.9613 
.9611 

.2756 
.2762 

74°0' 

58 

4 

.9779 

.2090 

56 

4 

.9700 

.2431 

56 

4 

.9609 

.2768 

56 

6 

.9778 

.2096 

54 

6 

.9699 

.2436 

54 

6 

.9608 

.2773 

54 

8 

.9777 

.2102 

52 

8 

.9697 

.2442 

52 

8 

.9606 

.2779 

52 

10 

.9775 

.2108 

50 

10 

.9696 

.2447 

50 

10 

.9605 

.2784 

50 

12 

.9774 

.2113 

48 

12 

.9694 

.2453 

48 

12 

.9603 

.2790 

48 

14 

•9773 

.2119 

46 

14 

.9693 

.2459 

46 

14 

.9601 

.2795 

46 

16 

.9772 

.2125 

44 

16 

.9692 

.2464 

44 

16 

.9600 

.2801 

44 

18 

.9770 

.2130 

42 

18 

.9690 

.2470 

42 

18 

.9598 

.2807 

42 

20 

.9769 

.2136 

40 

20 

.9689 

.2476 

40 

20 

.9596 

.2812 

40 

22 

.9768 

.2142 

38 

22 

.9687 

.2481 

38 

22 

.9595 

.2818 

38 

24 

.9767 

.2147 

36 

24 

.9686 

.2487 

36 

24 

.9593 

.2823 

36 

26 

.9765 

.2153 

34 

26 

.9684 

.2493 

34 

26 

.9591 

.2829 

34 

28 

.9764 

.2159 

32 

28 

.9683 

.2498 

32 

28 

.9590 

.2835 

32 

30 

.9763 

.2164 

30 

30 

.9681 

.2504 

30 

30 

.9588 

.2840 

30 

32 

.9762 

.2170 

28 

32 

.9680 

.2509 

28 

32 

.9587 

.2846 

28 

34 

.9760 

.2176 

26 

34 

.9679 

.2515 

26 

34 

.9585 

.2851 

26 

36 

.9759 

.2181 

24 

36 

.9677 

.2521 

24 

36 

.9583 

.2857 

24 

38 

.9758 

.2187 

22 

38 

.9676 

.2526 

22 

38 

.9582 

.2862 

22 

40 

.9757 

.2193 

20 

40 

.9674 

.2532 

20 

40 

.9580 

.2868 

20 

42 

.9755 

.2198 

18 

42 

.9673 

.2538 

18 

42 

.9578 

.2874 

18 

44 

.9754 

.2204 

16 

44 

.9671 

.2543 

16 

44 

.9577 

.2879 

16 

46 

.9753 

.2210 

14 

46 

.9670 

.2549 

14 

46 

.9575 

.2885 

14 

48 

.9751 

.2215 

12 

48 

.9668 

.2554 

12 

48 

.9573 

.2890 

12 

50 

.9750 

.2221 

10 

50 

.9667 

.2560 

10 

50 

.9572 

.2896 

10 

52 

.9749 

.2227 

8 

52 

.9665 

.2566 

8 

52 

.9570 

.2901 

8 

54 

.9748 

.223" 

6 

54 

.9664 

.2571 

6 

54 

.9568 

.2907 

6 

56 

.9746 

,22.>8 

4 

56 

.9662 

.2577 

4 

56  • 

.9566 

.2913 

4 

58 

.9745 

.2244 

2 

58 

.9661 

.2583 

2 

58 

.9565 

.2918 

2 

13°0' 

.9744 

.2250 

77°0' 

150Q' 

.9659 

.2588 

75°0' 

L7°0- 

.9563 

.2924 

7300* 

2 

.9742 

.2255 

58 

2 

.9658 

.2594 

58 

2 

.9561 

.2929 

58 

4 

.9741 

.2261 

56 

4 

.9656 

.2599 

56 

4 

.9560 

.2935 

56 

6 

.9740 

.2267 

54 

6 

.9655 

.2605 

54 

6 

.9558 

.2940 

54 

8 

.9738 

.2272 

52 

8 

.9653 

.2611 

52 

8 

.9556 

.2946 

52 

10 

.9737 

.2278 

50 

10 

.9652 

.2616 

50 

10 

.9555 

.2952 

50 

12 

.9736 

.2284 

48 

12 

.9650 

.2622 

48 

12 

.9553 

.2957 

48 

14 

.9734 

.2289 

46 

14 

.9649 

.2628 

46 

14 

.9551 

.2963 

46 

16 

.9733 

.2295 

44 

16 

.9647 

.2633 

44 

16 

.9549 

.2968 

44 

18 

.9732 

.2300 

42 

18 

.9646 

.2639 

42 

18 

.9548 

.2974 

42 

20 

.9730 

.2306 

40 

20 

.9644 

.2644 

40 

20 

.9546 

.2979 

40 

22 

.9729 

.2312 

38 

22 

.9642 

.2650 

38 

22 

.9544 

.2985 

38 

24 

.9728 

.2317 

36 

24 

.9641 

.2656 

36 

24 

.9542 

.2990 

36 

26 

.9726 

.2323 

34 

26 

.9639 

.2661 

34 

26 

.9541 

.2996 

34 

28 

.9725 

.2329 

32 

28 

.9638 

.2667 

32 

28 

.9539 

.3002 

32 

30 

.9724 

.2334 

30 

30 

.9636 

.2672 

30 

30 

.9537 

.3007 

30 

32 

.9722 

.2340 

28 

32 

.9635 

.2678 

28 

32 

.9535 

.3013 

28 

34 

.9721 

.2346 

26 

34 

.9633 

.2684 

26 

34 

.9534 

.3018 

26 

36 

.9720 

.2351 

24 

36 

.9632 

.2689 

24 

36 

.9532 

.3024 

24 

38 

.9718 

.2357 

22 

38 

.9630 

.2695 

22 

38 

.9530 

.3029 

22 

40 

.9717 

.2363 

20 

40 

.9628 

.2700 

20 

40 

.9528 

.3035 

20 

42 

.9715 

.2368 

18 

42 

.9627 

.2706 

18 

42 

.9527 

.3040 

18 

44 

.9714 

.2374 

16 

44 

.9625 

.2712 

16 

44 

.9525 

.3046 

16 

46 

.9713 

.2380 

14 

46 

.9624 

.2717 

14 

46 

.9523 

.3051 

14 

48 

.9711 

.2385 

If 

48 

.9622 

.2723 

12 

48 

.9521 

.3057 

12 

50 

.9710 

.2391 

10 

50 

.9621 

.2728 

10 

50 

.9520 

.3062 

10 

52 

.9709 

.2397 

8 

52 

.9619 

.2734 

8 

52 

.9518 

.3068 

8 

54 

.9707 

.2402 

6 

54 

.9617 

.2740 

6 

54 

.9516 

.3074 

6 

56 

.9706 

.2408 

4 

56 

.9616 

.2745 

4 

56 

.9514 

.3079 

4 

58 

.9704 

.2414 

2 

58 

.9614 

.2751 

2 

58 

.9512 

.3085 

2 

14°0' 

.9703 

.2419 

76°0' 

16°0' 

.9613 

.2756 

74°0' 

18°0' 

.9511 

.3090 

72°0' 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

or 

or 

or 

or 

or 

or 

E.  W. 

N.S. 

E.  W. 

N.S. 

E.  W. 

N.S. 

TRAVEESE   TABLE. 


85 


Traverse  Table  for  a  Distance  =:  1.  (CONTINUED.) 


Lat. 
or 
N.  S. 

Dep. 
or 
E.  W. 

Lat. 

or 

N.  S. 

Dep. 
or 
E.  W. 

t 

Lat. 
or 
N.  S. 

Dep. 
or 
E.W. 

18°0' 

.9511 

.3090 

72°0' 

20rO' 

.9397 

.3420 

70°0 

22°0 

.9272 

.3746 

68°0' 

2 

.9509 

.3096 

58 

2 

.9395 

.3426 

58 

2 

.9270 

.3751 

58 

4 

.9507 

.3101 

56 

4 

.9393 

.3431 

56 

4 

.9267 

.3757 

56 

6 

.9505 

.3107 

54 

6 

.9391 

.3437 

54 

6 

.9265 

.3762 

54 

8 

.9503 

.3112 

52 

8 

.9389 

.3442 

52 

8 

.9263 

.3768 

52 

10 

.9502 

.3118 

50 

10 

.9387 

.3448 

50 

10 

.9261 

.3773 

50 

12 

.9500 

.3123 

48 

12 

.9385 

.3453 

48 

12 

.9259 

.3778 

48 

14 

.9498 

.3129 

4<i 

14 

.9383 

.3458 

46 

14 

.9257 

.3784 

46 

16 

.9496 

.3134 

44 

16 

.9381 

.3464 

44 

16 

.9254 

.3789 

44 

18 

.9494 

.3140 

42 

18 

.9379 

.3469 

42 

18 

.9252 

.3795 

42 

20 

.9492 

.3145 

40 

20 

.9377 

.3475 

40 

20 

.9250 

.3800 

40 

22 

.9491 

.3151 

38 

22 

.9375 

.3480 

38 

22 

.9248 

.3805 

88 

24 

.9489 

.3156 

36 

24 

.9373 

.3486 

36 

24 

.9245 

.3811 

36 

26 

.9487 

.3162 

34 

26 

.9371 

.3491 

34 

26 

.9243 

.3816 

34 

28 

.9485 

.3168 

32 

28 

.9369 

.3497 

32 

28 

.9241 

.3821 

82 

30 

.9483 

.3173 

30 

30 

.9367 

.3502 

30 

30 

.9239 

.3827 

30 

32 

.9481 

.3179 

28 

32 

.9365 

.3508 

28 

32 

.9237 

.3832 

28 

34 

.9480 

.3184 

26 

34 

.9363 

.3513 

26 

34 

.9234 

.3838 

26 

36 

.9478 

.3190 

24 

36 

.9361 

.3518 

24 

36 

.9232 

.3843 

24 

88 

.9476 

.3195 

22 

38 

.9359 

.3524 

22 

38 

.9230 

.3848 

22 

40 

.9474 

.3201 

20 

40 

.9356 

.3529 

20 

40 

.9228 

.3854 

20 

42 

.9472 

.3206 

18 

42 

.9354 

.3535 

18 

42 

.9225 

.3859 

18 

44 

.9470 

.3212 

16 

44 

.9352 

.3540 

16 

44 

.9223 

.3864 

16 

46 

.9468 

.3217 

14 

46 

.9350 

.3516 

14 

46 

.9221 

.3870 

14 

48 

.9466 

.3223 

12 

48 

.9348 

.3551 

12 

48 

.9213 

.3875 

12 

50 

.9465 

.3228 

10 

50 

.9346 

.3557 

10 

50 

.9216 

.3881 

10 

52 

.9463 

.3234 

8 

52 

.9344 

.3562 

8 

52 

.9214 

.3886 

8 

64 

.9461 

.3239 

6 

54 

.9342 

.3567 

54 

.9212 

.3891 

6 

56 

.9459 

.3245 

4 

56 

.9340 

.3573 

4 

56 

.9216 

.3897 

4 

58 

.9457 

.3250 

2 

58 

.9338 

.3578 

2 

58 

.9207 

.3902 

2 

|9°0' 

.9455 

.3256 

71  °0' 

n°o 

.9336 

.3584 

690Q 

3°0' 

.9205 

.3907 

67°0' 

2 

.9453 

.3261 

58 

2 

.9334 

.3589 

58 

2 

.9203 

.3913 

58 

4 

.9451 

.3267 

56 

4 

.9332 

.3595 

56 

4 

.9200 

.3918 

56 

6 

.9449 

.3272 

54 

6 

.9330 

.3600 

54 

6 

.9198 

.3923 

54 

8 

.9448 

.3278 

52 

8 

.9327 

.3605 

8 

.9196 

.3929 

52 

10 

.9446 

.3283 

50 

10 

.9325 

.3611 

50 

10 

.9194 

.3934 

50 

12 

.9444 

.3289 

48 

12 

.9323 

.3616 

48 

12 

.9191 

.3939 

48 

14 

.9442 

.3294 

46 

14 

.9321 

.3622 

46 

14 

.9189 

.3945 

46 

16 

.9440 

.3300 

44 

16 

.9319 

.3627 

44 

16 

.9187 

.3950 

44 

18 

.9438 

.3305 

42 

18 

.9317 

.3633 

42 

18 

.9184 

.3955 

42 

20 

.9436 

.3311 

40 

20 

.9315 

.3638 

40 

20 

.9182 

.3961 

40 

22 

.9434 

.3316 

38 

22 

.9313 

.3643 

38 

22 

.9180 

.3966 

38 

24 

.9432 

.3322 

36 

24 

.9311 

.3649 

36 

24 

.9178 

.3971 

36 

26 

.9430 

.3327 

34 

26 

.9308 

.3654 

34 

26 

.9175 

.3977 

34 

28 

.9428 

.3333 

32 

28 

.9306 

.3660 

32 

28 

.9173 

.3982 

32 

30 

.9426 

.3338 

30 

30 

.9304 

.3665 

30 

30 

.9171 

.3987 

30 

32 

.9424 

.3344 

28 

32 

.9302 

.3670 

28 

32 

.9168 

.3993 

28 

34 

.9423 

.3349 

26 

34 

.9300 

.3676 

26 

34 

.9166 

.3998 

26 

36 

.9421 

.3355 

24 

36 

.9298 

.3681 

24 

36 

.9164 

.4003 

24 

38 

.9419 

.3360 

22 

38 

.9296 

.3687 

22 

38 

.9161 

.4009 

22 

40 

.9417 

.3365 

20 

40 

.9293 

.3692 

20 

40 

.9159 

.4014 

20 

42 

.9415 

.3371 

18 

42 

.9291 

.3697 

18 

42 

.9157 

.4019 

18 

44 

.9413 

.3376 

16 

44 

.9289 

.3703 

16 

44 

.9  54 

.4025 

16 

46 

.9411 

.3382 

14 

46 

.9287 

.3708 

14 

46 

.9  52 

.4030 

14 

48 

.9409 

.3387 

12 

48 

.9285 

.3714 

12 

48 

.9  50 

.4035 

12 

50 

.9407 

.3393 

10 

9o 

.9283 

.3719 

10 

50 

.9  47 

.4041 

10 

62 

.9405 

.3398 

8 

52 

.9281 

.3724 

8 

52 

.9  45 

.4046 

8 

14 

.9403 

.3404 

6 

54 

.9278 

.3730 

6 

54 

.9  43 

.4051 

6 

56 

.9401 

.3409 

4 

56 

.9276 

.3735 

4 

56 

.9  40 

.4057 

4 

58 

.9399 

.3415 

2 

58 

.9274 

.3741 

2 

58 

.9138 

.4062 

2 

20°0' 

.9397 

.3420 

70°0' 

>2^0' 

.9272 

.3746 

68^0' 

4°0' 

.9135 

.4067 

B6C0 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

or 

or 

or 

or 

or 

or 

E.W: 

N.  S. 

E.  W. 

N.  S. 

E.W. 

N.  S. 

86 


TRAVERSE   TABLE. 


Traverse  Table  for  a  Distance  =  1.    (CONTINUED.) 


Lat. 
or 

N.S. 

Dep. 
or 
E.W. 

Lat. 

or 
N.S. 

Dep. 
or 
E.W. 

Lat. 
N?S. 

Dep. 
or 
E.  W. 

24^0' 
2 

.9135 
.9133 

.4067 
.4073 

66^0' 

58 

26°0' 
2 

.8988 
.8985 

J4384 
.4389 

64°0' 

58 

28°0' 
2 

.8829 

.8827 

.4695 

.4700 

62°0/ 

58 

4 

.9131 

.4078 

56 

4 

.8983 

.4394 

56 

4 

.8824 

.4705 

56 

6 

.9128 

.4083 

54 

6 

.89«0 

.4399 

54 

6 

.8821 

.4710 

54 

8 

.9126 

.4089 

52 

8 

.8978 

.4405 

52 

8 

.8819 

.4715 

52 

10 

.9124 

.4094 

50 

10 

.8975 

.4410 

50 

10 

.8816 

.4720 

50 

12 

.9121 

.4099 

48 

12 

.8973 

.4415 

48 

12 

.8813 

.4726 

48 

14 

16 

'.9116 

.'4110 

44 

16 

.8967 

.4425 

44 

16 

.'8808 

.4731 
.4736 

44 

18 

.9114 

.4115 

42 

18 

.8965 

.4431 

42 

18 

.8805 

.4741 

42 

20 

.9112 

.4120 

40 

20 

.8962 

.4436 

40 

20 

.8802 

.4746 

40 

22 

.9109 

.4126 

38 

22 

.8960 

.4441 

38 

22 

.8799 

.4751 

38 

24 

.9107 

.4131 

36 

24 

.8957 

.4446 

36 

24 

.8796 

.4756 

36 

26 

.9-504 

.4136 

34 

26 

.8955 

.4452 

34 

26 

.8794 

.4761 

34 

28 

.9102 

.4142 

32 

28 

.8952 

.4457 

32 

28 

.8791 

.4766 

32 

30 

.9100 

.4147 

30 

30 

.8949 

.4462 

30 

30 

.8788 

.4772 

30 

32 

.9097 

.4152 

28 

32 

.8947 

.4467 

28 

32 

.8785 

.4777 

28 

34 

.9095 

.4158 

26 

34 

.8944 

.4472 

26 

34 

.8783 

.4782 

26 

36 

.9092 

.4163 

24 

36 

.8942 

.4478 

24 

36 

.8780 

.4787 

24 

38 

.9090 

.4168 

22 

38 

.8939 

.4483 

22 

38 

.8777 

.4792 

22 

40 

.9088 

.4173 

20 

40 

.8936 

.4488 

20 

40 

.8774 

.4797 

20 

42 

.9085 

.4179 

18 

42 

.8934 

.4493 

18 

42 

.8771 

.4802 

18 

44 

.9083 

.4184 

16 

44 

.8931 

.4498 

16 

44 

.8769 

.4807 

16 

46 

.9080 

.4189 

14 

46 

.8928 

.4504 

14 

46 

.8766 

.4812 

14 

48 

.9078 

.4195 

12 

48 

.8926 

.4509 

12 

48 

.8763 

.4818 

12 

50 

.9075 

.4200 

10 

50 

.8923 

.4514 

10 

50 

.8760 

.4823 

10 

52 

.9073 

.4205 

8 

52 

.8921 

.4519 

8 

52 

.8757 

.4828 

8 

54 

.9070 

.4210 

6 

54 

.8918 

.4524 

6 

54 

.8755 

.4833 

6 

56 

.9068 

.4216 

4 

56 

.8915 

.4530 

4 

56 

.8752 

.4838 

4 

58 

.9066 

.4221 

2 

58 

.8913 

.4535 

2 

58 

.8749 

4843 

2 

25°0' 

.9063 

.4226 

65°0' 

27°0 

.8910 

,4540 

6300' 

29°0' 

.8746 

.4848 

61  °0' 

2 

.9061 

.4231 

58 

2 

.8907 

.4545 

58 

2 

.8743 

.4853 

58 

4 

.9058 

.4237 

56 

4 

.8905 

.4550 

56 

4 

.8741 

.4858 

56 

6 

.9056 

.4242 

54 

6 

.8902 

.4555 

54 

6 

.8738 

.4863 

54 

8 

.9053 

.4247 

52 

8 

.8899 

.4561 

52 

8 

.8735 

.4868 

52 

10 

.9051 

.4253 

50 

10 

.8897 

.4566 

50 

10 

.8732 

.4874 

50 

12 

.9048 

.4258 

48 

12 

.8894 

.4571 

48 

12 

.8729 

.4879 

48 

14 

.9046 

.4263 

46 

14 

.8892 

.4576 

46 

14 

.8726 

.4884 

46 

16 

.9043 

.4268 

44 

16 

.8889 

.4581 

44 

16 

.8724 

.4889 

44 

18 

.9041 

.4274 

42 

18 

.8886 

.4586 

42 

18 

.8721 

.4894 

42 

20 

.9038 

.4279 

40 

20 

.8884 

.4592 

40 

20 

.8718 

.4899 

40 

22 

.9036 

.4284 

38 

22 

.8881 

.4597 

38 

22 

.8715 

.4904 

38 

24 

.9033 

.4289 

36 

24 

.8878 

.4602 

36 

24 

.8712 

.4909 

36 

26 

.9031 

.4295 

34 

26 

.8875 

.4607 

34 

26 

.8709 

.4914 

34 

28 

.9028 

.4300 

32 

28 

.8873 

.4612 

32 

28 

.8706 

.4919 

32 

30 

.9026 

.4305 

30 

30 

.8870 

.4617 

30 

30 

.8704 

.4924 

30 

32 

.9023 

.4310 

28 

82 

.8867 

.4623 

28 

32 

.8701 

.4929 

28 

34 

.9021 

.4316 

26 

34 

.8865 

.4628 

26 

34 

.8698 

.4934 

26 

36 

.9018 

.4321 

24 

36 

.8862 

.4633 

24 

36 

.8695 

.4939 

24 

38 

.9016 

.4326 

22 

38 

.8859 

.4638 

22 

38 

.8692 

.4944 

22 

40 

.9013 

.4331 

20 

40 

.8857 

.4643 

20 

40 

.8689 

.4950 

20 

42 

•9011 

.4337 

18 

42 

.8854 

.4648 

18 

42 

.8686 

.4955 

18 

44 

.9008 

.4342 

16 

44 

.8851 

.4654 

16 

44 

.8683 

.4960 

16 

46 

.9006 

.4347 

14 

46 

.8849 

.4659 

14 

46 

.8681 

.4965 

14 

48 

.9003 

.4352 

12 

48 

.8846 

.4664 

12 

48 

.8678 

.4970 

12 

50 

.9001 

.4358 

10 

50 

.8843 

.4669 

to 

50 

.8675 

.4975 

10 

52 

.895)8 

.4363 

8 

52 

.8840 

.4674 

8 

52 

.8672 

.4980 

8 

54 

.8996 

.436S 

6 

54 

.8838 

.4679 

6 

54 

.8669 

.49^5 

6 

56 

.81)93 

.4373 

4 

56 

.8835 

.4684 

4 

56 

.8666 

.4990 

4 

58 

.8990 

.4378 

2 

58 

.8832 

.4690 

2 

58 

.8663 

.4995 

2 

26°0' 

.8988 

.4384 

64  °0' 

.'8°0' 

.8829 

.4695 

62°0' 

30°0' 

.8660 

.5000 

60°0' 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

or 

or 

or 

or 

E.W. 

N.S. 

E.W. 

N.S. 

E.W. 

N.S. 

TRAVERSE   TABLE. 


87 


Traverse  Table  for  a  Distance  =  1.     (CONTINUED.) 


Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

or 

or 

or 

N.S. 

E.W. 

N.S. 

E.W. 

N.S. 

E.W. 

S0°0' 
2 

.8660 
.8657 

.5000 
.5005 

60°0' 
58 

32CQ' 
2 

•8480 
•8477 

.5299 
•  5304 

58^0' 
58 

34°0' 
2 

.8290 
.8287 

.5592 
.5597 

56^0' 
58 

4 

.8654 

.5010 

56 

4 

•8474 

•5309 

56 

4 

.8284* 

.5602 

56 

6 

.8652 

.5015 

54 

6 

•8471 

•5314 

54 

6 

.8281 

.5606 

54 

8 

.8649 

.5020 

52 

8 

•8468 

•5319 

52 

8 

.8277 

.5611 

52 

10 

.8646 

.5025 

50 

10 

•8465 

•5324 

50 

10 

•8274 

.5616 

50 

12 

.8643 

.5030 

48 

12 

•8462 

•5329 

48 

12 

.8271 

.5621 

48 

14 

.8640 

.5035 

46 

14 

•8459 

•  5334 

46 

14 

.8268 

.5626 

46 

16 

.8637 

.5040 

44 

16 

•8456 

•5389 

44 

16 

.8264 

.5630 

44 

18 

.8634 

.5045 

42 

18 

•8453 

•5344 

42 

18 

.8261 

.5635 

42 

20 

.8631 

.5050 

40 

20 

•8450 

•5348 

40 

20 

.8258 

.5640 

40 

22 

.8628 

.5055 

38 

22 

•8446 

•5353 

38 

22 

.8254 

.5645 

38 

24 

.8625 

.5060 

36 

24 

•8443 

•5358 

36 

24 

.8251 

•5650 

36 

28 

.8622 

.&065 

34 

26 

•8440 

•5363 

34 

26 

.8248 

.5654 

34 

28 

.8619 

.5070 

32 

28 

•8437 

•  5368 

32 

28 

.8245 

•5659 

32 

30 

.8616 

.5075 

30 

30 

•8434 

•5373 

30 

30 

.8241 

.5664 

30 

32 

.8613 

.5080 

28 

32 

•8431 

•5378 

28 

32 

.8238 

.5669 

28 

34 

.8610 

.5085 

26 

34 

•8428 

•5383 

26 

34 

.82^5 

.5674 

26 

36 

.8607 

.5090 

24 

36 

•8425 

•5388 

24 

36 

.8231 

.5678 

24 

38 

.8604 

.5095 

22 

38 

•8421 

•5393 

22 

38 

.8228 

.5683 

22 

40 

.8601 

.5100 

20 

40 

•8418 

•5398 

20 

40 

.8225 

.5688 

20 

42 

.8599 

.5105 

18 

42 

•8415 

•5402 

18 

42 

.8221 

.5693 

18 

44 

.8596 

.5110 

16 

44 

•8412 

•5407 

16 

44 

.8218 

.5698 

16 

46 

.8593 

.5115 

14 

46 

•8409 

•5412 

14 

46 

.8215 

.5702 

14 

48 

.8590 

.5120 

12 

48 

•8406 

•5417 

12 

48 

.8211 

•5707 

12 

50 

.8587 

.5125 

10 

50 

•8403 

•5422 

10 

50 

.8208 

•5712 

10 

52 

.8584 

.5130 

8 

52 

•8399 

•5427 

8 

52 

.8205 

.5717 

8 

54 

,8581 

.5135 

6 

54 

•8396 

•5432 

6 

54 

.8202 

•5721 

6 

56 

.8578 

.5140 

4 

56 

•8393 

•5437 

4 

56 

.8198 

•5726 

4 

58 

.8575 

.5145 

2 

58 

•8390 

•5442 

2 

58 

.8195 

•5731 

2 

31°0' 

.8572 

.5150 

5900' 

d3°0' 

•8387 

•  5446 

57°0' 

35°0' 

.8192 

.5736 

5500' 

2 

.8569 

.5155 

58 

2 

•8384 

•5451 

58 

2 

.8188 

.5741 

58 

4 

.8566 

.5160 

56 

4 

•8380 

•5456 

56 

4 

.8185 

•5745 

56 

6 

.8563 

.5165 

54 

6 

•8377 

•  5461 

54 

6 

.8181 

.5750 

54 

8 

.8560 

.5170 

52 

8 

•8374 

•5466 

52 

8 

.8178 

.5755 

52 

10 

.8557 

.5175 

50 

10 

•8371 

•5471 

50 

10 

.8175 

.5760 

50 

12 

.8554 

.5180 

48 

12 

•8368 

•5476 

48 

12 

.8171 

.5764 

48 

14 

.8551 

.5185 

46 

14 

•8364 

•5480 

46 

14 

.8168 

.5769 

46 

16 

.8548 

.5190 

44 

16 

•8361 

•5485 

44 

16 

.8165 

.5774 

44 

18 

.8545 

.5195 

42 

18 

•8358 

•5490 

42 

18 

.8161 

.5779 

42 

20 

.8542 

.5200 

40 

20 

•8355 

•5495 

40 

20 

.8158 

.5783 

40 

22 

.8539 

.5205 

38 

22 

•8352 

.5500 

38 

22 

.8155 

.5788 

38 

24 

.8536 

.5210 

36 

24 

•8348 

.5505 

36 

24 

.8151 

.5793 

36 

26 

.8532 

.5215 

34 

26 

•8345 

.5510 

34 

26 

.8148 

.5798 

34 

28 

.8529 

.5220 

32 

28 

•8342 

.5515 

32 

28 

.8145 

.5802 

32 

30 

.8526 

.5225 

30 

30 

•8339 

.5519 

30 

30 

.8141 

.5807 

30 

32 

.8523 

.5230 

28 

32 

•8336 

.5524 

28 

32 

.3138 

.5812 

28 

34 

.8520 

.5235 

26 

34 

•8332 

.5529 

26 

34 

.8134 

.5816 

26 

36 

.8517 

.5240 

24 

36 

•8329 

.5534 

24 

36 

.8131 

.5821 

24 

38 

.8514 

.5245 

22 

38 

•8326 

.5539 

22 

38 

.8128 

.5826 

22 

40 

.8511 

.5250 

20 

40 

•8323 

.5544 

20 

40 

.8124 

.5831 

20 

42 

.8508 

.5255 

18 

42 

•8320 

.5548 

18 

42 

.8121 

.5835 

18 

44 

.8505 

.5260 

16 

44 

•8316 

.5553 

16 

44 

.8117 

.5840 

16 

46 

.8502 

.5265 

14 

46 

•  8313 

.5558 

14 

46 

.8114 

.5845 

14 

48 

.8499 

.5270 

12 

48 

•8310 

.5563 

12 

48 

.8111 

.5850 

12 

50 

.8496 

.5275 

10  • 

50 

•8307 

.5568 

10 

50 

.8107 

.5854 

10 

52 

.8493 

.5279 

8 

52 

•  8303 

.5573 

8 

52 

.8104 

.5859 

8 

54 

.8490 

.5284 

6 

54 

•8300 

.5577 

6 

54 

.8100 

.5864 

6 

56 

.8487 

.5289 

4 

56 

•  8297 

.5582 

4 

56 

.8097 

.5868 

4 

58 

.8484 

.5294 

2 

58 

.8294 

.5587 

2 

58 

.8094 

.5873 

2 

3200' 

.8480 

.5299 

58°0' 

34=0' 

•  8290 

.5592 

56°0' 

36°0' 

.8090 

.5878 

54°0' 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

or 

or 

or 

or 

or 

or 

E.W. 

N.S. 

E.W. 

N.S. 

E.  W. 

N.S. 

TKAVERSE   TABLE. 


Traverse  Table  for  a  Distance  =  1.   (CONTINUED.) 


Lat. 
or 

N.  S. 

Dep. 
or 
E.  W. 

Lat. 

N.  S. 

Dep. 
or 
E.  W. 

Lat. 

or 

N.  S. 

Dep. 
E.  W. 

36°0' 
2 

.8090 
.8087 

.5878 
.5883 

54^0' 

58 

i8°0' 
2 

.7880 
.7877 

.6157 
.6161 

52°0' 
58 

10°0' 

2 

.7660 
;7657 

.6428 
.6432 

50°0' 
58 

4 

.8083 

'  .5887 

56 

4 

.7873 

.6166 

56 

4 

.7653 

.6437 

56 

6 

.8080 

.5892 

54 

6 

.7«69 

.6170 

54 

6 

.7649 

.6441 

54 

8 

.8076 

.5897 

52 

8 

.7866 

.6175 

52 

8 

.7645 

.6446 

52 

10 

.8073 

.5901 

50 

10 

.7862 

.6180 

50 

10 

.7642 

.6450 

50 

12 

.M)70 

.5906 

48 

12 

.7859 

.6184 

48 

12 

.7638 

.6455 

48 

14 

.8066 

.5911 

46 

14 

.7855 

.6189 

46 

14 

.7634 

.6459 

46 

16 

.80<>3 

.5915 

44 

16 

.7851 

.6193 

44 

16 

.7630 

.6463 

44 

18 

.8059 

.5920 

42 

18 

.7848 

.6198 

42 

18 

.7627 

.6468 

42 

20 

.8056 

.5925 

40 

20 

.7844 

.6202 

40 

20 

.7623 

.6472 

40 

22 

.8052 

.59,'iO 

38 

22 

.7841 

.6207 

38 

22 

.7619 

.6477 

38 

24 

.8049 

.5934 

36 

24 

.7837 

.621  1 

36 

24 

.7615 

.6481 

36 

26 

.8045 

.59:59 

34 

26 

.7833 

.6216 

34 

26 

.7612 

.6486 

34 

28 

.8042 

.5944 

32 

28- 

.7830 

.6221 

32 

28 

.7608 

.6490 

32 

30 

.8039 

.5948 

30 

30 

.7826 

.6225 

30 

30 

.7604 

.6494 

30 

32 

.8035 

.5953 

28 

32 

.7822 

.6230 

28 

32 

.7600 

.6499 

2b 

34 

.8032 

.5958 

26 

34 

.7819 

.6234 

26 

34 

.7596 

.6503 

26 

36 

.8028 

.5962 

24 

36 

.7815 

.6239 

24 

36 

.7593 

.6508 

24 

38 

.8025 

.5967 

22 

38 

.7812 

.6243 

22 

38 

.7589 

.6512 

22 

40 

.8021 

.5972 

20 

40 

.7808 

.6248 

20 

40 

.7585 

.6517 

20 

42 

.8018 

.5976 

18 

42 

.7804 

.6252 

18 

42 

.7581 

.6521 

18 

44 

.8014 

.5981 

16 

44 

.7801 

.6257 

16 

44 

.7578 

.6525 

16 

46 

.8011 

.5986 

14 

46 

.7797 

.6262 

14 

46 

.7574 

.6530 

14 

48 

.8007 

.5990 

12 

48 

.7793 

.62(56 

12 

48 

.7570 

.6534 

12 

50 

.8004 

.5995 

10 

50 

.7790 

.6271 

10 

50 

.7566 

.6539 

10 

52 

.8000 

.6000 

8- 

52 

.7786 

.6275 

8 

52 

.7562 

.6543 

8 

54 

.7997 

.6004 

6 

54 

.7782 

.6280 

6 

54 

.7559 

.6547 

6 

56 

.7993 

.6009 

4 

56 

.7779 

.6284 

4 

56 

.7555 

.6552 

4 

58 

.7990 

.6014 

2 

58 

.7775 

.6289 

2 

58 

.7551 

.6556 

2 

37°0' 

.7986 

.6018 

53°0' 

9°0  ' 

.7771 

.6293 

5iq) 

n°0' 

.7547 

.6561 

49°0' 

2 

.7983 

.6023 

58 

2 

.7768 

.6298 

A58 

2 

.7543 

.6565 

58 

4 

.7979 

.6027 

56 

4 

.7764 

.6302 

56 

4 

.7539 

.6569 

56 

6 

.7976 

.6032 

54 

6 

.7760 

.6307 

54 

6 

.7536 

.6574 

54 

8 

.7972 

.6037 

52 

8 

.7757 

.6311 

52 

8 

.7532 

.6578 

52 

10 

.7969 

.6041 

50 

10 

.7753 

.6316 

50 

10 

.7528 

.6583 

50 

12 

.7965 

.6046 

48 

12 

.7749 

.6320 

48 

12 

.7524 

.65H7 

48 

14 

.7962 

.6051 

46 

14 

.7746 

.6325 

46 

14 

.7520 

.6591 

46 

16 

.7958 

.6055 

44 

16 

.7742 

.6329 

44 

16 

.7516 

.6596 

44 

18 

.7955 

.6060 

42 

18 

.7738 

.6334 

42 

18 

.7513 

.6600 

42 

20 

.7951 

.6065 

40 

'20 

.7735 

.6338 

40 

20 

.7509 

.6604 

40 

22 

.7948 

.6069 

38 

22 

.7731 

.6343 

38 

22 

.7505 

.6609 

38 

24 

.7944 

.6074 

36 

24 

.7727 

.6347 

36 

24 

.7501 

.6613 

36 

26 

.7941 

tows 

34 

26 

.7724 

.6352 

34 

26 

.7497 

.6617 

34 

28 

.7937 

.6083 

32 

28 

.7720 

.6356 

32 

28 

.74!)3 

.6622 

32 

30 

.7934 

.6088 

30 

30 

.7716 

.6361 

30 

30 

.7490 

.6626 

30 

32 

.7930 

.6092 

28 

32 

.7713 

.6365 

28 

32 

.7486 

.6631 

28 

34 

.7926 

.6097 

26 

34 

.7709 

.6370 

26 

34 

.7482 

.6635 

26 

36 

.7923 

.6101 

24 

36 

.7705 

.6374 

24 

36 

.7478 

.6639 

24 

38 

.7919 

.6106 

22 

38 

.7701 

.6379 

22 

38 

.7474 

.6644 

22 

40 

.7916 

.6111 

20 

40 

.7698 

.6383 

20 

40 

.7470 

.6648 

20 

42 

.7912 

.6115 

18 

42 

.7694 

.6388 

18 

42 

.7466 

.6652 

18 

44 

.7909 

.6120 

16 

44 

.7690 

.6392 

16 

44 

.7463 

.6657 

16 

46 

.7905 

.6124 

14 

46 

.7687 

.6397 

14 

46 

.7459 

.6661 

14 

48 

.7902 

.6129 

12 

48 

.7683 

.6401 

12 

48 

.7455 

.6665 

12 

50 

.7898 

.6134 

10 

50 

.7679 

.6406 

10 

50 

.7451 

.6670 

10 

52 

.7894 

.6138 

8 

52 

.7675 

.6410 

8 

52 

.7447 

.6674 

8 

54 

.7891 

.6143 

6 

54 

.7672 

.6414 

6 

54 

.7443 

.6678 

6 

56 

.7887 

.6147 

4 

56 

.7668 

.6419 

4 

56 

.7439 

.66b3 

4 

58 

.7884 

.6152 

2 

58 

.7664 

.6423 

2 

58 

.7435 

.6687 

2 

»8°0' 

.7880 

.6157 

52°0' 

0°0' 

.7660 

.6428 

50°0' 

2°0' 

.7431 

.6691 

48°0' 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

or 

or 

or 

or 

E.  W. 

N.  S. 

E.  W. 

N.  S. 

E.  W. 

N.  S. 

TRAVERSE   TABLE. 


Traverse  Table  for  a  Distance  =  I.    (CONCLUDED.) 


Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

or 

or 

or 

or 

or 

or 

N.  S. 

E.  W. 

N.S. 

K.  W. 

N.S. 

E.W. 

12°0 
2 

.7431 

.7428 

.6691 
.6696 

48°0' 
58 

W°0' 
2 

.7314 
.7310 

.6820 
.6824 

47°0' 
58 

44°0' 
2 

.7193 
.7189 

.6947 
.6951 

46°0' 

58 

4 

.7424 

.6700 

56 

4 

.7306 

.6828 

56 

4 

.7185 

.6955 

56 

6 

.7420 

.6704 

54 

6 

.7302 

.6833 

54 

6 

.7181 

.6959 

54 

8 

.7416 

.6709 

52 

8 

.7298 

.6837 

52 

8 

.7177 

.6963 

52 

10 

.7412 

.6713 

50 

10 

.7294 

.6841 

50 

0 

.7173 

.6967 

59 

12 

.7408 

.6717 

48 

12 

.7290 

.6845 

48 

2 

.7169 

.6972 

48 

14 

.7404 

.6722 

46 

14 

.7286 

.6850 

46 

4 

.7165 

.6976 

46 

16 

.7400 

.6726 

44 

16 

.7282 

.6854 

44 

6 

.7161 

.6980 

44 

18 

.7396 

.6730 

42 

18 

.7278 

.6858 

42 

8 

.7157 

.6984 

42 

20 

.7392 

.6734 

40 

20 

.7274 

.6862 

40 

20 

.7153 

.6988 

40 

22 

.7388 

.6739 

38 

22 

.7270 

.6867 

38 

22 

.7149 

.6992 

38 

24 

.7385 

.6743 

36 

24 

.7206 

.6871 

36 

24 

.7145 

.6997 

36 

26 

.7381 

.6747 

34 

26 

.7262 

.6875 

34 

26 

.7141 

.7001 

34 

28 

.7377 

.6752 

32 

28 

.7258 

.6879 

32 

28 

.7137 

.7005 

32 

30 

.7373 

.6756 

30 

30 

.7254 

.6884 

30 

30 

.7133 

.7009 

30 

32 

.73(59 

.6760 

28 

32 

.7250 

.6888 

28 

32 

.7128 

.7013 

28 

34 

.7365 

.6764 

26 

34 

.7246 

.6892 

26 

34 

.7124 

.7017 

26 

36 

.7361 

.6769 

24 

36 

.7242 

.6896 

24 

36 

.7120 

.7021 

24 

38 

.7357 

.6773 

22 

38 

.7238 

.6900 

22 

38 

.7116 

.7026 

22 

40 

.7.353 

.6777 

20 

40 

.7234 

.6905 

20 

40 

.7112 

.7030 

20 

42 

.7349 

.6782 

18 

42 

.7230 

.6909 

18 

42 

.7108 

.7034 

18 

44 

.7345 

.6786 

16 

44 

.7226 

.6913 

16 

44 

.7104 

.7038 

16 

46 

.7341 

.6790 

14 

46 

.7222 

.6917 

14 

46 

.7100 

.7042 

14 

48 

.7337 

.6794 

12 

48 

.7218 

.6921 

12 

48 

,7096 

.7046 

12 

50 

.7333 

.6799 

10 

50 

.7214 

.6926 

10 

50 

.7092 

.7050 

10 

52 

.7329 

.6803 

8 

52 

.7210 

.6930 

8 

52 

.7088 

.7055 

8 

54 

.7325 

.6807 

6 

54 

.7206 

.6934 

6 

54 

.7083 

.7059 

6 

56 

.7321 

.6811 

4 

56 

.7201 

.6938 

4 

56 

.7079 

.7063 

4 

58 

.7318 

.6816 

2 

58 

.7197 

.6942 

2 

58 

.7075 

.7067 

2 

t3°0' 

.7314 

.6820 

47°0' 

44°0' 

.7193 

.6947 

46°0' 

45°0' 

.7071 

.7071 

45°0' 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

or 

or 

or 

or 

E.  W. 

N.S. 

E.  W. 

N.S. 

E.  W. 

N.S. 

When  the  anffle  exceeds  45°,  the  lats  and  deps  are  read  upward  from  the  bottom. 
Rent. — Since  these  lats  and  deps  are  for  a  dist  1,  we  may  proceed  as  follows  for  greater  dists. 
Thus,  let  the  dist  be  856.1.    Add  together  800  times,  50  times,  6  times,  and  ^  time  the  corresi>ond- 
ing  lats  and  deps  of  the  table. 

Ex.— What  is  the  lat  and  dep  for  856.1  feet ;  the  angle  being  43°  ? 
Here  for  43°  we  have  from  the  table,  lat  .7314;  dep  .6820. 

Hence,  .7314  X  800  =  585.12;  and  .6820  X  800  =.  545.60 
.7314  X  50  r=  36.57  ;  and  .6820  X  50  =  34.10 
.7314  X  6=  4.39 ;  and  .6820  X  6=  4.09 
.7314  X  .1  =  .07  ;  and  .6820  X  -1  =  .07 

Lat  626. 15  Dep  583.86 

These  multiplications  may  be  made  mentally.     Or  we  may,  with  a  little  more  trouble,  mult  the  lat 
and  dep  of  the  table  by  the  given  dist.    Thus, 

.7314  X  856.1  =  626.15  lat ;  and  .6820  X  856.1  =  583.86  dep.* 


jasmuch  as  the  engineer  but  rarely  needs  a  traverse  table,  we  have  thought  it  best  to  give  a 
st  one,  rather  than  the  common  one  for  y±  degrees.    The  first  involves  more  trouble  in  using  it ; 

ie  last  is  entirely  unfit  for  other  than  the  rude  calculations  for  common  surveying  with  compass 

courses  taken  to  the  nearest  y±  degree. 

To  divide  a  scale  of  one  mile  into  feet,  first  cut  off  one-sixth  of  it; 

then  divide  the  remainder  into  four  equal  parts.    Each  of  these  parts  will  be  1100  feet. 


*  Inasmuch 
correc 
but  the 
courses 


90 


LAND   SURVEYING. 


LAND  SUKVEYING, 


IN  surveying  a  tract  ef 
ground,  the  sides  which  com* 
pose  its  outline  are  desig. 
nated  by  numbers  in  th» 
order  in  which  they  occur. 
That  end  of  each  side  whicb 
first  presents  itself  in  the 
course  of  the  survey,  may  be 
called  its  near  end  ;  aud  the 
other  its  far  end.  The  num- 
ber of  each  side  is  placed  at  iti 
far  end.  Thus,  in  Fig  1,  the 
survey  being  supposed  to 
commence  at  the  corner  6, 
and  to  follow  the  direction 

is  6,  1 ;  and  its  number  is 
placed  at  its  far  end  at  1 ; 
and  so  of  the  rest.  Let  N  S 
be  a  meridian  line,  that  is,  a 
north  and  south  line;  and 
E  W  an  east  and  west  line. 
Then  in  any  side  which  runs 
northwardly,  whether  due 
north,  or  northeast,  as  side  2;  or  northwest,  as  sides  5  and  1,  the  dist  in  a  due  north  direction 
between  its  near  end  and  its  far  end,  is  called  its  northing ;  thus,  a  1  is  the  northing  of  side  1 ;  1  ft 
the  northing  of  side  2;  4  c  of  side  5.  In  like  manner,  if  any  side  runs  in  a  southwardly  direction, 
whether  due  south,  or  southeastwardly,  as  side  3 ;  or  south westwardly,  as  sides  4  and  6,  the  corre- 
sponding dist  in  a  due  south  direction  between  its  near  end  and  its  far  end,  is  called  its  southing  ;  thus, 
d  3  is  the  southing  of  side  3 ;  3  e  of  side  4 ;  /  6  of  side  6.  Both  northings  and  southings  are  included  in  the 
general  term  Difference  of  Latitude  of  a  side ;  or  more  commonly,  but  erroneously,  its  latitude.  The  dist 
due  east,  or  due  west,  between  the  near  and  the  far  end  of  any  side,  is  in  like  manner  called  the  easting, 
or  westing  of  that  side,  as  the  case  may  be  ;  thus,  6  a  is  the  westing  of  side  1 ;  5/  of  side  6 ;  c5  of  side 
5  ;  e  4  of  side  4 ;  and  b  2  is  the  easting  of  side  2 ;  2  rf  of  side  3.  Both  eastings  and  westings  are  included 
in  the  general  term  Departure  of  a  side ;  implying  that  the  side  departs  so  far  from  a  north  or  south 
direction.  We  may  employ  the  directions  (or  courses,  or  bearings,  as  they  are  usually  called)  of  sides, 
as  verbs;  and  say  that  a  side  norths,  wests,  southeasts,  &c.  We  shall  call  the  northings,  southings, 
Ac,  the  Ns,  Ss,  Es,  and  Ws;  the  latitudes,  lats ;  and  the  departures,  deps.  The  preceding  Traverse 
Table  consists  of  the  lats,  (or  Ns  and  Ss  ;)  and  the  deps,  (or  Es  and  Ws,)  corresponding  to  diff  angles 
or  courses,  for  a  side  whose  length  is  1 ;  therefore,  to  obtain  the  actual  lat  and  dep  of  any  given  side, 
those  taken  from  the  table  must  be  mult  by  the  length  of  the  side.  Beyond  44°,  the  lats  and  deps 
of  this  table  must  be  read  upward  from  the  bottom  of  the  page.  The  angles  in  the  table  are  those 
whic*  the  course  or  bearing  of  any  side  would  make  with  a  meridian  line  drawn  through  either  end 
of  said  side ;  but  it  is  self-evident  that  what  would  be  N  from  one  end,  would  be  S  from  the  other ;  and 
so  of  E  and  W ;  in  other  words,  the  angle  is  the  same  at  both  ends  ;  but  the  direction  is  reversed. 

Perfect  accuracy  is  unattainable  in  any  operation  involving  the  measurements  of  angles  and  dists. 
That  work  is  accurate  enough,  which  cannot  be  made  more  so  without  an  expenditure  more  than  com- 
mensurate with  the  object  to  be  gained.  The  writer  conceives  that  the  accuracy  essential  to  constitute 
practically  fair  surveying  is  purely  a  matter  of  dollars  and  cents.  In  the  purchase  and  sale  of  tracts 
of  land,  such  as  farms,  &c,  an  uncertainty  of  about  1  part  in  200  respecting  the  content,  and  consequently 
respecting  the  price,  probably  never  prevents  a  transfer ;  and  on  this  principle  we  assume  that  a  survey 
which  proves  itself  within  that  limit,  may  ordinarily  be  regarded  as  accurate  enough.  There  is  no 
great  difficulty  in  attaining  this  limit,  which,  if  exceeded,  is  the  result  of  bad  work.  Many  circum- 
stances combine  to  render  trifling  errors  absolutely  unavoidable :  *  they  always  become  apparent 
when  we  come  to  work  out  the  field  notes  ;  and  since  the  map  or  plot  of  the  survev,  and  the  calcula- 
tions for  ascertaining  the  content,  should  be  consistent  within  themselves,  we  do  what  is  usually  called 
correcting  the  errors,  but  what  in  fact  is  simply  humoring  them  in.  no  matter  how  scientific  the  pro- 
cess may  appear.  We  distribute  them  all  around  the  survey.  Two  methods  are  used  for  this  purpose, 
both  based  upon  precisely  the  same  principle;  one  of  them  mechanical,  by  means  of  drawing;  the 
other  more  exact,  but  much  more  troublesome,  by  calculation.  We  shall  describe  both  :  but  will  state 
now,  that  by  proportioning  the  scale  of  the  plot  in  the  following  manner,  the  mechanical  method 
becomes,  in  the  hands  of  a  correct  draftsman,  sufficiently  exact  for  all  ordinary  purposes.  Add  all 
the  sides  in  feet  together ;  and  div  the  sum  by  their  number,  for  the  average  length.  Div  this  average 
by  8;  the  quot  will  be  the  proper  scale  in  feet  per  inch.  In  other  words,  take  about  8  ins  to  represent 
an  average  side.t  We  shall  take  it  for  granted,  that  an  engineer  does  not  consider  it  accurate  work  to 

*  A  100  ft  measuring-chain  may  vary  its  length  5  feet  per  mile,  between  winter  and  summer,  by 
mere  change  of  temperature ;  and  by  this  alone  we  shall  make  a  difference  of  about  1-jl  acres  in  a  lot 
1  mile  square,  which  contains  640  acres.  Even  this  error  amounts  to  1  acre  in  533.  Not  one  farmer 
in  a  hundred  would  dream  of  pavine  for  a  scrupulouslv  correct  survey  aitd  plot  of  his  property. 

t  It  will  seldom  happen  that  precisely  this  quotient  will  be  adopted  for  a  scale.  For  instance,  if  the 
quot  should  be  738  feet,  or  83  feet,  per  inch,  we  should  adopt  the  most  convenient  near  number,  if 
smaller  the  better,  as  700,  or  80;  or  rather  more  than  8  inches  to  a  side.  With  a  scale  so  proportioned, 
and  with  good  drawing  instruments,  the  error  in  protracting  (excluding  of  course  errors  of  the  field 
work)  will  rarely  exceed  about  ^JJ-Q  part  of  the  periphery  of  the  plot;  and  the  area  may  be  found 
mechanically  by  dividing  the  plot  into  triangles,  within  ^-i-ff  Part  °*  the  truth.  This  remark  applies 
particularly  to  such  plots  as  may  be  protracted,  and  computed  within  a  period  so  short  as  not 
to  allow  the  paper  to  contract  or  expand  appreciably  by  atmospheric  changes.  A  larger  scale 
will  insure  proportionally  greater  accuracy.  The  young  assistant  should  practise  plotting 
from  perfectly  accurate  data ;  as,  for  instance,  from  the  example  given  in  the  table,  p.  97,  or, 


LAND   SURVEYING. 


91 


measure  his  angles  to  the  nearest  quarter  of  a  degree,  which  is  the  usual  practice  among  land-survey 
ors.  They  can,  by  means  of  the  engineer's  transit,  now  in  universal  use  on  our  public  works,  be  readily 
measured  within  a  miuute  or  two ;  and  being  thus  much  more  accurate  than  the  compass  courses, 
(which  cannot  be  read  off  so  closely,  and  which  are  moreover  subject  to  many  sources  of  error,)  they 
serve  to  correct  the  latter  in  the  office.  The  noting  of  the  courses,  however,  should  not  be  confined  to 
the  nearest  quarters  of  a  degree,  but  should  be  read  as  closely  as  the  observer  can  guess  at  the  minutes. 
The  back  courses  also  should  be  taken  at  every  corner,  as  an  additional  check,  and  for  the  detection 
of  local  attraction.  It  is 
well  in  taking  the  com- 
pass bearings,  to  adopt 
as  a  rule,  always  to  point 
the  north  of  the  compass- 
box  toward  the  object 
whose  bearing  is  to  be 
taken,  and  to  read  off 
from  the  north  end  of  the 
needle.  A  person  who 
uses  indifferently  the  N 
and  the  S  of  the  box,  and 
of  the  needle,  will  be  very 
liable  to  make  mistakes. 
It  is  best  to  measure  the 
least  angle  (shown  by 
dotted  arcs,  Fig  2,)  at  the 
corners ;  whether  it  be 
exterior,  as  that  at  corner 
5;  or  interior,  as  all  the 
others ;  because  it  is  al- 
ways less  than  180° ;  so 
that  there  is  less  danger 
of  reading  it  off  incor- 
rectly, than  if  it  exceeded 
180°;  taking  it  for  grant- 

ed  that  the  transit  instrument  is  graduated  from  the  same  zero  to  180°  each  way  ;  if  it  is  graduated 
from  zero  to  360°  the  precaution  is  useless.  When  the  small  angle  is  exterior,  subtract  it  from  360° 
for  the  interior  one. 

Supposing  the  field  work  to  be  finished,  and  that  we  require  a  plot  from  which  the  contents  may 
be  obtained  mechanically,  by  dividing  it  into  triangles,  (the  bases  and  heights  of  which  may  be 
measured  by  scale,  and  their  areas  calculated  one  by  one,)  a  protraction  of  it  may  be  made  at  once 
from  the  field  notes,  either  by  using  the  angles,  or  by  first  correcting  the  bearings  by  means  of  the 
angles,  and  then  using  them.  The  last  is  the  best,  because  in  the  first  the  protractor  must  be  moved 
to  each  angle  ;  whereas  in  the  last  it  will  remain  stationary  while  all  the  bearings  are  being  pricked 
off.  Every  movement  of  it  increases  the  liability  to  errors.  The  manner  of  correcting  the  bearings 
is  explained  on  the  next  page. 

In  either  case  the  protracted  plot  will  certainly  not  close  precisely ;  not  only  in  consequence  of  errors  in 
the  field  work,  but  also  in  the  protracting  itself.  Thus  the  last  side.  No  6,  Fig  2,  instead  of  closing  in  at 
corner  6,  will  end  somewhere  else,  say,  for  instance,  at  «;  the  dist  t  6  being  the  closing  error,  which, 
however,  as  represented  in  Fig  2,  in  more  than  ten  times  as  great,  proportionally  to  the  size  of  the 
survey,  as  would  be  allowable  in  practice.  Now  to  humor-in  this  error,  rule  through  every  corner 
a  short  line  parallel  to  t  6;  and,  in  all  cases,  in  the  direction  from  t  (wherever  it  may  be)  to  the 
starting  point  6.  Add  all  the  sides  together  ;  and  measure  t  6  by  the  scale  of  the  plot,  then  begin- 
ning at  corner  1,  at  the  far  end  of  side  1,  say,  as  the 

Sum  of  all        .        Total  closing 
the  sides          •  error  1 6 

Lay  off  this  error  from  1  to  a.     Then  at  corner  2,  say,  as  the 

Sum  of  all        .        Total  closing        .  .  Sum  of 

the  sides          •  error  t  6  •  •       sides  1  and  2 

Which  error  lay  off  from  2  to  b  ;  and  so  at  each  of  the  corners;  always  using,  ns  the  third  term,  the 
sum  of  the  sides  between  the  starting  point  and  the  given  corner.  Finally,  join  the  poiuts  a,  b,  c, 
d,  e,  6;  and  the  plot  is  finished. 

The  correction  has  evidently  changed  the  length  of  every  side;  lengthening  some  and  shortening 
others.  It  has  also  changed  the  angles.  The  new  lengths  and  angles  may  with  tolerable  accuracy 
be  found  by  means  of  the  scale  and  protractor  ;  and  be  marked  on  the  plot  instead  of  the  old  ones. 

from  those  to  be  found  in  books  on  surveying.  This  is  the  only  way  in  which  he  can  learn  what  is 
meant  by  accurate  work.  His  semicircular'protractor  should  be  about  9  to  12  ins  in  diam  and  gradu- 
ated to  10  min.  His  straight  edge  and  triangle  should  be  of  metal:  w.e  prefer  German  silver,  which 
does  not  rust  as  steel  does;  and  thev  should  be  made  with  scrupulous  accuracy  by  a  skilful  instru- 
ment maker.  A  very  fine  needle,  with  a  sealing-wax  head,  should  be  used  for  pricking  off  dists  and 
angles  ;  it  must  be  held  vertically  ;  and  the  eye  of  the  draftsman  must  be  directly  over  it.  The  lead 
pencil  should  be  hard  (Faber's  No.  4  is  pood  for  protracting),  and  must  be  kept  to  a  sharp  point  by 
rubbing  on  a  fine  file,  after  using  a  knife  for  removing  the  wood.  The  scale  should  be  at  least  as  long 
as  the  longest  side  of  the  plot,  and  should  be  made  at  the  edge  of  a  strip  of  the  same  paper  as  the  plot 
is  drawn  on.  This  will  obviate  to  a  considerable  extent,  errors  arising  from  contraction  and  expan- 
sion. Unfortunately,  a  sheet  of  paper  does  not  contract  and  expand  in  the  same  proportion  length- 
wise and  crosswise,  thus  preventing  the  paper  scale  from  being  a  perfect  corrective.  In  plots  of  com- 
mon farm  surveys,  &c,  however,  the  errors  from  this  source  may  be  neglected.  For  such  plots  as  may 
be  protracted,  divided,  and  computed  within  a  time  loo  short  to i  admit  of  appreciable  change,  the  ordi- 
nary scales  of  wood,  ivorv  or  metal  mav  be  used  ;  but  satisfactory  accuracv  cannot  be  obtained  with 
them  on  plots  requiring  several  days,  if  the  air  be  meanwhile  alternately  moist  and  dry.  or  subject  to 
considerable  variations  in  tempera'ture.  What  is  called  parchment  paper  is  worse  in  this  respect  thau 
good  ordinary  drawing-_paper. 

With  the  foregoing  precautions  we  may  work  from  a  drawing,  with  as  much  accuracy  as  is  usually 
attaimed  in  the  field  work. 


Error 
for  side  1. 


Error 
for  side  2. 


92 


LAND   SURVEYING. 


When  the  plot  has  many  sides,  this  calculating  the  error  for  each  of  them  becomes  tedious ;  and 
since,  in  a  well-performed  survey  and  protraction,  the  entire  error  will  be  but  a  very  small  quantity, 
(it  should  not  exceed  about  ^^  part  of  the  periphery,)  it  may  usually  be  divided  among  the  sides  by 
merely  placing  about  34,  %,  and  %  of  it  at  corners  about  J4,  *4,  and  %  way  around  the  plot;  and  at 
n  ;  .  •  intermediate  corners  proper- 

-,  •">  1 "    1 u      tion  it  by  eye.    Or  calculation 

I J? 1      "  "I       may   be   avoided    entirely   by 

J  U  drawing  a  line  a  6  of  a  length 
equal  to  the  united  lengths 
of  all  the  sides  ;  dividing  it 
into  distances  a,  1;  1,  2  ;  &c,  equal  to  the  respective  sides.  Make  b  c  equal  to  the  entire  closing  error  ; 
join  a  c;  and  draw  1,1';  2,  2' ,  &c,  which  will  give  the  error  at  each  corner. 

When  the  plot  is  thus  completed,  it  may  be  divided  by  fine  pencil  Hues  into  triangles,  whose 
bases  and  heights  may  be  measured  by  the  scale,  in  order  to  compute  the  contents.  With  care  in 
both  the  survey  and  the  drawing,  the  error  should  not  exceed  about  -R-^TT  Part  of  the  true  area.  At 
least  two  distinct  sets  of  triangles  should  be  drawn  and  computed,  as  a  guard  against  mistakes ;  and  if 
the  two  sets  differ  in  calculated  contents  more  than  about  -^ ^-^  part,  they  have  not  been  as  carefully 
prepared  as  they  should  have  been.  The  closing  error  due  to  imperfect  field- work,  may  be  accurately 
calculated,  as  we  shall  show,  and  laid  down  on  the  paper  before  beginning  the  plot ;  thus  furnishing 
a  perfect  test  of  the  accuracy  of  the  protraction  work,  which,  if  correctly  done,  will  not  close  at  the 
point  of  beginning,  but  at  the  point  which  indicates  the  error.  But  this  calculation  of  the  error,  by 
a  little  additional  trouble,  furnishes  data  also  for  dividing  it  by  calculation  among  the  diff  sides  : 
besides  the  means  of  drawing  the  plot  correctly  at  once,  without  the  use  of  a  protractor  ;  thus  ena- 
bling us  to  make  the  subsequent  measurements  and  computations  of  the  triangles  with  more  cer- 
tainty. 

We  shall  now  describe  this  process,  but  would  recommend  that  even  when  it  is  employed,  and 
especially  in  complicated  surveys,  a  rough  plot  should  first  be  made  and  corrected,  by  the  first  of  the 
two  mechanical  methods  already  alluded  to.  It  will  prove  to  be  of  great  service  in  us'ing  the  method 
by  calculation,  inasmuch  as  it  furnishes  an  eye  check  to  vexatious  mistakes  which  are  otherwise  apt 
to  occur  ;  for,  although  the  principles  involved  are  extremely  simple,  and  easily  remembered  when 
once  understood,  yet  the  continual  changes  in  the  directions  of  the  sides  will,  without  great  care, 
cause  us  to  use  Ns  instead  of  Ss ;  Es  instead  of  Ws,  &c. 

We  suppose,  then,  that  such  a  rough  plot  has  been  prepared,  and  that  the  angles,  bearings,  and 
distances,  as  taken  from  the  field  book,  are  figured  upon  it  in  lead  pencil. 

Add  together  the  interior  angles  formed  at  all  the  corners  :  call  their  sum  a.  Mult  the  number  of 
sides  by  180° ;  from  the  prod  subtract  360°  :  if  the  remainder  is  equal  to  the  sum  o,  it  is  a  proof  that 
the  angles  have  been  correctly  measured.*  This,  however,  will  rarely  if  ever  occur ;  there  will 
always  be  some  discrepancy  ;  but  if  the  field  work  has  been  performed  with  moderate  care,  this  will 
not  exceed  about  two  min  for  each  angle.  In  this  case  div  it  in  equal  parts  among  all  the  angles, 
adding  or  subtracting,  as  the  case  may  be,  unless  it  amounts  to  less  than  a  min  to  each  angle,  when 
it  may  be  entirely  disregarded  in  common  farm  surveys.  The  corrected  angles  may  then  be  marked 
on  the  plot  in  ink,  and  the  pencilled  figures  erased.  We  will  suppose  the  corrected  ones  to  be  as 
shown  in  Fig  3. 

Next,  by  means  of  these 
corrected  angles,  correct  the 
bearings  also,  thus,  Fig  3  ; 
Select  some  side  (the  longer 
the  better)  from  the  two  ends 
of  which  the  bearing  and  the 
reverse  bearing  agreed  ;  thus 


bearing 
fluenced 


was  probably  not  influ 
by  local  attraction.  Let  side 
2'be  the  one  so  selected  ;  as- 
sume its  bearing,  N  75°  32'  E, 
as  taken  on  the  ground,  to  be 
correct;  through  either  end 
of  it,  as  at  its  far  end  2,  draw 
the  short  meridian  line  ;  par- 
allel to  which  draw  others 
through  every  corner.  Now, 
having  the  bearing  of  side  2, 
N  75°  32'  E.  and  requiring 
that  of  side  3,  it  is  plain  that 
the  reverse  bearing  from  cor- 
ner 2  is  S  75°  32'  W  ;  and 
that  therefore  the  angle  1,  2, 
m,  is  75°  32'.  Therefore,  if  we 
take  75°  32'  from  the  entire 
corrected  angle  1  ,  2,  3,  or  144° 
57',  the  rem  69°  25'  will  be 
the  angle  m  23  ;  consequently 
the  bearing  of  side  3  must  be 

8  69°  25'  E.    For  finding  the  bearing  of  side  4,  we  now  have  the  angle  23  a  of  the  reverse  bearing  of 
Bide  3,  also  equal  to  69°  25'  ;  aud  if  we  add  this  to  the  entire  corrected  angle  234,  or  to«      32  ,  we  ha.\ 
'  '  '  ° 


, 

the  angle  a  34=69°  25'  4-  69°  32'  =  138°  57'  ;  which  taken  from  180°,  leaves  the  angle 
the  bearing  of  side  4  must  be  S  41°  3'  W.     For  the  bearing  of  side  owe  n 

' 


,  . 

4: 


*  Because  in  every  straight-lined  figure  the  sum  of  all  its  interior  angles  is  equal  to  twice  as  many 
right  angles  as  the  figure  has  sides,  minus  4  right  angles,  or  360°.. 


LAND   SURVEYING. 


careful  observation  Is  necessary  to  see  ho-w  the  several  angles  are  to  be  employed  at  each  corner. 
Rules  are  sometime*  given  for  this  purpose,  but  unless  frequently  used,  they  are  soon  forgotten. 
The  plot  mechanically  prepared  obviates  the  necessity  for  such  rules,  inasmuch  as  the  principle  of 
proceeding  thereby  becomes  merely  a  matter  of  sight,  and  tends  greatly  to  prevent  error  from  using 
the  wrong  bearings  ;  while  the  protractor  will  at  once  detect  any  serious  mistakes  as  to  the  angles, 
and  thus  prevent  their  being  carried  further  along.  After  having  obtained  all  the  corrected  bearings, 
they  may  be  figured  on  the  plot  instead  of  those  taken  in  the  field.  They  will,  however,  require  a 
still  further  correction  after  a  while,  since  they  will  be  affected  by  the  adjustment  of  the  closing  error. 
We  now  proceed  to  calculate  the  closing  error  tf>  of  Fig  2,  which  is  done  on  the  principle  that  in  a 
correct  survey  the  northings  will  be  equal  to  the  southings,  and  the-  eastings  to  the  westings.  Pre- 
pare a  table  of  7  columns,  as  below,  and  in  the  first  3  cols  place  the  numbers  of  the  sides,  and  their  cor- 
rected courses ;  also  the  dists  or. lengths  of  the  sides,  as  measured  on  the  rough  plot,  if  such  a  one 
has  been  prepared ;  but  if  not,  then  as  measured  on  the  ground.  Let  them  be  aa  follows  : 


Side. 

Bearing. 

Dist.  Ft. 

Latitudes. 

Departures. 

N. 

S. 

E. 

W. 

1 
2 
3 
4 
5 
tt 

N  16°  40'  W 
N  75°  32'  E 
S  69°  25'  E 
S  41°    3'  W 
N  79°  40'  W 
S  53°  30'  W 

1060 
1202 
1110 
850 
802 
705 

1015.5 
300.3 

143.9 

390.2 
641. 

419.3 

1163.9 
1039.2 

304. 

558.2 
789. 
566.7 

1459.7 
1450.5 

1450.5 

Error  in 
Lat. 

2203.1 

Error  in 
Dep. 

2217.9 
2203.1 

9.2 

14.8 

Now  find  the  N,  S,  E.  W,  of  the  several  sides,  and  place  them  in  the  corresponding  four  columns, 
thus :  By  means  of  the  Traverse  Table  find  out  the  lat  and  dep  for  the  angle  of  each  course.  Mult  each 
of  them  by  the  length  of  the  side ;  and  place  the  prod  in  the  corresponding  col  of  N,  S,  E,  W.  Thus, 
for  side  1,  which  is  1060  feet  long,  the  latitude  from  the  traverse  table  for  16°  40'  is  .9580;  and  the 
departure  is  .2868:  and  .«5hO  X  1060  =  1015.5  lat;  which,  since  the  side  norths,  we  put  in  the  N 
col.  Again,  .2868  X  1060  =  30*  dep;  which,  since  the  side  wests,  we  put  in  the  W  col.  Proceed 
thus  with  all.  Add  up  the  four  cols ;  find  the  diff  between  the  N  and  S  cols ;  and  also  between 
the  E  and  W  ones.  In  this  instance  we  find  that  the  Ns  are  9.2  feet  greater  than  the  Ss  ;  and  that 
the  Ws  are  14.8  ft  greater  than  the  Es  ;  in  other  words,  there  is  a  closing  error  which  would  cause  a 
correct  protraction  of  our  first  three  cols,  to  terminate  9.2  feet  too  far  north  of  the  starting  point ;  and 
14.8  feet  too  far  west  of  it.  So  that  by  placing  this  error  upon  the  paper  before  beginning  to  protract, 
we  should  have  a  test  for  the  accuracy  of  the  protracting  work  ;  but,  as  before  remarked,  a  little  more 
trouble  will  now  enable  us  to  div  the  error  proportionally  among  all  the  Ns,  Ss,  Es,  and  Ws,  and  thereby 
give  as  data  for  drawing  the  plot  correctly  at  once,  without  using  a  protractor  at  all. 

To  divide  the  errors,  prepare  a  table  precisely  the  same  as  the  foregoing,  except  that  the  hor  spaces 
are  farther  apart;  and  that  the  addings-up  of  the  old  N,  S,  E,  W  columns  are  omitted.  The  addition* 
here  noticed  are  made  subsequently. 

The  new  table  is  on  the  next  page. 

REMARK.  The  bearing  and  the  reverse  bearing:  from  the  two  ends 
of  a  line  will  not  read  precisely  the  same  angle ;  and  the  difference  varies  with  the 
latitude  and  with  the  length  of  the  line,  but  not  in  the  same  proportion  with  either. 
It  is,  however,  generally  too  small  to  be  detected  by  the  needle,  being,  according  to 
Gummere,  only  three  quarters  of  a  minute  in  a  line  one  mile  long  in  lat  40°.  In 
higher  lats  it  is  more,  and  in  lower  ones  less.  It  is  caused  by  the  fact  that  meridians 
or  north  and  south  lines  are  not  truly  parallel  to  each  other;  but  would  if  extended 
meet  at  the  poles. 

Hence  the  only  bearing*  that  can  be  run  in  a  straight  line, 

•with  strict  accuracy,  is  a  true  N  and  S  one ;  except  on  the  very  equator,  where  alone  a  due  E  and  W 

one  will  also  be  straight.    But  a  true  curved  JE  and  W  line  may  be  found 

anywhere  with  sufficient  accuracy  for  the  surveyor's  purposes  thus.  Having  first  by  means  of  the  N 
star  (p  99)  or  otherwise  got  a  true  N  and  S  bearing  at  the  starting  point,  lay  off  from  it  90°,  for  a  true 
E  and  W  bearing  at  that  point.  This  E  and  W  bearing  will  be  tangent  to  the  true  E  and  W  curve. 
Run  this  tangent  carefully  :  aud  at  intervals  (say  at  the  end  of  each  mile)  lay  oft'  from  it  (towards 
the  N  if  in  N  lat,  or  vice  versa)  an  offset  whose  length  in  feet  is  equal  to  the  proper  one  from  the 
following  table,  multiplied  by  the  square  of  the  distance  in  miles  from  the  starting  point.  These 
offsets  will  mark  points  in  the  true  K  and  W  curve. 


15°         20° 


Latitude  N  or  S. 

25°          30°          35°          40°          45° 


Offsets  in  ft  one  mile  from  starting  point. 

.058        .118        .179        .243        .311        .385        .467         .559        .667          .795          .952         1.15         1.43 

Or,  any  offset  in  ft  =  .6666  X  Total  Dist  in  miles*  X  Nat  Tang  of  Lat. 

A   rhtimb  line  is  any  one  that  crosses  a  meridian  obliquely,  that  is,  is 
neither  due  N  and  S,  nor  E  and  W. 


94 


LAND   SURVEYING. 


Side. 

Bearing. 

Dist.  Ft. 

Latitudes. 

Departures. 

N. 

S. 

E. 

W. 

1 
2 
3 
4 

5 

N  16°  40'  W 
N  75°  32'  E 
S  G9°  25'  E 
S  41°    3'  W 

N  79°  4(K  W 
S  53°  30'  W 

1060 
1202 
1110 
850 

802 
705 

1015.5 
1.7 

304.0 
2.7 
...    301.3 

1013  8 

300.3 
1.9 

1163.9 
3.1 

558.2 
2.2 
556  o 

298.4 

143.9 
1.3 

...  1167.0 

1039.2 
2.9 

390.2 
1.8 

392    ... 

641.0 
1.3 
642.3... 

...  1042.1 

789.0 
2.1 

1426 

...    7869 

419.3 
1.1 

566.7 
1.8 

420.4... 

564.9 

5729 
Sum  of 
Sides. 

1454.8 
Cor'd  Ns. 

1454.7 
Cor'd  Ss. 

2209.1 
Cor'd  Es. 

2209.1 
Cor'd  Ws. 

Now  we  have  already  found  by  the  old  table  that  the  Ns  and  the  Wa  are  too  long ;  consequently 
they  must  be  shortened  ;  while  the  Ss,  and  Es,  must  be  lengthened;  all  in  the  following  proportions: 
As  the 

Sura  of  all     .     Any  given     ..     Total  err  of    .    Err  of  lat,  or  dep, 
the  sides      •          side         •  •      lat  or  dep      •        of  given  side. 
Thus,  commencing  with  the  lat  of  side  1,  we  have,  as 

Sum  of  all  the  sides.     .     Side  1.     .  .     Total  lat  err.     .    Lat  err  of  side  1. 

5729  •        1060       •  •  9.2  •  1.7 

Now  as  the  lat  of  side  1  is  north,  it  must  be  shortened ;  hence  it  becomes  =  1015.5  — 1.7  =  1013.8,  as 
figured  out  in  the  new  table.  Again  we  have  for  the  departure  of  side  1, 

Sum  of  all  the  sides.     .     Sidel.     ..     Total  dep  err.     .     Dep  err  of  side  1. 

5729  •       1060       •  •  14.8  •  2.7 

Now  as  the  dep  of  side  1  is  west,  it  must  be  shortened ;  hence  it  becomes  304  —  2.7  =  301.3,  as  figured 
out  in  the  uew  table. 

.,  Proceeding   thus   with  each 

N  side,  we  obtain  all  the  corrected 

lats  and  deps  as  shown  in  the 
new  table ;  where  they  are  con- 
nected with  their  respective 
sides  by  dotted  lines;  but  in 
practice  it  is  better  to  cross  out 
the  original  ones  when  the  cal- 
culation is  finished  and  proved. 
If  we  now  add  up  the  4  cols  of 
corrected  N,  S,  E,  W,we  find  that 
the  Ns  =  the  Ss ;  and  the  Es  = 
theWs;  thus  proving  that  the 
work  is  right.  There  is,  it  is 
true,  a  discrepancy  of  .1  of  a  ft 
between  the  Ns.  and  the  Ss ;  but 
this  is  owing  to  our  carrying 
out  the  corrections  to  only  one 
decimal  place ;  and  is  too  small 
to  be  regarded.  Discrepancies 
of  3  or  4  tenths  of  a  foot  will 
sometimes  occur  from  this 
otouse ;  but  may  be  neglected. 
The  corrected  lats  and  depa 
must  evidently  change  the 
bearing  and  distance  of  every 
irvey  by  means  of  the  corrected 


Ftp"  4 

•ide ;  but  without  knowing  either  of  these,  we  can  now  plot  the 


LAND   SURVEYING. 


95 


hits  and  deps  alone.    The  principle  is  self-evident,  explaining  itself.    First  draw  a  meridian  line 
N  S,  Fig  4  ;  and  upon  it  fix  on  a  point  1,  to  represent  the  extreme  west*  corner  of  the  survey. 

Then  from  the  point  1,  prick  off  by  scale,  northward,  the  dist  1,  2  =the  corrected  northing  298.4 
of  side  2,  taken  from  the  last  table  ;  from  2'  southward  prick  off  the  dist  2',  3',  the  corrected  south- 
iug  392  of  side  3;  from  3'  southward  prick  off  3',  4',  =  southing  642.3  of  side  4;  from  4'  northward 
prick  off  4',  5'  =  northing  142.6  of  side  5;  from  5'  prick  off  southward  5',  6'  =  southing  of  side  6.t 
Then  from  the  points  2',  3',  4',  5',  6',  draw  indefinite  lines  due  eastward,  or  at  right  angles  to  the 
meridian  line.  Make  by  scale,  2',  2  =  corrected  departure  of  side  2;  and  join  1,  2.  Make  3',  3  =  dep 
of  side  2-j-depof  side  3 ;  and  join  2,  3;  make  4',  4  =  3',  3  — dep  of  side  4;  and  join  3,  4;  make  5',  5 
=  4',  4  — dep  of  side  5;  andjoiu4,5;  make  6',  6=5',  5  — dep  of  side  6;  and  join  5,  6}  Finally  join 
6.  1 ;  and  the  plot  is  complete.  If  scrupulous  accuracy  is  not  required,  the  contents  may  be  found  by 
tne  mechanical  method  of  triangles;  the  bearings,  by  the  protractor;  and  the  lengths  of  the  sides, 
by  the  scale  ;  all  with  an  approximation  sufficient  for  ordinary  purposes;  and  perhaps  quite  as  close 
as  by  the  method  by  calculation,  when,  as  is  customary,  the  bearings  are  taken  only  to  the  nearest 

quarter  of  a  degree.  We  have  already  said  that  with  a  scale  of  feet  per  inchr=  — 

the  error  of  area  need  not  exceed  the  YO^th  part. 

But  if  it  is  required  to  calculate  the  area  of  the  corrected  survey  with  rigorous  exactness,  it  may 
be  done  on  the  following 
principle,  (see  Fig  5.)  If  a  ti 
meridian  line  N  S  be  sup-  N 
posed  to  be  drawn  through 
the  extreme  west  corner  1  of 
a  survey;  and  lines  (called 
middle  distances)  drawn  (as 
the  dotted  ones  in  the  Fig) 
at  right  angles  to  said  me- 
ridian, from  the  center  of  1 
each  side  of  the  survey; 
then  if  each  of  th«  middle 
dists  of  such  sides  as  have 
northings,  be  mult  by  the 
corrected  northing  of  its  cor- 
responding side ;  and  if  each 
of  the  middle  dists  of  such 
sides  as  have  southings,  be 
mult  by  the  corrected  south- 
ing of  its  corresponding 
side ;  if  we  add  all  the  north 
prods  into  one  sum ;  and  all 
the  south  prods  into  another 
<um;  and  subtract  the  least 

>f  these  sums  from  the  great- 

5st,  the  rein  will  be  the  area 


Fid  5 


*  The  extreme  east  corner  would  answer  as  well,  with  a  slight  change  in  the  subsequent  oper- 
ations, as  will  become  evident. 

t  Instead  of  pricking  off  these  northings  and  southings  in  succession,  from  each  other,  it  will  be 
more  correct  in  practice  to  prepare  first  a  table  showing  how  far  each  of  the  points  2',3',  &c,  is  north 
or  south  from  1.  This  being  done,  the  points  can  be  pricked  off  north  or  south  from  1,  without  mov- 
ing the  tcale  each  time ;  and  of  course  with  greater  accuracy.  Such  a  table  is  readily  formed.  Rule 
it  as  below;  and  in  the  first  three  columns  place  the  numbers  of  the  sides  (starting  with  side  2  from 
point  1 ;)  and  their  respective  corrected  northings  and  southings.  The  formation  of  the  4th  and  5th 
cols  by  means  of  the  3d  and  4th  ones,  explains  itself.  Its  accuracy  is  proved  by  the  final  result 
being  0. 


Side. 

N.  lat. 

S.  lat. 

Dist  N  or  S  f 

N. 

rom  Point  1. 
S. 

2 
3 
4 
5 
6 
1 

298.4 

142.6 
1013.8 

392. 
612.3 

420.4 

298.4 
000.0 

93.6 
735.9 
593.3 
1013.7 
000.0 

t  A  similar  table  should  be  prepared  beforehand  for  the  dists  of  the  points  2,  3,  4,  &c,  east  from  the 
meridian  line.  It  is  done  in  the  same  manner,  but  requires  one  col  less,  as  all  the  dists  are  on  the 
«ame  side  of  the  mer  line.  Thus,  starting  from  point  1,  with  side  2 : 


Side. 

E.  dep. 

W.  dep. 

Dist  east  from 
meridian  line. 

2 
3 
4 
5 
6 
1 

1167.0 
1042.1 

556.0 
786.9 
564.9 
301.3 

1167.0 
2209.1 
1653.1 
866.2 
301.3 
000.0 

'  This  work  likewise  proves  itself  by  the  final  result  being  0. 


96 


LAND  SURVEYING. 


of  the  survey.*  The  corrected  northings  and  southings  we  have  already  found  ;  as  also  the  eastings 
and  westings.  The  middle  dists  are  found  by  means  of  the  latter,  by  employing  their  halves ;  adding 
half  eastings,  and  subtracting  half  westings.  Thus  it  is  evident  that  the  middle  dist  2'  of  side  2,  is 
equal  to  half  the  easting  of  side  2.  To  this  add  the  other  half  easting  of  side  2,  aud  a  half  easting 
of  side  3;  and  the  sum  is  plainly  equal  to  the  middle  dist  3'  of  side  3.  To  this  add  the  other  half 
easting  of  side  3,  and  subtract  a- half  westing  of  side  4,  for  the  middle  di^t  4'  of  side  4.  From  this 
subtract  the  other  half  westing  of  side  4,  and  a  half  westing  of  side  5,  for  the  middle  dist  5'  of  .side 
5;  and  so  on.  The  actual  calculation  may  be  made  thus  : 

Half  easting  of  side  2  =    ^-  =    583.5  E  =  mid  dist  of  side  2. 
2  583.5  E 


1042.1      1167.0  E 

Half  easting  of  side  3  = =    521.0  E 

2  

1688.0  E  =  mid  dist  of  side  3. 
521.0  E 

556          2209.0  E 
Half  westing  of  side  4  =  —     =    278.0  W 

1931.0  E  =  mid  dist  of  side  4. 
278.0  W 

786.9        1653.0  E 
Half  westing  of  side  5  =  — —  =     393.5  W 

2  

1259.5  E  =  mid  dist  of  side  5. 
393.5  W 


564.9        866.0  E 

Half  westing  of  side  6  =  =    282.4  W 

2 


583.6  E  =  mid  dist  of  side  6. 
282.4  W 


_.*- — '  301.3        301.2  E 

Half  westing  of  side  1  =  — -  =    150.6  W 

150.6  E  =  mid  dist  of  side  1. 

The  work  always  proves  itself  by  the  last  two  results  being  equal. 

Next  make  a  table  like  the  following,  in  the  first  4  cols  of  which  place  the  numbers  of  the  sidea, 
the  middle  dists,  the  northings,  and  southings.  Mult  each  middle  dist  by  its  corresponding  northing 
or  southing,  and  place  the  products  in  their  proper  col.  Add  up  each  col ;  subtract  the  least  from  the 


Side. 

Middle  dist. 

Northing. 

Southing. 

North  prod. 

South  prod. 

1 

2 
3 
4 
5 

6 

150.6 
583.5 
1688 
1931 
1259.5 
583.6 

1013.8 

298.4 

142.6 

392 
642.3 

420.4 

152678 
174116 

179605 

661696 
1240281 

245345 

506399 
435 

2147322 
506399 

30)1640923(37.67  Acres. 

*  Proof.  To  illustrate  the  principle  upon  which  this 
rule  is  based,  let  ab,  be,  and  c  a,  Fig  6,  represent  in 
order  the  3  sides  of  the  triangular  plot  of  a  survey,  with 
a  meridian  line  d/drawn  through  the  extreme  westcor- 
ner,  a.  Let  lines  b  d  and  c/  be  drawn  from  each  corner, 
perp  to  the  meridian  line ;  also  from  the  middle  of  each 
side  draw  lines  we,  mn,  so,  also  perp  to  meridian  ;  and 
representing  the  middle  dists  of  the  sides.  Then  since 
the  sides  are  regarded  in  the  order  ah,  be,  ca,  it  is 
plain  that  a  d  represents  the  northing  of  the  side  aft; 
fa  the  northing  of  ca;  and  df  the  southing  of  &  c. 
Now  if  we  mult  the  northing  ad  of  the  side  a  b,  by  its 
mid  dist  ew,  the  prod  is  the  area  of  the  triangle  abd. 
In  like  manner  the  northing  fa  of  the  side  ca,  mult  by 
its  mid  dist  s  o,  gives  the  area  of  the  triangle  a  ef.  Again, 
the  southing  df  of  the  side  be,  mult  by  its  mid  dist  mn, 
gives  the  area  of  the  entire  fig  dbcfd.  If  from  this 
area  we  subtract  the  areas  of  the  two  triangles  ab  d, 
and  a  cf,  the  rem  is  evidently  the  area  of  the  plot  a  b  c. 
80  with  any  other  plot,  however  complicated. 


lAND  SURVEYING. 


97 


rrcatest.    The  rem  will  be  the  area  of  the  survey  in  sq  ft ;  which,  div  by  43560,  (the  number  of  sq  ft 
ID  an  acre,)  will  be  the  area  in  acres ;  in  this  instance,  37.67  ac. 

It  now  remains  only  to  calculate  the  corrected  bearings  and  lengths  of  the  sides  of  the  survey,  all 
o"f  which  are  necessarily  changed  by  the  adoption  of  the  corrected  lats  and  deps.  To  find  the  bearing 
of  any  side,  div  its  departure  (E  or  W)  by  its  lat  (N  or  S) ;  in  the  table  of  nat  tang,  tiud  the  quot ; 

the  angle  opposite  it  is  the  reqd  angle  of  bearing.    Thus,  for  the  course  of  side  2,  we  have  - — ^ 

=  .2972  — nat  tang ;  opposite  which  in  the  table  is  the  reqd  angle,  16°  33' ;  the  bearing,  therefore,  is 
N  163  33'  W. 

Again :  for  the  dist  or  length  of  any  side,  from  the  table  of  nat  secants*  take  the  sec  opposite  to  the 
angle  of  the  corrected  bearing ;  mult  it  by  the  corrected  lat  (N  or  fc>)  of  the  side.     Thus,  for  the  dist 
•f  side  1,  we  find  opposite  16°  33',  the  sec  1.0432.    And 
Sec.  Lat. 

1.0432  X  1013.8  =  1057.6  the  reqd  dist. 
Thfl  following  table  contains  all  the  corrections  of  the  foregoing  survey ;  consequently,  if  the  bear. 


Side. 

Bearing. 

Dist.  Ft. 

1 
2 
3 
4 
5 
6 

N  16°  33'  W 
N  76°  39'  E 
S  69°  23'  E 
S  40°  53'  W 
N  79°44'W 
S  53°  21'  W 

1057.6 
1204.0 
1113.3 
849.6 

800.1 
704.3 

* , *...* 


ings  and  dists  are  correctly  plotted,  they  will  close  perfectly.  The  young  assistant  is  advised  to 
practise  doing  this,  as  well  as  dividing  the  plot  into  triangles*  and  computing  the  content.  In  this 
manner  he  will  soon  learn  what  degree  of  care  is  necessary  to  insure  accurate  results. 

Under  the  heads  Mensuration,  Geometry,  and  Trigonometry  will  be  found  much  pertaining  to  land 
surveying.  See  Remark  after  Parallelograms.  The  following  hints  may  often  be  of  service. 
1st.  Avoid  taking  bearings  and 
dists  along  a  aircuitous  bound- 
ary line  like  a  b  c,  Fig  7  ;  but  run 
the  straight  line  a  c;  and  at 
right  angles  to  it,  measure  off- 
sets to  the  crooked  line.  2d. 
Wishing  to  survey  a  straight 
line  from  a  to  c,  but  being  una- 
ble to  direct  the  instrument 
precisely  toward  c,  on  account 
of  intervening  woods,  or  other 
obstacles ;  first  run  a  trial  line, 
as  a  m,  as  nearly  in  the  proper 
direction  as  can  be  guessed  at. 
Measure  m  c,  and  say,  as  a  m  is  to  w  c,  so  Is  100  ft  to  ?  Lay  off  a  o  equal  to  100  ft,  and  o  *  equal 
to  ?  ;  and  run  the  final  Hne  a  *  c.  Or.  if  m  c  is  quite  small,  calculate  offsets  like  o  g  for  every  100  ft 
along  a  m,  and  thus  avoid  the  necessity  for  running  a  second  line.  8d.  When  c  is  visible  from  a,  but 
the  intervening  ground  difficult  to  measure  along,  on  account  of  marshes,  &c.  extend  the  side  y  a 
to  good  ground  at  t :  then,  making  the  angle  y  t  d  equal  to  y  a  c,  run  the  line  t  n  to  that  point  d  at 
which  the  angle  n  d  c  is  found  by  trial  to  be  equal  to  the  angle  a  t  d.  It  will  rarely  be  necessary  to 
make  more  than  one  trial  for  this  point  d;  for,  suppose  it  to  be  made  at  x,  see  where  it  strikes  a  c  at 
t;  measuret  c,  and  continue  from  x,  making  x  d  =tc.  4th.  In  case  of  a  very  irregular  piece  of 
land,  or  a  lake,  Fig  8,  surround  it  by  straight  lines.  Survey  these,  and  at  right  angles  to  them, 
measure  offsets  to  the  crooked  boundary.  5th.  Surveying  a  straight  line  from  w  toward  y,  Fig  9, 


»n  Obstacle,  o,  is  met.  To  pass  it,  lay  off  a  right  angle  wtu;  measure  any  t  u ;  make  t  u  v  = 
90° ;  measure  u  v ;  make  u  v  i  ~  90°  ;  make  v  i  ~  t  u ;  make  viy  =  90°.  Then  is  e  t  ~  «  v ;  and 
iy  u  in  the  straight  line.  Or,  with  less  trouble,  at  g  make  t  g  n  ~  60 ;  measure  any  g  a;  make 
g  a  «=r60° ;  and  a  s  ~  g  a:  make  a  s  i  —  60°.  Then  is  g  s  r=  g  a  or  as;  nnd  i  s,  continued  toward 
y,  is  in  the  straight  line.  6th.  Being  between  two  objects,  m  and  n,  and  wishing  to  place  myself  in 
range  with  them,  I  lay  a  straight  rod  c  b  on  the  ground,  and  point  it  to  one  of  the  objects  m  ;  then 
going  to  the  end  c,  I  find  that  it  does  not  poit)t  to  the  other  object.  By  successive  trials,  I  find  the 
position  e  d,  in  which  it  points  to  both  objects,  and  consequently  is  in  Vange  with  them.  If  no  rod 

*  Our  table  does  not  con  tain  nat  secants;  but  the  nat  sec  of  any  angle  is  readily  found,  thus  :  Di« 
vide  1  by  the  nat  fcosine  of  the  angle. 

7 


LAND   SURVEYING. 


is  at  hand,  two  stones  will  answer,  or  two  chain-pins.    A  plumb-line  (a  pebble  tied  to  a  piece  of 
thread)  will  add  to  the  accuracy  of  ranging  the  rod,  or  stones,  &c. 


THE  FOl^OWING  TABLE 

gives  deductions  or  additions  to  be  made  every  10O  ft  as  actually  chained  along  sloping 
ground,  iu  order  to  reduce  the  sloping  measurements  to  Horizontal  ones.  Even  when  it  is  so  nearly 
level  that  the  eye  cannot  detect  the  slope,  an  over-measurement  of  an  inch  or  two  in  100ft  may 
readily  occur,  it  is  plain,  that,  if  we  measure  all  the  undulations  of  the  ground,  we  shall  get 
greater  totals  than  if  we  measure  hor,  as  is  supposed  always  to  be  done ;  but  since  few  surveyors 
pretend  to  measure  hor  until  the  slope  becomes  apparent  to  the  eye,  their  lines  are  usually  -too  long 
by  from  one  to  two  ins  in  100  feet.  To  counteract  this  to  some  extent,  chains  are  frequently  made 
from  one  to  two  ins  longer  than  100  feet;  and  for  ordinary  purposes  the  precaution  is  a  pood  one. 
When  greater  accuracy  is  required  the  chainmen  should  be  attended  by  a  third  person,  with  a  rod 
and  slope-level,  for  taking  the  inclinations  of  the  ground.  These  deductions  being  made,  the  remain- 
der will  be  the  actual  hor  dist. 

For  example,  in  Fig  10^,  each  100  f t  a  o  measured  np  or  down  the 
slo|>e  ae  plainly  corresponds  to  the  shorter  horizontal  distance  a  c;  the 
difference  or  deduction  being  c  n.  Taking  a  o  as  Rad.  then  a  c  is  the 
cosine,  and  c  n  the  versed  sine  of  the  angle  e  a  n  of  the  slope.  Thore- 
fore  a  o  multiplied  by  the  nat.  cosine  of  the  ansrte  e  n  n  rives  the  reduced 
hor  dist  n  r. ;  which  tak^O  from  ao  gives  tho  deduction  en  of  oar  table. 
But  If  while  chaining  along  the  slope  n  «  we  wish  to  drive 
stakes  that  shall  correspond  with  hor  dist*  u  n  of  100  ft.  it  is  evident 
that  we  must  add  c  n  to  each  100  ft  a  o,  as  shown  ;it  x  e ;  and  the  stake 
mu«t  be  driven  at  e,.  instead  of  at  o.  Observe  that  x  e  =.  C  n  must 
be  measured  horizontally. 
When  the  ground  is  very  sloping,  alt  this  calcula-iou  may  be  avoided  where  great  accuracy  is  not 
required,  by  actually  holding  the  chain  horizontal,  as  nearly  as  can  he  judged  by  ey  ,  and  finding, 
by  means  of  a  plumb-line,  where  its  raised  end  would  strike  the  ground.  A  whole  chain  at  a  time 
cannot  be  measured  in  this  way;  but  shorter  distances  must  be  taken  as  the  ground  requires;  at 
times,  on  very  steep  ground,  not  more  than  5  or  10  feet.  See  note,  p  40. 

Table  of  Deductions  or  Additions  to  be  made  per  1OO  feet, 
iu  chaining  over  sloping  ground. 

IX  ORDER  TO  REDUCE  THE  INCLINED  MEASUREMENTS  TO  HORIZONTAL  ONKB 

See  p  629  for  another  table. 


c 


/Slope 
in 
Deg. 

Deduct 
Feet. 

Rise  in 
100ft 
hor. 

Slope 
Deg. 

Deduct 
Feet. 

Rise  in 
100  ft 
hor. 

Slope 
Deg. 

Deduct 
Feet. 

Rise  in 
100  ft 
hor. 

Slope 
in 
Deg. 

Deduct 
Feet. 

Rise  in 

.  ido  ft 

hor. 

H 

.001 

.436 

K 

.420 

9.189 

y± 

1.596 

1808 

H 

3.521 

27.26 

9 

.004 

.873 

i^ 

.460 

-   9.629 

H 

1.675 

18.53 

H 

3.637 

27.73 

H 

.009 

1.309 

H 

.503 

10.07 

H 

1.755 

18.99 

34 

3.754 

28.20 

i 

.015 

1.746 

6 

.548 

10.51 

11 

1.837 

19.44 

16 

3.874 

28.67 

X 

.024 

2.182 

% 

.594 

10.95 

% 

1.921 

19.89 

H 

3.995 

29.15 

K 

.034 

2.619 

% 

.643 

11.39 

M 

2.008 

20.35 

X 

4.118 

29.62 

% 

.047 

3.055 

H 

.693 

11.84 

H 

2.095 

20.80 

% 

4.243 

30.10 

2 

.061 

3.492 

7 

.745 

12.28 

12 

2.185 

21.26 

17 

4.370 

30.57 

/4 

.077 

3.929 

K 

.800 

12.72 

M 

2.277 

21.71 

y* 

4.498 

31.05 

y^ 

.095 

4.366 

N 

.856 

13.17 

*A 

2.370 

22.17 

4.628 

31.53 

K 

.115 

4.803 

% 

.913 

13.61 

H 

2.466 

22.63 

*4 

4.7HO 

32.01 

« 

.137 

5.241 

8 

.973 

14.05 

13 

2.563 

23.01) 

18 

4.894 

32.49 

34 

.161 

5.678 

y* 

1.035 

14.50 

2.662 

23  55 

H' 

5.030 

32.98 

i^ 

.187 

6.116 

y* 

1.098 

14.95 

% 

2.763 

24.01 

X 

5.168 

33.46 

h 

.214 

6.554 

H 

1.164 

15.39 

5i 

2.86B 

24.47 

H 

5.307 

3395 

4 

.244 

6.993 

9 

1.231 

15.84 

14 

2.970 

24.93 

19 

5.448 

34.43 

X 

.275 

7.431 

H 

1  .300 

16.29 

y* 

3.077 

25.40 

« 

5.591 

34.92 

% 

.308 

7.870 

% 

1.371 

16.73 

y* 

3.185 

25.86 

X 

5.736 

35.41 

H 

.343 

8.309 

X 

1.444 

17.18 

H 

3.295 

26.33 

% 

5.882 

35.90 

5 

.381 

8.749 

10 

1.519 

17.63 

15 

3  407 

26.79 

20 

I-   6.031 

36.40 

Chain  and  Pins. 


Engineers  have  abandoned  the  Gunter's  chain  of  66  ft,  div  into  100  links  of  7.92  ins  in  length  ;  and 
use  one  of  100  ft.  with  links  1  ft  long ;  and  calculate  areas  in  sq  ft ;  which,  div  by  43560,  reduces  them 
to  acres  and  decimal  parts,  instead  of  roods  and  perches.  Both  the  chain  and  thevchaiu  pins.  Fig.  11, 
•hould  be  of  good  strong  steel;  and  there  should  be  a  stout  leather  bag  for  carrying  them.  To  bear  ham- 
mering into  hard  ground,  the  pins  may  be  of  this  shape 
and  size,  11  or  12  ins  long,  H  inch  thick,  %  wide,  head 
'2%  wide,  with  a  circular  hole  of  1  %  diam.  Each  pin 
should  have  a  strip  of  bright  red  flannel  tied  to  its  top, 
th.it  it  may  be  readily  found  among  grass,  &c.  by  the 
hind  c'hainman.  The  length  of  the  chain  should  be 
tested  every  few  days;  and  the  target-rod  may  be  used 
for  this  purpose.  While  locating,  it  is  well  to  have  the  chain  one  or  two  ins  longer  than  100  feet. 
Steel  wire,  No  11  or  12,  is  a  good  size  for  a  100  ft  chain.  This  is  scant  H  inch  diam. 


41 
1  1 
'1  • 


LAND   SURVEYING. 


99 


Nat  Sines  of  Polar  Dists  of  Polaris  or  X.  Star. 


Year. 

1880 

1881 
1882 

Sine. 

Year. 

Sine. 

Year. 

1886 

1887 
1888 

Sine. 

|  Year. 

Sine. 

Year. 

Sine. 

Year. 

Sine. 

.0232 
.0231 

.0230 

1883 

1884 
1885 

.0229 
.0229 
.0228 

.0227 
.0226 
.0225 

1889 
1890 
1891 

.0224 
.0223 
.0222 

1892 
1893 
1894 

.0221 
.0220 
.0219 

1895 
1896 
1897 

.0218 
.0217 
.0216 

Nat  Secants  of  North  Latitudes. 


Lat. 

Sec. 

Lat. 

Sec. 

Lat. 

Sec. 

Lat. 

Sec. 

Lat. 

Sec. 

Lat. 

Sec. 

0° 

1.000 

24° 

1.095 

1^0 

1.196 

/f 

1.296 

3/0 

1.433 

52° 

1.624 

2 

1.001 

72 

1.099 

IX 

1.199 

M 

1.301 

46 

1.440 

74 

1.633 

4 

1.002 

25 

1.103 

ax 

1.203 

1.305 

IX 

1.446 

7? 

1.643 

5 

1.004 

IX 

1.108 

34 

1.206 

i 

1.310 

17 

1.453 

7/t 

1.652 

6 

1.006 

26 

1.113 

IX 

1.210 

IX 

1.315 

74 

1.460 

53 

1.662 

7 

1.008 

IX, 

1.117 

IX 

1.213 

&2 

1.320 

47 

1.466 

/4 

1.671 

8 

1.010 

27 

1.122 

74 

1.217 

41 

1.325 

IX. 

1.473 

72 

1.681 

9 

1.013 

X2 

1.127 

35 

1.221 

ix. 

1.330 

17 

1.480 

74 

1.691 

10 

1.016 

28 

1.133 

i£ 

1.225 

/I 

1.335 

74 

1.487 

54 

1.701 

11 

1.019 

IX 

1.138 

IX 

1.228 

74 

1.340 

48 

1.495 

IX 

1.712 

12 

1.022 

29 

1.143 

37 

1.232 

42 

1.346 

1.502 

72 

1.722 

13 

1.026 

1.149 

36 

1.236 

I/ 

1.351 

ll 

1.509 

74 

1.733 

14 

1.031 

30 

1.155 

IX 

1.240 

IX 

1.356 

M 

1.517 

55 

1.743 

15 

1.035 

IX 

1.158 

7^ 

1.244 

?i 

1.362 

49 

1.524 

i4 

1.754 

16 

1.040 

IX 

1.161 

xl 

1.248 

43 

1.367 

74 

1.532 

1.766 

17 

1.046 

3X 

1.164 

37 

1.252 

74" 

1.373 

17 

1.540 

74 

1.777 

18 

1.052 

31 

1.167 

IX 

1.256 

1.379 

% 

1.548 

56 

1.788 

19 

1.058 

14 

1.170 

IX 

1.261 

74 

1.384 

50 

1.556 

74 

1.800 

20 

1.064 

ix 

1.173 

74 

1.265 

44 

1.390 

iX^ 

1.564 

IX 

1.812 

21 

1.071 

74 

1.176 

38 

1.269 

17 

1.396 

YL 

1.572 

74 

1.824 

1.075 

32 

1.179 

ix/ 

1.273 

1^ 

1.402 

74 

1.581 

57 

1.836 

1.079 

ix_ 

1.182 

72 

1.278 

74 

1.408 

51 

1.589 

74 

1.849 

72 

1.082 

IX 

1.186 

74 

1.282 

45 

1.414 

Yi 

1.598 

17 

1.861 

23 

1.086 

74 

1.189 

39 

1.287 

IX 

1.420 

17 

1.606 

37 

1.874 

^ 

1.090 

33 

1.192 

xi 

1.291 

8 

1.427 

» 

1.615 

58 

1.887 

To  find  a  Meridian  Line  (a  true  North  and  South  line)  by 
means  of  the  North  Star.    (Polaris.) 

The  north  star  appears  to  describe  a  small  circle,  n  n',  &c,  Fig  14,  around  the  true  north  point,  OP 
north  pole,  as  a  center.  The  rad  of  this  circle  is  estimated  bv  the  angle  between  the  star  and  the 
pole,  as  measured  from  the  earth  ;  and  is  called  the  polar  diet  of  the  star.  This  polar  dist  be- 
comes 19 yjy  seconds,  or  very  nearly  ^  of  a  minute  less  every  year.  On  Jan  1,  1880,  it  is,  approxi- 
mately, 1°  19  51".  On  the  first  of  1890,  it  will  be  about  1°  16'  41",  &c.  When,  in  its  revolution,  the 
star  is  farthest  castor  west  from  the  pole,  as  at  n'  or  n",  it  is  said  to  be  at  its  greatest  E  or 
W  elongation.  Then  its  apparent  motion  for  several  min  is  nearly  vert,  and  consequently 
affords  the  best  opportunity  for  an  observation  in  the  simple  manner  here  described.  The  arrows  in 
Fig  14  show  the  direction  in  which  the  stars  appear  to  move  from  east  to  west  when  the  spectator 
faces  the  north. 

The  latitude  of  the  place  must  be  known  approximately.  Taking  it  at  the  closest  one  in  our  forego- 
ing table,  the  error  will  not  exceed  half  a  min  in  lat  57°,  or  one-quarter  min  in  lat  40° ;  and  still  less 
in  lower  lats. 

About  3  ft  above  ground  fix  firmly,  perfectly  level,  and  as  nearly  east  and  west  as  may  be,  a 
smooth  narrow  piece  of  board,  about  3  ft  long,  to  serve  as  a  kind  of 
table.  Also  prepare  another  piece  a  a,  about  a  foot  long ;  and  fasten 
toil,  at  right  angles,  a  compass-sight,  or  astripof  thin  metal,  with 
a  straight  slit,  < shown  by  a  black  line  in  the  fig.)  about  6  ins  long 
and  JL  inch  wide.  This  piece  of  board  is  to  be  slid  along  the 
table,  as  the  observer  follows  the  motions  of  the  star  toward  the 
east  or  west:  looking  at  it  through  the  vert  slit.  Plant  a  stout 
pole,  about  20  ft  long,  firmly  in  the  ground,  with  its  top  as  nearly 

north  as  possible  from  the  middle  of  the  table.     Its  top  should  ±tq.12 

lean  2  or  3  ft  toward  either  the  east  or  the  west ;  and  a  plumb- 
line  must  be  suspended  from  its  top,  with  a  bob  weighing  one  or  two  Ibs,  -which  may  swing  in  a 
bucket  of  water  plnced  orf  the  ground.     This  is  to  prevent  the  line  from  being  so  easily  moved  by 
slight  currents  of  air  ;  and  for  further  steadiness,  the  pole  itself  should  be  well  braced  from  within,  a 


100 


LAND   SURVEYING. 


few  feet  below  Its  top.  The  proper  dist  a  o,  of  the  pole  p  o,  from  the  table  t  a,  may  be  found 
thus :  Make  an  angle  n  m  s,  equal  approximately  to  the  lat  of  the  place.  Open  a  pair  of  dividers 
to  equal,  by  any  convenient  scale,  the  height  t  a  of  the  table ;  and  draw  t  a.  Then  take,  by  the 
same  scale,  the  height,  p  o,  of  the  pole  above  ground  ;  and  place 
it  upon  the  sketch,  so  that  the  top  p  shall  be  by  scale  a  ft  01 
two  above  m  n.  Then  a  o,  by  the  same  scale,  will  be  about  th#- 
dist  reqd ;  probably  from  3  to  5  yards.  A  deviation  of  a  ft  or  so 
from  this  will  not  be  important. 

The  correct  clock  time  at  any  place,  for  the  elongation,  may  be 
found  within  a  few  min  from  the  following  table. 

Instead  of  a  pole  and  plumb-line,  the  writer  would  suggest  a 
planed,  straight-edged  board  planted  vert  and  braced;  its  aid* 
toward  the  observer. 

The  observer  should  be  at  his  station  at  least  %  of  an  hour  in 
advance  of  the  time.  Placing  the  board  a  a,  upon  the  table  and 
in  range  with  the  plumb-line  and  star,  he  will  watch  both  of  them 
—5  through  the  vert  slit ;  sliding  the  board  along  the  table,  so  as  to 
keep  the  slit  in  the  range  as  long  as  the  star  continues  to  move 
toward  the  east  or  west,  as  the  case  may  be.  An  assistant  must 
hold  a  caudle,  or  lantern,  on  a  pole  near  the  plumb-line,  to  enable 
the  observer  to  see  the  latter.  As  the  star  approaches  its  elonga- 
tion, it  will  appear  to  move  nearly  vert  for  several  min.  so  that  it  can  be  seen  without  moving  the 
slit.  When  certain  by  this  that  the  star  has  reached  its  elongation,  confine  the  sliding  board  to  the 
table  by  sticking  a  few  tacks  around  its  edges.  Then  let  a  third  person,  with  another  caudle,  go  off 
some  dist,  (a  hundred  yards  or  more  if  convenient,)  in  a  direction  toward  the  star  ;  and  then  drive 
a  stake  as  directed  by  the  observer,  who  will  take  care  that  it  is  exactly  in  range  with  the  slit  and 
plumb-line.  Another  stake  must  then  be  driven  exactly  under  either  the  slit  or  the  plumb-line. 
Having  thus  placed  the  two  stakes  in  the  range  of  the  elongation,  defer  the  remainder  of  the  operation 
•ntil  morning.  From  the  tables  given  above  take  out  the  sine  of  the  polar  dist,  and  also  the  secant 
of  the  lat.  Mult  these  together.  The  prod  will  be  the  nat  sine  of  an  angle  called  the  azimuth 
of  the  star.  Find  the  sine  in  table,  p  102,  &c.  and  the  angle  which  corresponds  to  it.  This  azimuth 
angle  will  be  between  1°  20'  and  2°  30',  according  to  lat.  Place  an  instrument  over  the  S  stake,  sight 
to  the  N  one,  and  lav  off  this  angle  to  the  E  if  the  elong  was  W,  or  vice  versa,  and  drive  a  stake  to 
mark  it.  This  last  direction  is  true  N  and  S.  It  might  be  supposed  that  after  driving  the  first  two 
stakes,  a  true  meridian  could  be  had  by  merely  laying  off  the  polar  dist,  by  means  of  a  compass  or 
transit ;  but  this  is  not  so.  Place  the  compass  over  the  south  stake,  and  take  sight  to  the  north  one. 
If,  then,  the  north  end  of  the  needle  points  east  of  the  line,  the  variation  of  the  compass  is  east,  and 
rice  versa. 

Times  by  a  correct  clock  of  Elongations  of  the  N.  Star. 

Deduced  from  U.  S.  Coast  Survey  table,  calculated  for  April  1,  1883,  to  April  1,  1884,  and  for  lat 
38°  N  ;  but  will  answer  within  about  5  minutes  for  any  lat  up  to  60°  N,  and  until  1890. 

Times  of  Eastern  Elongations. 


Dav  of 
Month. 

Apr. 

May. 

June. 

July. 

AUK. 

Sep. 

7 
13 
19 
25 

H.  M. 
6  37  A  M 
6  14  " 
5  50  " 
5  26  " 
53" 

H.  M. 

4  39  A  M 
4  16  " 
3  52  " 
3  28  " 
35" 

H.  M. 
2  37  A  M 
2  14  " 
1  50  " 
1  26  " 
13" 

H.  M. 

12  39  A  M 
12  16  " 
11  52  P  M 
11  29  " 
11   5  " 

H.  M. 

10  37  P  M 
10  14  " 
9  50  " 
9  27  " 
93" 

H.  M. 
8  36  P  M 
8  12  " 

7  48  " 
7  25  " 
7   1  " 

Times  of  Western  Elongations. 


Dav  of 

Month. 

Oct. 

Nov. 

Dee. 

Jan. 

Feb. 

Mar. 

1 
7 
13 
19 
25 

H.    M. 
6    27AM 
64" 
5    40   " 
5    17   " 
4    53   " 

H.    M. 
4    25AM 
4       2    " 
3    38   " 
3    15   " 
2    51    " 

H.   M. 

2     28AM 
24" 
1     40    " 
1     17    " 
12    53    " 

H.    M. 
12     26  A  M 
12      2    " 
11     39  PM 
11     15     " 
10    51     " 

H.    M. 
10    24PM 
10    00     " 
9    36     " 
9     13     " 
8    49     " 

H.    M. 
8    30  P.  M. 
86" 
7    43      " 
7     19      " 
6    55      " 

For  days  of  the  month  intermediate  of  those  in  the  table,  it  will  be  near  enough  to  make  the  time 
4  min  earlier  each  succeeding  day. 

During  nearly  all  of  the  four  months,  March,  April.  September,  and  October,  the  elongations  take 
place  in  daylight ;  so  that  this  method  cannot  then  be  used.  Nor  can  it  be  used  at  an  v  time  in  places 
south  of  about  4°  north  of  the  equator,  because  there  the  north  star  is  not  visible.  But  during  that 
time  a  meridian  may  be  found  by  recollecting  that  when  the  north  star  n  is  on  the  meridian,  or,  in 
other  words,  is  truly  north  from  us,  the  star  Alioth,  a,  is  very  nearly  vertically  above  it,  if  the 
north  star  n  is  on  the  meridian  below  the  pole:  or  below  it,  as  at  a'",  if  the  north  star,  n'"  is  on 
the  meridian  above  the  pole.  When  the  north  starn'  is  at  its  east  elongation,  Alioth  is  horizon- 
tally west  of  it.  as  at  a':  and  when  the  north  star.  n".  is  at  its  west  elongation,  Alioth  is  horizontally- 
east  of  it,  as  at  o".  All  that  is  necessary,  therefore,  is  to  prepare  an  arrangement  of  table,  pole, 
plumb  line,  &c,  precisely  as  before  ;  except  that  the  plumb-line  must  be  nearer  the  observer,  as  he 
will  have  to  watch  Alioth  above  the  north  star.  Watch  through  the  movable  slit  until  Alioth  is  on 
the  same  vert  line  with  the  north  star.  Then  put  in  two  stakes  as  before,  and  they  will  be  nearly 
in  a  true  north  and  south  line.-  But  to  be  more  exact,  either  lay  off  (to  the  E  if  Alioth  is  above 


LAND    SURVEYING, 


There  can  be  no 
difficulty  in  fiud- 
ing  Alioth,  as  it 
is  one  of  the  7 
bright  stars  in 
the  fine  constel- 

known  as  the 
Greiit  Bear,  or 
the  Wagon  aud 
Horses.  Alioth  is 
the  horse  nearest 
tothe  fore-wheels 
of  the  wagon. 
The  two  hind- 
wheel  a  t  t  are 
known  to  every 
schoolboy  as  the 
"Pointers,"  he- 
cause  they  point 
nearly  in  the  di- 
rection  to  the 
North  Star.  The 
relative  positions 
of  these  7  stars, 
as  shown  in  Pig 
14,  are  tolerably 
correct. 


REMARK  OS*  THE  FOLLOWING  TAHI<E  OF  WATURAI, 

Ac. 


The  following  table  does  not  contain  nat  versed  sines,  co-versed  sines,  secants,  nor  cosecants, 
but  these  may  be  found  thus  :  for  any  angle  not  exceeding  90  degrees. 

Versed  Sine.     From  1  take  the  nat  cosine. 

Co-versed  Sine.     From  1  take  the  nat  sine. 

Secant.     Divide  1  by  the  nat  cosine. 

Cosecant.    Divide  1  by  the  nat  sine. 

For  angles  exceeding  9O°  ;  to  find  the  sine,  cosine,  tangent,  cotang,  secant,  or  cosec,  (but  not 
the  versed  sine  or  co-versed  sine),  take  the  angle  from  180°  :  if  between  180°  and  270°  take  180°  from 
the  angle  ;  if  bet  270°  and  360°,  take  the  angle  from  360°.  Then  in  each  case  take  from  the  table  the 
sine,  cosine,  tang,  or  cotang  of  the  remainder.  Find  its  secant  or  cosec  as  directed  above.  For  the 
versed  sine  ;  if  between  90°  and  270°,  add  cosine  to  1  ;  if  bet  270°  and  360°,  take  cosine  from  1.  (  The 
engineer  seldom  needs  sines,  Ac,  exceeding  180°. 

To  find  the  nat  sine,  eosine,  tang,  secant,  versed  sine,  drc, 

of  an  mijjl<*  containing  seconds.     First  find  that  due  to  the  given  deg 

and  min  ;  then  the  next  greater  one.     Take  their  dift'.     Then  as  60  sec  are  to  this  diff,  so  are  the  sec 

only  of  the  given  angle  to  a  dec  quantity  to  be  added  to  the  one  first  taken  out 
if  it  is  a  sine,  tang,  secant,  &c  ;  or  to  be  subtracted  from  it  if  it  is  a  cosine, 
cotang,  cosecant,  &c. 

The  tangents  in  the  table  are  strict  trigonometrical  ones;  that  is, 
tangents  to  given  angles,  as  described  on  p  63;  and  which  must  extend  to  meet  the 
secants  of  the  angles  to  which  they  belong. 

Ordinary,  or  geometrical  tangents,  as  those  on  p  66,  may  extend  as 
far  as  we  please. 

Ill  the  field  practice  of  railroad  curves,  two  trigonometrical  tan- 
gents terminate  where  they  meet  each  other;  and  the  corresponding  secants  would, 
if  drawn,  terminate  at  that  same  point. 


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CONTOUR   LINES.  147 

REM.  The  table,  pp.  608-612,  is  to  furnish  the  means  of  laying  down  angles  on 
paper  more  accurately  than  by  an  ordinary  protractor.  To  do  this,  after  having  drawn 
and  measured  the  first  side  (say  ac)  of  the  figure  that  is 

to  be  plotted ;  from  its  end  c  as  a  center,  describe  an  arc     -  - 

ny  of  a  circle  of  sufficient  extent  to  subtend  the  angle  at 
that  point.  The  rad  en  with  which  the  arc  is  described 
should  be  as  great  as  convenience  will  permit ;  and  it  is  to 
be  assumed  as  unity  or  1 ;  and  must  be  decimally  divided, 
and  subdivided,  to  be  used  as  a  scale  for  laying  down  the 
chords  taken  from  the  table,  in  which  their  lengths  are 
given  in  parts  of  said  rad  1.  Having  described  the  arc,  find 
in  the  table  the  length  of  the  chord  n  t  corresponding  to 
the  angle  act.  Let  us  suppose  this  angle  to  be  45°;  then 
we  find  that  the  tabular  chord  is  .7654  of  our  rad  1.  There- 
fore  from  n  we  lay  off  the  chord  n  t,  equal  to  .7654  of  our  radius-scale ;  and  the  lint 
cs  drawn  through  the  point  t  will  form  the  reqd  angle  act  of  45°.  And  so  at  each 
angle.  The  degree  of  accuracy  attained  will  evidently  depend  on  the  length  of  the 
rad,  and  the  neatness  of  the  drafting.  The  method  becomes  preferable  to  the  com- 
mon protractor  in  proportion  as  the  lengths  of  the  sides  of  the  angles  exceed  the  rad 
of  the  protractor.  With  a  protractor  of  4  to  6  ins  rad,  and  with  sides  of  angles  not 
much  exceeding  the  same  limits,  the  protractor  will  usually  be  preferable.  The  di- 
viders in  boxes  of  instruments  are  rarely  fit  for  accurate  arcs  of  more  than  about  6 
iris  diam.  In  practice  it  is  not  necessary  to  actually  describe  the  whole  arc,  but 
merely  the  portion  near  t,  as  well  as  can  be  judged  by  eye.  We  thus  avoid  much  use 
of  the  India-rubber,  and  dulling  of  the  pencil-point.  For  larger  radii  we  may  dis- 
pense with  the  dividers,  and  use  a  straight  strip  of  paper  with  the  length  of  the  rad 
marked  on  one  edge ;  and  by  laying  it  from  c  toward  s,  and  at  the  same  time  placing 
another  strip  (with  one  edge  divided  to  a  radius-scale)  from  n  toward  /,  we  can 
by  trial  find  their  exact  point  of  intersection  at  the  required  point  t.  In  such  mat- 
ters, practice  and  some  ingenuity  are  very  essential  to  satisfactory  results.  We  can- 
not devote  more  space  to  the  subject. 


CONTOUB  LINES, 


A  CONTOUR  LINE  is  a  curved  hor  one,  every  point  in  which  represents  the  same  level ; 
thus  each  of  the  contour  lines  88c,  9lc,  94c,  &c,  Fig  1,  indicates  that  every  point  in 
the  ground  through  which  it  is  traced  is  at  the  same  level ;  and  that  that  level  or 
height  is  everywhere  88,  91 ,  or  94  ft  above  a  certain  other  level  or  height  called 
datum ;  to  which  all  others  are  referred. 

Frequently  the  level  of  the  starting  point  of  a  survey  is  taken  as  being  0,  or  zero, 
or  datum ;  and  if  we  are  sure  of  meeting  with  no  points  lower  than  it,  this  answers 
every  purpose.  But  if  there  is  a  probability  of  many  Idwer  points,  it  is  better  to 
assume  the  starting  point  to  be  so  far  above  a  certain  supposed  datum,  that  none  of 
these  lower  points  shall  become  minus  quantities,  or  below  said  supposed  datum  or 
zero.  The  only  object  in  this  is  to  avoid  the  liability  to  error  which  arises  when 
some  of  the  levels  are  -{-,  or  plus  ;  and  some  — ,  or  minus.  Hence  we  may  assume 
the  level  of  the  starting  point  to  be  10,  100,  1000,  &c,  ft  above  datum,  according  to 
circumstances. 

The  vert  diets  between  each  two  contour  lines  are  supposed  to  be  equal ;  and  in 
railroad  surveys  through  well-known  districts,  where  the  engineer  knows  that  his 
actual  line  of  survey  will  not  require  to  be  much  changed,  the  dist  may  be  1  or  2  ft 
only  ;  and  the  lines  need  not  be  laid  down  for  widths  greater  than  100  or  200  ft  on 
each  side  of  his  center-stakes.  But  in  regions  of  which  the  topography  is  compara- 
tively unknown ;  and  where  consequently  unexpected  obstacles  may  occur  which 
require  the  line  to  be  materially  changed  for  a  considerable  dist  back,  the  observa- 
tions should  extend  to  greater  widths ;  and  for  expedition  the  vertical  dists  apart 
may  be  increased  to  3,  5,  or  even  10  ft,  depending  on  the  character  of  the  country, 
&c.  Also,  when  a  survey  is  made  for  a  topographical  map  of  a  State,  or  of  a  county, 
vert  dists  of  5  or  10  ft  will  generally  suffice. 

Let  the  line  A  B,  Fig  1,  starting  from  O,  represent  three  stations  (S  1,  S  2,  S  3,)  of 
the  center  line  of  a  railroad  survey ;  and  let  the  numbers  100,  103,  101,  104,  along 
that  line  denote  the  heights  at  the  stakes  above  datum,  as  determined  by  levelling. 
Then  the  use  of  the  contour  lines  is  to  show  in  the  ofnc«  what  would  be  the  effect 
of  changing  the  surveyed  center  line  A  B,  by  moving  any  part  of  it  to  the  right  or 


148 


CONTOUR   LINES. 


left  hand.*  Thus,  if  it  should  be  moved  100  ft  to  the  left,  the  starting  point  0  would? 
be  on  ground  about  6  ft-higher  than  at  present ;  inasmuch  as  its  level  would  then 
be  about  106  ft  above  datum,  instead  of  100.  Station  1  would  be  about  7  ft  higher 
or  110  ft  instead  of  103.  Station  2  would  be  about  7  ft  higher,  or  108  it  instead  of 
101.  If  the  line  be  thrown  to  the  right,  it  will  plainly  be  on  lower  ground. 

The  field  observations  for  contour  lines  are  sometimes  made  with  the  spirit-level ; 
but  more  frequently  by  a  slope-man,  with  a  straight  12-ft  graduated  rod,  and  a  slope 
instrument,  or  clinometer.  At  each  station  he  lays  his  rod  upon  the  ground,  as 


Rgl. 


nearly  at  right  angles  to  the  center  line  A  B  as  he  can  judge  by  eye ;  and  placing 
the  slope  instrument  upon  it,  he  takes  the  angle  of  the  slope  of  the  ground  to  the 
nearest  l/±  of  a  degree.  He  also  observes  how  tar  beyond  the  rod  the  slope  continue* 
the  same ;  and  with  the  rod  he  measures  the  dist.  Then  laying  down  the  rod  at  that 
point  also,  he  takes  the  next  slope,  and  measures  its  length  ;  and  so  on  as  far  as  may 
be  judged  necessary.  His  notes  are  entered  in  his  field-book  as  shown  in  Fig  2 ;  the 
angles  of  the  slopes  being  written  above  the  lines,  and  their  lengths  below ;  and 
should  be  accompanied  by  such  remarks  as  the  locality  suggests ;  such  as  wood$ 
rocks,  marsh,  sand,  field,  garden,  across  small  run,  &c,  &c. 

*  In  thus  using  the  words  right  and  left  we  are  supposed  to  have  our  backs  turned  to  the  starting 

point  of  the  survey.  In  a  river,  the  rig-lit  bank  or  shore  is  that  which 
is  on  the  right  hand  as  we  descertd  it,  that  is,  in  speaking  of  its  right  or  left 
bank,  we  are  supposed  to  have  our  backs  turned  towards  its  head,  or  origin ;  and  so  with  a  survey. 


CONTOUR    LINES. 


149 


It  Is  not  absolutely  necessary  to  represent  the  slopes  roughly  in  the  field-book,  as 
in  Fig  2;  for  by  using  the  sign  +  to  signify  "up;"  — "down;"  and  =  "level,'* 
the  slopes  may  be  writ- 
ten in  a  straight  line, 
as  in  Fig  2U. 

The  notes  naving  been 
taken,  the  preparation 
of  the  contour  lines  by 
means  of  them,  is  of 
course  office-work ;  and 
is  usually  done  at  the 
same  time  as  the  draw- 
ing of  the  map,  &c.  The  v 

16+  .      10-t-         20  + 3*     |5- 


Fid  a 


4--    .   26- 
84-     '  70 


field  observations  at  each 

station   are   then    sepa-    ( 

rately  drawn  by  protrac-  91  58 

tor  and  scale,  as  shown 
in  Fig  3  for  the  starting 
point  0.  The  scale  should  not  be  less  than  about  •fg  inch  to  a  ft,  if  anything  like 
accuracy  is  aimed  at.  Suppose  that  at  said  station  the  slopes  to  the  right,  taken  in 
their  order,  are,  as  in  Fig  2,  15°,  4°,  and  26°;  and  those  to  the  left,  20°,  10°,  and  16° ; 
and  their  lengths  as  in  the  same  Fig.  Draw  a  hor  line  h  o,  Fig  3 ;  and  consider  the 
center  of  it  to  be  the  station-stake.  From  this  point  as  a  center,  lay  off  these  angles 
with  a  protractor,  as  shown  on  tho  arcs  in  Fig  3.  Then  beginning  say  on  the  right 
hand,  with  a  parallel  ruler  draw  the  first  dist  a  c,  at  its  proper  slope  of  15° ;  and  of 
its  proper  length,  45  ft,  by  scale.  Then  the  same  with  c  y  and  y  t.  Do  the  same  with 
those  on  the  left  hand.  We  then  have  a  cross-section  of  the  ground  at  Sta  0.  Then 
on  the  map,  as  in  Fig  1,  draw  a  line  as  ra  w,  or  h  w,  at  right  angles  to  the  line  of  road, 
and  passing  through  tha  station-stake.  On  this  line  lay  down  the  hor  dists  ad,ds,svt 
a  e,  e  g,  g  k,  marking  them  with  a  small  star,  as  is  done  and  lettered  in  Fig  1,  at  Sta  0. 

When  extreme  accuracy  is  pretended  to,  these  hor  dists  must  be  found  by  measure 
on  Fig  3 ;  but  as  a  general  rule  it  will  be  near  enough,  when  the  slopes  do  not  ex- 
ceed 10°,  to  assume  them  to  be  the  same  as  the  sloping  dists  measured  in  the  field. 
Next  ascertain  how  high  each  of  the  points  c  y  1 1  n  i  is  above  datum.  Thus,  measura 
by  scale  the  vert  dist  d  c.  Suppose  it  is  found  to  be  5  ft ;  or  in  other  words,  that  c 
is  5  ft  below  station-stake  0.  Then  since  the  level  at  stake  0  is  100  ft  above  datum, 
that  at  c  must  be  5  ft  less,  or  100  —  5  =  95  ft  above  datum ;  which  may  be  marked  in 
light  lead-pencil  figures  on  the  map,  as  at  d,  Fig  1.  Next  for  the  point  y,  suppose 
we  find  s  y  to  be  11  ft,  or  y  to  be  11  ft  below  stake  O ;  then  its  height  above  datum 
must  be  100  —  11  =  89 :  which  also  write  in  pencil,  as  at  s.  Proceed  in  the  same 
way  with  t.  Next  going  to  the  left  hand  of  the  station-stake,  we  find  el  to  be  say 
2  ft ;  but  I  is  above  the  level  of  the  station-stake,  therefore  its  height  above  datum  is 


100  +  2  =  102  ft,  as  figured  at  e  on  the  map.  Let  ng  be  5  ft;  then  is  n,  100  -f  5  =» 
105  ft  above  datum,  as  marked  at  g ;  and  so  on  at  each  station.  When  this  has  been 
done  at  several  stations,  we  may  draw  in  the  contour  lines  of  that  portion  by  hand 
thus :  Suppose  they  are  to  represent  vert  heights  of  3  ft.  Beginning  at  Station  O 
(of  which  the  height  above  datum  is  100  ft)  to  lay  down  a  contour  line  103  ft  above 
datum,  we  see  at  once  that  the  height  of  103  ft  must  be  at  t,  or  at  %  the  dist  from  « 
tog.  Make  a  light  lead-pencil  dot  at  t ;  and  then  go  to  the  next  Station  1.  Here 
we  see  that  the  height  of  103  ft  coincides  with  the  station-stake -itself;  place  a  dot 
there,  and  go  to  Sta  2.  The  level  at  this  stake  is  101 ;  therefore  the  contour  for  103 


150 


DIALLING. 


ft  must  evidently  be  2  ft  higher,  or  at  t,  %  of  the  dist  from  Sta  2  to  -f  104 ;  therefore 
make  a  dot  at  i.  Then  go  to  Sta  3.  Here  the  level  being  104  above  datum,  the  con- 
tour of  103  must  be  at  y,  or  }  of  the  dist  from  Sta  3  to  +99  ;  put  a  dot  at  y.  Finally 
draw  by  hand  a  curving  line  through  t,  SI,  t,  and  y-  and  the  contour  line  of  103  it 
is  done.  All  the  others  are  prepared  in  the  same  way,  one  by  one.  The  level  of  each 
must  be  figured  upon  it  at  short  intervals  along  the  map,  as  at  103  c,  106 c,  &c. 

Or,  instead  of  first  placing  the  +  points  on  the  map,  to  denote  the  slope  dists  actu- 
ally measured  upon  the  ground,  we  may  at  once,  and  with  less  trouble,  find  and  show 
those  only  which  represent  the  points  t,  S  1,  t,  y,  &c,  of  the  contours  themselves. 
Thus,  say  that  at  any  given  station-stake,  Fig  4,  the  level  is  104;  that  the  cross-sec- 
tion c  s  of  the  ground  has  been  prepared  as  before  ;  and  that  we  want  the  hor  dists 
from  the  stake,  to  contour  lines  for  94,  97, 100  ft,  <fec,  3  ft  apart  vert 


X2"  incn  to  a  side,  for  drawing  the  cross-sections  upon. 

When  the  ground  is  very  steep,  it  is  usual  to  shade  such  portions  of  the  map  to 
represent  hill-side.    The  closer  together  the  contours  come,  the  steeper  of  course  is 


DIALLING, 


To  make  a  horizontal  Sun-dial, 

DRAW  a  line  a  b ;  and  at  right  angles  to  it,  draw  C6.  From  any  convenient  point,  as  c, 
in  a  6,  draw  the  perp  c  o.  Make  the  angle  c  a  o  equal  to  the  lat  of  the  place  ;  also 

*  The  preparing  of  contour  lines  is  a  slow  and  tedious  office- work  ;  and  the  writer  considers  them 
of  but  little  value  in  many  cases ;  as  when  they  are  taken  foronly  about  100  ft  on  each  side  of  the 
line,  with  reference  to  slight  changes  of  direction.  He  conceives  that,  ordinarily,  every  useful  purpose 
is  fulfilled  if  the  leveller  or  the  topographer  enters  into  his  field-book  at  each  station,  notes  similar 
to  the  following : 

StaGO —  3. IE.    +2.1L. 

61 +  2.2R.    — 1.3L. 

62 r:l.K.  +4.  2  L. 

63 "  " 

Which  means  that  at  station  60,  the  slope  of  the  ground  on  the  right,  as  nearly  as  he  can  judge  by 
eye,  or  by  his  hand-level,  is  about  3  ft  downward,  for  1  chain,  or  100  ft;  uud  on  the  Jeft,  about  2  ft 
upward  in  1  chain.  At  61,  2  ft.  up,  in  2  chains  to  the  right;  and  1  ft  down  in  3  chains  to  the  left. 
At  62,  level  for  1  chain  to  the  right ;  and  ascending  4  ft  in  2  chains  to  the  left.  At  63,  the  same  as  at 
62.  At  some  spots  it  will  be  well  to  add  a  sketch  of  a  cross-section,  like  Fig  2  ;  only,  instead  of  the 
angles,  use  ft  of  rise  or  fall,  to  indicate  the  slopes,  as  judged  by  eye,  or  by  a  hand-level.  By  this 
method,  the  result  at  every  station  will  be  somewhat  in  error;  but  these  small  errors  will  balance 
each  other  so  nearly  that  the  total  may  be  regarded  as  sufficiently  correct  for  all  the  purposes  of  n 
preliminary  estimate  of  the  cost  of  a  road.  When  the  final  stakes  for  guiding  the  workmen  are  placed, 
the  slopes  should  be  carefully  taken,  in  order  to  calculate  the  quantity  of  excavation  accurately  for 
payment. 


PAPER. 


151 


the  angle  c  o  e  equal  to  the  same  ;  join  o  e.    Make  e  n  equal  to  o  e;  and  from  n  as  a 
center,  with  the  rad  e  w,  describe  a  quadrant  e  *;  and  div  it  into  6  equal  parts.  Draw  e 
y,  parallel  to  6,  6;  and 
from  n,  through   the  5  DIAL  Y 

points  on  the  quadrant,  \X 

draw  lines  n  t,  n  i,  &c, 
term  inating  in  e  y .  From 
a  draw  lines  a  5,  a  4,  &c, 
passing  through  £,  t,  Ac. 
From  any  convenient 
point,  as  c,  describe  an 
arc  r  m  h,  as  a  kind  of  fin- 
ish or  border  to  half  the 
dial.  All  the  lines  may 
now  be  effaced,  except 
the  hour  lines  a  6,  a  5, 
a  4,  <fcc,  to  a  12,  or  a  h ; 
unless,  as  is  generally 
the  case,  the  dial  is  to 
be  divided  to  quarters 
of  an  hour  at  least.  In 
this  case  each  of  the 
divisions  on  the  quad- 
rant e  .?,  must  be  subdivided  into  4  equal  parts ;  and  lines  drawn  from  n,  through 
the  points  of  subdivision,  terminating  in  ey.  The  quarter-hour  lines  must  be  drawn 
from  a,  as  were  the  hour  lines.  Subdivisions  of  5  min  may  be  made  in  the  same 
way ;  but  these,  as  well  as  single  min,  may  usually  be  laid  off  around  the  border,  by 
eye.  About  8  or  10  times  the  size  of  our  Fig  will  be  a  convenient  one  for  an  ordi- 
nary dial.  To  draw  the  other  half  of  the  Fig,  make  a  d  equal  to  the  intended  thick- 
ness of  the  gnomon,  or  style,  of  the  dial;  and  draw d  12,  parallel, and  equal  toa!2;  and 
draw  the  arc  xg  w,  precisely  similar  to  the  arc  r  m  h.  Between  x  and  w,  on  the  arc  xg  w, 
space  off  divisions  equal  to  those  on  the  arc  rmh\  and  number  them  for  the  hours, 
as  in  the  Fig.  The  style  F,  of  metal  or  stone,  (wood  is  too  liable  to  warp,)  will  be 
triangular ;  its  thickness  must  throughout  be  equal  to  a  d  or  h  w ;  its  base  must 
cover  the  space  adhw,  its  point  will  be  at  ad;  and  its  perp  height  h  u,  over  hw, 
must  be  such  that  lines  vd,ua,  drawn  from  its  top,  down  to  a  and  d,  will  make  the 
angles  uah,vd  w.  each  equal  to  the  lat  of  the  place.  Its  thickness,  if  of  metal,  may 
conveniently  be  from  %  to  %  inch ;  or  if  of  stone,  an  inch  or  two,  or  more,  according 
to  the  size  of  the  dial.  Usually,  for  neatness  of  appearance,  the  back  //  u  v  w  of  the 
style  is  hollowed  inward.  The  upper  edges,  ua,  vd,  which  cast  the  shadows,  must 
be  sharp  and  straight.  The  dial  must  be  fixed  in  place  hor,  or  perfectly  level ;  a  h 
and  dw  must  be  placed  truly  north  and  south  ;  ad  being  south, and  hw  north.  The 
dial  gives  only  sun  or  solar  time ;  but  clock  time  can  be  found  by  means  of  the  "  fast 
or  slow  of  the  sun,"  as  given  by  all  almanacs.  If  by  the  almanac  the  sun  is  5  min, 
&c,  fast,  the  dial  will  be  the  same ;  and  the  clock  or  watch,  to  be  correct,  must  be  5 
min  slower  than  it ;  and  vice  versa. 


PAPEE. 


Ins. 

Antiquarian 52^  X  30V 

Double  Elephant   39 1|  X  26> 

Atlas....: 33       X  26 

Columbier   3i%  X  23 

Elephant  28       X  23 


24  sheets  1  quire.        20  quires  1  ream. 
Sizes  of  drawing?  papers. 


Ins.  Tns. 

Imperial 29  V£  X  21U 

Super  Royal 2?C|  X   w/L 

Royal 23U  X  19 

Medium 22||  X  H^ 

Demy  l9)|  X  1^| 

The  English  drawing-papers  are  stronger  and  superior  to  the  American.  Thosa 
by  Whatman  have  a  high  reputation  ;  they  are,  however,  of  different  qualities.  When 
paper  is  pasted  on  muslin,  the  difference  in  quality  is  not  so  important. 

Both  white  ami  tinted  papers,  for  the  use  of  engineers,  are  made  in 
continuous  rolls,  without  seams.  Widths  40,  54,  and  60  ins ;  usual  lengths  up  to  40 
yds ;  but  can  be  had  to  order  to  400  yds  or  more.  These  may  also  be  purchased  ready 
pasted  on  muslin,  in  rolls  up  to  10  yds  long.  This  last,  on  account  of  its  strength. 


152  THE  LEVEL. 

should  be  used  for  all  drawings  which  undergo  frequent  rolling  and  unrolling;  or 
other  hard  usage ;  particularly  working-drawings.  For  the  last  purpose,  strong  car- 
tridge, or  pattern  paper  answers  very  well.  It  is  for  sale  in  long  rolls;  and  can  be 
had  in  lengths  of  10  yds,  pasted  on  muslin  ;  widths  24,  40,  50  ins.  Color,  a  light  buff. 

Tracing1  paper.  Most  of  that  sold,  whether  domestic  or  foreign,  tears  so 
readily  as  to  be  of  comparatively  little  service,  except  for  tracings  to  be  enclosed  in 
letters  for  mailing.  Some  of  what  is  called  French  vegetable  tracing-paper,  is,  how- 
ever, quite  stout  and  strong,  and  good  for  line  drawings ;  but  it  shrivels  badly 
under  broad  washes  of  color,  even  when  stretched,  forming  little  puddles,  which 
make  it  difficult  to  produce  a  uniform  tint.  Sizes  18  X  24,  and  27  X  40  ins ;  also  in 
rolls  of  11  and  22  yds. 

Tracing  cloth,  usually  called  tracing  muslin,  and  sometimes  vellum  cloth,  is 
altogether  preferable  to  tracing  paper,  on  account  of  its  great  strength.  Widths  18, 
30,  33,  36,  40,  and  42  ins.;  lengths  to  24  yds. 

Common  inks  dry  pale  on  either  tracing  muslin  or  tracing  paper;  therefore  use 
India  ink.  Neither  the  muslin  nor  the  paper  takes  colors  as  kindly  as  drawing  paper. 

Profile  paper  has  hitherto  been  made  in  single  sheets  ;  but  can  now  be  had 
in  long,  continuous  rolls;  thus  avoiding  the  trouble  of  pasting  separate  sheets  to- 
gether. This  is  a  great  improvement,  and  must  at  once  bring  the  new  paper  into 
general  use  among  engineers.*  Used  for  profiles  of  railroads,  &c. 

Ruled  squares.  Paper  carefully  ruled  in  small  squares,  so  that  the  divisions 
answer  for  a  scale  for  the  drawing,  is  exceedingly  useful  for  sketching  out  plans,  &c. 
It  is  sometimes  ruled  on  both  sides  of  the  sheet. 


bl 
Ro 


Koman  ochre,  sap  green.  Newman  s,  Ackerm;m  s,  or  Osoorne  s  colors  are  among  the 
best  in  use.  Purchase  none  but  the  very  best  India  ink.  Cakes  of  colors  should  always 
be  wiped  dry  on  paper,  after  bSTng  rubbed  in  water ;  and  but  little  water  should  be 
used  while  rubbing;  more  being  added  afterward. 

Lead  pencils.  Genuine  A.  W.  Faber's  Nos.  2,  3,  and  4,  are  very  good.  The 
hardness  increases  with  the  number.  Nos  3  and  4  are  good  for  field-book  use :  which 
to  prefer,  will  depend  on  the  character  of  the  paper;  No.  3  for  smooth,  and  No.  4  for 
the  coarser  or  more  granular  papers.  The  office  draughtsman  should  have  a  flat  file, 
upon  which  to  rub  his  lead  to  a  fine  point  readily,  after  using  the  knife. 

Hover's  patent  carbonized  writing,  letter,  and  note  papers  possess  the 
important  quality  of  at  once  imparting  a  clear  black  color  to  writing  done  with  pale 
common  inks,  if  the  latter  contains,  as  usual,  sulphate  of  iron  and  gallic  acid.  These 
papers  do  not  differ  in  appearance  from  ordinary  ones ;  nor  is  there  much  difference 
in  cost. 


THE  LEVEL. 

ALTHOUGH  the  levels  of  different  makers  vary  somewhat  in  their  details,  still  their 
principal  parts  will  be  understood  from  the  following  figure.t  The  telescope  T  T 
rests  upon  two  supports  Y  Y,  called  Ys ;  out  of  which  it  can  be  lifted,  first  removing 
the  pins  *  5  which  confine  the  semicircular  clips  n  n,  and  then  opening  the  clips. 
The  pins  should  be  tied  to  the  Ys,  by  pieces  of  string,  to  prevent  their  being  lost. 
The  slide  of  the  object-glass  O,  is  moved  backward  or  forward  by  a  rack  and  pinion, 
by  means  of  the  milled  head  A.  The  slide  of  the  eye-glass  E,  is  moved  in  the  same 
way  by  the  milled  head  e.  A  cylindrical  tube  of  brass,  called  a  shade,  is  usually 
furnished  with  each  level.  It  is  intended  to  be  slid  on  to  the  object-end  O  of  the 
telescope,  to  prevent  the  glare  of  the  sun  upon  the  object-glass,  when  the  sun  is 
low.  At  B  is  an  outer  ring  encircling  the  telescope,  and  carrying  4  small  capstan- 
headed  sere  ws ;  two  of  which, pp,  are  at  top  and  bottom;  while  the  other  two, 
of  which  i  is  one,  are  at  the  sides,  and  at  right  angles  to^  p.  Inside  of  this  outer 
ring  is  another,  inside  of  the  telescope,  and  which  has  stretched  across  it  two 
spider-webs,  usually  called  the  CROSS-HAIRS.  These  are  much  finer  than  they  ap- 
pear to  be,  being  considerably  magnified.  They  are  at  right  angles  to  each  other ; 
and,  in  levelling,  one  is  kept  vert,  and  the  other  hor.  They  are  liable  at  times  to  be 

*  Made  and  for  sale  by  James  W.  Queen  &  Co,  dealers  in  philosophical  apparatus,  engineers'  sta- 
tionery, &c,  No-  924  Chestnut  St,  Philadelphia.  It  can  be  had  also  pasted  on  muslin,  in  rolls  of  10  yds. 

t  The  price  of  a  first-class  level,  by  Heller  &  Brightly,  is  $145.  It  is 
bad  economy  to  buy  inferior  instruments. 


THE   LEVEL. 


153 


thrown  out  of  this  position  by  a  partial  revolution  of  the  telescope,  when  carrying 
the  level,  or  when  setting  the  tripod  down  suddenly  upon  the  ground ;  but  since,  in 
levelling,  the  intersection  of  the  hairs  is  directed  to  the  target-rod,  this  derangement 
does  not  affect  the  accuracy  of  the  work.  Still  it  is  well  to  keep  them  nearly  vert 
and  hor,  by  keeping  the  BUBBLE-TUBE  D  D  as  nearly  directly  over  the  bar  V  F  as  can 
be  judged  by  eye.  This  enables  the  leveller  to  see  that  the  rod-man  holds  his  rod 
nearly  vert,  which  is  absolutely  essential  for  correct  levelling.  If  perfect  verticality 
is  desired,  as  is  sometimes  the  case,  when  staking  out  work,  it  may  be  obtained  (if 
the,  instrument  is  in  perfect  adjustment,  and  levelled)  by  sighting  at  a  plumb-line,  or 
other  vert  object,  and  then  turning  the  telescope  a  little  in  its  Ys,  so  as  to  bring  the 
hair  to  correspond.  When  this  is  done,  a  short  continuous  scratch  may  be  made  on 
the  telescope  and  Y,  to  save  that  trouble  in  future.  Heller  &  Brightly,  however, 
provide  their  levels  with  a  small  projection  inside  of  the  Ys,  and  a  corresponding 
stop  on  the  telescope,  the  contact  of  which  insures  the  verticality  of  the  hair. 
Should  the  hairs  be  broken  by  accident,  they  may  be  replaced  as  directed  here- 
after. 
The  small  holes  around  the  heads  of  the  4  small  capstan-screws  p,  i,  just  referred  to, 


are  for  admitting  the  end  of  a  small  steel  pin.  or  leyer,  for  turning  them.  If  first 
the  upper  screw  p  be  loosened,  and  then  the  lower  one  tightened,  the  interior  ring 
will  be  lowered,  and  the  horizontal  hair  with  it.  But  on  looking  through  the  tele- 
scope they  will  appear  to  be  raised.  If  first  the  lower  one  be  loosened,  aud  the  upper 
one  tightened,  the  hor  hair  will  be  actually  raised,  but  apparently  lowered.  This  is 
because  the  glasses  in  the  eye-piece  E  reverse  the  apparent  position  of  objects  inside 
of  the  telescope  ;  which  effect  is  obviated,  as  regards  exterior  objects,  by  means  of 
the  object-glass  0.  This  must  be  remembered  when  adjusting  the  cross-hairs  ;  for  if  a 


hair  appears  to  strike   too  high,  it  must  be  raised  still  higher;  if  it  appears  to  be 

already  too  far  to  the  right  or  left,  it 

direction. 


t  must  be  actually  moved  still  more  in  the  same 


. 

This  remark,  however,  does  not  apply  to  telescopes  which  make  objects  appear 
inverted. 

There  is  no  danger  of  injuring  the  hairs  by  these  motions,  inasmuch  as  the  four 
screws  act  against  the  ring  only,  and  do  not  come  in  contact  with  the  hairs  them- 
selves. 

Under  the  telescope  is  the  BUBBLE-TUBE  D  D.  One  end  of  this  tube  can  be  raised  or 
owered  slightly  by  means  of  the  two  capstan-headed  nuts  n  n,  one  of  which  must 
je  loosened  'before  the  other  is  tightened.  On  top  of  the  bubble-tube  are  scratches 


154  THE   LEVEL. 

for  showing  when  the  bubble  is  central  in  the  tube.  Frequently  these  scratches,  or 
marks,  are  made  on  a  strip  of  brass  placed  above  the  tube,  as  in  our  fig.  There  are 
several  of  them,  to  allow  for  the  lengthening  or  shortening  of  the  bubble  by  changes 
of  temperatuie.  At  the  other  end  of  the  bubble-tube  are  two  small  capstan-screws, 
placed  on  opposite  sides  horizontally.  The  circular  head  of  one  of  them  is  shown 
near  t.  By  moans  of  these  two  screws,  that  end  of  the  tube  can  be  slightly  moved 
hor,  or  to  right  or  left.  Under  the  bul»ble-tube  is  the  BAR  V  F ;  at  one  end  of  which, 
as  at  V,are  two  large  capstan-nuts  w  w,  which  operate  upon  a  stout  interior  screw 
which  forms  a  prolongation  of  the  Y.  The  holes  in  these  nuts  are  larger  than  the 
others,  as  they  require  a  larger  lever  for  turning  them.  If  the  lower  nut  is  loosened 
and  the  upper  one  tightened,  the  Y  above  is  raised ;  and  that  end  of  the  telescope 
becomes  farther  removed  from  the  bar;  and  vice  versa.  Some  makers  place  a  similar 
screw  and  nuts  under  both  Ys ;  while  others  dispense  with  the  nuts  entirely,  and 
substitute  beneath  one  end  of  the  bar  a  large  circular  milled  head,  to  be  turned  by 
the  fingers.  This,  however,  is  exposed  to  accidental  alteration,  which  should  be 
avoided. 

When  the  portions  above  m  are  put  upon  w,  and  fastened  by  the  screw  Y,  all 
the  upper  part  may  be  swung  round  hor,  in  either  direction,  by  loosening  the 
clamp-screw  H  ;  or  such  motion  may  be  prevented  by  tightening  thatscrew. 
It  frequently  happens,  after  the  telescope  has  been  sighted  very  nearly  upon  an 
object,  and  then  clamped  by  H,  that  we  wish  to  bring  the  cross-hairs  to  coincide 
more  precisely  with  the  object  than  we  can  readily  do  by  turning  the  telescope  by 
hand;  and  in  this  case  we  use  the  tangent-screw  6,  by  means  of  which  a 
slight  but  steady  motion  may  be  given  after  the  instrument  is  clamped.  For 
fuller  remarks  on  the  clamp  and  tangent-screws,  see  "Transit." 

The  parallel  plates  m  and  S  are  operated  by  four  levclling-screws; 
three  of  which  are  seen  in  the  figure,  at  K  K.  The  screws  work  in  sockets  K ; 
which,  as  well  as  the  screws,  extend  above  the  upper  plate.  When  the  instrument 
is  placed  on  the  ground  for  levelling,  the  lower  parallel  plate  S  is  never  hor,  unless 
by  accident;  nor  is  it  necessary  that  it  should  be.  But  for  levelling,  the  upper 
one  m  must  always  be  made  hor  (as  indicated  by  the  bubble)  by  the  levelling- 
scrcws.  The  lower  plate  S,  and  the  brass  parts  below  it,  are  together  called  the 
tripod-bead ;  and,  in  connection  with  three  wooden  legs  Q  Q  Q,  constitute 
the  tripod.  In  the  figure  are  seen  the  heads  of  wing-nuts  J  which  confine  the 
legs  to  tiie  tripod-head.  Under  the  center  of  the  tripod-head  should  always  be 
placed  a  small  ring,  from  which  a  plumb-bob  may  be  suspended.  This  is  not 
needed  in  ordinary  levelling,  but  becomes  useful  when  ranging  center-stakes,  &c. 

To  adjust  a  Level. 

This  is  a  quite  simple  operation,  but  requires  a  little  patience.  Be  careful  to  avoid 
straining  any  of  the  screws.  The  large  Y  nuts  ww  sometimes  require  some  force  to 
start  them ;  but  it  should  be  applied  by  pressure,  and  not  by  blows.  Before  begin- 
ning to  adjust,  attend  to  the  object-glass,  as  directed  in  the  first  sentence  under  "To 
adjust  a  plain  transit,"  p.  159. 

Three  adjustments  are  necessary ;  and  must  be  made  in  the  following  order: 

First,  that  of  tile  cross-hairs;  to  secure  that  their  intersection  shall 
continue  to  strike  the  same  point  of  a  distant  object,  while  the  telescope  is  being 
turned  round  a  complete  revolution  in  its  Ys.  This  is  called  adjusting  the  line 
Of  colliniation,  or  sometimes,  the  line  of  sight;  but  it  is  not  strictly  the  line 
of  sight  until  all  the  adjustments  are  finished;  for  until  then,  the  line  of  collimation 
will  not  serve  for  taking  levelling  sights. 

Second,  that  of  the  bubble-tube  B  D,  to  place  it  parallel  to  the  line 


THE   LEVEL.  155 

of  collimation,  previously  adjusted  ;  so  that  when  the  bubble  stands  at  the  centre  of 
its  tube,  indicating  that  it  is  level,  we  know  that  our  sight  through  the  telescope 
is  hor. 

Third,  that  of  the  Ys,  by  which  the  telescope  and  bubble-tube  are  supported ; 
yo  that  the  bubble-tube,  and  line  of  sight,  shall  be  perp  to  the  vert  axis  of  the  instru- 
ment; so  as  to  remain  hor  while  the  telescope  is  pointed  to  objects  in  diff  directions, 
as  when  taking  back  and  fore  sights. 

To  make  the  first  adjustment,  or  that  of  the  cross-hairs,  plant  the 
tripod  firmly  upon  the  ground.  In  this  adjustment  it  is  not  necessary  to  level  the 
instrument.  Open  the  clips  of  the  Ys;  unclamp;  draw  out  the  eye-glass  E,  until 
the  cross-hairs  are  seen  perfectly  clear ;  sight  the  telescope  toward  some  clear  dis- 
tant point  of  an  object;  or  still  better,  toward  some  straight  line,  whether  vert  or 
not.  Move  the  object-glass  0,  by  means  of  the  milled  head  A,  so  that  the  object  shall 
be  clearly  seen,  without  parallax,  that  is,  without  any  apparent  dancing 
about  of  the  cross-hairs,  if  the  eye  is  moved  a  little  up  or  down  or  sideways.  To 
secure  this,  the  object-glass  alone  is  moved  to  suit  different  distances ;  the  eye-glass 
is  not  to  be  changed  after  it  is  once  properly  fixed  upon  the  cross-hairs.  The  neglect 
of  parallax  is  a  source  of  frequent  errors  in  levelling.  Clamp ;  and,  by  means  of  the 
tangent-screw  6,  bring  either  one  of  the  cross-hairs  to  coincide  precisely  with  the 
object.  Then  gently,  and  without  jarring,  revolve  the  telescope  half-way  round  in 
its  Ys.  When  this  is  done,  if  the  hair  still  coincides  precisely  with  the  object,  it  is 
in  adjustment;  and  we  proceed  to  try  the  other  hair.  But  if  it  does  not  coincide, 
then  by  means  of  the  4  screws  jo,  t,  move  the  ring  which  carries  the  hairs,  so  as  to 
rectify,  as  nearly  as  can  be  judged  by  eye,  only  one-half  of  the  error;  remembering 
that  the  ring  must  be  moved  in  the  direction  opposite  to  what  appears  to  be  the 
right  one ;  unless  the  telescope  is  an  inverting  one.  Then  turn  the  telescope  back 
again  to  its  former  position;  and  again  by  the  tangent-screw  bring  the  cross-hair  to 
coincide  with  the  object.  Then  again  turn  the  telescope  half-way  round  as  before. 
The  hair  will  now  be  found  to  be  more  nearly  in  its  right  place,  but,  in  all  probabil- 
ity, not  precisely  so ;  inasmuch  as  it  is  difficult  to  estimate  one-half  the  error  accu- 
rately by  eye.  Therefore  a  little  more  alteration  of  the  ring  must  be  made ;  and  it 
may  be  necessary  to  repeat  the  operation  several  times,  before  the  adjustment  is 
perfect.  Afterward  treat  the  other  hair  in  precisely  the  same  manner.  When  both 
are  adjusted,  their  intersection  will  strike  the  same  precise  spot  while  the  telescope 
is  being  turned  entirely  round  in  its  Ys.  This  must  be  tried  before  the  adjustment 
can  be  pronounced  perfect;  because  at  times  the  adjustment  of  the  second  hair, 
slightly  deranges  that  of  the  first  one;  especially  if  both  Were  much  out  in  the  be- 
ginning. 

To  make  the  second  adjustment,  or  to  place  the  bubble-tube  parallel 
to  the  line  of  collimation.  This  consists  of  two  dis- 
tinct adjustments,  one  vert,  and  one  hor.  The  first 
of  these  is  effected  by  means  of  the  two  nuts  «  n  on 
the  vert  screw  at  one  end  of  the  tube  ;  and  the  second 
by  the  two  hor  screws  at  the  other  end,  t,  of  the  tube. 
Looking  at  the  bubble-tube  endwise,  from  t  in  the 
foregoing  Fig,  its  two  hor  adjusting-screws  1 1  are 
seen  as  in  this  sketch.  The  larger  capstan-headed 
nut  below,  has  nothing  to  do  with  the  adjustments; 
it  merely  holds  the  end  of  the  tube  in  its  place. 

To  make  the  vert  adjustment  of  the  bubble-tube,  by  means  of  the  two  nuts  nn.  Place 
the  telescope  over  a  diagonal  pair  of  the  levelling-screws  K  K ;  and  clamp  it  there. 
Open  the  clips  of  the  Ys ;  and  by  means  of  the  levelling-screws  bring  the  bubble  to 
the  center  of  its  tube.  Lift  the  telescope  gently  out  of  the  Ys,  turn  it  end  for  end,  and 
put  it  back  again  in  its  reversed  position.  This  being  done,  if  the  bubble  still  remains 
at  the  center  of  its  tube,  this  adjustment  is  in  order  ;  but  if  it  moves  toward  one  end, 
that  end  is  too  high,  and  must  be  lowered ;  or  else  the  other  end  must  be  raised. 
This  must  be  done  by  means  of  the  two  small  capstan-headed  nuts  nn.  If  the  end  nn 
is  to  be  raised,  the  upper  nut  must  first  be  loosened,  then  the  lower  one  tightened, 
and  vice  versa ;  one-half  the  error  is  to  be  corrected  by  ihis  first  process.  Then  correct 
the  other  half  by  means  of  the  levelling-screws  K  K.  Having  thus  brought  the  bub- 
ble to  the  middle  again,  again  lift  the  telescope  out  of  its  Ys;  turn  it  end  for  end, 
and  replace  it.  The  bubble  will  now  settle  nearer  the  center  than  it  did  before,  but 
will  probably  require  still  further  adjustment.  If  so,  correct  half  the  remaining 
error  by  the  nuts  as  before :  and  half  by  the  levelling-screws;  and  so  continue  to  re- 
peat the  operation  until  the  bubble  remains  in  the  center  in  both  positions.  That 
part  also  of  its  adjustment  will  then  be  complete  ;  and  the  bubble  will  remain  at  the 
center  (after  the  instrument  is  levelled)  while  the  telescope  is  pointed  in  any  direction. 


156  THE    LEVEL. 

To  make  the  hor  adjustment  of  the  bubble-tube,  place  the  telescope  over  two  of  the 
levelling-screws  K  K,  which  stand  diagonally  to  each  other;  clamp  it;  and  by  the 
same  two  levelling-screws  bring  the  bubble  to  the  centre  ;  at  the  same  time  seeing 
that  the  bubble-tube  appears,  as  nearly  as  may  be,  to  be  directly  under  the  telescope, 
or  over  the  bar.  Then  gently  revolve  the  telescope  a  little  to  one  side  in  its  Ys,  say 
about  y±  inch,  so  that  the  bubble-tube  shall  stand  out  on  that  side,  from  over  the 
centre  of  the  bar,  or  from  under  the  telescope.  If  the  bubble  then  remains  at  the 
centre  of  the  tube,  the  hor  adjustment  is  correct;  but  if  it  runs  toward  one  end, 
that  end  is  too  high  ;  and  one-half  of  the  error  of  the  bubble  must  be  corrected  by  the 
hor  screws  1  1  ;  raising  the  end  that  is  lowest,  or  lowering  that  which  is  highest, 
as  the  case  may  require.  This  done,  turn  the  telescope  back  again,  until  the 
bubble-tube  is  over  the  bar;  and  bring  the  bubble  again  to  the  center  of  its 
tube,  by  means  of  the  levelling-screws  K.  Then  again  turn  the  telescope  as  before,  so 
as  to  make  the  bubble-tube  stand  out  from  over  the  bar.  As  it  is  only  by  chance 
that  this  adjustment  is  perfected  at  the  first  trial,  the  bubble  will  most  probably 
again  leave  the  centre  of  the  tube,  but  not  as  much  as  before.  If  so,  again  rectify 
half  the  error  by  means  of  the  screws  tt\  turn  the  telescope  back  again  ;  re-level  the 
tube  ;  and  so  repeat  until  the  bubble  remains  at  the  center,  both  when  under  the 
bar,  and  when  standing  out  from  it  a  short  dist.  This  hor  adjustment  is  usually 
ready  made  in  new  instruments  ;  and  is  not  at  all  liable  to  derangement,  except  by 
accidental  blows. 

To  make  the  third  adjustment,  or  to  adjust  the  heights  of  the  Ys,  so 
as  to  make  the  line  of  colliraation  parallel  to  the  bar  V  F,  or  perp  to  the  vert  axis 
of  the  instrument.  The  other  adjustments  being  made,  fasten  down  the  clips  of  the 
Ys.  Make  the  instrument  nearly  level  by  means  of  all  four  of  the  levelling-screws 
K.  Place  the  telescope  over  two  of  the  levelling-screws  which  stand  diagonally; 
and  leave  it  there  undamped.  Then  bring  the  bubble  to  the  center  of  its  tube,  by 
the  two  levelling-screws.  Swing  the  upper  part  of  the  instrument  half-way  around, 
so  that  the  telescope  shall  ^gain  stand  over  the  same  two  screws;  but  end  for  end. 
This  done,  if  the  bubble  leaves  the  center,  bring  it  half-way  back  by  the  large  cap- 
stan nuts  w,  w  ;  and  the  other  half  by  the  two  levelling-screws.  Remember  that  to 
raise  the  Y,  and  the  end  of  the  bubble  over  to,  w,  the  lower  w  must  be  loosened  ;  and 
the  upper  one  tightened  ;  and  vice  versa.  Now  place  the  telescope  over  the  other 
diagonal  pair  of  levelling-screws:  and  repeat  the  whole  operation  with  them.  Hav- 
ing completed  it,  again  try  with  the  first  pair;  and  so  keep  on  until  the  bubble  re- 
mains at  the  center  of  its  tube,  in  every  position  of  the  telescope. 

Correct  levelling  may  -be  performed  even  if  all  the  foregoing  adjustments  are 
out  of  order;  provided  each  fore-sight  be  taken  at  precisely  the  same  distance  frrnn 
the  instrument  as  the  back-sight  is.  But  a  good  leveller  will  keep  his  instrument  always 
in  adjustment;  and  will  test  the  adjustments  at  least  once  a  day  when  at  work.  As 
much,  however,  depends  upon  the  rodman,  or  target-man,  as  upon  the  leveller.  A  rod- 
man  who  is  careless  about  holding  the  rod  vert,  or  about  reading  the  sights  correctly, 
should  be  discharged  without  mercy. 

Forms  for  level  note-books.  When  the  distance  is  short,  so  as  not  to 
require  two  sets  of  books,  the  following  is  perhaps  as  good  as  any. 


it  best,  both  with  the  level  and  with  the  transit,  to  consider  the  term  "    TATION     to 
apply  to  the  whole  dist  between  two  consecutive  stakes;  and  that  its  number  shall 
be  that  written  on  the  last  stake.     Thus,  with  the  transit,  Station  6  means  the  dist 
stake  6;  that  it  has  a  bearing  or  course  of  so  and  so;  and  its  length 
nd  with  the  level,  Station  6  also  means  the  dist  from  stake  5  to  stake 

- 


from  stake  5  to  st 


the  foregoing  one  as  being  perfectly  simple,  and  free  from  liability  to  mistakes.     It 
does  not  interfere  with  designating  any  stake  by  its  number  also,  as  stake  No  6,  &c, 

NOTE.  The  levelling-screws  in  manv  instruments  become  very  hard  to  turn  if  dirty.  Clean  with 
water  and  a  tooth-brusn.  Don't  use  oil  on  neld  instruments.  Heiier  &  Brightly's  screws  are  pro- 
tected against  dust. 


THE  ENGINEER'S  TRANSIT. 


157 


THE  ENGINEER'S  TRANSIT, 


158 


THE  ENGINEER'S  TRANSIT. 


THE  details  of  the  transit,  like  those  of  the  level,  are  differently  arranged  by 
diff  makers,  and  to  suit  particular  purposes.  We  describe  it  in  its  modern  form, 
as  made  by  Heller  and  Brightly,  of  Philada.  Without  the  lon»  bubble- tube 
F  F,  Fig  1,  under  the  telescope,  and  the  graduated  arc  g,  it  is  their  plain 
transit.  WTith  these  appendages,  or  rather  with  a  graduated  circle  in  place  of 
the  arc,  it  becomes  virtually  a  Complete  Theodolite.* 

B  D  D,  Fig  2,  is  the  tripod-head.  The  screw-threads  at  v  receive  the  screw 
of  a  wooden  tripod-head-cover  when  the  instrument  is  out  of  use.  S  S  A  is  the 
lower  parallel  plate.  After  the  transit  has  been  set  very  nearly  over  the 
center  of  a  stake,  the  shifting-plate,  dd  ce,  enables  us,  by  slightly  loosening 
the  levelling-SCrews  K,  to  shift  the  upper  parts  horizontally  a  trifle,  and 
thus  bring  the  plumb-bob  exactly  over  the  center  with  less  trouble  than  by  the 
usual  method  of  pushing  one  or  two  of  the  legs  further  into  the  ground,  or  spread- 
ing them  more  or  less.  The  screws,  K,  are  then  tightened,  thereby  pushing  up- 
ward the  upper  parallel  plate  mmmxx,  and  with  it  the  halt-ball  6,  thus 
pressing  c  c  tightly  up  against  the  under  side  of  S.  The  plumb-line  passes 


through  the  vert  hole  in  6.  Screw-caps,  /  g,  protect  the  levelling-screws  from 
dust,  &c.  The  feet,  i,  of  the  screws,  work  in  loose  sockets, .7,  made  flat  at  bottom, 
to  preserve  S  from  being  indented.  The  parts  thus  far  described  are  generally 
left  attached  to  the  legs  at  all  times.  Fig  1  shows  the  method  of  attachment. 

To  set  the  upper  parts  upon  the  parallel  plates.  Place  the 
lower  end  of  U  U  in  x  x,  holding  the  instrument  so  that  the  three  blocks  on  m  m 
(of  which  the  one  shown  at  F  is  movable)  may  enter  the  three  corresponding 

*  The  price  of  a  first-class  plain  transit  with  shifting-plate  and  plumb-bob,  by  Heller  &  Brightly, 
in  1882,  is  $185.  One  with  vert  arc  g  aud  long  bubble-tube  F  F,  $220. 


THE  ENGINEER'S  TRANSIT.  159 

recesses  in  a,  thus  allowing  a  to  bear  fully  on  m,  upon  which  the  upper  parts 
then  rest.  (The  inner  end  of  the  spring-catch,  I,  in  the  meantime  enters  a  groove 
around  U,  just  below  a,  and  prevents  the  upper  parts  from  falling  off',  if  the  in- 
strument is  now  carried  over  the  shoulder.)  Revolve  the  upper  parts  horizontally 
a  trifle,  in  either  direction,  until  they  are  stopped  by  the  striking  of  a  small  lug 
on  a  against  one  of  the  blocks  F.  The  recesses  in  a  are  now  clear  of  the  blocks. 
Tighten  <?,  thereby  pushing  inward  the  movable  block  F,  which  clamps  the 
bevelled  flange  a  between  it  and  the  two  fixed  blocks  on  m  m,  and  confines  the 
spindle  U  to  the  fixed  parallel  plates.  It  remains  so  clamped  while  the  instrument 
is  being  used. 

To  remove  the  upper  parts  from  the  parallel  plates.  Loosen 
9,  bring  the  recesses  in  a  opposite  the  blocks  F.  Hold  back  /,  and  lift  the  upper 
parts,  which  are  then  held  together  by  the  broad  head  of  the  screw  inserted  into 
the  foot  of  the  spindle  w. 

T  T  is  the  outer  revolving-  spindle,  cast  in  one  with  the  support- 
ing-plate Z  Z,  to  which  is  fastened  the  graduated  limb  O  O.  The  limb 
extends  beyond  the  compass-box,  and  thus  admits  of  larger  graduations  than 
would  otherwise  be  obtainable,  w  w  is  the  inner  revolving1  spindle.  At 
its  top  it  has  a  broad  flange,  to  which  is  fastened  the  vernier  plate  P  P.  To 
the  latter  are  fastened  the  compass-box  C,  one  of  the  bubble-tubes  M  M 
(the  one  shown  in  Fig  2),  the  dust-box  W  W,  the  standards  V  V,  supporting 
the  telescope,  &c.  Each  bubble-tube  is  supported  and  adjusted  by  two  capstan- 
screws,  one  at  each  end.  One  is  shown  at  r.  The  bent  strip  curving  over  the 
tube  protects  the  glass. 

The  clamp-screw,  H,  presses  the  split  collar,  tt,  tightly  against  the  fixed  spindle, 
U,  but  not  against  Z  or  T.  The  set-screws,  G  G,  working  in  nuts  that  are  cast  in 
one  with  Z,  hold  between  them  the  tongue,  y,  which  projects  from  1 1,  and  the 
graduated  limb  is  thus  held  fast,  except  that  by  moving  the  screws,  G,  it  may  be 
made  to  revolve  slightly. 

In  Fig  1,  the  tangent-screw,  6,  is  seen  passing  through  two  towers,  in  which  it 
works.  One  of  the  towers  is  fast  to  the  lower  one  of  the  two  small  pieces  at  the 
foot  of  the  clamp-screw  e,  Fig  2.  When  e  is  tightened,  it  draws  the  two  small 
pieces  together,  confining  between  them  an  edge  of  the  graduated  limb,  which  is 
thus  made  fast  to  the  above-mentioned  tower.  The  other  tower  is  fast  to  the 
vernier-plate;  and  the  tangent-screw,  &,  holds  the  towers  at  a  fixed  dist  apart. 
The  clamping  of  e  thus  prevents  the  vernier-plate  from  revolving  over  the  gradu- 
ated limb,  except  that  it  may  be  moved  slightly  by  turning  6,  and  thus  changing 
the  dist  apart  of  the  towers.  In  Heller  and  Brightly's  instruments,  the  screw,  6, 
is  provided  with  means  for  taking  up  its  "wear,"  or  "lost-motion." 

There  are  two  verniers.  One  is  shown  at  p,  Fig  1.  Both  may  be  read,  and 
their  mean  taken,  when  great  accuracy  is  required.  Ivory  reflectors,  c,  facilitate 
their  reading.  Before  the  instrument  is  moved  from  one  place  to  another,  the 
compass-needle.  &,  Fig  2,  should  always  be  pressed  up  against  the  glass  cover 
of  the  compass-box  by  means  of  the  upright  milled-head  screw  seen  on  the  ver- 
nier-plate in  Fig  1,  just  to  the  right  of  the  nearest  standard.  The  pivot-point  is 
thus  protected  from  injury. 

R,  Fig  1,  is  a  ring  with  a  clamp  (the  latter  not  shown)  for  holding  the  telescope 
in  any  required  position.  It  is  best  to  let  the  eye-end,  e,  of  the  telescope  revolve 
downward,  as  otherwise  the  shade  on  O,  if  in  use,  may  fall  off.  The  tangent-screw, 
d,  moves  a  vert  arm  attached  to  R,  and  is  thus  used  for  slightly  changing  the 
elevation  of  the  telescope.  In  the  arm  is  a  slit  like  that  seen  in  the  vernier-arm 
I.  When  0°  of  the  vernier  is  placed  at  30°  on  the  arc.  g,  and  the  index  of  the 
opposite  arm  is  placed  over  a  small  notch  on  the  horizontal  brace  (not  seen  in  our 
figs)  of  the  standards,  the  two  slits  will  be  opposite  each  other,  and  may  be  used 
for  laying  off  offsets,  &c,  at  right-angles  to  the  line  of  sight. 

One  end,  R,  of  the  telescope  axis  rests  in  a  movable  box,  under  which  is  a  screw. 
By  means  of  the  screw,  the  box  may  be  raised  or  lowered,  and  the  axis  thus  ad- 
justed, for  very  slight  derangements  of  the  standards.  For  E,  B,  O,  and  A,  see 
Level,  p  152.  «  is  a  dust-guard  for  the  object-slide. 

Stadia  Hairs.  Immediately  behind  the  capstan  -screw,  p,  Fig  1,  is  seen  a 
smaller  one.  This  and  a  similar  one  on  the  opposite  side  of  the  telescope,  work 
in  a  ring  inside  the  telescope,  and  hold  the  ring  in  position.  Across  the  ring  are 
stretched  two  additional  horizontal  hairs,  called  stadia  hairs,  placed  at  such  a 
distance  apart,  vertically,  that  they  will  subtend  1  foot  on  a  rod,  Ac,  placed  100 
ft  from  the  plumb-bob/1%  ft  at  150  ft,  &c.  They  are  thus  used  for  measuring 
distances. 

The  long-  bubble-tube,  F  F,  Fig  1,  enables  us  •  o  use  the  transit  as  a  level, 
although  it  is  not  so  well  adapted  as  the  latter  to  this  purpose. 


160  THE  ENGINEER'S  TRANSIT. 

To  adjust  a  plain  Transit. 

When  either  a  level  or  a  transit  is  purchased,  it  is  a  good  precaution  (but  one 
which  the  writer  has  never  seen  alluded  to)  to  first  screw  the  object-glass  firmly  home 
to  its  place ;  and  then  make  a  short  continuous  scratch  upon  the  ring  of  the  glass,  and 
upon  its  slide;  so  as  to  be  able  to  see  at  any  time  when  at  work,  that  the  glass  is 
always  in  the  same  position  with  regard  to  the  slide.  For  if,  after  all  the  adjustments 
are  completed,  the  position  of  the  glass  should  become  changed,  (as  it  is  apt  to  be  if 
unscrewed,  and  afterward  not  screwed  up  to  the  same  precise  spot,)  the  adjustments 
may  thereby  become  materially  deranged ;  especially  if  the  object-glass  is  eccentric, 
or  not  truly  ground,  which  is  often  the  case.  Such  scratches  should  be  prepared  bj 
the  maker.  In  making  adjustments,  as  well  as  when  using  a  transit  or  level,  be 
careful  that  the  eye-glass  and  object-glass  are  so  drawn  out  that  there  shall  be  no 
parallax.'  The  eye-glass  must  first  be  drawn  out  so  as  to  obtain  perfect  distinctness 
of  the  cross-hairs ;  it  must  not  be  disturbed  afterward ;  but  the  object-glass  must 
be  moved  for  different  distances. 

First,  to  ascertain  that  the  bubble-tubes,  M  91,  are  placed 
parallel  to  the  vernier-plate,  and  that  therefore  when  both  bubbles  are  in 
the  centers  of  their  tubes  the  axis  of  the  inst  is  vert.  By  means  of  the  four  levelling- 
screws,  K,  bring  both  bubbles  to  the  centers  of  their  tubes  in  one  position  of  the 
inst ;  then  turn  the  upper  parts  of  the  inst  half-way  round.  If  the  bubbles  do  not 
remain  in  the  center,  correct  half  the  error  by  means  of  the  two  capstan-screws 
rr;  and  the  other  half  by  the  levelling-screws  K.  Repeat  the  trial  until  both 
bubbles  remain  in  the  center  while  the  inst  is  being  turned  entirely  around  on 
its  spindle. 

Second,  to  see  that  the  standards  have  suffered  no  derange- 
ment ;  that  is,  that  they  are  of  equal  height  and  perpendicular  to  the  vernier- 
plate,  as  they  always  are  when  they  leave  the  maker's  hands.  Level  the  inst 
perfectly ;  then  direct  tlie  intersection  of  the  hairs  to  some  point  of  a  high  object 
(as  the  top  of  a  steeple)  near  by ;  clamp  the  inst  by  means  of  screws  H  and  e, 
and  lower  the  telescope  until  the  intersection  strikes  some  point  of  a  low  object. 
(If  there  is  none  such  drive  a  stake  or  chain-pin,  Ac,  in  the  line.)  Then  un- 
clamp  either  H  or  e,  and  turn  the  upper  parts  of  the  inst  half-way  round  ;  fix  the 
intersection  again  upon  the  high  point;  clamp;  lower  the  telescope  to  the  low 
point.  If  the  intersection  still  strikes  the  low  point,  the  standards  are  in  order. 
If  not,  correct  one-quarter  of  the  diff  (same  principle  as  in  Fig  4)  by  means  of  the 
adjusting-block  and  screw  at  the  end,  R,  of  the  telescope  axis,  Fig  1,  and  repeat 
the  trial  de  novo,  resetting  the  stake  or  chain-pin  at  each  trial.  If  the  inst  has  no 
ad  just  ing- block  for  the  axis,  it  should  be  returned  to  the  maker  for  correction  of 
any  derangement  of  the  standards. 

A  transit  may  be  used  for  running  straight  lines,  even  if  the  standards  become 
slightly  bent,  by  the  process  described  at  the  end  of  the  fourth  adjustment. 

Third,  to  see  that  the  cross-hairs  are  truly  vert  and  hor 
when  the  inst  is  level.  When  the  telescope  inverts,  the  cross-hairs  are 
nearer  the  eye-end  than  when  it  shows  objects  erect.  The  maker  takes  care  to  place 
the  cross-hairs  at  right-angles  to  each  other  in  their  ring,  or  diaphragm  ;  and  gene- 
rally he  so  places  the  ring  in  the  telescope,  that  when  levelled,  they  shall  be  vert 
and  hor.  Sometimes,  however,  this  is  neglected ;  or  if  attended  to,  the  ring  may  by 
accident  become  turned  a  little ;  especially  if  the  novice  permits  all  the  screws  to  be 
loosened  at  one  time.  To  be  certain  that  one  hair  is  vert,  (in  which  case  the  other 
must,  by  construction,  be  hor,)  after  having  adjusted  the  bubble-tubes,  level  the  in- 
strument carefully,  and  take  sight  with  the  telescope  at  a  plumb-line,  or  other  vert 
straight  edge.  If  the  vert  hair  coincides  with  this  object, 
it  is,  so  far,  in  adjustment;  but  if  not,  then  loosen  slightly 
only  two  adjacent  screws  of  the  four,  pp  i  t,  Fig  1;  and 
with  a  knife,  key,  or  other  small  instrument,  tap  very 
gently  against  the  screw-heads,  so  as  to  turn  the  ring  a 
little  in  the  telescope;  persevering  until  the  hair  be- 
comes truly  vertical.  When  this  is  done,  tighteu  the 
screws;  and  in  future  take  care  that  but  two  are  loosened 
at  once.  In  the  absence  of  a  plumb-line,  or  vert  straight 
edge,  sight  the  cross-hair  at  a  very  small  distinct  point; 
and  see  if  the  hair  still  cuts  that  point,  when  the  tele- 
scope is  raised  or  lowered  by  revolving  it  on  its  axis. 
.ff.  3.  The  mode  of  performing  the  foregoing  will  be  readily 

understood  from  this  Fijr,  which  represents  a  section  across  the  top  part  of  the  tele-, 
scope,  and  at  the  cross-hairs.  The  hair-ring,  or  diaphragm,  a;  vert  hair,  v;  tele- 
scope tube,  g ;  ring  outside  of  telescope  tube,  d ;  6  is  one  of  the  four  capstan-headed 
screws  which  hold  the  hair-ring,  a,  in  its  place,  and  also  serve  to  adjust  it.  The 
lower  ends  of  these  screws  work  in  the  tnickness  of  the  hair-ring;  so  that  when 
they  are  loosened  somewhat,  they  do  not  lose  their  hold  on  the  ring.  Small  loose 


THE  ENGINEER'S  TRANSIT.  161 

washers,  e,  are  placed  under  the  heads  b  of  the  screws.  A  space  y  y  is  left  around 
each  screw  where  it  passes  through  the  telescope  tube,  to  allow  the  screws  and  ring 
together  to  be  moved  slightly  sideways  when  the  screws  b  are  slightly  loosened. 

Fourth,  to  see  that  the  vertical  hair  is  in  the  line  of  colli- 
maf  son.  so  that  it  shall  strike  in  the  same  straight  line  in  both  directions  from 
the  instrument,  when  the  telescope  is  revolved  vert  for  taking  both  a  back  and  a  fore 
sight.  Plant  the  tripod  firmly  upon  the  ground,  as  shown  by  the  three  dots  at  a. 
Level  the  instrument;  clamp 

it;  and  direct  the  vert  hair  by  ..'-•£ 

means  of  the  tangent-screws  ^--'~ 

G  (see  figure  of  transit)  upon          -.  *\  ^--'* 

some  convenient  object  6;  or          b  «V '"-/•**'* 

if  there  is  none  such,  drive  a  »  •  4^'  •  «Q 

thin  stake,  or  a  chain-pin.  '*»,, 

Then  revolving  the  telescope  *-*,^  9~v 

Vert  on  its  axis,  observe  some 

object,  as  c,  where   the  vert  "Flfi*    4  "^"Itl 

hair  now  strikes;  or  if  there  o* 

is  none,  place  a  second  pin.  Then  unclamp  the  instrument  by  the  clamp-screw  H ; 
and  turn  the  whole  upper  part  of  it  around  until  the  vert  hair  again  strikes  b. 
Clamp  again ;  and  again  revolve  the  telescope  vertically  on  its  axis.  If  the  vert  hair 
now  strikes  c,  as  it  did  before,  it  shows  that  c  is  really  at  o;  and  that  6,  a,  c,  are  in 
the  same  straight  line ;  and  therefore  this  adjustment  is  in  order.  But  if  it  is  not  so, 
observe  also  where  it  does  strike  when  the  telescope  is  revolved  the  second  time,  say  at 
rn,  (the  dist  a  m  being  taken  equal  to  a  c,)  and  place  a  pin  there  also.  Measure  m  c ;  and 
place  a  pin  at  r,  in  the  line  m  c,  making  m  v  equal  to  one-fourth  of  m  c.  Also  put  a 
pin  at  o,  hall-way  between  m  and  c,  or  in  range  with  a  and  b.  By  means  of  the  two 
hor  screws  that  move  the  ring  carrying  the  cross-hairs,  adjust  the  vert  hair  until  it 
cutsu;  in  doing  which,  remember  that  the  hair  must  be  moved  in  the  direction 
opposite  to  what  appears  to  be  the  right  one,  unless  the  telescope  is  an  inverting 
one.  Now  repeat  the  entire  operation  ;  and  persevere  until  the  telescope,  after  being 
directed  to  6,  shall  strike  the  same  object  o,  both  times,  when  revolved  on  its  axis. 

See  whether  the  movement  of  the  ring  in  this  fourth  adjustment  has  disturbed  the 
vertically  of  the  hair ;  and  if  it  has,  repeat  the  third  adjustment.  Then  repeat  the 
fourth,  if  necessary  ;  and  so  on  until  both  adjustments  are  found  to  be  right  at  the 
same  time.  * 

Hence  it  is  seen  that  a  straight  line  may  be  run,  even  if  the  hairs  are  out  of 
adjustment ;  but  with  somewhat  more  trouble.  For  at  each  station,  as  at  a,  two 
back-sights,  and  two  fore-sights,  a  c  and  a  m,  may  be  taken,  as  when  making  the 
adjustment;  and  the  point  o,  half-way  between  c  and  m,  will  be  in  the  straight  line. 
The  instrument  may  then  be  moved  to  the  station  o,  and  the  two  back-sights  be 
taken  to  a  ;  and  so  on. 

Angles  measured  by  the  transit,  whether  vert  or  hor,  will  evidently  not  be  affected 
by  the  hairs  being  out  of  adjustment,  provided  either  that  the  vert  hair  is  truly  vert, 
or  that  we  use  the  intersection  of  the  hairs  when  measuring. 

The  foregoing1  are  all  the  adjustments  needed,  unless  the  transit 
is  required  for  levelling,  in  which  case  the  following  one  must  be  attended  to: 

To  adjust  the  long'  bubble-tube,  F  F,  Fig  1,  we  must  first  place  the  line 
of  sight  of  the  telescope  level ;  and  then  make  the  tube  parallel  to  it.  Drive  two 
pegs,  a  and  6,  with  their  tops  at  pre- 
cisely the  same  level  (see  Rem  be- 
low), and  not  less  than  about  100  feet 
apart;  300  feet  or  more  will  be  better. 
Then  plant  the  instrument  firmly,  in 
range  with  them,  as  at  c.  Level  it  by 
the  four  levelling-screws ;  then,  byre- 
peated  trials,  by  moving  the  target- 
rod  from  stake  to  stake,  altering  the  height  of  the  target,  and  revolving  the  tele- 
scope up  or  down  a  little  to  correspond,  find  that  position  in  which  the  hairs  cut  at 
precisely  the  same  heights  on  the  rod  at  both  stakes.  The  line  of  sight  through  the 
telescope  is  then  evidently  level,  or  hor.  Keep  the  target  at  the  same  height,  stand- 
ing on  the  farthest  peg  a,  as  a  guide  for  now  adjusting  the  bubble-tube  parallel  to 
the  line  of  sight.  This  is  done  by  moving  the  two  small  nuts  nn,  at  one  end  of  the 
tube,  Fig  1,  until  the  bubble  stands  at  the  center  of  the  tube;  while  the  line  of 
sight  at  the  same  time  strikes  the  target  at  s.  The  zeros  of  the  vert  circle,  and  of 
its  vernier,  may  now  be  adjusted,  if  they  require  it,  by  loosening  the  vernier 
screws;  and  then  moving  the  vernier  until  the  two  coincide. 

Rem.  If  no  level  is  at  hand  for  levelling  the  two  pegs  a  and  6,  it  may  be  done 
by  the  transit  itself,  thus:  Carefully  level  the  two  short  bubbles,  by  means  of  the 

11 


162  THE   THEODOLITE. 

le veiling-screws  K.  Drive  a  peg  w,  from  100  to  300  feet  from  the  instrument  o. 
Then  placing  a  target-rod  on  m,  clamp  the  target  tight  at  whatever  height,  as  ev, 
the  hor  hair  happens  to  cut  it;  it  being  of  no  i ra- 


the vernier  also  is  made 


m 

•*!§"•  O-  nearly  or  quite  half  way.    Place  another  peg'w, 

at  precisely  the  same  dist  from  the  instrument  that  m  is ;  and  continue  to  drive  it  un- 
til the  hor  hair  cuts  the  target  placed  on  it,  and  still  kept  clamped  to  the  rod,  at  the 
same  height  as  when  it  was  on  m.  When  this  is  done,  the  tops  of  the  two  pegs  are 
on  a  level  with  each  other,  and  are  ready  to  be  used  as  before  directed. 

When  a  transit  is  intended  to  be  used  for  surveying  farmf,  &c,  or  for  retracing 
lines  of  old  surveys,  it  is  very  useful  to  set  the  compass  so  as  to  allow  for  the  "va- 
riation" during  the  interval  between  the  two  surveys.  For  this  purpose  a 
*'  variation- vernier"  is  added  to  such  transits;  and  also  to  the  compass. 

When  the  graduations  of  a  transit  are  figured,  or  numbered,  so  as  to  read  both 

10  0  10 

•ways  from  zero,  thus,   M  i  I  n  i  i  I  i  i  t  i  I  i  i  i  i !  it  i  i  I  i  i  i 

double ;  that  is,  it  also  is  graduated  and  numbered  from  its  zero  both  ways.  In  this 
case,  if  the  angle  is  measured  from  zero  toward  the  right  hand,  the  reading  must  ba 
made  from  the  right  hand  half  of  the  vernier;  and  vice  versa.  If  the  figuring  is 
single,  or  only  in  one  direction,  from  zero  to  360°,  then  only  the  single  vernier  is 
necessary,  as  the  angles  are  then  measured  only  in  the  direction  that  the  figuring 
counts.  Engineers  differ  iir  their  preferences  for  various  manners  of  figuring  the 
graduations.  The  writer  prefers  from  zero  each  way  to  180°,  with  two  double  ver- 
niers. 

To  replace  cross-hairs  in  a  level,  or  transit.  Take  out  the  tube 
from  the  eye  end  of  the  telescope.  Looking  in,  notice  which  side  of  the  cross- 
hair diaphragm  is  turned  toward  the  eye  end.  Then  loosen  the  four  screws  which 
hold  the  diaphragm,  so  as  to  let  the  latter  fall  out  of  the  telescope.  Fasten  on  new 
hairs  with  beeswax,  varnish,  glue,  or  gum-arabic  water,  &c.  This  requires  care. 
Then,  to  return  the  diaphragm  to  its  place,  press  firmly  into  one  of  the  screw-holes 
on  the  circumf  of  the  diaphragm  itself,  the  end  of  a  piece^of  stick,  long  enough  to 
reach  easily  into  the  telescope  as  far  as  to  where  the  diaphragm  belongs.  By  this 
stick,  as  a  handle,  insert  the  diaphragm  edgewise  to  its  place  in  the  telescope,  and  hold 
it  there  until  two  opposite  screws  are  put  in  place  and  screwed.  Then  draw  the  stick 
out  of  the  hole  in  the  diaphragm ;  and  with  it  turn  the  diaphragm  until  the  same 
side  presents  itself  toward  the  eye  end  as  before;  then  put  in  the  other  two  screws. 

The  so-called  cross-hairs  are  actually  spider-web,  so  fine  as  to  be  barely  visible  to 
the  naked  eye.  Heller  &  Brightly  use  very  fine  platina  wire,  which  is  much  better. 
Human  hair  is  entirely  too  coarse. 

To  replace  a  spirit-level,  or  bubble-glass.  Detach  the  level  from 
the  instrument;  draw  off  its  sliding  ends;  push  out  the  broken  glass  vial,  and  the 
cement  which  held  it ;  insert  the  new  one,  with  the  proper  side  up  (the  upper  side 
is  always  marked  with  a  file  by  the  maker);  wrapping  some  paper  around  its  ends, 
if  it  fits  loosely.  Finally,  put  a  little  putty,  or  melted  beeswax  over  the  ends  of  the 
vial,  to  secure  it  against  moving  in  its  tube. 

In  purchasing  instruments,  especially  when  they  are  to  be  used  far  from  a  maker, 
it  is  advisable  to  provide  extras  of  such  parts  as  may  be  easily  broken  or  lost;  such 
as  glass  compass -covers,  and  needles;  adjusting  pins;  level  vials;  magnifiers,  &c. 


THE  THEODOLITE. 


THE  adjustments  are  performed  like  those  of  the  level  and  transit. 

1st.  That  of  the  cross-hairs;  the  same  as  in  the  level. 

2d.  The  long  bubble-tube  of  the  telescope ;  also  as  in  the  level. 

3d.  The  two  short  bubble-tubes ;  as  in  the  transit. 

4th.  The  vernier  of  the  vert  limb;  as  in  the  transit  with  a  vert  circle. 

5th.  To  see  that  the  vert  hair  travels  vertically;  as  in  the  fourth  adjustment  of  the 
transit.  In  some  theodolites,  as  in  most  transits,  no  adjustment  is  provided  for  this ; 
but  in  large  ones  there  is.  It  is  accomplished  by  screws  under  the  feet  of  the 
standards. 

Sometimes  a  second  telescope  is  added;  it  is  placed  below  the  hor  limb,  and  is 


THE  BOX  OR  POCKET  SEXTANT. 


163 


called  a  watcher.  It  has  its  own  clamp,  and  tangent-screw.  Its  use  is  to  ascertain 
whether  the  zero  of  that  limb  has  moved  during  the  measurement  of  hor  angles. 
When,  previously  to  beginning  the  measurement,  the  zero  and  upper  telescope  are 
directed  toward  the  first  object,  point  the  lower  telescope  to  any  small  distant 
object,  and  then  clamp  it.  During  the  subsequent  measurement,  look  through  it, 
from  time  to  time,  to  be  sure  that  it  still  strikes  that  object;  thus  proving  that  ne 
slipping  has  occurred. 


THE  BOX  OR  POCKET  SEXTANT, 


THE  portability  of  the  pocket  sextant,  and  the  fact  that  it  reads  to  single  minutes, 
render  it  at  times  very  useful  to  the  engineer.*  By  it,  angles  can  be  measured  while 
in  a  boat,  or  on  horseback;  and  in  many  situations  which  preclude  the  use  of  a 
transit.  It  is  useful  for  obtaining  latitudes,  by  aid  of  an  artificial  horizon.  When 
closed,  it  resembles  a  cylindrical  brass  box,  about  3  inches  in  diameter,  and  1% 
inches  deep.  This  box  is  in  two  parts ; 
by  unscrewing  which,  then  inverting 
one  part,  and  then  screwing  them  to- 
gether again,  the  lower  part  becomes  a 
handle  for  holding  the  instrument. 
Looking  down  upon  its  top  when  thus 
arranged,  we  see,  as  in  this  figure,  a 
movable  arm  I  C,  called  the  index, 
which  turns  on  a  center  at  C,  and  car- 
ries the  vernier  V  at  its  other  end.  Q 
(j  is  the  graduated  arc  or  limb.  It 
actually  subtends  about  73°,  but  is  di- 
vided into  about  146°.  Its  zero  is  at 
one  end.  Its  graduations  are  not  shown 
in  the  Fig. 

Attached  to  the  index  is  a  small  mov- 
able lens,  (not  shown  in  the  figure,) 
likewise  revolving  around  C,  for  read- 
ing the  fine  divisions  of  the  limb.  When 
measuring  an  angle,  the  index  is  moved 
by  turning  the  milled-head  P  of  a 
pinion,  which  works  in  a  rack  placed  within  the  box.  The  eye  is  applied  to  a  cir- 
cular hole  at  the  side  of  the  box,  near  A.  A  small  telescope,  about  3  inches  long, 
accompanies  the  instrument ;  but  may  generally  be  dispensed  with.  When  so,  the 
eye-hole  at  A  should  be  partially  closed  by  a  slide  which  has  a  very  small  eye-hole 
in  it ;  and  which  is  moved  by  the  pin  ft,  moving  in  the  curved  slot.  Another  slide, 
at  the  side  of  the  box,  carries  a  dark  glass  for  covering  the  eye-hole  when  observing 
the  sun.  When  the  telescope  is  used,  it  is  fastened  on  by  the  milled-head  screw  T. 
The  top  part  shown  in  our  figure,  can  be  separated  from  the  cylindrical  part,  by 
removing  3  or  4  small  screws  around  its  edge ;  and  the  interior  can  then  be  exam- 
ined, and  cleaned  if  necessary.  Like  nautical,  and  other  sextants,  this  one  has 
two  principal  glasses,  both  of  them  mirrors.  One,  the  index-glass,  is  attached 
to  the  underside  of  the  index,  at  C;  its  upper  edge  being  indicated  by  the 
two  dotted  lines.  The  other,  the  horizon-glass,  (because,  when  meas- 
uring the  vert  angles  of  celestial  bodies,  it  is  directed  toward  the  horizon,)  is  also 
within  the  box;  the  position  of  its  upper  edge  being  shown  by  the  dotted  lines  at 
R.  The  horizon-glass  is  silvered  only  half-way  down ;  so  that  one  of  the  observed 
objects  may  be  seen  directly  through  its  lower  half,  while  the  image  of  the  other 
object  is  seen  in  the  upper  half,  reflected  from  the  index-glass.  That  the  instrument 
may  be  in  adjustment,  ready  for  use,  these  two  glasses  must  be  at  right  angles  to  th« 
plane  of  the  instrument ;  that  is,  to  the  under  side  of  the  top  of  the  box,  to  which  they 
are  attached;  and  must  also  be  parallel  to  each  other,  when  the  zeros  of  the  vernier 
and  of  the  limb  coincide.  The  index-glass  is  already  permanently  fixed  by  the 
maker,  and  requires  no  other  adjustment.  But  the  horizon-glass  has  two  adjust- 
ments, which  are  made  by  a  key  like  that  of  a  watch,  and  having  a  milled-head  K. 
It  is  screwed  into  the  top  of  the  box,  so  as  to  be  always  at  hand  for  use.  When 
needed,  it  is  unscrewed.  This  key  fits  upon  two  small  square-heads,  (like  that  for 

•Price,  with  telescope,  about  $50.    Made  by  Stackpole  &  Bro..  41  Pulton  St.,  New  York. 


164 


THE   COMPASS. 


winding  a  watch ;)  one  of  which  is  shown  at  S ;  while  the  other  is  near  it,  but  on  the 
SIDE  of  the  box.  These  squares  are  the  heads  of  two  small  screws.  If  the 
horizon  glass  II  should,  v  in  this  sketch,  (where  it  is  shown  endwise,)  not  be  at 
right  angles  to  the  top  U  J  of  the  box,  it  is  brought  right  by  turning  the  square- 
head S  of  the  screw  S  T;  and  if,  after  being  so  far  rectified,  it  still  is  not  parallel  to 
the  index-glass  when  the  zeros  coincide,  it  is  moved 
a  little  backward  or  forward  by  the  square  head 
at  the  side. 

To  adjust  a  box  sextant,  bring  the  two 
zeros  to  coincide  precisely;  then  look  through  the 
eye-hole,  and  the  lower  or  unsilvered  part  of  the 
horizon-glass,  at  some  distant  object.  If  the  instru- 
ment is  in  adjustment,  the  object  thus  seen  directly, 
will  coincide  precisely  with  its  reflected  image, 
seen  at  the  same  time,  at  the  same  spot.  But  if  it 
is  not  in  adjustment,  the  two  will  appear  separated 
either  hor  or  vert,  or  both,  thus,  *  * ;  in  which  case 


H 


B 


j ,  _.  ju  a  icvei  wiin  eauii  omer,  mus,  "  *•.  ±nen  appiy  me  Key  10  me  square- 
head in  the  side  of  the  box;  and  by  turning  it  slightly,  bring  the  two  to  coincide 
perfectly.  The  instrument  is  then  adjusted. 

Tri  some  instruments,  -the  hor  glass  has  a  hinge  at  v,  to  allow  it  play  Avhile  being 
adjusted  by  the  single  screw  S  T ;  but  others  dispense  with  this  hinge,  and  use  two 
screws  like  S  on  top  of  the  box,  in  addition  to  the  one  in  the  side. 

If  a  sextain  is  used  lor  measuring  vert  angles  by  means  of  an  artificial 
horizon,  the  actual  altitude  will  be  but  one-half  of  that  read  off  on  the 
limb ;  because  we  then  read  at  once  both  the  actual  and  the  reflected  angle.  The 
great  objection  to  the  sextant  for  engineering  purposes,  is  that  it  does  not  measure 
angles  horizontally,  as  the  transit  does;  unless  when  the  observer,  and  the  two  ob- 
jects happen  to  be  in  the  same  hor  plane. 
Thus  an  observer  with  a  sextant  at  A,  if 
measuring  the  angle  subtended  by  the 
mountain-peaks  B  and  C,  must  hold  the 
graduated  plane  of  the  sextant  in  the 
plane  of  A  B  C ;  and  must  actually  meas- 
ure the  angle  BAG;  whereas  what  he 
wants  is  the  hor  angle  n  A  m.  This  is 
greater  than  BAG,  because  the  dists  A  n 
and  A  m  are  shorter  than  A  B  and  A  C. 
The  transit  gives  the  hor  angle  n  A  m,  be- 
cause its  graduated  plane  is  first  fixed  hor  by  the  levell ing-screws :  and  the  subse- 
quent measurement  of  the  angle  is  not  affected  by  his  directing  merely  the  line  of 
sight  upward,  to  any  extent,  in  order  to  fix  it  upon  B  and  C.  1'or  more  on  this  sub- 
ject; and  for  a  method  of  partially  obviating  this  objection  to  the  sextant,  see  the 
note  to  Example  2,  Gase  4,  of  "  Trigonometry,"  page  40. 

The  nautical  sextant,  used  on  ships,  is  constructed  on  the  same  principle 
as  the  box  sextant ;  and  its  adjustments  are  very  similar.  In  it,  also,  the  index- 
glass  is  permanently  fixed  by  the  maker;  and  thje  horizon-glass  has  the  two  adjust- 
ments of  the  box  sextant.  It  also  has  its  dark  glasses  for  looking  at  the  sun ;  and 
a  small  sight-hole,  to  be  used  when  the  telescope  is  dispensed  with. 


THE  COMPASS. 


To  adjust  a  Compass. 

The  first  adjustment  is  that  of  the  bubbles.  Plant  firmly  ;  and  level  the 
instrument,  in  any  position ;  that  is,  bring  the  bubbles  to  the  centers  of  their  tubes. 
Then  turn  the  instrument  half-way  round.  If  the  bubbles  then  remain  at  the  cen* 
ters,  they  are  in  adjustment;  but  if  not,  correct  one-half  the  diff  in  each  Bubble, 
by  means  of  the  adjusting-screws  of  the  tubes.  Level  the  instrument  again;  turi? 
it  half  round;  and'if  the  bubbles  still  do  not  remain  at  the  center,  the  adjusting- 
screws  must  be  again  moved  a  little,  so  as  to  rectify  half  the  remaining  diff.  Gener- 


THE   COMPASS.  165 

ally  several  ti-ials  must  be  thus  made,  until  the  bubbles  will  remain  at  the  center, 
while  the  compass  is  being  turned  entirely  around. 

Second  adjustment.  Level  the  compass,  and  then  see  that  the  needle  is 
hor ;  and  if  not,  make  it  so  by  means  of  the  small  piece  of  wire  which  is  wrapped 
around  it ;  sliding  the  wire  toward  the  high  end.  A  needle  thus  horizontally  ad- 
justed at  one  place,  will  not  remain  so  if  re-moved  far  north  or  south  from  that  place. 
If  carried  to  the  north,  the  north  end  will  dip  down ;  and  if  to  the  south,  the  south 
end  will  do  so.  The  sliding  wire  is  intended  to  counteract  this. 

Third  adjustment.  This  is  always  fixed  right  at  first  by  the  maker;  that 
is,  the  sights,  or  slits  for  sighting  through,  are  placed  at  right  angles  to  the  compass 
plate ;  so  that  when  the  latter  is  levelled  by  the  bubbles,  the  sights 
are  vert.  To  test  whether  they  are  so,  hang  up  a  plumb-line ;  and 
having  levelled  the  compass,  take  sight  at  the  line,  aud  see  if  the 
slits  coincide  with  it.  If  one  or  both  slits  should  prove  to  be 
out  of  plumb,  as  shown  to  an  exaggerated  extent  in  this  sketch, 
it  should  be  unscrewed  from  the  compass,  and  a  portion  of  its  foot 
on  the  high  side  be  filed  or  ground  off,  as  per  the  dotted  line ;  or 
as  a  temporary  expedient,  a  small  wedge  may  be  placed  under  the  ^ 
low  side,  so  as  to  raise  it. 

Fourth  adjustment,  to  straighten  the  needle,  if  it  should  become  bent. 
The  compass  being  levelled,  and  the  needle  hor,  and  loose  on  its  pivot,  see  whether 
its  two  ends  continue  to  point  to  exactly  opposite  graduations,  (that  is,  graduations 
180°  apart;)  while  the  compass  is  turned  completely  around.  If  it  does,  the  needle 
is  straight ;  and  its  pin  is  in  the  center  of  the  graduated  circle  ;  but  if  it  does  not, 
then  one  or  both  of  these  require  adjusting.  First  level  the  compass.  Then  turn  it 
until  some  graduation  (say  90°)  comes  precisely  to  the  north  end  of  the  needle.  If 
the  south  end  does  not  then  point  precisely  to  the  opposite  90°  division,  lilt  off  tho 
needle,  and  bend  the  pivot-point  until  it  does ;  remembering  that  every  time  said 
point  is  bent,  the  compass  must  be  turned  a  hairsbreadth  so  as  to  keep  the  north  end 
of  the  needle  at  its  90°  mark.  Then  turn  the  compass  half-way  round,  or  until  the 
opposite  90°  mark  comes  precisely  to  the  north  end  of  the  needle.  Make  a  fine  pen- 
cil mark  where  the  south  end  of  the  needle  now  points.  Then  take  off  the  needle, 
and  bend  it  until  its  south  end  points  half-way  between  its  90°  mark  and  the  pencil 
mark,  while  its  north  end  is  kept  at  90°  by  moving  the  compass  round  a  hairsbreadth. 
The  needle  will  then  be  straight,  and  must  not  be  altered  in  making  the  following 
adjustment,  although  it  will  not  yet  cut  opposite  degrees. 

Fifth  adjustment,  of  the  pivot-pin.  After  being  certain  that  the  needle  is 
straight,  turn  the  compass  around  until  a  part  is  arrived  at  where  the  two  ends  of  the 
needle  happen  to  cut  opposite  degrees.  Then  turn  the  compass  quarter  way  around, 
or  through  90°.  If  the  needle  then  cuts  opposite  degrees,  the  pivot-point  is  already 
in  adjustment;  but  if  the  needle  does  not  so  cut,  bend  the  pivot-point  until  it  does. 
Repeat,  if  necessary,  until  the  needle  cuts  opposite  degrees  while  being  turned  entirely 
around. 

Care  and  nicety  of  observation  are  necessary  in  making  these  adjustments  properly ; 
because  the  entire  error  to  be  rectified  is,  in  itself,  a  minute  quantity;  and  the  novice 
is  very  apt  to  increase  his  trouble  by  not  knowing  how  to  use  his  magnifier, 
when  looking  at  the  end  of  the  needle  and  the  corresponding  graduations.  The  mag- 
nifier must  always  be  held  with  its  center  directly  over  the  point  to  be  examined;  and 
it  must  be  held  parallel  to  the  graduated  circle.  Otherwise  annoying  errors  of 
several  minutes  will  be  made  in  a  single  observation ;  and  the  accumulation  of  two 
or  three  such  errors,  arising  from  a  cause  unknown  to  him,  may  compel  him  to 
abandon  the  adjustments  in  despair.  This  suggestion  applies  also  to  the  reading  of 
angles  taken  by  the  transit,  &c ;  although  the  errors  are  not  then  likely  to  be  so 
great  as  in  the  case  of  the  compass.  In  purchasing  a  magnifier  for  a  compass,  see 
that  no  part  of  it,  as  hinges,  or  rivets,  are  made  of  iron ;  for  such  would  change  the 
direction  of  the  needle. 

If  the  sight-slits  of  a  compass  are  not  fixed  by  the  maker  in  line  with  the  two 
opposite  zeros,  the  engineer  cannot  remedy  the  defect.  This  can  be  ascertained  by 
passing  a  piece  of  fine  thread  through  the  slits,  and  observing  whether  it  stands 
precisely  over  the  zeros. 

Variation  of  the  Compass.* 

The  numerous  disturbing  influences  to  which  the  compass  is  subject,  render  its 

*  For  full  information  on  this  subject  see  that  useful  little  book  "  Magnetic  Variation  iu  the  U  S,* 
by  J  B  Stone,  C  E,  1678.  It  is  invaluable  iu  retracing  old  lines. 


166  THE   HAND-LEVEL. 

indications  of  bearings  or  courses  very  unreliable.  The  daily  variation  itself  some' 
times  amounts  to  ^  of  a  degree ;  and  always  to  at  least  several  minutes.  It  is  almost 
incessantly  changing  the  direction  of  the  needle,  to  one  side  or  the  other,  at  the  rate 
of  1  or  2  minutes  per  hour,  especially  in  summer.  Local  attraction,  from  iron  in  the 
soil,  ferruginous  gravel,  trap  rocks,  &c,  is  another  source  of  inaccuracy;  as  are  also 
the  annual  and  the  secular  variations.  Electricity,  either  atmospheric,  or  excited 
by  rubbing  the  glass  cover,  sometimes  gives  trouble.  It  may  be  removed  by  touch- 
ing the  glass  with  the  moist  tongue,  or  finger.  It  is  plain  that  none  of  these  causes 
(except  the  last)  will  aifect  the  measurement  of  angles  by  the  compass. 

A  straight  line  drawn  across  the  United  States,  passing  through  Cleveland,  Ohio, 
and  Wilmington,  N  C,  is  nearly  a  line  of  110  variation,,  in  1875;  that  is. 
a  compass  placed  anywhere  in  the  vicinity  of  that  line,  will  point  nearly  due  north 
and  south.  To  the  eastward  of  this  line,  the  variation  is  westward ;  and  vice  versa ; 
becoming  greater,  the  farther  the  place  is  from  the  line ;  until  in  some  parts  of  Maine 
and  along  the  Pacific  coast  it  is  as  great  as  18°  to  21°.  This  line  is  moving  westward, 
at  an  average  rate  of  about  3  or  4  minutes  per  year. 

The  needle,  if  of  soft  metal,  sometimes  loses  part  of  its  magnetism,  and  consequently 
does  not  work  well.  It  may  be  restored  by  simply  drawing  the  north  pole  of  a 
common  magnet  (either  straight  or  horseshoe)  about  a  dozen  times,  from  the  center 
to  the  end  of  the  south  half  of  the  needle :  and  the  south  pole,  in  the  same  way,  along 
the  north  half;  pressing  the  magnet  gently  upon  the  needle.  After  each  stroke, 
remove  the  magnet  several  inches  from  the  needle,  while  bringing  it  back  to  the 
center  for  making  another  stroke.  Each  half  of  the  needle  in  turn,  while  being  thus 
operated  on,  should  be  held  flat  upon  a  smooth  hard  surface.  Sluggish  action  of  the 
tieedle  is,  however,  more  generally  produced  by  the  dulling  or  other  injury  of  the 
point  of  the  pivot.  Remagnetizing  will  throw  the  needle  out  of  balance ;  which  must 
be  counteracted  by  the  sliding  wire. 

In  order  to  prevent  mistakes  by  reading  sometimes  from  one  end, 
and  sometimes  from  the  other  end  of  the  needle,  it  is  best  to  ALWAYS  point  the  N  of 
the  compass-box  toward  the  object  whose  bearing  is  to  be  taken ;  and  to  read  off 
from  the  north  end  of  the  needle.  This  is  also  more  accurate. 


THE    HAND-L.EVEL,. 


THIS  very  useful  little  instrument,  as  arranged  by  Professor  Locke,  of  Cincinnati,  is 
but  about  five  or  six  inches  long.  Simply  holding  it  in  one  hand,  and  looking  through 
It  in  any  direction,  we  can  ascertain  at  once,  approximately,  what  objects  are  at  the 
same  level  with  the  eye.  E  is  the  eye  end:  and  0  the  object  end.  L  is  a  small 
level,  enclosed  in  a  kind  of  brass  boxing  t  g,  the  bottom  of  which  is  open,  with  a  cor- 
responding opening  under  it,  through  the  top  of  the  main  tube  E  0.  Immediately 
at  the  bottom  of  the  small  level  L,  is  a  cross-wire,  stretched  across  said  opening,  and 
carried  by  a  small  plate,  which,  for  adjusting  the  wire,  can  be  pushed  backward  a 
trifle  by  tightening  the  screw  t,  or  pushed  forward  by  a  small  spring  within  the  box- 
ing, near  <7,  when  the  screw  t  is  loosened.  At  m  is  a  small  semicircular  mirror  a  a, 
silvered  on  the  back  m.  This  is  placed  at  an  angle  of  45°,  and  occupies  one-hall  the 
width  of  the  tube  E  0.  Through  the  forementioned  openings,  the  images  ot  the 
cross-wire  and  of  the  level-bubble  are  reflected  down  on  the  unsilvered  face  a  a  of 
the  mirror,  and  thence  to  the  eye,  as  shown  by  the  single  dotted  lines  c  and  w;  and 
when  the  instrument  is  adjusted,  and  held  level,  the  wire  will  appear  to  be  at  the 
center  of  the  bubble.  At  /,:  is  one-half  of  a  plano-convex  lens,  at  the  inner  end  of  a 
short  tube  k  «,  which  may  be  moved  backward  or  forward  by  a  pm  w,  projecting 
through  a  short  slit  in  the  main  tube.  By  this  means  the  image  of  the  cross-wire  w 
rendered  distinct, ;  and  the  half  lens  mush  be  moved  until,  when  viewing  an  object 
the  wire  shall  show  no  parallax ;  but  appear  steady  against  the  object  when  the  ey» 


LEVELLING   BY   THE    BAROMETER. 


167 


Is  slightly  moved  up  or  down.  At  each  end  of  the  tube  E  O  is  a  circular  piece  of 
plain  glass  for  excluding  dust. 

To  adjust  the  hand-level,  first  fix  two  precisely  level  marks,  say  from 
50  feet  to'lOO  yards  apart.  This  being  done,  rest  the  instrument  against  one  of  the 
level  marks,  and  take  sight  at  the  other.  If,  then,  the  wire  does  not  appear  to  be 
precisely  at  the  center  of  the  bubble,  move  it  slightly  backward  or  forward,  as  the 
case  may  be,  by  the  screw  t,  until  it  does  so  appear. 
The  two  level  marks  may  be  fixed  by  means  of  the 
hand-level  itself,  even  if  it  is  entirely  out  of  adjust- 
ment, thus  :  First,  by  the  pin  n  arrange  the  half  lens 
A-,  so  as  to  show  the  wire  distinctly  and  without  paral- 
lax. Then  holding  the  level  steadily,  at  any  selected 
object,  as  a,  so  that  the  wire  appears  to  cut  the  center 

of  the  bubble,  see  where  it  cuts  any  other  convenient  object,  as  6.  Then  go  to  &, 
and  from  it,  in  like  manner,  sight  back  toward  a.  If  the  instrument  is  in  adjust- 
ment, the  wire  will  cut  a  ;  but  if  not,  it  will  strike  either  above  it  or  below  it,  as  at  c. 
In  either  case,  make  a  mark  m,  half-way  between  c  and  a.  Then  b  and  m  will  be  the 
two  level  marks  required.  With  care,  these  adjustments,  when  once  made,  will 
remain  in  order  for  years.  The  instrument  generally  has  a  small  ring  r,  for  hanging 
it  around  the  neck  :  it  is  not  adapted  to  very  accurate  work,  but  admirably  so  for 
exploring  a  route.  The  height  of  a  bare  hill  can  be  found  by  beginning  at  the  foot, 
and  sighting  ahead  at  any  little  chance  object  which  the  cross-wire  may  strike,  as  a 
pebble,  twig,  &c;  then  going  forward,  stand  at  that  object,  and  fix  the  wire  on 
another  one  still  farther  on,  and  so  to  the  top.  At  each  observation  we  plainly  rise 


a  height  equal  to  that  of  the  eye,  say  5^  feet,  or  whatever  it  may  be.  Whether 
going  up  or  down  it,  if  the  hill  is  covered  with  grass,  bushes,  &c,  a  target  rod  must 
be  used  for  the  fore-sights  ;  and  the  constant  height  of  the  eye  may  be  regarded  as 


the  back-sight  at  each  station.  An  attachment  may  be  made  for  screwing  the  level 
to  a  small  ball  and  socket  on  top  of  a  cane,  or  of  a  longer  stick,  for  occasional  use, 
when  rather  more  accuracy  is  desired.  The  hand-level  deserves  much  more  notice 
from  the  profession  than  it  has  received,  especially  for  preliminary  examinations  of 

utes,  or  for  taking  topography,  &c.    No  assistant  engineer,  or  leader  of  a  field 

rty,  should  be  without  one.* 


pa 


To  adjust  a  builder's  plumb- 
level,  t'b  d\  stand  it  upon  any  two  sup- 
ports m  and  w,  and  mark  where  the  plumb- 
line  cuts  at  o.  Then  reverse  it.  placing  the 
foot  t  upon  w,  and  d  upon  w,  and  mark  where 
the  line  now  cuts  at  c.  Half-way  between  o 
and  c  make  the  permanent  mark.  Whenever 
the  line  cuts  this,  the  feet  t  and  d  are  on  a 
level. 


To  adjust  a  slope-instrument,  or  clinometer.  As  usually  made, 
the  bubble-tube  is  attached  to  the  movable  bar  by  a  screw  near  each  end,  and  the 
head  of  one  of  the  screws  conceals  a  small  slot  in  the  bar,  which  allows  a  slight  vert 
motion  to  the  screw  when  loose,  and  with  it  to  that  end  of  the  tube.  Therefore,  in 
order  to  adjust  the  bubble,  this  screw  is  first  loosened  a  little,  and  then  moved  up 
or  down  a  trifle,  as  may  be  reqd.  It  is  then  tightened  again. 


LEVELLING-  BY  THE  BAROMETER. 

1.  Many  circumstances  combine  to  render  the  results  of  this  kind  of  levelling  un- 
reliable where  great  accuracy  is  required.  This  fact  was  most  conclusively  proved 
by  the  observations  made  by  Captain  T.  J.  Cram,  of  the  U.  8.  Coast  Survey.  See 
Report  of  U.  S.  C.  S.,  vol.  for  1854.  It  is  difficult  to  read  off  from  an  aneroid  (the 
kind  of  barora  generally  employed  for  engineering  purposes)  to  within  from  two  to 
five  or  six  ft,  depending  on  its  size.  The  moisture  or  dry  ness  of  the  air  affects  the 
results;  also  winds,  the  vicinity  of  mountains,  and  the  daily  atmospheric  tides, 
which  cause  incessant  and  irregular  fluctuations  in  the  barom.  A  barom  hanging 
quietly  in  a  room  will  often  vary  y\j-  of  an  inch  within  a  few  hours,  corresponding 
to  a  diff  of  elevation  of  nearly  100  ft.  No  formula  can  possibly  be  devised  that  shall 
embrace  these  sources  of  error.  The  variations  dependent  upon  temperature,  lati- 

*  Price,  about  $10  or  $12 ;  with  about  $3  more  for  attachable  ball  and  socket. 


168 


LEVELLING   BY   THE   BAROMETER. 


tude,  &c,  are  in  some  measure  provided  for ;  so  that  with  very  delicate  instruments,  a 
skilful  observer  may  measure  the  diff  of  altitude  of  two  points  close  together,  such 
as  the  bottom  and  top  of  a  steeple,  with  a  tolerable  confidence  that  he  is  within  two 
or  three  feet  of  the  truth.  But  if  as  short  an  interval  as  even  a  few  hours  elapses 
between  his  two  observations,  such  changes  may  occur  in  the  condition  of  the  atmo- 
sphere that  he  may  make  the  top  of  the  steeple  to  be  lower  than  its  bottom ;  or  at 
least,  cannot  feel  by  any  means  certain  that  he  is  not  ten  or  twenty  ft  in  error;  and 
this  may  occur  without  any  perceptible  change  in  the  atmosphere.  Whenever  prac- 
ticable, therefore,  there  should  be  a  person  at  each  station,  to  observe  at  both  points 
at  the  same  time.  Single  observations  at  points  many  miles  apart,  and  made  on  dif- 
ferent days,  and  in  different  states  of  the  atmosphere,  are  of  little  value.  In  such 
cases  the  mean  of  many  observations,  extending  over  several  days,  weeks,  or  months, 
and  made  when  the  air  is  apparently  undisturbed,  will  give  tolerable  approximations 
to  the  truth.  In  the  tropics  the  range  of  .the  atmospheric  pres  is  much  less  than 
in  other  regions,  seldom  exceeding  %  inch  at  any  one  spot ;  also  more  regular  in 
time,  and,  therefore,  less  productive  of  error.  Still,  the  barometer,  especially  either 
the  aneroid,  or  Bourdon's  metallic,  may  be  rendered  highly  useful  to  the  civil  engi- 
neer, in  cases  where  great  accuracy  is  not  demanded.  By  hurrying  from  point  to 
point,  and  especially  by  repeating,  he  can  form  a  judgment  as  to  which  of  two  sum- 
mits is  the  lowest.  Or  a  careful  observer,  keeping  some  miles  ahead  of  a  surveying 
party,  may  materially  lessen  their  labors,  especially  in  a  rough  country,  by  select- 
ing the  general  route  for  them  in  advance.  The  accounts  of  the  agreement  within 
a  few  inches,  in  the  measurements  of  high  mountains,  by  diff  observers,  at  diff 
periods  ;  and  those  of  ascertaining  accurately  the  grades  of  a  railroad,  by  means  of 
an  aneroid,  while  riding  in  a  car,  will  be  believed  by  those  only  who  are  ignorant 
of  the  subject.  Such  results  can  happen  only  by  chance. 

When  possible,  the  observations  at  different  places  should  be  taken  at  the  same 
time  of  day,  as  some  check  upon  the  effects  of  the  daily  atmospheric  tides ;  and  in 
yery  important  cases,  a  memorandum  should  be  made  of  the  year,  month,  day,  and 
hour,  as  well  as  of  the  state  of  the  weather,  direction  of  the  wind,  latitude  of  the 
place,  &c,  to  be  referred  to  an  expert,  if  necessary. 

The  effects  of  latitude  are  not  included  in  nny  of  our  formulas.  When 
reqd  they  may  be  found  in  the  table  page  170.  Several  other  corrections  must  be 
made  when  great  accuracy  is  aimed  at ;  but  they  require  extensive  tables. 

In  rapid  railroad  exploring,  however,  such  refinements  may  be  neglected,  inas- 
much as  no  approach  to  such  accuracy  is  to  be  expected ;  but  on  the  contrary,  errors 
of  from  1  to  10  or  more  feet  in  100  of  height,  will  frequently  occur. 

As  a  very  rough  average  we  may  assume  that  the  barometer  falls  y1^ 
inch  for  every  90  feet  that  we  ascend  above  the  level  of  the  sea,  up  to  1000  ft.  But 
in  fact  its  rate  of  fall  decreases  continually  as  we  rise ;  so  that  at  one  mile  high  it 
falls  ^  inch  for  about  106  ft  rise.  Table  2,  p  171,  shows  the  true  rate. 

Corrections  for  temperature;  to  be  used  in  connection  with 
Rule  3,  when  greater  accuracy  is  necessary.  Also  in  con- 
nection with  Table  2  when  the  temp  is  not  32°. 


Mean 

Mean 

Mean 

Mean 

temp 

Mult 

temp 

Mult 

temp 

Mult 

temp 

Mult 

in  the 

by 

in  the 

by 

in  the 

by 

in    the 

by 

shade. 

shade. 

shade. 

shade. 

Zero. 

.933 

28° 

.992 

56° 

1.050 

84° 

1.108 

2° 

.937 

30 

.996 

58 

1.054 

86 

1.112 

4 

.942 

32 

1.000 

60 

1058 

88 

1.117 

6 

.946 

34 

1.004 

62 

1.062 

90 

1.121 

8 

.950 

36 

1.008 

64 

1.066 

92 

1.125 

10 

.954 

38 

1.012 

66 

1071 

94 

1.129 

12 

.958 

40 

1.016 

68 

1.075 

96 

1.133 

14 

.9*i2 

42 

1.020 

70 

1.079 

98 

1.138 

16 

.967 

44 

1.024 

72 

1.083 

100 

1.142 

18 

.971 

46 

1.028 

74 

1.087 

102 

1.146 

20 

.975 

48 

1.032 

76 

1.091 

104 

1.150 

22 

.979 

50 

1.036 

78 

1.096 

1U6 

1.154 

24 

.9S3 

52 

1.041 

80 

1.100 

108 

1.158 

26 

.987 

54 

1.046 

82 

1.104 

110 

1.163 

LEVELLING   BY   THE   BAROMETER. 


169 


To  ascertain  the  diff  of  height  between  two  points. 

RULE  1.  Take  readings  of  the  barom  and  therm  (Fah)  in  the  shade  at  both 
stations.  Add  together  the  two  readings  of  the  barom,  and  div  their  sum  by  2,  for 
their  mean  ;  which  call  6.  Do  the  same  with  the  two  readings  of  the  therm oin,  and 
call  the  mean  t.  Subtract  the  least  reading  of  the  barom  from  the  greatest ;  and  call 
the  diff  d.  Then  mult  together  this  diff  d ;  the  number  from  the  next  Table  No.  1, 
opposite  t;  and  the  constant  number  30.  Div  the  prod  by  6.  Or 

Height       Diff  (d)  of  ^  Tabular  number  opposite  v  rw,  „*„»,«•  ^n 
in  feet  =     barom       *        mean  (Q  of  thermom       XL 

mean  (6)  of  barom. 

EXAMPLE.  Reading  of  the  barom  at  lower  station,  26.64  ins  ;  and  at  the  upper 
sta  20.82  ins.  Thermom  at  lowest  sta,  70°;  at  upper  sta,  40°.  What  is  the  difl'  in 


height  of  the  two  stations?     Here, 
Barom,  26.64 
"       20.82 

2)47.46 


Also, 


Therm,  70° 

"       40° 

2)110 


23.73  mean  of  bar,  or  b. 


55°  mean  of 

therm,  or  t. 
The  tabular  number  opposite  55°,  is  917.2. 

Bar.        Bar. 
Again,  26.64  —  20.82  =  5.82,  diff  of  bar ;  or  d.    Hence, 

d,      Tab  No.  Con. 

Height  _  5.82  X  917.2  X  30  __  160143.12        6T48  5  ft     aMWen 
m  feet  23.73  (or  6)  23.73 

Then  correct  for  latitude,  if  more  accuracy  is  reqd,  by  rule  on  next  page. 
The  screw  at  the  back  is  for  adjusting  the  index  by  a  standard  barom ; 
and  after  this  has  been  done  it  must  by  no  means  be  meddled  with.  In 
some  instruments  specially  made  to  order  with  that  intention,  this  screw  may  be 
used  also  for  turning  the  index  back,  after  having  risen  to  an  elevation  so  great  that 
the  index  has  reached  the  extreme  limit  of  the  graduated  arc.  After  thus  turning 
it  back,  the  indications  of  the  index  at  greater  heights  must  be  added  to  that  at- 
tained when  it  was  turned  back. 

TABLE  1.    For  Rule  1. 


Mean 

Meau 

Mean 

Mean 

of 

No. 

of 

No. 

of 

No. 

of 

No. 

Ther. 

Ther. 

Ther. 

Ther. 

0° 

801.1 

30° 

864.4 

60° 

927.7 

90° 

991.0 

1 

803.2 

31 

866.5 

61 

929.8 

91 

993.1 

2 

805.3 

32 

868.6 

62 

931.9 

92 

995.2 

3 

807.4 

33 

870.7 

63 

934.0 

93 

997.3 

4 

809.5 

34 

872.8 

64 

936.1 

94 

999.4 

5  » 

811.7 

35 

874.9 

65 

938.2 

95 

1001.6 

6 

813T8 

36 

877.0 

66 

940.3 

96 

1003.7 

7 

815.9 

37 

879.2 

67 

942.4 

97 

1005.8 

8 

818.0 

38 

881.3 

68 

944.5 

98 

1007.9 

9 

820.1 

39 

883.4 

69 

946.7 

99 

1010.0 

10 

822.2 

40 

885.4 

70 

948.8 

100 

1012.1 

11 

824.3 

41 

887.5 

71 

950.9 

101 

1014.2 

12 

826.4 

42 

889.6 

72 

953.0 

102 

1016.3 

13 

828.5 

43 

891.7 

73 

955.1 

103 

1018.4 

14 

830.6 

44 

893.8 

74 

957.2 

104 

1020.5 

15 

832.8 

45 

896.0 

75 

959.3 

105 

1022.7 

16 

834.9 

46 

898.1 

76 

961.4 

106 

1024.8 

17 

837.0 

47 

900.2 

77 

963.5 

107 

1026.9 

18 

839.1 

48 

902.3 

78 

965.6 

108 

1029.0 

19 

841.2 

49 

904.5 

79 

967.7 

109 

1031.1 

20 

843.3 

50 

906.6 

80 

969.9 

110 

1033.2 

21 

845.4 

51 

908.7 

81 

972.0 

111 

1035.3 

22 

847.5 

52 

910.8 

82 

974.1 

112 

1037.4 

23 

849.6 

53 

913.0 

83 

976.2 

113 

1039.5 

24 

851.8 

54 

915.1 

84 

978.3 

114 

1041.6 

25 

853.9 

55 

917.2 

85 

980.4 

115 

1043.8 

26 

856.0 

56 

919.3 

86 

982.6 

116 

1045.9 

27 

858.1 

57 

921.4 

87 

984.7 

117 

1048.0 

28 

860.2 

58 

923.5 

88 

986.8 

118 

1050.1 

29 

862.3 

69 

925.6 

89 

988.9 

119 

1052.2 

170 


LEVELLING   BY   THE   BAROMETER. 


RULE  2.  Belville's  short  approx  rule  is  the  one  best  adapted  to  rapid 
field  use,  namely,  add  together  the  two  readings  of  the  barom  only.  Also  find  the 
diff  between  said  two  readings;  then,  as  the  sum  of  the  two  readings 
is  to  their  diff,  so  is  55OOO  feet  to  the  reqd  altitude. 

Ex.   Same  as  before.    Here  the  sum  =  26.64  +  20.82  =  47.46  ins ;  and  the  diff  = 

Sum.    Diff.      Feet.      Feet. 
Hence,  as  47.46  :  5.82  : :  55000  :  6744.6  reqd  alt ;  instead  of  the  6748.5  of  Rule  1. 

Correction  for  latitude.  The  lat  of  the  place  affocts  the  result  to  some 
extent;  but  is  usually  omitted  wheu  great  accuracy  is  not  reqd.  To  apply  the  correction  first  find 
the  altitude  by  the  rule,  as  before.  Then  divide  it  by  the  number  iu  the  following  table  opp  the  lat 
of  the  place.  Add  the  quotient  to  the  alt  if  the  lat  is  less  than  45° ;  or  subtract  it  if  the  lat  is  more 
than  45°.  If  the  two  places  are  in  diff  lats,  use  their  mean. 

Table  of  corrections  for  latitude. 


Lat. 

Lat. 

Lat. 

• 

Lat. 

Lat. 

Lat. 

0° 

352 

14o 

399 

28^ 

630 

42° 

3367 

54° 

1140 

68° 

490 

2 

354 

16 

416 

30 

705 

44 

10101 

56 

941 

70 

460 

4 

356 

18 

436 

32 

804 

45 

0 

58 

804 

72 

436 

6 

360 

20 

460 

34 

941 

46 

10101 

60 

705 

74 

416 

8 

367 

22 

490 

36 

1140 

48 

3367 

62 

630 

76 

399 

10 

375 

24 

527 

38 

1458 

50 

2028 

64 

572 

78 

386 

12 

386  ' 

26 

572 

40 

2028 

52 

1458  1 

66 

527  ' 

80 

375 

Levelling  by  Barometer;  or  by  the  boiling  point. 

RULE  3.  The  following  table,  No.  2,  enables  us  to  measure  heights  either  by  means 
of  boiling  water,  or  by  the  barom.  The  third  column  shows  the  approximate  alti- 
tude above  sea-level  corresponding  to  diff  heights,  or  readings  of  the  barom;  and  to 
the  diff  degrees  of  Fahrenheit's  thermom,  at  which  water  boils  in  the  open  air.  Thus 
when  the  barom,  under  iindisturbed  conditions  of  the  atmosphere,  stands  at  24.08 
inches,  or  when  pure  rain  or  distilled  water  boils  at  the  temp  of  201°  Fah  ;  the  place 
is  about  5764  ft  above  the  level  of  the  sea,  as  shown  by  the  table.  It  is  therefore 
very  easy  to  find  the  diff  of  altitude  of  two  places.  Thus  :  take  out  from  table  No  2, 
the  altitudes  opposite  to  the  two  boiling  temperatures ;  or  to  the  two  barom  readings. 
Subtract  the  one  opposite  the  lower  reading,  from  that  opposite  the  upper  reading. 
The  rem  will  be  the  reqd  height,  as  a  rough  approximation.  To  correct  this,  add 
together  the  two  therm  readings ;  and  div  the  sum  by  2,  for  their  mean.  From  table 
for  temperature,  p  168,  take  out  the  number  opposite  this  mean.  Mult  the  ap- 
proximate height  just  found,  by  this  tabular  number.  Then  correct  for  lat  if  reqd. 

Ex.  The  same  as  preceding ;  namely,  barom  at  lower  sta,  26.64 ;  and  at  upper  sta, 
20.82.  Thermom  at  lower  sta,  70°  Fuh ;  and  at  the  upper  one,  40°.  What  is  the  diff 
of  height  of  the  two  stations  ? 

Alt. 

Here  the  tabular  altitudes  are,  for  20.82 9579 

and  for  26.64 3115 


To  correct  this,  we  have 


70°  -f  40°      110° 


6464  ft,  approx  height. 
=  55°  mean ;  and  in  table  p  168,  opp  to 


55°,  we  find  1.048.    Therefore  6464  X  1.048  =  6774  ft,  the  reqd  height. 

This  is  about  26  ft  more  than  by  Rule  1  ;  or  nearly  .4  of  a  ft  in  each  100  ft. 

At  70°  Fah,  pure  water  will  boil  at  1°  less  of  temp,  for  an  average  of  about  550  ft 
of  elevation  above  sea-level,  up  to  a  height  of  %  a  mile.  At  the  height  of  1  mile,  1° 
of  boiling  temp  will  correspond  to  about  560  ft  of  elevation.  In  table  p  168  the 
mean  of  the  temps  at  the  two  stations  is  assumed  to  be  3'2°  Fah  ;  at  which  DO  correc- 
tion for  temp  is  necessary  in  using  the  table  ;  hence  the  tabular  number  opposite  32° 
is  1. 

This  diff  produced  in  the  temp  of  the  boiling  point,  by  change  of  elevation,  must 
not  be  confounded  with  that  of  the  atmosphere,  due  to  the  same  cause.  The  air  be- 
comes cooler  as  we  ascend  above  sea-level,  at  the  rate  (very  roughly)  of  about  1°  Fah 
for  every  200  ft  near  sea-level,  to  350  ft  at  the  height  of  1  mile.  See  "-Atmosphere." 

The  following1  table,  K"o.  2,  (so  far  as  it  relates  to  the  barom,)  was  de- 
duced by  the  writer  from  the  standard  work  on  the  barom  by  Lieut.-Col.  R.  S.  Wil- 
liamson, U.  S.  army.* 

*  Published  by  permission  of  Government  in  1868  by  Van  Nostrand,  N.  Y. 


OR   BY   THE    BOILING    POINT. 


171 


TABLE  2. 

Levelling1  by  Barometer;  or  by  the  boiling1  point. 

Assumed  temp  in  the  shade  32°  Fah.    If  not  32°,  mult  barom  alt  as  per  Table,  p  168. 


Boil 

Altitude 

Boil 

Altitude 

Boil 

Altitude 

Boil 

Altitude 

point 

Barom. 

above 

poiut 

Barom. 

above 

point 

Barom. 

above 

K>int 

Barom. 

above 

in  deg 

ea  level 

ndeg 

sea  level 

n  deg 

sea  level 

ndeg 

sea  level 

Fah 

Ins. 

Feet. 

Fah. 

Ins. 

Feet. 

Fah. 

Ins. 

Feet. 

Fah. 

Ins. 

Feet. 

184° 

16.79 

15221 

.3 

19.66 

11083 

.6 

22.93 

7048 

.9 

26-59 

3164 

.1 

16.83 

15159 

.4 

19.70 

11029 

.7 

22.98 

6991 

206 

26.64 

3115 

.2 

16.86 

15112 

.5 

19.74 

10976 

.8 

23.02 

6945 

.1 

26-69 

3066 

.3 

16.90 

15050 

.6 

19.78 

10923 

.9 

23.07 

6888 

.2 

26-75 

3007 

.4 

1693 

15003 

.7 

19.82 

10870 

199 

23.11 

6843 

.3 

26-80 

2958 

.5 

1697 

14941 

.8 

19.87 

10804 

.1 

23.16 

6786 

.4 

26.86 

2899 

.6 

17.00 

14895 

.9 

19.92 

10738 

.2 

23.21 

6729 

.5 

26.91 

2850 

.7 

1704 

14833 

192 

19.96 

10685 

.3 

23.26 

6673 

.6 

26.97 

2792 

.8 

17.08 

14772 

.1 

20.00 

10633 

.4 

23.31 

6617 

.7 

27.02 

2743 

.9 

17.12 

14710 

.2 

20.05 

10567 

.5 

23.36 

6560 

.8 

27.08 

2685 

185 

17.16 

14649 

.3 

20.10 

10502 

.6 

23.40 

6516 

.9 

27.13 

2637 

.1 

17.20 

14588 

.4 

20.14 

10450 

.7 

2345 

6460 

207 

27.18 

2589 

.2 

17.23 

14543 

.5 

20.18 

10398 

.8 

23.49 

6415 

.1 

27.23 

2540 

.3 

17.27 

14482 

.6 

20.22 

10346 

.9 

23.54 

6359 

.2 

27.29 

2483 

.4 

17.31 

14421 

.7 

20.27 

10281 

200 

23.59 

8M« 

.3 

27.34 

2435 

.5 

17.35 

14361 

.8 

20.31 

10230 

.1 

23  64 

6248 

.4 

27.40 

2377 

.6 

17.38 

14315 

.9 

20.35 

10178 

.2 

23.69 

6193 

.5 

27.45 

2329 

.7 

17.42 

14255 

193 

20.39 

10127 

.3 

23.74 

6137 

.6 

27.51 

2272 

.8 

17.46 

14195 

.1 

20.43 

10075 

.4 

23.79 

6082 

.7 

27.56 

2224 

.9 

17.50 

14135 

.2 

20.48 

10011 

.5 

23.84 

6027 

.8 

27.62 

2167 

186 

17.54 

14075 

.3 

20.53 

9947 

.6 

23.89 

5972 

.9 

27.67 

2120 

.1 

17.58 

14015 

.4 

20.57 

9896 

.7 

23.94 

5917 

208 

27.73 

2063 

.2 

17.62 

13956 

.5 

20.61 

9845 

.8 

23.98 

5874 

.1 

27.78 

2016 

.3 

17.66 

13896 

.6 

20.65 

9794 

.9 

24.03 

5819 

.2 

27.84 

1959 

.4 

17.70 

13837 

.7 

20.69 

9743 

201 

24.08 

5764 

.3 

27.89 

1912 

.5 

17.74 

13778 

.8 

20.73 

9693 

.1 

24.13 

5710 

.4 

27.95 

1856 

.6 

17.78 

13718 

.9 

20.77 

9642 

.2 

24.18 

5656 

.5 

28.00 

1809 

.7 

17.82 

13660 

194 

20.82 

9579 

.3 

24.23 

5602 

.6 

28.06 

1753 

.8 

17.86 

13601 

.1 

20  87 

9516 

.4 

24.28 

5547 

.7 

28.11 

1706 

.9 

17.90 

13542 

.2 

2091 

9466 

.5 

24.33 

5494 

.8 

28.17 

1650 

187 

17.93 

13498 

.3 

20.96 

9403 

.6 

24.38 

5440 

.9 

28.23 

1595 

.1 

17.97 

13440 

.4 

21  00 

9353 

.7 

24.43 

5386 

209 

28.29 

1539 

.2 

18.00 

13396 

.5 

21.05 

9291 

.8 

24.48 

5332 

.1 

28.35 

1483 

.3 

18.04 

13338 

.6 

21.09 

9241 

.9 

24.53 

5279 

.2 

28.40 

1437 

.4 

18.08 

13280 

.7 

21.14 

9179 

202 

24.58 

5225 

.3 

28.45 

1391 

.5 

18.12 

13222 

.8 

21.18 

9130 

.1 

24.63 

5172 

.4 

28.51 

1336 

.6 

18.16 

13164 

.9 

21.22 

9080 

.2 

24.68 

5119 

.5 

28.56 

1290 

.7 

18.20 

13106 

195 

21.26 

9031 

.3 

24.73 

5066 

.6 

28.62 

1235 

.8 

18.24 

13049 

.1 

21.31 

8969 

.4 

24.78 

5013 

.7 

28.67 

1189 

.9 

18.28 

12991 

.2 

21.35 

8920 

.5 

24.83 

4960 

.8 

28.73 

1134, 

188 

18.32 

12934 

.3 

21.40 

8859 

.6 

24.88 

4907 

.9 

28.79 

1079 

.1 

18.36 

12877 

.4 

21.44 

8810 

.7 

24.93 

4855 

210 

28.85 

1025 

.2 

18.40 

12820 

.5 

21.49 

8749 

.8 

24.98 

4802 

.1 

2b.9l 

970 

.3 

18.44 

12763 

.6 

21.53 

8700 

.9 

25.03 

4750 

.2 

28.97 

916 

.4 

18.48 

12706 

.7 

21.58 

8639 

203 

25.08 

4697 

.3 

29.03 

862 

.5 

18.52 

12649 

.8 

21.62 

8590 

.1 

25.13 

4645 

.4 

29.09 

808 

.6 

18.56 

12593 

.9 

21.67 

8530 

.2 

•25.18 

4593 

.5 

29.15 

754 

.7 

18.60 

12536 

196 

21.71 

8481 

.3 

25.23 

4541 

.6 

29.20 

709 

.8 

18.64 

12480 

.1 

21.76 

8421 

.4 

25.28 

4489 

.7 

29.25 

664 

.9 

18.68 

12424 

.2 

21.81 

8361 

.5 

25.33 

4437 

.8 

29.31 

610 

189 

18.72 

12367 

.3 

21.86 

8301 

.6 

25.38 

4386 

.9 

29.36 

565 

.1 

18.76 

12311 

.4 

21.90 

8253 

.7 

25.43 

4334 

211 

29.42 

512 

.2 

18.80 

12256 

.5 

21.95 

8193 

.8 

25.49 

4272 

.1 

29.48 

458 

.3 

18.84 

12200 

.6 

2199 

8145 

.9 

25.54 

4221 

.2 

29.54 

405 

.4 

18.88 

1-2144 

.7 

22.04 

8086 

204 

25.59 

4169 

.3 

29.60 

352 

.5 

18.92 

12089 

.8 

22.08 

8038 

.1 

25.64 

4118 

.4 

29.65 

308 

.6 

18.96 

12033 

.9 

22.13 

7979 

.2 

25.70 

4057 

.5 

29.71 

255 

.7 

19.00 

11978 

197 

22.17 

7932 

.3 

25.76 

3996 

.6 

29.77 

202 

.8 

19.04 

11923 

.1 

•22.22 

7873 

.4 

25.81 

3945 

.7 

29  83 

149 

.9 

19.08 

11868 

.2 

22.27 

7814 

.5 

25.86 

3894 

.8 

29.88 

105 

190 

19.13 

11799 

.8 

22.32 

7755 

.6 

25.91 

3844 

.9 

29.94 

52 

.1 

19.17 

11745 

.4 

22.36 

7708 

.7 

25.96 

3793 

212 

30.00     sealev=0 

.2 

19.21 

11690 

.5 

22.41 

7649 

.8 

26.01 

3742 

Below  sea  level. 

.3 

19.25 

11635 

.6 

22.45 

7602 

.9 

26.06 

3892 

.1 

30.06 

—  52 

.4 

19.29 

11581 

.7 

22.50 

7544 

205 

26.11 

3642 

.2 

30.12 

—104 

.5 

19.33 

11527 

.8 

22.54 

7498 

.1 

26.17 

3582 

.3 

30.18 

—156 

.6 

19.37 

11472 

.9 

22.59 

7439 

.2 

26.22 

3532 

.4 

30.24 

—209 

.7 

19.41 

11418 

198 

22.64 

7381 

.3 

26.28 

3472 

.5 

30.30 

—261 

.8 

19.45 

11364 

.1 

22.69 

7324 

.4 

26.33 

3422 

.6 

30.35 

—304 

.9 

19.49 

11310 

.2 

22.74 

7266 

.5 

26.38 

8372 

.7 

30.41 

—356 

191 

19.54 

11243 

.3 

22.79 

7208 

.6 

26.43 

3322 

.8 

30.47 

—408 

.1 

19.58 

11190 

.4 

22.84 

7151 

.7 

26.48 

3273 

.9 

30.53 

—459 

.2 

19.62 

11136 

.5 

22.89 

7093 

.8 

26.54 

3213 

213 

30.59 

—611 

172 


PENDULUMS. 


pl 
is 


DESCENT  ON  INCLINED  PLANES,* 

For  more  on  Inclined  Planes,  see  p  484,  &c. 

IT*  ALL  BELOW,  THE  RESISTANCE  OF  THE  AIR  IS  OMITTED. 

To  find  the  vel  in  ft  per  sec  acquired  in  sliding  down  an  inclined 
•       plane  a  6.     Divide  the  ver  ht  b  c  by  the  length  a  5.     The  quot  will  be  the  nat  sine  of 
the  angle  a.     Opposite  this  sine  in  the  table  of  nat  sines,  takeout  the  angle  a,  and 
i*8  nat  cosine.     Mult  this  cosine  by  the  proper  coefficient  of  sliding  friction  on  p  699 
'0     or  600,  as  the  case  may  be.     Take  the  prod  from  the  nat  sine.     Mult  the  remainder 
by  32.2.    Mult  this  last  prod  by  twice  the  length  a  6  in  ft.    Take  the  sq  rt.     If  the 
prod  is  greater  than  the  nat  sine,  body  will  not  move  down  the  plane. 

To  find  t.lie  time  of  the  descent  in  seconds.  First  by  the  foregoing 
find  the  acqd  vel  in  ft  per  sec.  Malt  the  length  a  6  by  2.  Divide  the  prod  by  the  acqd  vel. 

The  acceleration  of  gravity  on  a  body  falling  freely  is  32.2  ft  per  sec; 
or  in  other  words,  a  body  falling  freely  from  a  state  of  rest,  acquires  a  vel  of  32.2  ft  per  sec  for  each 
sec  of  its  descent.  But  on  an  inclined*  plane  this  acceleration  is  less  in  the  same  proportion  as  the 
actual  force  down  the  plane  (see  pp  485,  486)  is  less  than  the  wt  of  the  body.  Now  the  actual  force 
down  the  plane  is  equal  to  the  theoretical  force  down  the  plane,  minus  the  friction  of  the  body  on  the 
lane.  But  the  theoretical  force  down  the  plane  is  as  the  nat  sine  of  a;  and  the  friction  of  the  body 
s  as  its  pressure  on  the  plane,  mult  by  the  coef  of  friction  ;  or  as  the  nat  cos  of  a  mult  by  the  coef 

of  friction.  Hence  the  acceleration  down  the  plane,  or  in  other  words 
the  additional  vel  in  ft  per  sec  that  the  body  acquires  in  each  sec  of  its  descent,  becomes  reduced  to 
82.2  X  (nat  sine  of  a  —  (nat  cos  of  a  X  coef  of  friction.)  ) 

If  twice  the  height  in  ft  fallen  freely,  or  twice  the  length  a  b  of  an 

inclined  plane,  be  divided  by  the  final  acqd  vel,  the  quot  will  be  the  time  in  sec. 

All  the  foregoing  applies  also  closely  enough  to  cars  or 
wagons  rolling  down  planes;  except  that  the  coef  of  combined  axle  and  rolling 
friction,  p  602,  must  then  be  used  instead  of  that  for  sliding  friction.  But  it  does 
not  apply  to  cylinders  or  spheres. 

The  dist  in  ft  to  which  a  body  would  slide  or  roll  on  a  liori- 

_       _  .    _  Square  of  vel  at  starting,  in  ft  per  sec. 

P  Coef  of  friction  (p  599,  600,  602)  X  64.4 
and  the  time  in  sec,  _  Twice  the  dist  in  ft. 

before  coming  to  rest.        ~~"  Vel  at  starting,  in  ft  per  sec. 

Iii  all  the  above  the  resistance  of  the  air  is  omitted. 

To  find  the  actual  force  in  Ibs  which  tends  to  start  a  bo<3y 
clown  a  plane.  Mult  the  wt  of  the  body  in  Ibs  by  the  nat  sine  of  angle  a. 
Call  the  prod  p.  Then  mult  the  wt  of  the  body  by  the  nat  cos  of  angle  a  ;  and  mult  this  last  prod  by 
the  coef  of  friction.  Take  the  result  from  p.  The  rem  is  the  reqd  force.  For  p  is  the  moving  force 
down  the  plane  ;  whereas  wt  X  nat  cos  X  coef  is  the  retarding  force  of  friction,  and  if  it  is  greater 
than  p,  the  body  will  not  move  down  the  plane. 

As  the  wt  of  the  body,  is  to  this  actual  force  down  the  plane,  so  is 

32.2  ft  per  sec,  (the  accel  of  a  body  falling  freely,)  to  the  actual  accel  down  the  plane,  in  ft  per  sec. 
See  Gravity,  p  587. 


PENDULUMS, 


THE  numbers  of  vibrations  which  diff  pendulums  will  make  in  any  given  place  in 
a  given  time,  are  inversely  as  the  square  roots  of  their  lengths;  thus,  if  one  of  them 
is  4,  9,  or  16  times  as  long  as  the  other,  its  sq  rt  will  be  2,  3,  or  4  times  as  great ;  but 
its  number  of  vibrations  will  be  but  %,  %,  or  %  as  great.  The  times  in  which  diff 
pendulums  will  make  a  vibration,  are  directly  as  the  sq  rts  of  their  lengths.  Thus, 
if  one  be  4,  9,  or  16  times  as  long  as  the  other,  its  sq  rt  will  be  2,  3,  or  4  times  as 
great ;  and  so  also  will  be  the  time  occupied  in  one  of  its  vibrations. 

The  length  of  a  pendulum  vibrating  seconds  at  the  level  of  the  sea,  in  a  vacuum, 
in  the  lat  of  London  (51%°  North)  is  39.1393  ins ;  and  in  the  lat  of  N.  York  (40%° 
North)  39.1013  ins.  At  the  equator  about  y1^  inch  shorter;  and  at  the  poles,  about  Txff 
inch  longer.  Approximately  enough  for  experiments  which  occupy  but  a  few  sec, 
we  may  at  any  place  call  the  length  of  a  seconds  pendulum  in  the  open  air,  89  ins ; 
half  sec,  9%  iiis  ;  and  may  assume  that  long  and  short  vibrations  of  the  same  pen- 
dulum are  made  in  the  same  time  ;  which  they  actually  are,  very  nearly.  For  meas- 
uring depths,  or  dists  by  sound,  a  sufficiently  good  sec  pendulum  may  be  made  of  a 
pebble  (a  small  piece  of  metal  is  better)  and  a  piece  of  thread,  suspended  from  a 
common  pin.  The  length  of  39  ins  should  be  measured  from  the  centre  of  the  pebble. 

*  Text  books  generally  give  merely  theoretical  rules,  which  lead  only  to  error  in  practice. 


SOUND.  173 

In  starting  the  vibrations,  the  pebble,  or  6oft,  must  not  be  thrown  into  motion,  but 
merely  let  drop,  after  extending  the  string  at  the  proper  height. 

To  find  the  length  of  a  pendulum  reqd  to  make  a  given  number  of 
vibrations  in  a  min,  divide  375  by  said  reqd  number.  The  square  of  the  quot  will  be 
the  length  in  ina,  near  enough  for  such  temporary  purposes  as  the  foregoing.  Thus, 
for  a  pendulum  to  make  100  vibrations  per  min,  we  have  JJ g-  =  3.75 ;  and  the  square 
of  3.75  =  14.06  ins,  the  reqd  length. 

To  find  the  number  of  vibrations  per  min  for  a  pendulum  of 
given  length,  in  ins,  take  the  sq  rt  of  said  length,  and  div  375  by  said  sq  rt.  Thus, 

for  a  pendulum  14.06  ins  long,  the  sq  rt  is  3.75 ;  and  ^  =  100,  the  reqd  number. 

REM.  1.  By  practising  before  the  sec  pendulum  of  a  clock,  or  one  prepared  as  just 
stated,  a  person  will  soon  leai  n  to  count  5  in  a  sec,  for  a  few  sec  in  succession  ;  and  will 
thus  be  able  to  divide  a  sec  into  5  equal  parts ;  and  this  may  at  times  be  useful  for 
very  rough  estimating  when  he  has  no  pendulum. 

Centre  of  Oscillation  and  Percussion. 

REM.  2.  When  a  pendulum,  or  any  other  suspended  body,  is  vibrating  or  oscillating 
backward  and  forward,  it  is  plain  that  those  particles  of  it  which  are  far  from  the 
point  of  suspension  move  faster  than  those  which  are  near  it.  But  there  is  always 
a  certain  point  in  the  body,  such  that  if  all  the  particles  were  concentrated  at  it,  so 
thut  all  should  move  with  the  same  actual  vel,  neither  the  number  of  oscillations, 
nor  their  angular  vel,*  would  be  changed.  This  point  is  called  the  center  of  oscilla- 
tion. It  is  not  the  same  as  the  cen  of  grav,  and  is  always  farther  than  it  from  the 
point  of  suspension.  It  is  also  the  centre  of  percussion  of  the  suspended  vibrating 
body.  The  dist  of  this  point  from  the  point  of  susp  is  found  thus  :  Suppose  the  body 
to  be  divided  into  many  (the  more  the  better)  small  parts ;  the  smaller  the  better. 
Find  the  weight  of  each  part.  Also  find  the  cen  of  grav  of  each  part ;  also  the  dist 
from  each  such  cen  of  grav  to  the  point  of  susp.  Square  each  of  these  dists,  and 
mult  each  square  by  the  wt  of  the  corresponding  small  part  of  the  body.  Add  the 
products  together,  and  call  their  sump.  Next  mult  the  weight  of  the  entire  body 
by  the  dist  of  its  cen  of  grav  from  the  point  of  susp.  Call  the  prod  g.  Divide  p  by^. 
This/?  is  the  moment  of  inertia  of  the  body,  and  if  divided  by  the  wtof  the 
body  the  sq  rt  of  the  quotient  will  be  the  Radius  of  Gyration. 


SOUND. 


Its  velocity,  in  quiet  open  air,  has  been  experimentally  determined  to 
bo  very  approximately  1090  ft  per  sec,  when  the  temp  is  at  freezing  point,  or  32° 
Fah ;  and  that  it  increases  about  1J^  ft  per  sec  for  every  degree  above  32°,  or,  by 
Boiue  authorities,  1  ft  for  every  2°.  Tiie  first  would  make  it 

1090  ft  at  32°  1150  ft  at    80° 

1100      "   40°  1162%  "     90° 

1112%  "   500  1175      «    10Q® 

1125      "   60°  1187^  "   1100 

1137%  "    70°  1200      "   120° 

At  32°  sound  would  travel  a  mile  in  4.84  sec ;  at  80e,  in  4.6 ;  and  at  120°,  in  4.4  sec. 
If  the  air  iscaltn,  fogs,  or  rain  do  not  appreciably  affect  the  result;  but  winds  do.  There  is  son* 
reason  to  believe  that  very  loud  sounds  travel  somewhat  faster  than  low  ones.  The  watchword  of 
sentinels  has  been  heard  across  still  water,  on  a  calm  night,  10%  miles  ;  and  a  cannon  20  miles. 
Separate  sounds,  at  intervals  of  -i  of  a  sec,  cannot  be  distinguished,  but  appear  to  be  connected. 
The  dists  at  which  a  speaker  can  be  understood,  in  front,  on  one  side,  and  behind  him,  are  about  as  4, 
3,  and  1. 

Dr.  Charles  M.  Cresson  informs  the  writer  that,  by  repeated  trials,  he  found  that  in  a  Philadelphia 
gas  main  20  ins  diam,  and  16000  ft  long,  laid  and  covered  in  the  earth,  but  empty  of  gas,  and  having 
one  horizontal  bend  of  90°,  and  of  40  ft  radius,  the  sound  of  a  pistol-shot  travelled  16000  ft  in  pre- 
cisely 16  sec,  or  1000  ft  per  sec.  The  arrival  of  the  sound  was  barely  audible  ;  but  was  rendered  very 
apparent  to  the  eye  by  its  blowing  off  a  diaphragm  of  tissue-paper  placed  over  th«  end  of  the  main. 

Two  boats  anchored  some  dist  apart  may  serve  as  a  base  line  for  triang- 
ulating objects  along  the  coast;  the  dist  between  them  being  first  found  by  firing  guns  on  board  one 
of  them. 

In  water  the  vel  is  about  4708  ft  per  sec,  or  about  4  times  that  in  air.  In 
woods,  it  is  from  10  to  16  times;  and  in  metals,  from  4  to  16  times  greater 
than  in  air,  according  to  some  authorities. 

*  For  angular  vel,  see  footnote  page  447,  "  Force  in  Rigid  Bodies." 


174 


STKENGTH   OF   MATERIALS. 


STKENGTH  OF  MATEEIALS, 


.   1.    Endwise*  average   ultimate    crashing  loads   per 
re  inch,  for  wood,  in  pieces  whose  height  does  not  exceed  '2  or  3  times 


The  strengths  in  all  these 
tables  may  readily  vary  as 
much  as  one-third  part  more 
or  less  than  our  average. 

Weight 
per' 
cub.  ft. 

Pounds 
per 
sq.  inch. 

Weight 
per 
cub.  ft. 

Pounds 
per 
sq.  inch. 

Alder 

6900 
8600 
7200 
7700 
9300 
6000 
11600 
4500 
6500 
10000 
5700 
6500 
6500 
7000 
5700 
6500 
6700 
7300 
7400 
9900 
10000 
6500 
6800 
6000 
4500 
7300 
3200 
5500 
will  aver 

Mahogany  Spanish   . 

8200 
4200 
6000 
6500 
9500 
7700 
6800 
5300 
5400 
5400 
7500 
7500 
3100 
5100 
3700 
9300 
7000 
6500 
6800 
6000 
12000 
3200 
5600 

Ash          ..... 

43  to  63 
51 
53 
43 

Oak,  Quebec,  unseas.. 
"            "        seas 

54 

Bay  wood                       . 

Beech,  unseasoned  
u        seasoned  
Birch   Amer  unseas  . 

"     English       

58 

"            "      well  seas 
"    Dantzic,  very  dry 
Pine  pitch 

"       seas  
English,  unseas 
"           "       seas  

41 

"      Ameryel'w,  uns 
"        *•     seas 
"      red  unseas  

Box  dry 

Cedar,  unseas  

56 

Pear 

Crab-tree   green 

Poplar   unseas.  

48 
47 

"           seas               . 

Deal,    red,  unseas  

Plum  wet    

"        dry 

"                   v  .. 

Sycamore  or  Planetree 
Spruce,  or  Fir,  unseas 

"          "     "    seas  .  . 

"           "      seas 

43 

37 
37 

<4    ' 

"           "     "    Riga  .. 

Elm   seas    

Teak  

Larch    unseas        ..... 

"     "        "        seas 

"    Riga    

6000 
7200 
2900 
6100 

Hornbeam,  unseas  
"         seas  
Larch  green  

47 
50 

Willow   unseas 

"           geas  

"        dry 

age  5500  Ibs.  ;  hemloi 

»fe,  4500. 

Seasoned  Wh.  Pine 

-K  But  sldewise  the  crushing  strength  Is  far  less.  Thus  Hatfleld  found  for  ash  only  .57 
of  the  endwise  strength;  live  oak  .67;  oak,  maple,  hickory,  .333;  locust,  black  walnut,  cherry, 
•white  oak  .3;  Georgia  pine.  Ohio  pine,  whitewood  .25;  chestnut  .'2;  spruce,  white  pine  .125;  hem- 
lock .111.  He  gives  as  safe  equally  distributed  pressures,  which  will  make  but  a  slight  impression, 
for  spruce  250  Ibs  per  sq  inch  ;  white  pine  and  hemlock  300;  Georgia  pine  850 ;  oak  9oO. 

But  it  is  well  to  bear  in  mind  that  in  practice  perfectly  equable  pressure  is  rarely  secured. 

The  writer  found  that  the  resistance  was  greatest  when  the  position  of  the  annual  layers  as  seen 
in  a  cross  section  of  a  beam  was  vertical.  But  in  practice  a  knowledge  of  this  fact  is  seldom  of 
use,  because  in  large  beams  the  layers  are  generally  disposed  more  or  less  in  curves.  In  a  few  hur- 
ried trials  on  sidewise  compression,  with  fairly  seasoned  white  pine  blocks,  6  ius  high,  5  ins  long, 
and  2  ins  wide,  we  found  that  under  an  equally  distributed  pressure  of  5000  Ibs  total  or  500  Ibs  per 
sq  Inch,  they  compressed  about  from  ^  to  #  inch  ;  which  is  equal  to  from  %  to  %  inch  per  foot  of 
height ;  or  from  A  to  A-  of  the  height ;  the  mean  being  about  %  inch  to  a  foot,  or  3*5  of  the  height. 
Under  10000  Ibs  total,  or  1000  Ibs  per  sq  inch,  they  split  badly;  and  in  some  cases  large  pieces  flew 
off.  See  Bern,  p.  193. 


STRENGTH   OF    MATERIALS. 


175 


practice  \ve  may  take  its  safe  strain  at  from  1000  to  2000  Ibs  per  sq  inch,  depending; 
upon  the  character  of  the  structure,  &c.,  without  regard  to  the  length,  except  when 
this  is  so  great  that  two  or  more  pieces  have  to  be  spliced  together  to  make  it;  thus 
weakening  the  piece  very  much.  It  is  seen  from  the  table  that  seasoned  woods 
resist  crushing  much  better  than  green  ones;  in  many  cases,  twice  as  well.  This 
must  be  taken  into  consideration  when  building  bridges,  &c.,  of  timber  recently  cut. 
Art.  2.  Ultimate  average  crushing:  loads  in  tons,  per  square 
foot,  lor  stones,  «&c.  The  stones  are  supposed  to  be  ON  BKD,  and  the  heights 
of  all  to  be  from.  1.5  to  2  times  the  least  side.  Stones  generally  begin  to  crack  or 
split  under  about  one-half  of  their  crushing  loads.  In  practice,  neither  stone  nor 
brickwork  should  be  trusted  with  more  than  >£  to  T^th  of  the  crushing  load,  ac- 
cording to  circumstances.  When  thoroughly  wet  some  absorbent  sand- 
stones lose  fully  half  their  strength.  See  head  of  next  page. 


4 

Tons  per 
sq.  ft. 

300  to  1200 

Mean. 
Tons. 

Tons  per 

sq.  ft. 

Mean. 
Tons. 

Granites  and  Syenites. 
Basalt  

750 
700 

625 
175 

350 

200 
170 

25 
35 

60 
600 
135 
70 
25 

40 

Cement,    Portland, 
neat,U.  S.  or  Foreign, 
7  days  in  water  
Common  U.S.cements, 
neat,  7  days  in  water 
CoiicreteofPort. 
cement,    sand,    and 
gravel  or  brok  stone 
in  the  proper  propor- 
tions.rammedlmold 
6  months  old 

75  to  150 
15  to  30 

12  to  18 
48  to  72 
74  to  120 

100  to  150 
15  to  35 

1300to2300 
s  that  of  g 
|   12  to  18 

112.5 
22.5 

15 
60 
97 

125 
25 

1800 
ranite. 
15 

Limestones  and  Mar- 
blest  

250  to  1000 
100  to  250 

150  to  550 

Oolites  good 

Sandstonesfitforbuild- 
ing 

Sandstone,  red,  of  Con- 
necticut and  N.  Jer- 

Brick*  

40  to  300 
20  to  30 
30   to   40 

50   to    70 
400  to  800 
70  to  200 

Brickwork,    ordinary, 
cracks  with 

12  months  old  

With  good  common 
hyd      cements, 

abt  .2  to  .25  as  much 
Coignct  betoii,  3 

months  old  
Rubble      masonry, 
mortar,  rough  
Glass,  green,crowu  and 
flint  . 

Brickwork,  good,  in  ce- 
ment   

Brickwork,    first-rate, 
in  cement  

Slate 

Caen  Stone  

"        "      to  crack 

Chalk  hard  

20   to    30 

Plaster  of  Paris,  i  day 
old  

or  3  time 
Ice,  firm  

Crushing  height  of  Brick  and  Stone. 

If  we  assume  the  wt  of  ordinary  brickwork  at  112  Ibs  per  cub  ft,  and  that  it  would 
Crush  under  30  tons  per  sq  ft,  then  a  vert  uniform  column  of  it  600  ft  high,  would 
-.rush  at  its  base,  under  its  own  wt.  Caen  stone,  weighing  ISO  ft>s  per  cub  ft.  would 
/equire  a  column  1376  ft  high  to  crush  it.  Average  sandstones  at  145  fibs  per  cub  ft, 
would  require  one  of  4158  ft  high ;  and  average  granites,  at  165  Jbs  per  ciib  ft,  one 
of  8145  feet.  But  stones  begin  to  crack  and  splinter  at  about  half  their  ultimate 
Crushing  load ;  and  in  practice  it  is  not  considered  expedient  to  trust  them  with  more 
than  i^th  to  ^th  part  of  it,  especially  in  important  works ;  inasmuch  as  settle- 
ments, and  Imperfect  workmanship,  often  cause  undue  strains  to  be  thrown  on  cer- 
tain parts. 

The  Merchants'  shot-tower  at  Baltimore  is  246  ft  high ;  and  its  base  sustains  6U . 
tons  per  sq  ft.  The  base  of  the  granite  pier  of  Saltash  bridge,  (by  Brunei,)  of  solid 
masonry  to  the  height  of  96  ft,  and  supporting  the  ends  of  two  iron  spans  of  455  ft 
each,  sustains  y%  tons  per  sq  ft  The  base  of  a  brick  chimney  at  Glasgow,  Scotland, 
468  ft  high,  bears  9  tons  per  sq  ft;  and  Professor  liankine  considers  that  in  a  high 
gale  of  wind,  its  leeward  side  may  have  to  bear  15  tons.  The  highest  pier  of  Rocque- 
favour  stone  aqueduct,  Marseilles,  is  305  ft,  and  sustains  a  pressure  at  base  of  13}^ 
tons  per  sq  ft.  For  greater  pressures  on  arch  stones,  see  p  342. 
fTrials  at  St.  Louis  bridge  by  order  of  Capt  James  B.  Bads,  C.  E.,  showed 
that  some  magnesian  limestone  did  not  yield  under  less  than  1100  tons  per  sq  ft.  A 
column  8  ins  high  and  2  ins  diam  shortened  ^  inch  under  pressure ;  and  recovered 
when  relieved. 

*Some  whole  bricks  laid  flat,  crushed  by  Baldwin  Latham,  C.  E.,  with  4 

to  5  tons  per  sq  inch  ;  after  compressing  tn  about  one- half  of  their  original  thickness!  B.  Latham's 
"Sanitary  Engineering."    Can  this  be  correct} 


176 


STRENGTH   OF   MATERIALS. 


Sheet  lead  is  sometimes  placed  at  the  .joints  of  stone  col- 
umns, with  a  view  to  equalize  the  pressure,  and  thus  increase  the  strength  of  the  column.  But 
experiments  have  proved  that  the  effect  is  directly  the  reverse,  and  that  the  column  is  materially 
weakened  thereby.  Does  this  singular  fact  apply  to  cast  iron  and  other  materials  ? 

Art.  3.    Average  crushing-  load  for  Metals. 

It  must  be  remembered  that  these  are  the  loads  for  pieces  but  two  or  three  times  their  least  side  in 
height.  As  the  height  increases,  the  crushing  load  diminishes.  See  "  Strength  of  Pillars."  p  221. 


The  crushing  load  per  sq  inch,  of  any  material,  is  frequently 
called  its  constant,  coefficient,  or  modulus,  of  crushing  or  of  com- 
pression. 


Pounds  per 
sq.  inch. 


Tons  per 
sq.  inch. 


Cast  Iron,  usually 

It  is  usually  assumed  at  100000  fts,  or  say  45  tons  per  sq  inch.  Its 
crushing  strength  is  usually  from  6  to  7  times  as  great  as  its  tensile. 
Within  its  average  elastic  limit  of  about  15  tons  per  sq  inch,  average 
cast  iron  shortens  about  1  part  in  5555;  or  %  inch  in  58  ft  under  each 
ton  per  sq  inch  of  load;  or  about  twice  as  much  as  average  wrought 
iron.  Hence  at  15  tons  per  sq  inch  it  will  shorten  about  1  part  in  370; 
or  full  %  inch  in  4  feet.  Different  cast  irons  may  however  vary  10  to 
15  per  ct  either  way  from  this. 

U.  S.  Ordnance,  or  gun  metal  ;  Some 

'Wrong-lit  iron,  within  elastic  limit , 

Its  elastic  limit  under  pressure  averages  about  13  tons  per  sq  inch. 
It  begins  to  shorten  perceptibly  under  8  to  10  tons,  but  recovers  when 
the  load  is  removed.  With  from  18  to  20  tons,  itshortens  permanently, 
about  ^th  part  of  its  length  ;  and  with  from  27  to  30  tons,  about  y^g-tn 
part,  as  averages.  The  crushing  weights  therefore  in  the  table  are 
not  those  which  absolutely  mash  wrought  iron  entirely  out  of  shape, 
but  merely  those  at  which  it  yields  too  much  for  most  practical  build- 
ing purposes.  About  4  tons  per  sq  inch  is  considered  its  average  safe 
load,  in  pieces  not  more  than  10  diams  long ;  and  will  shorten  it  J^  inch 
in  30  ft.  average. 

Brass,  reduced  T^th  part  in  length,  by  51000;  and  %by 
Copper,  (cast,)  crumbles 

(wrought)  reduced  J^th  part  in  length,  by 


85000  to  125000  38  to  56 


175000 

22400  to  35840 


78.1 
10  to  16 


165000 

..117000 

103000 

15500 

7350 


.102050.. 


Tin,  (cast,)  reduced  y^th  in  length,  by  8800 ;  and  %  by 
Lead,  (cast,)  reduced  %  of  its  length,  by  7000  to  7700..*. 

"  By  writer.  A  piece  1  inch  sq,  2  ins  high ,  at  1200  Ibs  the  com- 
pression was  1-200  of  the  ht;  at  2000,  1-29 ;  at  3000,  1-8 ;  at 
5000,  1-3  ;  at  7000,  1-2  of  the  ht. 

Spelter  or  Zinc,  (cast.)    By  writer.    A  piece  1  inch 

square,  4  ins  high,  at  2000  fts  was  compressed  1-400  of  its  ht ;  at  4000, 
1-200 ;  at  6000,  1-100;  at  10000,  1-38  ;  at  20000,  1-15 ;  at  40000  yielded 
rapidly,  and  broke  into  pieces. 

Steel,  224000  Ibs  or  100  tons  shorten  it  from  .2  to  .4  part. 

"        American.     Black   Diamond   steel-works,   Pittsburg,    Penn. 

experiments  by  Lieut  W.  H.  Shock.  U.  S.  N.,  on  pieces  %  in 

square;  and  3}^  ins,  or  7  sides  long. 

u        TJntempered.  100100  to  104000 

"        Heated  to  light  cherrv  red,  then  plunged  into  oil  of  82°  Fab, 

173200  to  199200. . .  .* i 

11        Heated  to  light  cherry  red,  then  plunged  into  water  of  79°! 

Fah  ;  then  tempered  on  a  heated  plate,  325400  to  340800.. ..   333100 

"        Heated  to  light  cherry  red,  then  plunged  into  water  of  79°  j 

Fah,  275600  to  400000 337800 

Elastic  limit,  15  to  27  tons 47040 

"      Compression,  within  elas  limit  averages  abt 

1  part  in  13300,  or  .1  of  an  inch  in  111  ft  per  ton  per  sq  inch; 
or  .1  of  an  inch  in  5.3  ft  under  21  tons  per  sq  inch. 

Best  steel  knife  edges,  of  large  R  E,  weigh  scales 

are  considered  safe  with  7000  fts  pres  per  lineal  inch  of  edge ;  and 

solid  cylindrical  steel   rollers  under  bridges,  and 

rolling  on  steel,  safe  with  1/diam  in  ins  X  3  100  000,  in  Ibs  per  lineal 
inch  of  roller  parallel  to  axis.  And  per  the  same,  for 


73.6 
52.2 
46.0 
6.92 
3.28 


.186200., 


150.8 
21 


Solid  cast  Iron  wheels  rolling  on  wrought  iron,  J/Diam  ins  X  352  000. 
"          "        "  "  "          «'  cast  iron,         J/Diam  ins  X  222  222. 

"  "          "  steel,  ]/ Diam  ins  XI 300 000. 

'  "          "  wrought  iron,  }/Diamins  X  1024000. 


Solid  steel  ' 


"          "        "  "  "          "  cast  iron,         )/Diam  ins  X  850  000. 

•From  "  Specifications  for  Iron  Drawbridge  at  Milwaukee,"  by  Don  J.  Wbittemore,  C.  E. 


STRENGTH   OF   MATERIALS. 


177 


Art.  4.    Ultimate  average  tensile  or  cohesive  strength  of 
Timber, 

In  fts  per  sq  inch ;  being  the  weights  which,  if  attached  to  the  lower  end  of  a  vert 
rod  one  inch  square,  firmly  upheld  at  its  upper  end,  would  break  it  by  tearing  it 
apart.  For  large  timbers  we  recommend  to  reduce  these  constants  %  to  ^  part. 


The  strengths  in  all  these  tables  may 
readily  be  one-third  part  more  or  less 
than  oar  averages. 

Lbs  per 
sq.  inch. 

Lbs  per 
sq.  inch. 

Alder 

14000 

Mahogany  Honduras  

8000 

Ash    English    

16000 

16000 

"     American  (author)  abt  
Birch 

16500 
15000 

Mangrove,  white,  Bermuda.... 
Mulberry 

10000 
12000 

"    Amcr'n  black       «.. 

7000 

Oak  Amer'n  white  "] 

12000 

"        "         basket 

JBeech  English     

11500 

"        "         red  

Bamboo  

6000 

"    Dantzic  seasoned  > 

10000 

Box                         

20000 

«    Riga      

Cedar  Bermuda                   . 

7600 

"    English 

9500 

Chestnut                       .  . 

13000 

Pear 

10000 

10000 

Pine    Amer'u    white    red  1 

Cyprus 

6000 

and  Pitch  Memel  Riga    3 

10000 

Elder  

10000 

Plane  

11000 

Elm 

6000 

Plum                                   .... 

11000 

"    Canada  

13000 

Poplar    

7000 

Fir  or  Spruce 

10000 

7000 

10000 

Spruce  or  Fir.....  ...  

10000 

Hazel  

18000 

12000 

Holly  

16000 

Teak                         

15000 

20000 

Walnut  

8000 

Hickory,  Amer'n  

11000 

Yew 

8000 

Lignum  Vitfle  Amer'n 

11000 

23000 

»                               ^.m^                           .              flair 

2300 

Larch   Scotch  ..  . 

7000 

«          «         ff  p     i  ' 

1800 

Locust  

18000 

"         "        "  Larch  900  to 

1700 

Maple  

10000 

"        "  Fir,  &  Pines 

650 

THESE  ARE  AVERAGES.  The  strengths  vary  much  with  the  age  of  the  tree ;  the 
locality  of  its  growth ;  whether  the  piece  is  from  the  center,  or  from  the  outer  por- 
tions of  the  tree;  the  degree  of  seasoning ;  straightness  of  grain;  knots,  &c,  Ac.  Also, 
inasmuch  as  the  constants  are  deduced  from  experiments  with  good  specimens  of 
small  size,  whereas  large  beams  are  almost  invariably  more  or  less  defective  from 
knots,  crookedness  of  fibre,  &c,  it  is  advisable  in  practice  to  reduce  these  constants 
as  recommended  above. 

The  modulus  of  Elasticity,  and  its  ttse.    within  the  limit  of  elasticity, » 

uniform  rod  of  given  material  lengthens  or  shortens  equally  under  equal  additions  of  load.  If  thw 
were  also  the  case  beyond  said  limit,  it  is  plain  that  there  would  be  some  load  which  would  stretch  a 
uniform  bar  to  twice  its  original  length,  or  shorten  it  to  zero  or  0.  And  this  load  in  Ibs,  for  a  bar  of 
one  inch  square  cross  section,  is  the  mod  of  elas  for  the  given  material ;  or  is  the  E  of  authors  on 
Strength  of  Materials.  For  example,  a  one-inch  square  bar  of  wrought  iron  will,  within  the  limit 
of  elas,  stretch  on  an  average  about  1  part  in  12000  of  its  length  under  each  additional  load  of  2240  Ibs. 
Consequently,  if  the  same  rate  of  stretching  continued  beyond  the  limit  of  elas,  it  is  evident  that 
12000X2240,  or  26880000  Ibs,  would  stretch  the  bar  to  twice  its  original  length.  Hence  these  26880000 
Ibs  are  the  mod  of  elas  for  average  bar  iron.  And  so  with  any  other  material.  Hence  the  mod  of 
elas  is  a  load  which  bears  the  same  proportion  to  the  original  length  of  a  uniform  bar,  as  the  load 
which  will  produce  any  given  amount  of  stretch,  is  to  the  length  of  said  stretch.  This  fact  facilitate* 
certain  calculations ;  thus, 

Load  in  Ibs  reqd    ^          Beqd  stretch 

to  produce  a  given  I    _    lp  1Dches         v   m< 

stretch  within         f   ~    orig  length       *   eli 

elas  limit,  in  ins,  j          in  inches 


cross  MO 
in  sq  int 


Stretch  in  ins  ~i 
produced  by  any  I  _ 
load  within  elas  f  ~ 
limit,  J 


Load  in  Ibs  X  orig  length  in  ins 
mod  of  elas  X  cross  sec  in  sq  ins 


See  Table,  p  632. 


E  may  a  . 

ply  auy  load  within  its  elas  limit,  and  measure  its  deflection.     Then  as  expressed  by  writers  E  is  — 

WP  T-  4  A  bd3.     Which  means 

Coef  const,  or  Mod  — (Load  *D  fl)s  +  -625  wt  of  clear  8Pap  of  bean|)  X  cube  of  span  ins, 
E  m  fits  per  sq  iucu  4  X  Def,  ius  X  breadth,  ins  X  cube  of  depth,  ins. 

12 


178 


STRENGTH  OF   MATERIALS. 


Art.  5.    Average  ultimate    tensile  or  cohesive  strength    of 
Metals,  per  square  inch.* 


The  ultimate  tensile  or  pulling  load  per  square  inch  of  any 
material  is  frequently  called  its  constant,  coejjicient,  or  modulus  of 
tensioH,  or  of  tensile  strength. 


Pounds      Tons 

per           per 

sq.  inch.    sq.  in. 


Antimony,  cast 1000 

Bismuth,  cast 3200 

Brass,  cast  8  to  13  tons,  say  18000  to  29000  ibs 23500 

"    wire,  unannealed  or  hard,  80000.    Annealed 49000 

Bronze,  phosphor  wire,  hard,  150000.    Annealed 63000 

Copper,  cast  18000  to  30000 24000 

"        sheet 30000 

u        bolts,  28000  to  38000 33000 

"        wire  (annealed  16  tons);  unannealed 60000 

Gold,  cast 20000 

"      wire,  25000  to  30000 27500 

Gun  metal  of  copper  and  tin,  23000  to  55000 39000 

"           cast  iron,  U.  S.  ordnance,  36000  to  40000 38000 

Iron,  cast,  English 13400  to  22400 17900 

"        "      ordinary  pig.. 13000  to  16000 14500 

American  cast  iron  averages  one-fourth  more  than  the  above. 
Average  cast  iron,  when  sound,  stretches  about  .00018 ;  or  1  part 
in  5555  of  its  length  ;  or  %  inch  in  57.9  ft.  for  every  ton  of  ten- 
sile strain  per  scinch, up  to  its  elastic  limit,  which  is  at  about 
Yz  its  break-strain.  The  extent  of  stretching,  however,  varies 
much  with  the  quality  of  the  iron ;  as  in  wrought-iron. 

Cast,  malleable,  annealed  18  to  25  tons 48160 

Iron,  wrought,  rolled  bars,  40000  to  75000,  the  last  exceptional...  57500 

"    ordinary  average.    See  Rein  p  375 44800 

"    good              .  "     50400 

"    superior 60000 

"    best  American,  (exceptional) 76100 

"        "    Low  Moor,  English,  average 60000 

"    plates  for  boilers,  &c,  40000  to  60000 50000 

"        English  rivet  iron 55000  to  60000 57500 

**        wire,  annealed 80000  to  60000 45000 

"    unannealed,  or  hard. ..50000  to  100000 76000 

"    ropes,  per  sq  inch  of  section  of  rope 38000 

"        large  forgings,  30000  to  40000 35000 

In  important  practice,  good  bar  iron  should  not  be  trusted  per- 
manently with  more  than  about  5  tons  per  sq  inch ;  which  will 
stretch  it  about  %  inch  in  from  20  to  25  ft. 

Good  bar  iron  stretches  about  1  part  in  12000  of  its  length  ;  or 
about  1  inch  in  1000  ft ;  or  */£  inch  in  125  ft,  for  every  ton  of 
tensile  strain  per  sq  inch  of  section,  up  to  its  elastic  limit. 
This  limit  usually  ranges  between  8  and  13  tons  per  sq  inch,  or 
about  half  the  breaking  strain,  according  to  quality.  The, 
ultimate  stretching  of  rolled  bars  is  from  5  to  30  per  ct  of  the 
original  length ;  usually  15  to  20  per  cent.  Plates  and  angle 
iron  3  to  17  per  cent.  Heating,  even  up  to  500°  Fah,  does  not 
weaken  bar  iron  or  steel.  For  stretch  by  heat  see  p  310. 

Lead,  cast,  1700  to  2400 by  author...  2050 

"      wire,  1200  to  1600.    Pipe  1600  to  1700 "        "      ...  1650 

Platinum  wire,  annealed,  32000.    Unannealed 56000 

Steel,  plates,  range,  60000  to  103000 81500 

of  Hussey,  Wells  &  Co,  Pittsburg,  Pa,  91500  to  97400  94450 

Bessemer 98600 

Bessemer  tool 112000 

"      wire,  annealed  30  to  50  tons.    Unan,  50  to  90  tons 156800 

*Ijf*,rge  hars  of  metal  bear  less  per  sq  inch  than  small  ones.   In  cast  iron 

ones  1,  2  and  3  ins  sq,  the  strengths  per  sq  inch  were  about  as  1,  .85  and  .66 ;  and  wrought  iron  prob- 
ably averages  about  the  same. 

Iron  bars  re-rolled  cold  have  tensile  strength  increased  25  to  50  per  ct,  with  no  increase  of 
density.    They  are  said  to  lose  this  strength  if  reheated. 


STRENGTH    OF    MATERIALS. 


179 


Average  ultimate  tensile,  or  cohesive  strength  of  Metals 
per  square  inch.    (CONTINUED.) 


Capt.  James  B.  Eads  foand  that  forged  steel  bolts  for  the  St.  Louis  bridge,  59$ 


ins  diaiu,  and  22  to  36  ft.  long,  broke  short  with  only  30000  tbs  per  sq  inch  f  while 

bolts  of  but  %  inch  section,  cut  from  the  large  ones,  in  no  instance  broke  with    8C1>  mcn- 

100000  fts  per  sq  inch,  but  stretched  considerably. 

Steel,  cast,  Bessemer  ingots,  average 63000 

"        "      beat  American  Bessemer  ingots 86600 

"        "  "         rolled  arid  hammered,  120000 

to  130000 125000 

"        "         homogeneous,  Cammell  &  Co,  England,  No  1... 68240 

No  2 716SO 

"        "  "  "  "          No  3 76160 

"      puddled  bars,  rolled  and  hammered,  65000  to  135000 100000 

Steel.  Experiments  by  Lieut.  W.  S.  Shock, U.  S  N.,  at  Washington, 
on  steel  from  the  Black  Diamond  Steel- Works,  Pittsburg, 
Pa  All  the  pieces  were  cut  from  the  same  bar,  three 
pieces  for  each  exp.  They  were  turned  down  to  a  diani 
of  .62  of  an  inch  at  the  intended  point  of  fracture,  by  a 
groove,  in  shape  of  a  circular  segment,  with  a  chord  of 
about  I  inch 

"      the  bar  in  its  original  condition,  109500  to  131900 120700 

"      heated  to  light  cherry -red,  then   plunged  into  oil  of  82° 

Fah,  201 300  to  227500  214400 

"      heated  to  light  cherry-red,  then  plunged  into  water  of  79° 

Fah.    Then  tempered  on  aheated  plate,  152500  to  176100..       161300 
heated  to  light  cherry-red,  then  plunged  into  water  of  79° 

Fah,  132700  to  150500  141600 

Tempering  in  oil  usually  increases  the  strength  from  40  to 

80  per  cent. 

"  chrome,  made  at  Brooklyn,  N.  Y.,  and  tested  at  West  Point 
Foundry,  N.Y  ,  (specific  gr  7.816  to 7.956,)  163000  to  199000. 

Average  of  12  specimens 180000 

"  made  from  very  pure  Swedish  iron,  but  containing  differ- 
ent proportions  of  carbon.  The  bars  were  21%  ins  long, 
with  14  ins  of  this  length  turned  down  to  a  uniform  di- 
am  of  1  inch.  The  breaking  wts,  however,  in  the  table, 
are  per  sq  inch  : 

Mark  No.    2,  carbon    .33  per  ct,  stretched  1.37  ins 68100 

"      No.    4        "         .43        "  "         1.37   "  7*5160 

"      No.    5        "         .48         "  "          1.25   "  84000 

"      No.    6        "         .53        "  "         1.12   "  95200 

"      No.    7         "         .53         "  "          0.81   "  92960 

"      No.    8        "         .63         "  "          1.00   "  100800 

"      No.  10        "         .74        "  "         0.69   "  101920 

"      No.  12        "         .84        "  "         1.12  "  123200 

"      No.  15         ««       1.00         "  "          1.00   "  134400 

"      No.  20        «       1.25        "  "         0  62  "  154560 

With  more  than  about  1.5  per  ct  of  carbon  the  tensile  strength  of 
steel  diminishes.  A  bar  of  the  above  No.  15,  which  broke  at 
60  tons  per  sq  inch,  when  turned  down  for  14  ins  of  its  length  ; 
broke  with  79%  tons  per  sq  inch  when  turned  down  at  one 
point  only.  This  is  owing  to  the  fact  that  the  last  could  not 
stretch  as  much  as  the  first,  and  therefore  its  diam  could 
not  be  diminished  as  much  before  breaking.  All  its  fibres 
pulled  more  unitedly.  It  will  be  observed  that  the  steel  of 
greatest  strength  stretched  the  least  before  breaking.  This 
stronger  steel  would  break  under  a  suddenly  applied  force,  or 
impulse,  more  easily  than  a  weaker  one  would  ;  because  the 
weaker  one,  by  its  stretching,  gradually  breaks  the  force  of  the 
impulse,  on  the  same  principle  as  a  spring.  Hence  the  steel, 
iron,  &c,  which  is  strongest  against  a  gradually  applied  force 
or  strain,  may  be  unfit  for  uses  where  the  strain  comes  upon  it 
suddenly.  The  average  ultimate  tensile  strength  of  steel  is  about 
twice  that  of  wrought  iron.  Its  deflection  as  a  beam  within  the  elastic 
limits  is  about  $  that  of  wrought,  or  %  that  of  cast  iron.  Its  average  stretch 


Pounds 


180 


STRENGTH   OF    MATERIALS. 


is  about  .1  inch  in  111  ft  for  every  ton  per  sq  inch  of  load,  up  to  its  elastic  limit,  which  generally 
ranges  at  between  %  and  %  of  its  breaking  strength  ;  the  latter  being  for  the  harder,  stronger,  and 
less  stretchy  kinds.  A  uniform  bar  of  rolled  steel,  gradually  loaded,  will  stretch  from  -*-——  to  1- 
of  its  length  before  breaking;  or  from  y^  of  an  inch  to  2.4  ms  per  foot,  according  to  quality.  The 
mean  of  these  is  nearly  y1^  of  tbe  length,  or  ly^iuch  to  a  foot.  When  steel,  especially  if  hard,  has 
to  be  heated  to  softness  in  order  to  give  it  a  required  shape,  it  is  thereby  weakened. 

Average  ultimate   tensile  or  cohesive   strength  of  Metals 
per  square  inch.    (CONTINUED.) 


Pounds 

Tons 

per 

per 

sq.  inch. 

sq.  in. 

Silver,  cast  

41000 

18  3 

Tin,  English  block  ... 

4600 

2 

"    wire  

7000 

3  1 

Zinc,  cast.  ..3000  to  3700  ;  (the  last  by  author)  

3350 

1.5 

Art.  6.    Average  ultimate  tensile  or  cohesive  strength  of 
various  materials. 


The  strengths  in  all   these 
tables  may  readily  be  one-third 
part  more  or    less    than  our 
averages. 

Pounds 
per 
sq.  inch. 

Tons 
per 
sq.  ft. 

Pounds 
per 
sq.  inch. 

Tons 
per 
sq.  ft. 

Brick  40  to  400 

220 

141 

Marble  strong  wh  Italy  * 

1034 

665 

Caen  stone,  100  to  200  
Cement,  hydraulic,  Port- 

150 

9.7 

"        Champlain,  varie- 
gated *  

1666 

1071 

land,  pure,  7  days 
in  water       .• 

300 

193 

"       Glenn's  Flls.N.Y, 
blk  *  75()tol034 

892 

57  4 

"    6  months  old  

450 

289 

"       Montg'y    co    Pa 

"    1  year  old 

550 

354 

1175 

75  6 

Common    hyd    cements 
average  1-6  as  much. 
The  last,  neat,  adhere 
to  brick  and  stone  with 
from  15  to  50  Ibs  when 
only  1  month  old  

32 

2 

"       white*... 
**        Lee,Mass,white.* 
"        Manchester,  Yt,* 
550  to  800  
"       Tennessee,  varie- 
gated*    

734 
875 

675 
1034~ 

47.2 
56.3 

43.4 
66.5 

At  end  of  1  year,  3 

Oolites  100  to  200 

150 

97 

times  as  much  
See  "  Cement,"  p  506,  &c 
Glass,  2500  to  9000  (p  515) 

96 
5750 

6 

385.7 

Plaster  of  Paris,  well  set. 
Rope,  Manilla,  best  
"      hemp    best    

70 
12000 
15000 

4.5 

771 

965 

Glue  holds  wood  together 

Sandstone  Ohio* 

105 

6  75 

with  from  300  to  800... 
Horn  ox 

550 
9000 

35 
579 

"           Pictou,  N.  S.* 
"           Conn    red  * 

434 
590 

27.9 
37  9 

16000 

1029 

Slate  Lehigh  * 

2475 

159  1 

Leather    belts,    1500    to 
5000     Good   

3000 

193 

"      Peach  bot'm,*  3025 
to  4600 

r>81  '> 

245  1 

Mortar,  common,  6  mos 
old,  10  to  20  

15 

.96 

Stone,  Ransome's  artif.... 
Whalebone  

300 
7600 

19.3 
489 

To  find  the  diam  in  ins  of  a  round  rod  to  bear  safely  a  given  pull 
in  fibs. 


Diam 
in  ins 


/given  pull  X  coef  of 
=  V  /  ult  tensile  strength 
V     of  material  in  Ibs  pei 


safetv 


,7854. 


er  sq  inch  ' 
Iron  is  weakened  by  extreme  cold. 

The  belief  (originating  with  Styff  of  Sweden,)  is  gaining  ground  that  iron  and 
steel  are  not  rendered  more  brittle  by  intense  cold,  but  that  the  great  number  of 

*  By  the  author's  trials  with  one  of  ftiehle's  testing  machines.    Sections 

broken  1%  sq  inches. 


STRENGTH   OP   MATERIALS. 


181 


breakages  of  rails,  wheels,  axles,  <fec,  in  winter,  is  owing  to  the  more  severe  blows 
incident  to  the  frozen  and  unyielding  nature  of  the  earth  at  that  period  of  the  year. 
But  Sandberg's  experiments  show  conclusively  that  although  these  metals  may  per- 
haps bear  as  much  steady  force,  gradually  applied,  in  winter  as  in  summer,  yet  their 
resistance  to  impulse,  or  sudden  force,  is  not  more  than  %  or  %  as  great  in  severe 
cold;  which  renders  them  less  flexible  and  less  stretchy.  It  is' probable  that  this 
fact  does  not  receive  as  much  attention  as  it  should,  in  proportioning  iron  bridges,  &c. 

Some  experiments  with  good  wrought  iron  showed  that  even  at  23°  Fah,  or  only 
9°  colder  than  freezing  point,  there  was  a  loss  of  strength  of  from  2%  to  4  per 
cent. 

Art.  7.  Breaking  by  shearing.  Let 
abed,  Fig  1,  represent  a  beam,  with  its  ends 
resting  on  supports  SS;  with  a  load  I,  so  heavy 
as  to  break  it  by  forcing  its  entire  central  part, 
oo  g  g,  away  from  the  two  end  parts  a  dg  and  bco; 
so  that  while  the  two  latter  remain  in  their 
places,  the  central  part  slides  out ;  or  is,  as  it 
were,  punched  clean  out  from  between  them. 
This  peculiar  mode  of  fracture  is  called  shearing, 
or  delrusion.  The  force  required  to  produce  it, 
or  the  resistance  which  the  beam  opposes  to  such 
a  force,  may  practically  be  assumed  to  be  in  pro- 
portion to  the  area  of  the  sheared  section.  Thus, 
since  the  area  of  cross  section  of  a  beam  1  ft  sq 

is  4  times  as  great  as  that  of  a  beam  6  ins  sq,  the  former  will  present  4  times  as  great 
a  resistance  to  shearing;  or  will,  in  other  words,  require  4  times  as  great  a  load,  or 
pres,  to  shear  it  across.  In  Fig  1  the  total  sheared  area  is  equal  to  twice  the  trans- 
verse area  of  the  beam.  See  Remark  2,  p  293.  of  "  Trusses."  Bridge  chords  are  ex- 
posed to  great  shearing  force  where  they  rest  on  the  abuts,  but  it  becomes  less 
toward  the  center  of  the  span;  and  so  with  every  equally  loaded  beam.  See  p  642. 

We  have  very  little  experimental  data  on  this  subject. 

The  shearing  strength  of  white  pine,  spruce, 
and  hemlock,  parallel  to  the  fibres,  by  the  author,  250  to  500  !bs 
per  sq  inch ;  oak  400  to  700 ;  and  is  of  use  in  estimating  the 
resistance  along  the  line  c  c,  Fig.  2,  at  the  end  of  a  tie-beam; 
or  at  the  head  of  a  queen-post,  <fcc. 

Across  the  fibres  the  writer  found  for  spruce  about 
3250  ft>s;  white  pine  and  hemlock  2500;  yellow  pine  4300  to 
5600 ;  white  oak  4400.  For  others  see  Shearing,  Art  2,  p  642. 

Wrought  iron  is  stated  at  35000  to  55000  fts  per  sq 
inch;  cast  iron  20000  to  30000;  steel  45000  to  75000  fbs; 
copper  33000.  « 

The  shearing  strength  of  steel  and  wrought  iron  is  about  % 
part  less  than  the  tensile.  The  punching  of  rivet-holes  in 
iron  or  steel  plates,  is  an  example  of  shearing.  The  rivets  in 


J 


» 

tiy 


tubular  bridges  are  frequently  sheared  in  two,  in  time,  by  the  motion  of  the  platei 
through  which  they  are  driven.  In  punching  holes,  the  area  of  section  is  evidently 
found  by  mult  the  circumf  of  the  hole  by  the  thickness  of  the  plate  in  which  it  ia 
punched.  If  a  piece  of  material  be  supported  as  shown  in  Fig  •-'%  its  resistance  to 
shearing  will  be  3  times  as  great  as  in  Fig  1,  where  it  is  sheared  across  in  2  places 
only;  whereas  in  Fig  2%,  shearing  would  have  to  occur  at  6  places,  as  per  the  6 
dotted  lines. 

Art.  8.    Breaking  by  torsion,  or  twisting.    Let  n,  Fig  3,  be  a  vert 

cylindrical  rod  of  any  material,  1  inch  diam,  the  lower 

end  of  which  is  immovably  fixed ;  and  let  c  be  a  lever         L 

whose  leverage  (see  levers)  a  b,  measd  from  the  axis  of         ° «• 

the  cylindrical  rod,  is  I  ft.     Suppose  that  with  a  spring 

balance  attached  to  the  end  b  of  the  lever,  we  apply  force 

horizontally,  and  around  the  axis  of  the  rod  as  a  center, 

until  the  rod  breaks  by  being  twisted.     Then  if  we  mult 

together  the  leverage  n  b  in  feet,  and  the  amount  of  force 

shown  by  the  spring  balance  in  Ibs,  and  div  the  prod  by 

the  cube  of  the  diam  of  the  rod  in  ins  the  quot  will  be  a 

certain  number  of  foot-pounds  ;  and  will  be  what  is  called 

the  constant,  or  coefficient  for  torsion,  for  all  cylindrical  bars  of  that  material.    If  we 

use  a  square  bar,  we  shall  get  the  coef  tor  square  bars ;  and  so  with  any  other  shape. 

So  that  if  with  any  other  bar,  or  shaft,  we  mult  the  cube  of  its  diam'  in  inches  by 


t 


182  STRENGTH   OF   MATERIALS. 

said  constant,  and  div  the  prod  by  the  leverage  in  feet,  the  quot  will  be  the  force  in 
Ibs  which  will  twist  the  bar  in  two.     In  shape  of  formulas, 

M^MT  ^W^X  Constant     Brea*g 

=  Constant.    And - : — =  .lor^e      Also 

Cube  ot  diam  in  ins  Leverage  in  feet  m  Ibs. 

Cube  of  diam  ,.,  ,-.     c  Leverage  v  Brkg  force 

__m_ins _  Leverage  in  feet    ^        infos       _  Cube  of  diam 

Brkg  force  in  ft>s          ~    in  feet"  Constant  in  ins- 

Hence  we  see  that  the  torsional  resistance  of  a  cylinder  increases  directly  as  the 
cube  of  the  diam,  or  cube  of  one  side  if  square;  and  diminishes  as  either  the 
leverage  or  the  force  increases. 

The  constant  for  solid  cylinders  of  average  cast  iron  is  about  600; 
and  for  wrought  iron  800.  For  puddled  steel  about  700;  cast  steel  1000  to  1700. 
Wrought  copper  400.  All  may  vary  one  fourth  part  of  these  more  or  less. 

For  woods,  rough  averages.  W  pine  or  spruce  20  to  25  ft-lbs.  Y  pine  35. 
Ash  40.  W  oak  50.  Locust  75.  Hickory  85. 

To  find,  by  the  last  formula,  the  diam  of  a  rod  to  have  a  safety  of  3,  4,  5,  Ac, 
against  a  given  twisting  force  in  Ibs,  first  mult  said  force  by  3,  4,  5,  Ac,  as  the  case 
may  be,  and  use  the  prod  as  the  breaking  force.  The  diam  will  then  be  .the  safe  one. 

Any  angle  described  by  the  force  at  b,  when  made  to  revolve  around  the  axis  of 
the  rod  as  a  center,  during  the  twisting  process,  is  called  the  ANGLE  OF  TORSION.  The 
length  of  the  twisted  rod  or  shaft  does  not  affect  the  amount  of  force  reqd  to  produce 
rupture ;  but  the  longer  it  is,  the  greater  will  be  the  angle  of  torsion ;  or  in  other 
words,  the  greater  will  be  the  dist  through  which  the  force  must  revolve  around  the 
axis  before  fracture  takes  place.  Authorities  say  that  a  working  shaft  should  not 
twist  more  than  1°.  We  should  not  expose  it  to  more  than  .1  of  its  ult  strain.  \ 

If  we  know  the  force  in  ibs  per  so.  inch  reqd  for  shearing  any  material,  see  pre- 
ceding Art,  then  the  force  required  to  break  a  cylinder  of  it  by  torsion,   is 
Tnrsinn      One-half  the  shearing  ^  Q  \4-ta  \/  Cube  of  rad  of 
force  =  force  in  ibs  per  sq  inch  X3'141    X  cylinder  in  ins 
in  ibs  Leverage  in  inches. 

That  of  a  square  shaft  is  about  1^  times  that  of  a  round  one  whose  diam 
is  equal  to  a  side  of  the  square ;  or  about  4-  less  than  that  of  a  round  one  of  the 
same  transverse  area.  For  any  solid  rectangular  shaft 

.  One-third  of  the  shearing  v  The  square  of  v  The  square  of 

Breaking  force,  in  fts  per  so  in      A       one  side       A  the  other  side. 

Torsional  force  = — *     .  ., —m T • 

jn  jks  Square  root  of  the  sum  of  the  ^  Leverage 

above  two  squares  •*  in  inches. 

Hollow  shafts  resist  torsion  better  than  solid  ones  of  the  same  area  of 
metal.    Calling  the  outer  and  inner  diams  in  ins  D  and  d,  then 
Breaking          (D4  _  dt)  x  Constant 

Torsional  force  =  \= : — -,  ,,  _.- 

in  Ibs  Leverage  in  ft  X  D 

Strength  of  wrought -iron  shafting.  The  shafting  used  for  the 
transmission  of  power  to  the  diff  parts  of  machine-shops,  many  manufacturing  es- 
tablishments, &c,  is  subjected  to  twisting  strains.  It  is  usually  made  cylindrical, 
and  of  wrought  iron.  Experience  shows  that  we  may  safely  use  the  following  for 
shafts  of  iron  of  good  quality,  bearing  but  little  weight,  and  well  supported  at  proper 
intervals,  say  8  or  9  ft.,  by  self-adjusting  ball  and  socket  hangers. 


/   Horse-powers 

\/    Number  of  rev 
per  minute 


Diam  of  a  wrought  _  .     /  .  Horse-power^ 

iron  Shaft  in  ins.  ~    \   /     Niimhor  nf  r«vs  A  A^O. 


pei 

Or  in  words :  for  the  diameter  in  inches  div  the  number  of  horse-powers  that  are 
to  be  transmitted  along  the  shaft,  by  the  number  of  revs  which  the  shaft  is  reqd  to 
make  per  min.  Mult  the  quot  by  125.  Take  the  cube  root  of  the  product.  This 
cube  root  will  be  the  diameter  itself,  at  the  thinnest  part,  at  its  bearings. 

The  last  formula  shows  that  the  faster  a  shaft  revolves  under  the  same  number 
of  horse-powers,  the  less  is  the  torsional  strain  upon  it.  This  may  at  first  seem 
strange,  but  less  so  when  we  reflect  that  a  horse-power  is  made  up  of pres  and  dist; 
therefore,  the  faster  it  moves,  the  less  is  its  pressure.  Hence  many  horse-powers  re- 


STRENGTH   OP   MATERIALS. 


183 


i 


volving  rapidly  will  require  a  less  diam  than  a  small  number  revolving  slower  in 
proportion  than  its  number. 

Art.  9.  Transverse  (or 
across)  strength  of  mate- 
rials ;  or  that  by  which  they  re- 
sist breaking,  when  employed  as 
beams ;  as  for  instance  in  Fig  4. 

To  find  constants.  In 
beams  of  the  same  material,  and 
exactly  alike,  except  in  their 
breadths,  n  d,  the  strengths  vary 
in  the  same  proportion  as  those 
breadths ;  that  is,  if  one  is  2,  3,  or 
10  times  broader  than  the  other,  its  strength  will  be  2,  3,  or  10  times  as  great.  If 
they  are  alike,  except  in  their  clear  lengths  or  spans,  a  a,  between  the  points  of  sup- 
port, their  strengths  will  be  inversely  as  those  lengths;  that  is,  if  one  is  2,  3,  or  10 
times  longer  than  the  other,  it  will  be  but  %,  %,  or  -J^  part  as  strong.  If  they  are 
alike,  except  in  point  of  depth,  o  d,  measured  vert,  their  strengths  will  be  directly  as 
the  squares  of  their  depths;  that  is,  if  one  is  2,  3,  or  10  times  as  deep  as  the  other, 
it  will  also  be  4,  9,  or  100  times  as  strong ;  or  in  other  words,  will  require  4,  9,  or  100 
times  as  great  a  load  to  break  it.  See  Art.  11.  It  must  be  remembered  that  we  are 
now  speaking  only  of  strength,  or  resistance  to  breaking ;  and  not  of  stiffness,  or  resist- 
ance to  bending,  or  deflecting.  Stiffness  follows  laws  very  diff  from 
those  of  strength.  See  Art  26,  &c. 

Now,  if  we  combine  all  the  three  foregoing  elements  of  size,  namely,  length, 
breadth,  and  depth,  we  have  the  fact,  that  the  strength  of  any  beam,  of  any  size,  of 

breadth  X  the  square  of  its  depth. 

any  given  material,  is  in  proportion  to  its —  -  There- 

its  length 

fore,  if  WB  find  by  actual  trial,  what  center  load  will  break  any  beam  of  known  size ; 

breadth  X  sq  of  depth 

and  then  find  what  is  the  proportion  between  its : ,  and  its 

length 

breakg  load,  said  proportion  (or,  more  strictly  speaking,  ratio)  will  also  be  that 
which  any  similar  beam  has  to  its  breakg  load,  and  will  therefore  serve  to  calcu- 
late the  breakg  load  of  any  other  similar  beam  of  the  same  material.  For  instance, 
if  we  take  any  piece  of  average  good  white  pine,  say  6  ins  broad,  10  ins  deep,  and  12 
breadth  X  sq  of  its  depth  .  6  X  100 

,feet  clear  span,  we  find  that  its — is  equal  to —      •  •   =  50. 

length  12 

And  if  we  gradually  load  this  at  its  center  until  it  breaks,  we  shall  find  that  the 
breakg  load,  including  half  the  wt  of  the  clear  span  of  the  beam  itself,  amounts  to 

breadth  y  sq  of  depth 
22500  Ibs.    Therefore,  the  proportion  between  the     in  ins    A    in  ins  ana  the 

length  in  feet 

breakg  load,  is  as  50  to  22500 ;  which  is  the  same  as  I  to  450 ;  that  is,  the  breakg 
load  of  the  beam,  including  half  its  own  weight,  may  be  found  by  mult  its 
breadth  y  sq  of  depth 

.  *n  in3 in  ins          by  450.    And  in  this  same  manner  may  be  found  the  total 

length  in  feet 

center  breakg  load  of  any  rectangular  beam  of  average  quality  of  white  pine.  For 
the  neat  load  one-half  the  wt  of  the  clear  span  must  be  deducted.  It  is  self-evident 
that  the  weight  of  the  beam  assists  to  break  it,  as  well  as  the  neat  load;  and  the 
extent  to  which  it  does  so,  is  the  same  as  if  one-half  of  its  unsupported  wt  were 
concentrated  at  its  center.  Hence  the  rule.  On  this  principle  the  rule  in  Art  12  is 
based.  The  ratio  thus  found  for  any  material,  is  called  its  coef  for  ceii 
breakg  loads,  or  its  constant  for  the  same,  as  it  does  not  vary  with  the 
size  of  the  beam.  If  we  take  a  piece,  all  of  whose  dimensions  are  1,  as  1 

breadth  v  sq  of  depth 
inch  wide,  1  inch  deep  and  1  ft  span;   then  the    in  ins  x        in  ins      will  be 

1  X  sq  of  1 
1 


length  in  feet 
=  1 ;  and  the  breakg  load  (including  half  its  own  weight)  of  such  a  piece 

is,  at  once,  the  constant  reqd.  See  Remarks  1  to  4.  In  an  average  piece  of  white 
pine,  this  load  will  be  found  to  be  about  450  Ibs ;  or  the  same  as  the  constant  obtained 
from  the  large  beam.  Ho  the  student  may  find  them  for  himself; 

and  if  he  uses  materials  not  included  in  our  table,  Art  10,  it  will  be  well  to  supply 
the  deficiency,  by  inserting  his  own  results. 
The  foregoing  directions  for  finding  coefficients  may  be  more  briefly  expressed 


184  STRENGTH   OF   MATERIALS. 


by  a  form., 
one-half  th 


mla.    After  finding  the  neat  center  breakg  load,  by  experiment,  add  to  It 
;he  wt  of  the  clear  span  of  the  beam,  for  a  total  center  load;  then  the 

Span     v         Total  load 

Coef  for  breakg  1    =    injeet__l        in  lbs 

strength  j         Breadth  v  Square  of  depth 

in  ins    A          in  inches. 

Ill  a  cylinder,  as  the  breadth  and  the  depth  are  each  equal  to  the  diain, 
it  is  plain  that  B  X  1^*,  amounts  to  the  same  thing  as  diam3 ;  and  it  is  always  so 
exjressed.  See  Remark,  Art  29.  p  201. 

HEM.  1.  The  variation  in  strength  of  equal  beams  of  the  same  material  is  so  great, 
that  it  is  necessary  to  experiment  with  several  pieces,  in  order  to  find  an  average  for 
a  constant.  The  loads  given  in  the  preceding  tables  are  also  constants,  bnt  for  crush- 
ing and  tension.  They  are  averages  of  the  strengths  of  the  materials,  derived  from 
experiment.  The  actual  strength  of  any  particular  specimen,  if  of  superior  quality, 
may  be  considerably  greater  than  the  average ;  or  on  the  other  hand,  if  of  very  poor 
quality,  it  may  fall  as  much  below  it.  We  should  always  keep  this  in  mind  when 
referring  to  any  table  of  constants;  and  if  we  have  doubts  as  to  the  quality  of  the 
piece  of  material  which  we  are  about  to  employ,  we  should  make  a  corresponding 
deduction  from  the  constant  in  the  table. 

Reni.  2.  If,  instead  of  pine,  we  had  experimented  with  oak,  iron,  stone, 
the  process  for  finding  the  constant  would  have  been  precisely  the  same.  If  in- 
stead of  a  square  beam,  we  use  cylindrical,  or  triangular  ones,  or  any  other 
shape,  such  as  hollow  cylinders,  H,  T,  or  U  beams,  &c,  we  shall  in  the  same  way 
establish  constants  for  either  larger  or  smaller  beams  of  those  shapes,  and  of  precisely 
the  same  proportions  in  every  part.  See  Remark,  A rt  29.  Or  if,  iusteadof  supjport- 
ing  the  beam  at  both  ends,  we  secure  it  firmly  at  one  end,  and  load  it  at  the  other 
end  until  it  breaks,  we  shall  obtain  the  constant  for  beams  fixed  tit  one  end,  and  loaded 
at  the  other,  &c.  Remember  that  the  constants  are  for  loads  at 
rest.  If  they  are  liable  to  jars,  jolts,  vibrations,  &c,  a  large  margin  must  be 
left  for  safety.  Moreover,  the  constants  given  in  tables  are  generally  deduced  from 
small  specimens  free  from  important  defects ;  whereas  large  beams  of  any  kind  of 
material  usually  contain  irregularities,  which  diminish  their  strength;  and  on  this 
account  larger  allowances  for  safety  should  be  made  as  the  dimensions  of  the  beam 
increase. 

Rein.  3.  It  is  not  necessary  that  &  and  d  be  taken  in  ins,  and 
lengths  in  ft.  They  may  all  be  in  ins,  ft,  yds,  or  any  other  measure,  but  since  in 
every-day  practice  we  usually  speak  of  the  breadths  and  depths  of  beams  in  ins,  and 
of  their  lengths  in  ft,  it  becomes  more  convenient  so  to  consider  them.  If  other  meas- 
ures be  used,  the  constant  will  of  course  be  diff ;  but  it  will  still  be  such  that  if  the 
same  measure  be  used  for  calculating  the  strength  of  another  beam,  the  final  result 
will  be  the  same  as  before.  In  like  manner,  the  loads  may  all  be  taken  in  tons,  &c, 
instead  of  tbs ;  but  in  giving  the  rule,  it  must  be  stated  what  measures  have  been 
employed.  See  Remark,  Art  29. 

Rein.  4.  There  are  peculiarities  in  some  materials,  which  les- 
sen the  reliability  of  constants  derived  from  experimenting  with  small  pieces. 
Thus,  a  large  beam  of  cast  iron  will  break  with  a  less  load  in  proportion  than  a  small 
one ;  because,  in  the  interior  of  thick  masses  of  that  material,  more  time  to  cool  is 
required  than  in  the  outer  surfaces ;  in  consequence  of  which,  there  is  a  want  of  uni- 
formity in  the  arrangement  of  the  particles  of  iron,  and  this  conduces  to  w.eakness. 
All  we  can  do  in  such  cases,  is  to  exercise  judgment  and  caution  in  making  sufficient 
allowance  for  safety. 

Art.  1O.  Table  of  constants  or  coefficients  for  the  quiescent 
breaking?  loads  of  rectangular  beams,  supported  horizon- 
tally at  both  ends,  and  loaded  at  the  center;  being  the  average  qui- 
escent breaking  loads  in  fibs  (including  one-half  the  weight  of  the  beams  themselves) 
for  beams  1  inch  square,  and  1  foot  clear  length  between  the  supports.  For  safety 
in  practice,  not  more  than  about  %  to  %  of  these  constants  should  be  employed ;  de- 
pending upon  the  importance  of  the  structure,  its  temporary  or  permanent  charac- 
ter, and  the  degree  of  vibration  to  which  it  will  be  exposed.  Thus  a  roof  will  prob- 
ably be  as  safe  at  *^,  as  a  bridge  at  %.  Even  with  a  perfectly  safe  load,  a  beam  may 
bend  too  much.  See  Art  -6.  p  196. 

If  any  of  these  coefficients  be  mult  by  .589  (or  say  .6)  it  will  give  that  for  a  cylin- 
drical beam  whose  diam  =  side  of  the  square.    Or  if  mult  by  .71  it  will  give  that 
for  a  square  beam  with  its  diagonal  vertical.    Any  of  these  constants 
may  vary  one-third  part  either  more  or  less.    For  any  beam, 
Ceil.  Breaks  _    Breadth  (ins)  X  Square  of  depth  (ins)     Congtant 
load  in  lbs.  clear  span  in  feet. 


STRENGTH   OF   MATERIALS. 


185 


One  Third  part  of  any  of  these  constants  (except  those  for  wrought 
iron  and  steel),  may  be  taken  in  ordinary  practice  as  about  the  average  constant 
for  the  greatest  center  load  within  t3ie  elastic  limit.  The  loads 
here  given  for  wrought  iron  and  steel,  are  already  the  greatest  within  elastic  limits. 
See  p  198.  The  modulus,  coef,  or  constant  of  rupture  or  frac- 
ture of  writers  on  physics  is  18  times  the  loads  in  this  table.  See  p  195. 


1 
Transverse  Strength. 

Lbs. 

Transverse  Strength. 

Lbs. 

WOODS. 

Ash    English            .           ...     M 

650 

but  at  about  the  average  of  2250 
Ibs  its  elas  limit  is  reached 

"  '  Amer  White  (Author).     P 
"    Swamp               ^ 

650 
400 

Steel,  hammered  or  rolled  ;  elas 
destroyed  by  3000  to  7000 

5000 

«     Black      0 

300 

Under  heavy  loads  hard  steel 

Arbor  Vitse  Amer   O 

250 

snaps   like   cast  iron    and    soft 

Balsam  Canada  8* 

350 

steel  bends  like  wrought  iron 

Beech    English           M 

500 

"     '  Amer  White                 —  "* 

450 

"       Amer  Red           <:  ^ 

550 

STONES,  ETC. 

Birch  Amer  Black               .      • 

450 

Blue  stone  flagging,  Hudson  River 

125 

"       Amer  Yellow  ^  3 

450 

Brick,  common,  10  to  30.  .average 

20 

Cedar  Bermuda         .      .       y?  S 

400 

"       good  Amer  pressed    30  to 

600 

50       average 

40 

"       Amer  White             1  ^"  P 

Caen  Ston? 

25 

or  Arbor  Vit».'."!  J  -""S 
Chestnut                   **  o 

250 
450 

Cement,  Hydraulic,  English  Port- 
land   artificial, 

Elm  English  °  -• 

350 

7  days  in  water 

30 

*'    Hock  Canada  .          .    o>  * 

600 

1  year  in  water 

50 

Hemlock  .'  (by  Auth.)*g- 
Hickory,  Amer.,     "       "       .  ^  g- 
"             "    Bitter  nut  |-S 
Iron  Wood  Canada  •* 

400 
700 
500 
600 

"         "       Portland,  King- 
ston, N.  Y.,  7 
days  in  water. 
"         "       Saylor's  Port.,  7 

30 

Locust                                       ^  ss 

600 

days  in  water. 

26 

Lignum  Vitss  *    ' 

650 

"         "       Common    TJ.  S. 

Larch                                             »* 

400 

cements  7  dys 

Mahogany  \ 

450 

in  water  

5 

650 

The  followin^  hydraulic  ce- 

"       '  Black  "§ 

550 

ments  were  made  into  prisms,  in 

Maple  Black  .                             ?~ 

550 

vertical  moulds  under  a  pressure 

"       Soft  £ 

550 

of  32  fts  per  sq  inch  and  were 

Oak  English     

550 

kept  in  sea  water  for  1  year 

"     Amer  White    (by  Author). 
"        "      Red,  Black,  Basket... 
"    Live 

600 
550 
600 

Portland  Cement,  English,  pure, 
1  year  old... 
Roman  Cement  Scotch  pure   .... 

64 
23 

Pine,  Amer  White...  ^l>y  Author) 
"        "       Yellow     "         " 

450 
500 

American  Cements,  pure,  av  about 
(jrranite  50  to  150    average 

25 

100 

u        «<       pitch        "         " 

550 

"       Quincy 

100 

"     Memel 

450 

Glass  Millville  N  Jersey    thick 

Poplar   

550 

flooring  (by  Author). 

170 

P<xm                .... 

700 

M<Yrtar  of  lime  alone  60  days  old 

10 

Spruce                        (by  Author) 

450 

"      1  measure  of  slacked  lime 

"      Black    

360 

in  powder  1  sand     

8 

Sycamore 

500 

s<      1  measure  of  slacked  lime 

Tamarack  ..        

400 

in  powder  2  sand     

7 

Teak 

750 

Marble  Italian    White  (Author) 

116 

Walnut  

450 

"   Manchester,  Vt,  "           " 

95 

Willow 

350 

"  East  Dorset   Vt  "            " 

111 

METALS. 

Brass  

850 

"  Lee,  Mass,            "           " 
"  Montg'y  Co,  Pa,  Gray     " 
"      "          "      Clouded       " 

86 
103 
142 

Iron,  cast,  1500  to  2700  ...average 
"    common  pig  
"    castings  from  pig  .... 
"        "    employed    in   our  ta- 

2100 
2000 
2300 

"  Rutland.Vt,  Grav            " 
"  Glenn'sFalls,N.Y.Black  " 
"  Baltimore,  Md,  white, 
coarse  u 

70 
155 

102 

bles 

2025 

Oolites  20  to  50     .           ... 

35 

*'        "    for  castings  2%  or  3 
ins  thick 

1800? 

Sandstones,  20  to  70  average 
"        Red  of  Connecticut  and 

45 

Iron  wrought  1900  to  2600.  ...av 

2250 

New  Jersey  

45 

Wrought  iron  does  not  break  ; 

Slatf,  laid  on  its  bed,  200  to  450,  av 

325 

186  STRENGTH   OF   MATERIALS. 

Art.  11.  General  facts  respecting  the  breakg1  loads  of  a  uni- 
form beam  of  any  form  of  section.  Calling  the  breakg  load, 

When  the  beam  is  firmly  fixed  at  one  end,  and  loaded  at  the  other 1 

Then  when  so  fixed,  and  uniformly  loaded,  it  will  be 2 

When  merely  supported  at  both  ends,  and  loaded  at  the  center 4 

"  "  "  "      and  uniformly  loaded  8 

Firmly  fixed,  or  tightly  confined  at  both  ends,  and  loaded  at  the  center...     8 
"  "  <%  "      and  uniformly  loaded. 16 

By  some  authorities  the  last  two  are  given  at  6  and  12.  The  term  "  fixed"  in  such 
cases  is  rather  indefinite  Prof.  De  Volson  Wood  gives  12  for  the  last. 

REM.  1.  When  one  beam  of  any  form  of  section  is  2,  3,  or  4,  &c,  times  as  long,  broad, 
and  deep,  as  another,  its  weight  will  be  8,  27,  or  64,  &c,  times  as  great,  or  as  the  cubes 
of  the  linear  dimensions;  but  its  breaking  load  will  be  only  4,  9,  or  16,  &c,  times  as 
great ;  that  is,  the  strengths  of  similar  beams  are  to  each  other  as  the  squares  of 
their  respective  linear  dimensions.  But  in  these  breakg  loads  are  included  the 
weights  of  the  beams  themselves  ;  one-half  of  which  must  be  deducted  when  the 
load  is  all  at  the  middle  of  the  beam;  or  the  whole  of  it  when  equally  distributed. 
When  beams  are  of  the  moderate  dimensions  usually  employed  in  buildings,  their 
own  weight  is  usually  so  small  in  comparison  with  their  loads,  that  at  times  it  may 
safely  be  neglected ;  but  as  they  become  longer,  since  their  weight  increases  much 
more  rapidly  than  their  strength,  it  at  last  constitutes  too  important  an  item  to  be 
overlooked;  for  the  beam  may  become  so  great  as  to  break  under  its  own  weight; 
although  proportioned  precisely  like  a  small  one  which  may  safely  bear  many  times 
its  own  weight. 

When  a  square  beam  is  supported  on  its  edg-e.  instead  of  on  a 
side  (or  in  other  words  has  its  diag  vert),  it  will  bear  but  about  y<jths  as  gVeat  a 
breakg  load. 

The  deflections  or  bendings  of  beams  (see  p  196)  are  directly  in 
proportion  to  the  load,  and  to  the  cube  of  the  length ;  and  inversely  to  the  breadth, 
and  to  the  cube  of  the  depth,  all  while  within  limits  of  elasticity. 

REM.  2.  It  is  very  important  to  remember  that  a  beam  will  bear  a  much  greater 
load  placed  upon  it  in  small  amounts  at  a  time ;  or  (in  case  it  is  all  applied  together) 
if  its  pres  be  allowed  to  come  upon  the  beam  very  gradually,  than  if  it  be  placed 
upon  it  suddenly,  even  without  any  jarring,  and  without  any  previous  momentum. 
When  applied  in  small  amounts,  or  gradually,  the  bending  takes  place  slowly  and 
with  slight  momentum ;  but  when  applied  all  at  once,  the  great  load  descends  im- 
mediately and  rapidly  to  the  full  extent  of  the  bending  due  to  the  load  itself,  and  in 
so  doing  acquires  a  momentum  which  carries  it  still  further;  thus  producing  a  strain 
which  authorities  maintain  to  be  just  twice  as  great  as  in  the  former  case.  A  heavy 
train  coming  very  rapidly  upon  a  bridge,  presents  a  condition  intermediate  between 
the  two.  The  tables  of  constants  always  suppose  the  load  to  be  applied  very  grad- 
ually, but  as  this  is  frequently  not  the  case  in  practice,  an  allowance  must  be  made 
accordingly. 
Art.  12.  To  find  the  quiescent  breaUgr  load  of  a  nor  square. 

or  rectangular  beam  B.  Fij?  4,  of  any  material,  supported 

at  both  ends,  and  loaded  at  the  centre.    See  table,  p  206. 

RULE.  Mult  together  the  square  of  its  depth  d  o  in  ins,  its  breadth  n  d  in  ins,  and 
the  coef  from  the  table  p.  185.  Div  the  prod  by  the  clear  length  a  a  between  the 
supports,  in  ft.  The  quot  will  be  the  reqd  breakg  load  in  Ibs;  including,  however, 
half  the  wt  of  that  portion  of  the  beam  which  lies  between  the  supports,  arid  which 
must  be  deducted  in  order  to  get  the  neat  load. 

Ex.     What  will  be  the  center  breakg  load  of  a  beam  of  Connecticut  red  sandstone, 
15  ins  wide,  10  ins  deep,  and  12  ft  long  between  its  supports  ?     Here 
Depth3  ./  Breadth  ^  Constant 
in  ins    X     in  ins    >       p  185      _  100X15X45  =  67500  =  ^  ^ .    the  fereakg 

Clear  length  in  ft  12  '     12 

load  reqd. 

But  we  must  deduct  one-half  the  vrt  of  the  clear  length  of  tbe  beam.  Now  abeam 
of  red  sandstone,  of  15  ins,  by  10  ins,  by  12  ft,  contains  21600  cub  ins  — -  12%  cub  tt ; 
and  a  cub  ft  of  red  sandstone  weighs  about  140  ft»s ;  therefore  the  beam  weighs 
12.5  X  140  =  1750  ft»s ;  one-half  of  which,  or  875  fos,  must  be  taken  from  the  6626 
fts  of  breakg  load,  leaving  4750  Ibs  as  the  actual  extraneous,  or  neat  breakg  load. 

Rent.  1.  If  cylindrical,  first  find  the  breakg  load  of  a  square  beam, 
of  which  each  side  is  equal  to  the  diam  of  the  cylinder,  Mult  this  load  by  the  dec  .6, 
or  more  correctly,  .589.  Hence  a  square  one  is  1.7  times  as  strong  as  a  cylindrical 
one.  See  table,  p  207. 


STRENGTH   OP    MATERIALS. 


187 


If  oval,  or  elliptic,  first  find  the  load  for  a  rectangular  beam,  whose  sides 
are  respectively  equal  to  the  two  diams,  and  mult  it  by  .6. 

If  of  wood,  and  triangular,  and  its  base  (whether  up  or  down)  hor, 
first  find  the  breakg  load  for  a  rectangular  beam,  whose  breadth  is  equal  to  the 
base ;  and  its  depth  equal  to  the  perp  height  of  the  triangle ;  and  take  one-third 
of  the  result  as  an  approximation.  When  the  edge  is  down,  tbe  ends  must  rest  in 
triangular  notches  in  the  supports;  otherwise,  they  will  be  crushed  when  loaded. 

For  beams  of  such  sections  as  A  to  G,  the  following  rude  rules  of 
thumb  will  often  be  preferred  to  more  intricate  ones,  being  sufficiently  approxi- 
mate for  ordinary  purposes,  and  for  any  material.  See  near  end  of  Art  24,  p  193. 

For  tbe  closed  Figs  A,  B,  JD,  O,  (each  one  supposed  to  be  of  equal 
thickness  throughout,)  first  find  the  load  for  a  solid  beam  of  the  same  size  and  shape. 


Then  find  that  of  a  beam  of  the  size  of  the  hollow  part.  Subtract  the  last  from  the 
first.  Take  %  of  the  remainder. 

For  C,  (its  top  and  bottom  being  of  equal  size,)  first  find  for  a  rectangular  beam 
a  a  a  a.  Then  for  two  beams  corresponding  to  the  two  hollows  v  v.  Subtract  these 
last  from  the  first.  Take  %  of  the  remainder. 

For  E  or  F,  find  for  three  separate  beams  r  r,  i  i,  t  n,  and  add  them  together.  Take 
%.  T,  U,  and  many  other  forms  of  beams  in  common  use,  do  not  admit  of  any 
such  simple  approximate  rules.  The  writer  knows  of  no  alternative  in  such  case* 
except  to  experiment  with  a  model  made  of  the  given  material,  and  thus  find  the 
necessary  constant,  as  directed  in  Art  9.  See  Remark,  Art  29;  also  Art  24,  &c. 

For  I  beams,  see  Arts  37,  38;  and  for  Hodgkiiison  beams,  see  Art 


SIDE 


TL. 

Fi65 


35.p  208. 

Rein.  2.    In  this  case  we  may  remove  ^  part  of  the 
material  of  the  solid  square,  or  rectangular  beam,  without 

diminishing  its  breaking  strength,  although  it  will  bend  rj^ BL J 

more.  The  width  may  remain  uniform,  and  the  depth  be  re-    -; 

duced  either  at  top  or  bottom*  as  shown  by  the  dotted  lines     |p— — •       " 11 

at  m,  Fig  5,  strictly  two  parabolas  with  bases  at  load,  but  the     &l- '  ur -•• 

straight  ones  are  best  in  practice.     Or  the  depth  may  remain      • 
uniform,  and  the  breadth  be  reduced,  as  shown  by  the  dots  at 
n,  which  is  a  top  view  of  the  beam.    Theoretically,  the  dotted  _ 

lines  in  n  might  meet  at  the  ends  of  the  beam ;  but  in  prac- 
tice this  would  not  generally  leave  sufficient  material  at  the  ends  for  the  beam  to 
rest  upon  securely. 

Such  reductions  of  beams  are  rarely  made  when  they  are 
of  wood ;  but  in  iron  ones  much  expense  may  be  saved 
thereby. 

Rem.  3.  Load  at  one  end  of  the  beam, 
Fig  6.  the  other  end  fixed,  imagine  the  load  to 
be  at  the  center,  and  calculate  it  by  the  foregoing  rule. 
Then  div  the  result  by  4. 

In  this  case  the  lower  side  of  the  beam  may  be  cut  away 
in  the  form  of  a  parabola,  as  shown  by  the  dots.  To  draw 
this  curve,  see  "  Parabola."  Or  the  depth  may  be  left  uni- 
form, and  the  sides  be  cut  away,  as  shown  by  the  dots  at  £,  which  is  a  top  view. 

Art.  13.    When  the  load  is  equally  distributed  along?  the 
entire  clear  length  of  a  horizontal  beam.  lMI^.!^l^i%IMi^l 

supported  at  both  ends,  as  in  Fig  7,  instead  of  be-         [  i^i^i^u^i^  so__ 

ing  all  applied  at  the  center,  assume  it  to  be  at  the  center,    ..,,..  K' Vj^ 

and  proceed  precisely  as  in    the   foregoing   rule,   Art   12.        "vty\      -p.  -  ri     W^ 
Then  mult  the  load  by  '2.     But  in  this  case  the  wt  of  the  Jt  10  /     If 

entire,  clear  length  of  the  beam  is  to  be  deducted  for  the  ^ 

neat  load. 

Ex.     What  will  be  the  equally  distributed  breakg  load  of  the  beam  of  sandstone 
In  the  last  example  ?     Here  the  center  breakg  load  has  already  been  found  to  be 


188 


STRENGTH   OF   MATERIALS. 


6625  Ibs ;  and  5625  X  2  =  11250  Ibs,  the  reqd  distributed  load.  From  this  snbtrao 
the  wt  of  the  entire  12  feet  clear  length  of  beam,  or  1750  Ibs  ;  and  the  rem,9500  1U 
is  the  neat  extraneous  breakg  load.  About  -J^-  part  of  this,  or  950  Ibs,  is  quite  at, 
much  as  should  be  trusted  upon  so  variable  and  treacherous  a  material  as  red  sand, 
stone. 

REM.  1.  A  beam  requires  twice  as  much  breaking  load,  equally  distributed,  as  it 
will  at  its  center.  In  this  case  the  breakg  strength  of  the 
beam  will  not  be  diminished  if  the  top  be  cut  away  in  the  form 
of  a  true  semi-ellipse,  as  shown  by  the  dots  in  Fig  7.  Or  if  the 
depth  must  be  kept  uniform,  the  sides  may  be  trimmed  to  two 
parabolas  oc  o,  o  to,  Fig  8.  The  mode  of  drawing  these  figs 
to  any  span  and  height  will  be  found  under  their  respective 
heads;  but  in  practice  circular  segments  will  answer. 
Hem.  2.  Load  uniformly  distributed  ALONG  THE 

ENTIRE   CLEAR  LENGTH,  y  g,  Fig  9,  OF  A  HOR   RECTANGULAR  BEAM, 

FIRMLY  FIXED  AT  ONE  END  ONLY,  assume  it  to  be  at  the  center,  as 
in  Art  12,  and  calculate  it  by  the  rule  in  that  Art.  Then  div 
the  result  by  2. 

In  this  case,  theoretically,  we  may  cut  off  one-half  the  pro- 
jecting part  y  o  of  the  beam,  as  by  the  dotted  line  y  o,  without 
diminishing  its  breakg  strength.  But  in  practice  it  wTill  rarely 
be  advisable  to  reduce  it  to  a  mere  thin  edge  at  o.  Or  the  depth 
c  s  of  the  beam  may  be  left  uniform,  and  the  sides  be  cut  away, 
as  shown  by  the  two  semi-parabolas  a  c,  a  c,  at  t,  wftich  is  the 
top  of  the  beam.  If  a  c',  ac,  be  even  made  straight,  instead  of 
parabolas,  it  is  plain  that  there  would  still  be  a  considerable 
saving  of  expense,  if  the  beam  is  of  iron. 

Art.  14.   When  the   entire   breakg 

1  load  is  applied  at  any  point  o,  Fig  1O, 

JFt||  not  at  tBie  center;  first  find  by  Art  12,  what 

•Silt  would  be  the  center  breakg  load ;  and  deduct  half 
the  wt  of  the  beam.     Then  mult  together  the  two 
dists  o  a  and  o  g,  in  feet.    Call  the  prod  a.    Also 
square  half  the  clear  length,  c  g  or  c  a.    Call  this 
square  b.    Then  mult  the  center  load  by  6,  and  div 
the  prod  by  a;  the  quot  will  be  the  reqd  load. 
Such  rules  do  not  hold  good  if  the  load  rests  upon  the  beam  for  a  short 
distance  on  either  or  both  sides  of  the  point  o,  but  only  when  it  all  rests  at  that 
very  point  alone ;  if  it  does  not  the  load  may  be  increased. 

REM.  1.  The  beam  will  bear  less  at  its  center  than  at  any  other  point;  and  the 
breakg  load  at  the  center,  is  to  that  at  any  other  point,  inversely  as  the  square  of 
half  the  length  is  to  the  rectangle  of  the  two  parts  o  a  and  o  g. 

REM.  2.  This  beam  will  bear  to  be  reduced,  as  at  m  and  n,  Fig  5 ;  except,  that 
instead  of  reducing  from  its  center,  as  in  Fig  5,  we  must  do  so  from  where  the  load 
is  applied. 

Art.  15.   When  the  beam,  instead  of 
being  hor,  is  inclined,  as  in  Fig  11,  in 

any  of  the  foregoing  cases,  the  hor  dist  o  y  must 
be  taken  as  its  span,  instead  of  the  actual  clear 
length  o  c  ;  and  s  o,  s  y  instead  of  a  o  and  a  c.  This 
applies  also  to  beams  fixed  at  one  end,  and  whether 
the  inclination  is  upward  or  downward  from  the 
fixed  end. 

NOTE.   The  quantity  of  material  In  inclined  beams  may  be 
reduced,  in  the  same  manner  as  in  hor  ones. 

Art.  16.  Triangular  beams  of  wood,  according  to  Barlow's  experi- 
ments with  pine,  require  about  %  greater  breakg  loads  with  the  base  up,  than  when 
it  is  down.  Or  with  the  base  down,  about  ^  less  than  when  up.  Tredgold  considers 
them  about  equally  strong  in  either  position ;  and  that  to  find  the  center  breakg 
load,  we  may  first  calculate  it  by  Art  12,  as  if  the  beam  were  a  rectangular  one  witk 
the  same  base  and  perp  height  as  the  triangle  ;  and  take  ^of  the  result.  Hence, 
the  triangle  is  not  an  economical  shape  for  a  beam;  for  with  only  3^3  the  strength  of 
a  rectangular  one,  it  has  half  as  much  material. 

Hodgkinson,  with  cast-iron    triangular   beams,  base  up, 


Fig  10 


STRENGTH   OF   MATERIALS.  189 

made  the  breakg  loads  equal  to  %  of  those  of  rectangular  bars,  aa  in  wood.  Rennte's 
experiments  give  about  the  same  proportion,  with  the  base  up  ;  but  with  the  base 
down,  he  made  the  strength  nearly  twice  as  great,  or  about  -fa  that  of  a  rectangular 
beam  of  the  same  width  and  vertical  height.  The  comparative  strengths  in  the  two 
positions  will  vary  in  diff  materials,  inasmuch  as  it  is  affected  by  the  comparative 
resistances  which  any  given  material  presents  to  tension  and  compression.  Within 
the  limit  of  elasticity  the  beam  will  be  equally  strong,  whether  the  edge  or  base  be 
up;  and  will  bend  equally  in  either  case;  so  also  with  the  Hodgkinson,  or  any  other 
form  of  beam. 

Art.  17.  To  find  the  side  of  a  square  nor  beam  supported  at 
both  ends,  and  reqd  to  break  under  a  given  quiescent  center 
load. 

RULE.  Mult  the  clear  bearing  in  ft,  by  the  given  breakg  load  in  pounds.  Div  the 
prod  by  the  corresponding  constant  p  185.  Take  the  cube  root  of  the  quot.  This 
cube  root  will  be  the  reqd  depth  or  breadth  of  the  beam,  approximately,  in  ins. 
When  the  size  of  the  beam  is  so  great  that  its  wt  must  be  taken  into  consideration, 
increase  either  its  breadth,  as  directed,  in  Remark,  Art  20;  or  its  depth,  as  per  Art  21. 

The  breakg,  or  the  sate  load  of  a  square  beam,  if  mult  by  .6,  will  give  that  of  a 
Cylinder,  whose  diam  is  equal  to  a  side  of  the  square  one. 

Art.  18.  When  the  beam  is  reqd  to  bear  its  center  load 
Safely,  mult  the  given  safe  load  by  the  number  of  times  it  is  exceeded  by  the 
breakg  load.  Then  find,  by  Art  17,  the  side  of  a  square  beam  to  break  under  this  in- 
creased load.  The  beam  thus  found  will  evidently  be  approximately  the  safe  one  for 
the  actual  load ;  exclusive,  however,  of  the  wt  of  the  beam.  When  this  must  be  in- 
cluded, increase  the  breadth,  by  Remark,  Art  20. 

If  the  load  is  equally  distributed,  first  div  it  by  2,  then  proceed  precisely  as  before. 

Art.  19.  When  the  beam  is  cylindrical,  and  reqd  to  break 
under  its  center  load,  to  find  its  diam,  mult  the  load  by  1.7,  and  by  Art  17 
find  the  side  of  a  square  beam,  to  break  under  this  increased  load.  The  side  thus 
found  will  also  be  approximately  the  reqd  diam. 

If  to  be  borne  safely,  first  mult  it  by  the  number  of  times  it  is  to  be 
exceeded  by  the  breakg  load.  Then  mult  the  prod  by  1.7,  and  proceed  precisely 
as  before. 

REM.  1.  In  neither  case,  however,  is  the  wt  of  the  beam  itself  included.  When 
this  is  necessary,  first  find  the  approximate  diam  as  before.  Then  calculate  the  wt 
of  a  beam  having  this  diam.  Add  this  wt  to  the  given  center  load,  in  either  case ; 
and  with  this  increased  center  load,  repeat  the  whole  calculation.  The  resulting 
diam  will  be  the  required  one  very  approximately,  but  stili  a  mere  trifle  too  small. 

REM.  2.  If  the  load  is  equally  distributed,  first  take  one-half  of 
it  as  being  a  center  load,  and  with  this  proceed  precisely  as  before. 

Art.  2O.  To  find  the  breadth  of  a  hor  rectangular  beam, 
supported  at  both  ends,  to  break  under  a  g^iveii  quiescent 
center  load;  mult  the  center  load  in  ft>s  by  the  span  in  fret.  Mult  the  square 
of  the  depth  in  ins  by  the  constant  p  185.  Div  the  first  prod  by  the  last.  The 
quot  will  be  the  breadth  approximately.  Calculate  the  wt  of  a  beam  having  this 
breadth.  Then  say,  as  the  center  load  is  to  half  this  wt,  so  is  the  breadth  found,  to 
a  new  breadth  to  be  added  to  it.  It  will  still  be  somewhat  too  small,  owing  to  the 
neglect  of  the  wt  of  the  breadth  last  added.  This  may  readily  be  found,  and  its 
corresponding  breadth  added. 

REM.  1.  If  the  load  is  to  be  borne  safely,  (without  any  regard  to  the 
amount  of  deflection,)  first  mult  it  by  the  number  of  times  it  is  exceeded  by  the 
breakg  load. 

REM.  2.  If  in  either  case  equally  distributed,  take  half  of  it  as  if  a 
center  load,  and  proceed  precisely  as  before. 

Art.  21.  To  find  the  depth,  when  the  breadth  is  given,  mult 
the  load  in  ft>s  by  the  span  in  feet.  Mult  the  breadth  in  ins  by  the  constant  p  185. 
Div  the  first  prod  by  the  last;  take  the  sq  rt  of  the  quot  for  an  approximate 
depth.  Calculate  the  wt  of  a  beam  having  the  depth  just  found  ;  add  half  of  it  to  the 
given  center  load,  and  with  this  new  load  repeat  the  whole  calculation;  for  a  more 
approximate  depth,  but  still  somewhat  too  small,  owing  to  the  neglect  of  the  wt  of 
the  depth  last  added.  We  may  find  this,  and  repeat  the  whole  calculation,  or  wo 
may  merely  increase  the  breadth  by  Art  20. 

REM.  If  the  load  is  to  be  borne  safely,  or  if  It  is  equally  distributed,  see  Remarks, 
Art  20. 

Art.  22.  To  find  the  safe  dimensions  to  be  griven  to  a  rec- 
*uii£ular  beam  of  given  span,  supported  at  both  ends,  and 


190 


STRENGTH   OF   MATERIALS. 


which  BM  at  the  same  time  exposed  both  to  a  transverse 
strain  and  to  a  longitudinal  tensile  or  pulling  one,  or  a 
longitudinal  compressive  one.  The  writer  is  unable  to  suggest  any 
better  rules  than  the  following,  which  are  at  least  safe.  Namely,  when  the  longi- 
tudinal strain  is  tensile,  find  separately  the  safe  dimensions  as  if  for  a  beam  alone; 
and  as  if  for  a  tie  alone;  and  add  the  two  resulting  areas  together.  When  the  longi- 
tudinal strain  is  compressive,  find  separately  the  sale  dimensions  as  if  for  abeam 
alone ;  and  as  if  for  a  pillar  alone ;  and  add  the  two  resulting  areas  together. 

Example  1.  A  wrought  iron  rectangular  beam  of  10  ft  span  is  to  sustain 
with  a  safety  of  6,  an  equally  distx-ibuted  transverse  load  of  100000  Ibs ;  and  a  pulling 
strain  of  200000  Ibs.  Of  what  size  must  it  be? 

Here  the  distributed  load  of  100000  Ibs  is  equal  to  a  safe  center  one  of  50000  Ibs  ; 
or  to  a  breaking  center  one  of  50000  X  6  =  300000  Ibs. 

Now  first  we  may  assume  for  the  beam  some  probable  approx  depth,  say  12  ins. 
Then  we  find  by  Art  21  that  its  breadth  as  a  beam  alone  will  be 

Breakg  load  in  Ibs  X  span  in  ft  _  300000X10   _  3000000  _          . 
sq  of~defth  in  ius~X  coef,  p  185  ~~  ~144  X  ^500    ~     36000     = 

Again,  a  bar  to  bear  a  pull  of  200000  Ibs  with  a  safety  of  6,  should  not  break  with 
less  than  1200000  Ibs;  therefore  since  fair  bar  iron  breaks  with  about  50000  Ibs  per 
sq  inch,  we  have  1200000  -=-  50000  =  24  sq  ins  as  the  area  of  bar  for  the  pull  alone. 
We  may  add  all  of  this  to  the  width  of  the  beam,  making  it  24  -;-  12  =  2  ins  wider ; 
or  10.33  ins  wide  in  all.  Or  we  may  add  it  all  to  the  depth,  thus  making  the  beam 
24  -f-  8.33  =  2.88  ins  deeper,  or  14.88  ins  in  all.  Or  part  may  be  added  to  the  breadth, 
and  part  to  the  depth. 

Example  2.  A  wrought  iron  rectangular  beam  of  10  ft  span,  is  to  sustain 
with  a  safety  of  6,  an  equally  distributed  transverse  load  of  100000  Ibs,  and  a  com-" 
pressive  strain  of  200000  Ibs.  Of  what  size  must  it  be? 

Here  first  assuming  some  probable  approx  depth,  say  12  ins,  we  find  as  before  that 
its  breadth  as  a  beam  only  will  be  8.33  ins.  As  to  the  compressive  force,  it  is  plain 
that  a  pillar  for  sustaining  it  should  be  a  hollow  one  with  its  sides  as  wide  as  possi- 
ble; and  this  is  to  be  effected  by  placing  it  around  the  outside  of  our  beam.  The 
pillar  will  therefore  have  sides  of  about  8.33  and  12  ins  wide ;  and  its  breaking  load 
must  be  200000X6  =  1200000  Ibs,  or  say  536  tons.  Now  the  length  of  the  pillar 
measured  by  its  narrowest  side  is  120-7-8.33  =  14.4  sides;  and  by  table  8,  p  232,  we 
find  that  a  hollow  square  wrought  iron  pillar  14.4  sides  long,  breaks  with  15.5  tons 
per  sq  inch  of  its  metal  area.  Hence  we  require  536-7-15.5  =  34.6  sq  ins  metal  area 
for  our  pillar.  Now  the  circumf  of  the  pillar  is  8.33X2  +  12X2  =  40.66  ins. 
Hence  its  thickness  must  be  34.6  -4-  40.66  =  .85  of  an  inch.  Hence  both  the  breadth 
and  the  depth  of  the  beam  must  each  be  increased  twice  that  much,or  1.7  inch ;  thus 
making  it  10.03  ins  broad  and  13.7  ins  deep. 

It  is  plain  that  our  pillar  is  thicker  than  necessary,  because 
in  table  8  the  widths  are  supposed  to  be  by  outside  measure,  whereas  our  width  of  8.33 
ins  is  inside  measure.  The  final  outer  width  of  10.03  ins  would  make  the  pillar  only 
12  sides  long;  at  which  it  would  require  15.7  instead  of  15.5  tons  per  sq  inch  to 
break  it.  Other  considerations  too  abstruse  to  be  explained  here,  combine  to  make 
the  resulting  dimensions  la  both  examples  somewhat  in  excess. 


STRENGTH   OF   MATERIALS. 


191 


Art.  23.  Table  of  safe  quiescent  loads  for  horizontal  rec- 
tangular beams  of  white  pine  or  spruce,  one  inch  broad, 
supported  at  both  ends,  and  loaded  at  the  center;  together 
with  their  deflections  under  said  loads. 

Loads  applied  suddenly  will  double  the  deflections  in  the  table ;  as 
when,  for  instance,  if  a  load  is  held  by  hand,  just  touching  a  beam,  the  hold  should  be 
suddenly  loosed. 

Caution.  Inasmuch  as  this  table  was  based  upon  well  seasoned,  straight 
grained  pieces,  free  from  knots,  and  other  defects,  we  must  not  in  practice  take 
more  than  about  two-thirds  of  the  loads  in  the  table  for  a  safety  of  6  in  ordinary 
building  timber  of  fair  quality ;  and  with  these  reduced  loads  should  not  reduce 
the  deflections. 

Observe  also  that  our  table  is  for  safe  center  loads,  but  it  is  plain  that 
in  practice  we  cannot  always  apply  the  term  in  its  utmost  strictness  ;  otherwise 
the  load  would  have  to  be  sustained  by  a  mere  knife-edge,  at  the  very  center  of 
the  beam.  Now,  in  the  instance  Rem.  p.  192,  if  we  attempted  to  sustain  the  center 
load  of  6075  ft>s  upon  such  a  knife-edge,  it  would  at  once  cut  the  beam  in  two.  If 
we  even  applied  it  along  3  or  4  ins  of  the  length,  it  would  cut  into  it,  and  we  should 
not  have  a  safety  of  6  against  crushing  the  top  of  the  beam  until  as  in  the  case  of 
the  ends  we  distributed  the  load  along  full  46  ins  of  length,  or  about  32  ins  for  a 
safety  of  4. 

The  safe  load  is  here  %  of  the  breakgone;  and  the  last  at  450  Ibsatthe 
center  of  a  beam  1  inch  square,  and  1  foot  clear  length  between  its  supports.  For 
mere  temporary  purposes,  ^  part  may  be  added  to  the  loads  in  the  table,  thus  mak- 
ing them  equal  to  the  \^  of  the  breakg  load.  But  in  important  structures,  subject 
to  vibration,  one  ^£  Part;  should  be  deducted  from  the  tabular  loads,  thus  reducing 
them  to  %  of  the  breaking  load.  This  is  especially  necessary  if  the  timber  is  not 
well  seasoued. 


With  the  safe  loads  in  this  table  a  beam  may  bend  too 
much  for  many  practical  purposes.  When  this  is  the  case,  we  may,  by  reducing 
the  loads,  reduce  the  deflections  in  nearly  the  same  proportion ;  or  see  table,  p  204. 

For  the  neat  loads,  deduct  ^  the  wt  of  the  beam  itself.  The  deflections 
however  are  the  actual  ones ;  the  wts  of  the  beams  having  been  introduced  in  calcu- 
lating them,  by  the  rule  in  art  27,  p  199. 

All  the  loads  in  the  Table  are  superabundantly  safe  against  shearing*. 
Against  crushing:  at  the  ends,  &c,  see  "Cautions  "  below  the  Table, 
p  192.  Original. 


Depth 
of 

Span  4  ft. 

Span  6  ft 

Span  8  ft. 

Span  10  ft 

Span  12  ft 

Span  14  ft 

Span  16  tt 

Wt.  of 
10  ft  of 

beam. 

load 

def. 

!oad|  def. 

load 

def. 

load 

def. 

load 

def 

load 

def. 

load 

def. 

beam. 

Ins. 

fts. 

ins. 

ft>s. 

ins. 

ft)S. 

ins. 

ft>8. 

ins. 

fl>8. 

ins. 

Bt>8. 

ins. 

ft>8. 

ins. 

K)8. 

1 

19 

.39 

13 

92 

10 

1  8 

g 

3  0 

6 

4.4 

2 

2 

75 

.22 

50 

.45 

38 

!82 

30 

1.3 

25 

1.9 

21 

2.7 

19 

3.7 

4 

3 

170 

.13 

114 

.30 

85 

.53 

67 

.84 

57 

1.3 

48 

1.7 

42 

2.3 

6 

4 

300 

.10 

200 

.22 

150 

.39 

120 

.63 

100 

.92 

86 

1.3 

75 

1.7 

8 

5 

469 

.08 

312 

.18 

234 

.31 

187 

.50 

156 

.72 

134 

1.0 

117 

1.3 

10 

6 

675 

.06 

450 

.15 

337 

.26 

270 

.41 

225 

.60 

193 

.83 

168 

1.1 

12 

7 

919 

.06 

612 

.12 

460 

.22 

367 

.35 

306 

.51 

262 

.70 

230 

.93 

14 

8 

1200 

.05 

800 

.11 

600 

.19 

480 

.31 

400 

.45 

343 

.61 

300 

.81 

16 

9 

1520 

.04 

1014 

.10 

760 

.17 

607 

.27 

507 

.40 

434 

.54 

3801  .72 

18 

10 

1875 

.04 

1-250 

.09 

937 

.16 

750 

.24 

625 

.35 

536 

.49 

4681  .64 

20 

11 

2270  1  .04 

1514 

.08 

1135 

.14 

907 

.22 

757 

.32 

648 

.44 

567 

.58 

22 

12 

2700  1  .03 

1800 

.07 

1350 

.13 

1080 

.20 

900 

.29 

772 

.40 

675 

.53 

24 

14 

3675 

.03 

2450 

.06 

837 

.11 

1470 

.17 

1225 

.25 

1050 

.34 

918 

.45 

28 

16 

4800 

.02 

5200 

.05 

2400 

.10 

1920 

.15 

1600 

.2-2 

1372 

.30 

1200 

.40 

32 

18 

6075 

.02 

4050 

.05 

3037 

.09 

2430 

.14 

2025 

.20 

1736 

.27 

1518 

.35 

36 

20 

7500 

.02 

DOOO 

.04 

3750 

.08 

3000 

.12 

2500 

.18 

2145 

.24 

1875 

.31 

40 

22 

9075 

.02 

6050 

.04 

4537 

.07 

3630 

.11 

3025 

.16 

2593 

.22 

2268 

.29 

44 

24 

10800 

.02 

200 

.04 

5400 

.06 

4320 

.10 

3600 

.15 

3088 

.20 

2700|  .26 

48 

(Continued  on  next  page.) 

192 


STRENGTH   OF   MATERIALS. 


Table,  continued. 


(Original.) 


Depth 
of 

Span  18  ft 

Span  20  ft. 

Span  25  ft. 

Span  30  ft. 

Span  35  ft. 

Span  40  ft. 

Wt.  of 

beam 

load 

def 

load 

def. 

load 

def. 

load 

def. 

load 

def. 

load 

def. 

10  ft  of 
beam. 

Ins. 

Ibs. 

ins 

fts. 

ins. 

fts. 

ins. 

fts. 

ins. 

fts. 

ins. 

Ibs. 

ins. 

fts. 

6 

150 

1.4 

135 

1.8 

108 

2.9 

90 

4.5 

77 

6.5 

67 

9.2 

12 

7 

204 

1.2 

184 

1.5 

147 

2.5 

122 

3.9 

105 

5.8 

92 

7.6 

14 

8 

267 

1.0 

240 

1.3 

192 

2.1 

160 

3.2 

137 

4.6 

120 

6.4 

16 

9 

338 

.92 

304 

1.2 

243 

1.9 

202 

2.8 

174 

4.0 

152 

5.5 

18 

10 

417 

.82 

375 

1.0 

300 

1.7 

250 

2.5 

214 

3.5 

188 

4.9 

20 

11 

605 

.74 

454 

.93 

363 

1.5 

302 

2.2 

259 

3.2* 

227 

4.3 

22 

12 

600 

.68 

540 

.85 

432 

1.4 

360 

2.0 

308 

2.9 

270 

3.9 

24 

14 

817 

.58 

735 

.72 

588 

1.2 

490 

1.7 

420 

2.4 

367 

3.2 

28 

16 

1067 

.50 

960 

.63 

768 

1.0 

640 

1.5 

548 

2.1 

480 

2.8 

32 

18 

1350 

.45 

1215 

.56 

972 

.90 

810 

1.3 

694 

1.8 

607 

2.5 

36 

20 

1666 

.40 

1500 

.50 

1200 

.79 

1000 

1.2 

857 

1.6 

750 

2.2 

40 

22 

2017 

.37 

1815 

.45 

1452 

'.72 

1210 

1.1 

1037 

1.5 

907 

fO 

44 

24 

2400 

.33 

2160 

.41 

1728 

.65 

1440 

.96 

1234 

1.3 

1080 

1.8 

48 

26 

2817 

.31 

2526 

.38 

2018 

.60 

1684 

.88 

1449 

1.2 

1263 

1.6 

52 

28 

3267 

.28 

2940 

.35 

2352 

.55 

1960 

.81 

1680 

1.1 

1470 

1.5 

56 

30 

3750 

.26 

3375 

.33 

2700 

.50 

2250 

.76 

1928 

1.1 

1687 

1.4 

60 

32 

4267 

.25 

3840 

.30 

3072 

.45 

2560 

.71 

2194 

1.0 

1920 

1.3 

64 

34 

4817 

.23 

4335 

.29 

3468 

.44 

2890 

.67 

2477 

.92 

2167 

1.2 

68 

36 

5400 

.22 

4860 

.27 

3888 

.43 

3240 

.63 

2777 

.86 

2430 

1.1 

72 

White  oak,  and  best  Southern  pitch  pine  will  bear  loads  % 
greater. 

For  cast  iron,  mult  the  loads  in  the  table  by  4.5;  and  for  wrought  by 

5.3.    For  these  new  loads,  mult  the  dels  by  .4  for  cast ;  and  by  .3  for  wrought. 
If  the  load  is  equally  distributed  over  the  span,  it  may  be  twice  as 

freat  as  the  center  one,  and  the  defs  will  be  1^£  times  those  in  the  table.  If  the 
oads  in  the  table  be  equally  distributed  along  the  whole  beam,  the  defs  will 
be  but  five-eighths  as  great  as  those  in  the  table.  See  Art  2(i,  p  196.  When  more 
accuracy  is  reqd,  half  the  wt  of  the  beam  itself  must  be  deducted  from  the  center 
load;  and  the  whole  of  it  from  an  equally  distributed  load.  The  wt  of  the  beam,  in 
the  last  column,  supposes  the  wood  to  be  but  moderately  seasoned,  and  therefore  to 
weigh  28.8  Ibs  per  cub  ft. 

Uses  of  the  foregoing;  table.  Ex.  1.  What  must  be  the  breadth 
of  a  hor  rect  beam  of  wh  pine,  18  ins  deep,  supported  at  both  ends,  and  of  20  ft  clear 
length  between  its  supports,  to  bear  safely  a  load  of  5  tons,  or  11200  Tbs  at  its  center? 
Here,  opposite  the  depth  of  18  ins  in  the  table,  and  in  the  column  of  20  feet  lengths, 
we  find  that  a  beam  1  inch  thick  will  bear  1215  Ibs ;  consequently,  1120°  =  9.22  ins, 

the  reqd  breadth ;  for  the  strength  is  in  the  same  proportion  as  the  breadth. 

Ex.  2.  What  will  be  the  safe  toad  at  the  center  of  a  joist  of  white  pine,  18  ft  long, 
3  ins  broad,  and  12  ins  deep?  Here,  in  the  col  for  18  ft,  and  opposite  12  ins  in  depth, 
we  find  the  safe  load  for  a  breadth  of  1  inch  to  be  600  Ibs ;  consequently,  600  X  3  — 
1800  fts,  the  load  reqd. 

REM.  Cautions  in  the  u#e  of  the  above  table.  For  instance,  in 
placing  very  heavy  loads  upon  short,  but  deep  and  strong  beams,  we  must  take  care 
that  the  beams  rest  for  a  sufficient  dist  on  their  supports  to  prevent  all  danger  from 
crushing  at  the  ends.  Thus,  if  we  place  a  load  of  b075  fts  at  the  center  of  a  beam 
of  4  feet  span,  18  ins  deep,  and  only  1  inch  thick,  each  end  of  the  beam  sustains  a 

yert  crushing  force  of  ^^  =  3037  Ibs,  and  that  sidewise  of  the  grain,  in 

which  position  average  white  pine,  spruce,  and  hemlock  crush  under  about  800 
fts  per  sq  inch,  and  do  not  have  a  safety  of  6  until  the  pressure  is  reduced  to  about 
133  fts  per  sq  inch.  Therefore  our  beam,  in  order  to  have  a  safety  of  6  against 
crushing  at  its  ends, must  rest  on  each  support  3037  -±-  133  =  23  sq  ins:  or  for  a 
safety  of  4  nearly  16  sq  ins.  When  a  pressure  is  equally  distributed  side- 
wise  (that  is,  at  right  angles  to  the  general  direction  of  the  fibres)  over  the  entire 
pressed  surface  of  a  block  or  beam  (to  ensure  which,  the  opposite  surface  must  be 
supported  throughout  its  entire  length)  the  resulting  compression  might  readily 
escape  detection  unless  actually  measured.  But  when  a  considerable  pressure  is 
applied  to  only  a  portion  of  the  surface,  as  of  caps  and  sills  where  in  contact  with 
the  heads  and  feet  of  posts,  or  at  the  ends  of  loaded  joists  or  girders,  th3  com- 
pression becomes  evident  to  the  eye,  because  the  pressed  parts  sink  below  the 
impressed  ones,  in  conseqr.enceof  the  bending  or  breaking  of  the  adjacent  fibres 
What  in  the  first  case  (especially  if  slight)  would  be  called  compression,  would 


STRENGTH   OF   MATERIALS. 


193 


in  the  second  be  called  crushing1;  even  when  neither  might  be  so  great  as 
to  be  unsafe. 

Owing  to  the  resistance  which  said  adjacent  fibres  oppose  to  being  bent  or 
broken,  it  is  plain  that  a  given  pressure  per  sq  inch,  or  per  sq  foot,  &c., 
will  cause  somewhat  less  compression  or  crushing  when  applied  to  only  a  part  of 
a  surface,  than  when  to  the  whole  of  it. 

The  writer  has  seen  40  half  seasoned  hemlock  posts,  each  12  ins  square, 
footing  at  intervals  of  5  ft  from  center  to  center,  upon  similar  12  X  12  inch  hem- 
lock sills,  to  which  they  were  tenoned,  and  which  rested  throughout  their  entire 
length  on  stone  steps.  Each  post  was  gradually  loaded  with  32  tons,  or  equal  to 
say  500  Ibs  per  sq  inch ;  and  their  feet  all  crushed  into  the  sills  from  J£  to  }%  inch. 
Their  heads  crushed  into  the  caps  to  the  same  extent.  In  practice  the  pres- 
sure at  the  heads  and  feet  of  posts  is  rarely,  if  ever,  perfectly  equable ;  and  the 
same  remark  applies  to  the  ends  of  loaded  joists,  girders,  &c.,  in  which  a  slight 
bending  will  throw  an  excess  of  pressure  upon  the  inner  edges  of  their  supports. 

See  other  cautions  on  p.  191.    See  also  footnote,  p.  174. 

Art.    24.    Strength    of    hollow 
Beams. 

During  the  preliminary  investigations 
relative  to  the  construction  of  the  Menai 
tubular  bridge,  a  few  experiments  were 
made  on  the  strength  of  hollow  cast-iron 
beams  of  circular,  oval,  square,  and  rectan- 
gular cross-sections,  supported  at  both 
ends,  and  loaded  at  the  center.  The  clear 
span  between  the  supports  was  in  every 
fcase  6  ft ,  the  thickness  of  metal  in  each 
beam,  %  inch ;  area  of  solid  cross-section 


assistant  engineer  in  charge,  deduced  the  following  constants,  and  rules  for  center 
breakg  loads : 

Const  for  circ  tubes,  .95  ;  oval,  1;  square,  1.14;  rectangle,  .91.    Then,  first 
finding  the  area  of  the  solid  part  of  the  cross-section  in  sq  ins, 

Area  of  solid  .,  Mean  depth,  o  o,  ^,  Corresponding 

Center  breaking:  in  sq  ins      x  in  ins  constant. 

load  in  tons  Clear  span  in  feet. 

Ex.  Circular  beam,  mean  depth  o  o,  3%  ins ;  area  of  solid  ring,  4.12  sq  ins ;  clear 
span,  6  ft.    Here, 


Area.   Mean  depth.  Const. 
4.12     X      3.5      X       -95     : 


:  2.28  tons,  or  5107  Ibs,  breakg  load. 


6  (length.) 

The  thickness  of  the  cylinder  or  tube  is  about  ^  of  the  diam ;  and  as  a  mean 
of  3  trials,  it  broke  with  a  center  load  of  2.287  tons,  or  5122  Ibs ;  span  6  ft.  Hence 
we  derive  for  similar  tubes,  the  constant  475,  to  be  used  in  the  rule,  Art  12;  that  is, 
center  breakg  load  in  Ibs  of  cast-iron  tubes  with  a  thickness  of  1  of  the  outer 

diam  =  ~Qtear  gpan  in  feet     5  supposing  Mr.  Clark's  iron  to  have  been  of  average 
quality. 

The  average  breakg  load  of  3  square  beams  was  2.152  tons,  or  4820  Ibs;  of  the  rec- 
tangular ones,  2.3  tons,  or  5152  Ibs  ;  and  of  the  6  elliptic  ones,  3.207  tons,  or  7183  Ibs. 
To  all  the  foregoing  extraneous  loads  must  be  added  half  the  wt  of  the  beam  itself. 
See  Art  9,  p  183. 

.Our  rule  of  thumb  on  p  187  agrees  well  with  all  Mr.  Clark's  results, 
except  for  the  oval  beams,  in  which  said  rule  rives  a  breaks:  load  of  but  &  that, 
recorded  by  him. 

Hollow  beams  of  thin  wrought  iron  were  experimented  on  at  the 
same  time ;  and  for  these  Mr.  Clark  deduced  the  following  constants,  to  be  used  with 
his  foregoing  rule  for  cast-iron  ones : 

Constants  for  thin  riveted  tubes,  circular,  1.74  ;  oval,  1.85  ;  rectangular,  1.96. 
welded  tubes,         "         1.09;      "     1.27;  1.61. 

Art.  25.  The  following  experiments  on  riveted  sheet-iron  cylindri- 
cal beams  are  by  Fairbairn.  1st.  Cylinder  18  ft  long;  1  ft  outer  diam;  clear 
span  17  ft ;  thickness  of  iron  .037,  or  ^  of  an  inch ;  wt  of  tube  107  Ibs. 

18 


194 


STRENGTH    OF    MATERIALS. 


Center  load.  Def. 

Lbs.  IQB. 

1360 .32 

1920  41 

2114 .46 

2256  60 


Center  load. 
Lbs. 


2480  . 
2592. 
2704  . 


Def. 

Ins. 
..  .60 
.  .61 
.  .61 
.  .65 


After  bearing  2704  Ibs.  for  1%  minutes,  failed  by  crushing  at  top. 
2d.  Cyl  16  ft  10  ins  long ;  12.4  ins  outer  diarn;  clear  span  15  ft  7U  ins ;  thickness  of 
iron  .113,  or  full  .J.  inch ;  wt  of  tube  392  Ibs. 


Center  load. 
Lbs. 


2000  . 
4000  . 
6000 


Def. 
Ins. 

.17 
.  .34 
.  .52 


Center  load.  Def. 

Lbs.  Ins. 

8000 84 

10000  1.06 

11440 Broke. 


With  11440  broke  by  the  tearing  of  the  bottom  across  the  shackle-hole  from  which 
the  load  was  suspended. 

3d.  Cyl  25  ft  long;  17.68  ins  outer  diam ;  clear  span  23  ft  5  ins;  thickness  .0631,  or 
full  A  inch;  weight  of  tube  346  Ibs. 


Center  load. 
Lbs. 


1000  . 
2000  . 
3000  . 
4000  . 


Def. 
Ins. 

.  .12 
.  .21 
.  .30 
.  .40 


Center  load. 
Lbs. 


5000 
5280 
5840 
6120 


Def. 
Ins. 

48 

....  .51 

60 

71 


With  6400  broke  at  bottom  ;  25  ins  from  center,  by  tearing  through  the  rivet-holes. 
4tli.  Cyl  25  it  long;  18.18  ins  outer  diam;  clear  span  23  ft  5  ins;  thickness  .119,  or 
scant  }/Q  inch  ;  wt  of  tube  777  Ibs. 

Center  load.  Def.  Center  load.  Def. 


Lbs. 
2000  . 
4000  , 
6000  . 
8000. 


Ins. 
.  .15 
,.  .30 
,.  .43 

i.  .59 


Lbs.  Ins. 

10000 82 

KOOO  95 

13000 1.04 

14240 Broke. 


Broke  through  the  rivet-holes  3  ft  3  ins  from  center,  after  sustaining  the  load  for 
half  a  min. 

The  tubes  were  composed  of  sheets  about  2^  ft  wide ;  and  so 
long  that  a  single  sheet  sufficed  to  form  the  entire  circumf  of  the 
tube.  They  were  united  by  double-riveted  lap-joints.  The  loads 
were  placed  on  a  platform,  supported  by  a  rod  r,  Fig  11,  which 
passed  through  a  hole  A  in  the  bottom  of  the  tube  s.  This  rod  was 
attached  at  its  upper  end  to  a  block  of  wood  w,  rounded  at  its 
lower  surface,  so  as  to  fit  the  tube. 

I  v       -v       Circular  blocks  of  wood  were  fitted  into  the  ends  of  the  tubes,  to 
JtlO  14  "1   k       prevent  them  from  crushing  at  those  parts  under  their  loads ;  and 
t»  the  ends  rested  upon  blocks  hollowed  out  to  correspond  with  their 

cylindrical  shape,  to  a  depth  equal  to  about  ^  part  of  their  diam. 
Art.  25%.  Breaking  load  for  a  beam  of  any  form  of  cross- 
section,  item.  Scientists  give  the  following,  which  however  often  differs  so 
much  from  experiment  (see  example,  p  196 ;*  as  to  prove  that  the  only  reliable 
mode  of  finding  the  strengths  and  deps  of  beams  is  by  experiment  with  models 
of  the  same  form  and  material ;  as  has  been  already  done  with  most  of  those  in 
common  use. 

If  the  beam  is  supported  at  both  eiids,  with  the  load  at  its  center,  half  its  wt 
must  be  deducted ;  or  all  of  it  for  an  equally  distributed  load. 

Moment  of  inertia  of  the  cross-section 

of  the  beam,  with  respect  to  a  neutral  v  Constant  for  rupture  of  the  material 
axis  G  G,  passing  hor  through  its  cen  *        of  which  the  beam  consists,  fts. 
of  grav  O,  in  ins.  _^__^__ 


Breakg 
in  Jbt 


load,  _ 


Dist  o  g  from  neutral  axis  ;  (and  at        ( 
right  angles  to  it)  to  the  fibre  g,  the  X 
farthest  one  from  the  axis,  in  ins. 


r  span  of  ^ 

i,  in  ins   x  m 


But  Prof  De  Volsoii  Wood  in  his  "  Resistance  of  Materials,"  p  150, 
says  that  instead  of  the  farthest  fibre  g  in  all  cases,  we  should  use  that  one  (either 
g  or  z)  which  is  on  the  side  that  would  yield  most  readily.  If  so  the  disagreement 
with  experiment  is  still  too  great ;  and  moreover  the  choice  of  fibre  would  often 

*  A  part  of  tlie  discrepancy  is  of  course  due  at  times  to  difference  of 

quality  of  material. 


STRENGTH   OF    MATERIALS. 


195 


be  difficult,  as  it  would  not  always  be  that  one  which  by  trial-calculation  would 
give  the  least  load.  On  the  other  hand  the  formula  cannot  always  be  correct  as 
it  stands,  for  then  a  Hodgkinson  beam,  p  208,  would  be  equally  strong  with  either 
flange  uppermost.  A  reliable  formula  of  this  kind  would  be  of  very  great  value, 
but  is  probably  an  impossibility. 

Here  in  is  either  1,  %,  ^,  %,  or  ^,  according  to  the  arrangement  of  the  beam 
and  its  load,  as  referred  to  in  Art  11,  p  186.  Thus,  if  supported  at  both  ends 
and  loaded  at  the  center,  it  will 
be  %  or  .25. 

The  Constant  for  Rup- 
ture is  18  times  the  constants 
for  center  breakg  loads  given  p 
185,  with  the  exception  (Ran- 
kine),  that  for  open-work  cast- 
iron  beams  it  is  but  9  times  the    "— 
constant  in  p  185.    Or  it  is  = 
3  ( wt  of  beam  +  2  brkg  load )  X  span 
4~X  breadth  X  sq  of  depth 

The  position  of  the 
hor  neutral  axis  G  G,  may 
be  found  by  cutting  out  a  correct 
figure,  4  or  5  ins  long,  of  the  sec- 
tion drawn  on  thick  paper  or  tin, 
and  balancing  it  over  a  straight 
edge.  The  line  at  which  it  bal- 
ances is  G  G.  When  this  has  been 
done,  the  dist  o  g,  in  ins,  to  the  farthest  fibre,  (which  may  be  either  above  or  below 
G  G,  according  to  the  shape  of  the  Fig,)  can  be  measured  in  ins. 
*> Moment  of  inertia  of  the  cross- section,  with  respect  to  the  neutral  axis  G  G, 
means  the  sum  which  results  from  adding  together  the  prods  found  by  mult  together 
the  infinitely  small  area  of  each  fibre,  as  y,  of  the  section,  by  the  square  of  its  dist 
6  y  from,  and  at  right  angles  to  G  G.  Such  multiplication  cannot  be  performed  by 
ordinary  arithmetic,  but  an  approximation,  sufficiently  close  for  all  practical  pur- 
poses, may  readily  be  made  thus :  Both  above  and  below  G  G,  and  parallel  to  it, 
draw  lines  ,;'  Ic,  I  m,  &c,  dividing  the  section  into  narrow  strips.  If  these  lines  are 
equidistant,  the  subsequent  calculations  will  in  some  cases  be  easier;  but  otherwise 
it  is  immaterial  whether  they  are  so  or  not.  If  they  .are  drawn  no  closer  together, 
proportionally  to  the  size  of  tfie  .figure,  than  in  Fig  14%,  the  approximation  will  be 
near  enough  for  practical  purposes.  The  closer  they  are  the  more  accurate  will  be 
the  result ;  but  however  close  they  may  be,  it  will  always  be  a  trifle  too  small.  Begin 
by  finding  the  area  in  sq  ins,  of  the  first  strip  x  xj  k,  below  G  G.  Mult  this  area  by 
the  square  of  the  dist  Or  to  the  cen  of  grav  of  the  strip.  Then  proceed  to  the  next 
strip./  klm;  find  its  area ;  and  mult  it  by  the  square  of  the  dist  o  s  to  its  center.  Do 
the  same  with  each  strip.  Add  all  the  prods  together,  and  if  the  section  has  the 
same  shape,  size,  and  position  above  G  G  as  below  it,  (as  would  be  the  case  with  a 
square,  rectangle,  or  circle,)  mult  their  sum  by  2.  The  prod  will  be  the  reqd  moment. 
But  if,  as  in  Fig  14%,  the  section  above  G  G  differs  from  the  portion  below  it,  we 
must  div  it  also  into  strips,  and  proceed  as  with  the  lower  part.  The  sum  of  all  the 
products  on  both  sides  of  the  neutral  axis  will  then  be  the  moment. 

Moments  of  Inertia  of  a  few  well  known  figs  are  given  below.  Those 
of  similar  figs  are  to  each  other  as  their  breadths  X  cubes  of  depths. 

Square  or  rectangle.  (Breadth  X  cube  of  depth)  H-  12,  whether  any  side 
or  diagonal  is  vertical. 

*  When  a  beam  just  breaks  under  its  load,  the  strain  in  fbs  per  sq  inch  on  the  fibers  g,  Fig  14% 
farthest  at  right  angles  from  the  neutral  axis  G  G,  (or  according  to  Prof  Wood,  the  farthest  on  the 
side  that  will  yield  first,  whether  those  at  g  or  those  at  z,  as  the  case  may  be)  is  called  the  Mod- 
ulus, Constant,  or  Coefficient  of  Rupture.  Writers  usually  denote  it  by  C ;  and  its  value  for 

hor  square  or  rectangular  beams  supported  at  both  ends  and  loaded  at  the  middle  i*  C  =  — . -  -—  ;  where 

2  u  Q 
v  is  the  load,  and  1  the  span.    In  other  words 

(center  load  ,   H»  wt  of  clear  \  v  clear  span 
in  tts       '  span  of  b«am/  *       iu  ins 
Breadth  in  ins  X  square  of  depth  in  ins. 

Which  gives  the  same  result  as  the  above  ;  and  both  of  them  show  how  to  find  C  for  any  given  ma* 
terial  by  experiment,  which  ia  the  only  possible  way.  See  Table,  p.  135. 


196 


STRENGTH   OF   MATERIALS. 


Hollow  square  or  rectangle.    (B  X  D3  —  6  X  d3)  -*•  12.  * 
Circle.    Rad<  X  -7854.    Semicircle.    Rad*  X  .1098. 
Ring.    (Outer  rad*  —  Inner  rad^j  X  .7854. 

Ellipse.     Long  diam  vert.    Half  short  diam  X  half  long  diam*  X  .7854. 
Elliptic  ring.    Long  diam  vert.    Let  L,  S,  I,  s,  be  half  the  long  and  short 
diams.    Then  (S  X  L8  —  s  X  l*>  X  .7854. 


.0.1113.      j.jit;u  ^o  A  w  —  S  A  l  )  A.  .You* 

Any  triangle.    (Base  X  Perp  Ht3)  -*-  36. 


1.  (B  XV*  -  2  6  X  &)  -*-  12.  2.  (B  X  D8  +  2  b  X  d3)  -=-  12.  3.  (6  X  d3  +  &' 
X  ?*  -  (&'  -  6)  X  d"»)  -*-  3.  4.  (6  X  d3  -  (fr-jfc)  X  (d-c?  -}-  V  X  d'*  -  (b'-k)  X  («*'- 
c  }«)  -f-  6. 

Example  of  Formula  011  page  194. 

What  is  the  center  breaking  load  of  a  solid  cast-iron  beam  4  ins  square,  and  6 
ft,  or  72  ins,  clear  span,  supported  at  both  ends? 


4*  X  4a          16  V  16 
Here  the  moment  of  inertia  is  -—  "—   =  -  ~-  —    = 


256 
— 
12 


=  21.333.    The  con- 


stant  for  rupture,  or  18  times  our  constant  for  cast  iron  on  p  185  is  2025  X  18  =» 
36450.  Since  the  beam  is  supported  at  both  ends,  and  loaded  at  the  middle,  wi,  (see 
Art  11)  is  i^.  The  dist  off  of  the  farthest  fibre  from  the  neutral  axis  must  in  a  square 
be  equal  to  %  of  one  side;  consequently  it  is  here  '2  ins.  The  clear  span  is  72  ins. 
Hence, 


Breakg  _ 


Mom  of  In  X  Con  for  Rup 
71  X     Uwf/flbre""  X  8pan 


—  2^-333  X  36450    _  777588  _ 
X  X  2  X  72 


=  21600  fts  =  9.64  tons. 


By  our  table,  p  206,  a  beam  of  average  cast  iron,  4  ins  deep,  1  inch  broad,  and  6  ft 
span,  breaks  with  2.41  tons;  consequently,  four  such,  or  one  4  ins  square,  would 
break  with  2.41  X  4  —  9.64  tons:  thus  confirming  the  accuracy  of  the  foregoing. 
Applied  in  the  same  way  to  solid  cylinders,  the  result  corresponds  equally  well 
with  experiment  and  with  our  table  on  p  207. 

But  for  Mr  Clark's  hollow  squares,  p  193,  the  formula  gives  3.06  tons  in- 
stead of  the  actual  2.15;  and  for  his  hollow  cylinders  2.980  instead  of  2.287.  See 
footnote,  p  194.  A  true  liodgkinson  beam,  p  208,  with  top  flange  of  1  by 
3  ins,  bottom  flange  1.5  by  12  ins,  vert  web  .75  inch  thick,  total  depth  15  ins,  clear 
span  20  ft,  has  a  moment  of  inertia  of  780;  dist  from  neutral  axis  to  upper  fibre 
10.7  ins,  and  to  the  lowest  one  4.3  ins.  By  Hodgkinson  it  would  yield  at  the  lower 
flange,  and  by  his  rule  with  a  center  load  of  29.24  tons.  By  the  formula  it  would 
be  19.8  tons;  and  by  Prof  Wood  49.2  tons.  Beam  I,  p  210,  actually  broke  with 
about  52  tons ;  by  the  formula  it  would  be  40,  and  by  Prof  Wood  59  tons. 

Art.  26.  Deflections,  or  beiidiiigs  of  beams,  under  their 
loads.  The  foregoing  relates  to  the  strength  of  beams,  or  their  resistance  to 
breakg ;  the  following  to  their  stiffness,  or  resistance  to  bendg.  The  two  follow  very 
diff  laws. 

It  is  with  the  defs  within  the  elastic  limit  that  the  engineer  is  chiefly  interested. 
They  then  are  directly  as  the  load  and  as  the  cube  of  the  span  ;  and  inversely  as 
the  breadth,  and  as  the  cube  of  the  depth ;  and  this,  with  the  following,  applies 
not  only  to  all  rectangular  beams,  but  to  all  others  of  whatever  cross  section, 
provided  the  sections  are  similar.  See.  p  61,  llth  line  of  Geometry. 

With  same  span,  breadth,  and  load,  the  deflections  within  elas  limits 
are  in  all  cases  inversely  as  the  cubes  of  the  depths.  Hence  the  depths  are 
inversely  as  the  cube  roots  of  the  deflections. 

With  same  span,  breadth,  and  deflection,  the  depths  are  directly 
as  the  cube  roots  of  the  loads.  Hence  the  loads  for  equal  deflections  are  as 
the  cubes  of  the  depths. 

Under  greatest  load  within  limit  of  elas,  the  defs  are  as  the  square 
of  the  span ;  and  inversely  as  the  depth  and  breadth. 

jjc  In  a  rectangle,  B  and  b  are  respectively  the  outer  and  inner  dimensions  parallel  to  the  neutral 
axis,  whether  said  axis  be  lengthwise  or  crosswise  of  the  figure. 


STRENGTH    OF    MATERIALS.  197 

If  the  deflection  of  a  beam  supported  at  both  ends  and  loaded  at  the 

center  be  called 1. 

Then  that  of  the  same  beam,  with  the  same  load  uniformly  dis- 
tributed, will  be 625,  or  % 

Firmly  fixed  at  both  ends,  and  loaded  at  the  center,  by  Moseley 2,     or   $ 

"    uniformly  loaded 125,  or  y% 

Fixed  at  one  end,  and  loaded  at  the  other 16. 

"       uniformly  loaded 6. 

The  extent  to  which  a  beam  may  bend  under  even  a  perfectly  safe  load,  may  be 
too  great  for  mainy  purposes  in  every -day  practice.  Tredgold  and  others  assume, 
that  in  order  not  to  be  observed,  or  that  it  may  not  cause  the  plaster  of  ceilings 
to  crack,  &c,  a  beam  should  not  deflect  at  its  center  more  than  the  z&gth  part  of 
its  span,  or  ^yth  of  an  inch  per  ft.  Thus,  if  its  span  be  20  ft,  it  should  not  bend 
more  than  f  gths,  or  y>  of  an  inch,  which  is  also  x40th  of  20  ft.  For  such  cases 
see  Art  29,  p  201. 

Shafts  of*  wheels  in  machinery  should  not  deflect  more  than  half  of  this,  nor  a 
bridge  more  than  about  %  of  it,  or  say  y^1^-  of  its  span,  or  y^  inch  per  loot,  under 
its  heaviest  loud. 

We  shall  allude  first  to  defs  within  the  limits  of  safety,  or  of  the  elasticity  of  the 
beam ;  and  afterward  of  those  not  exceeding  ^1^.  of  the  span.  After  the  elastic 
limit  is  passed,  the  defs  increase  irregularly,  and  more  rapidly  than  before ;  and  the 
beam  becomes  unsafe.  As  a  general  rule,  the  elasticity  of  a  wooden  beam  is  not 
injured  for  practical  purposes,  if  the  quiescent  load  does  not  rxcenl  about  %  of  the 
breaking  one.  Within  the  elastic  limirs,  the  defs  theoretically  vary  din'ctly  in  pro- 

Sortion  to  the  load,  and  also  to  the  cube  of  the  span,  or  clear  length;  and  inversely 
a  proportion  to  the  breadth,  and  to  the  cube  of  the  depth.    That  is, 

Deflection  within    >     is  in  proportion  to     (     Load  *  cube  of  8Pan 
^  elastic  Hmit3      $          (not  equal  to)         }  breadth  X  cube  of  depth. 

Therefore,  constants  for  the  bending*  of  beams  of  diff  shapes, 
within  the  limit  of  elasticity,  may,  like  those  of  transverse  strength,  (see  Art  9,) 
be  readily  found  by  experiment.  Thus,  at  the  center  of  any  rectangular  beam, 
placed  lior  upon  supports  at  e-ich  end,  place  any  load  that  is  within  its  elastic  limit, 
and  measure  the  resulting  def  in  ins.  Mult  the  wt  of  the  span  of  the  beam  by 
.625  ;  add  the  prod  to  the  neat  load,  for  a  total  load.  Mult  together  the  total  load 
in  Ibs.  and  the  cube  of  the  span  in  feet.  Also  mult  together  the  breadth  in  ins,  and 
the  cube  of  the  depth  in  ins.  Div  the  first  prod  by  the  last  one.  Div  the  def  by  the 
quot.  The  last  quot  will  be  the  reqd  constant  for  any  rectangular  beam  of  the  same 
kind  and  quality  of  material,  whether  wood,  metal,  stone,  &c.  That  is, 


The  constant  for^i  f  Total  1< 

def  within       1  Def  in  ins    1      in  ft* 

elastic,  or  safe    f  divided  by    T  R^eaSt! 

limits  of  beam.  J  l1*™^ 


Total  load    v    Cube  of  spaa 
'     feet 


Cube  of  depth 
in  ins. 

We  add  to  the  experimental  neat  load,  the  .G.'o,  or  %  of  the  wt  of  the  clear  span 
of  the  beam  itself,  because  the  wt  of  the  beam  equally  distributed  throughout  its 
span,  also  aids  in  producing  the  def;  and  it  does  so  to  the  same  extent  that  %,  of  it 
would  do,  if  collected  at  the  center  of  an  imaginary  beam  having  the  same  strength 
throughout  as  the  real  one,  but  of  no  wt  except  at  its  very  center.  Ihereibre,  in 
applying  the  constants  for  def  to  beams  intended  for  actual  use,  we  must  not  omit 
to  add  %  of  the  wt  of  the  span,  to  the  intended  center  load,  for  an  equivalent  total 
center  load,  before  making  the  calculations  for  def.  The  weights  of  similar  beams 
(that  is,  beams  proportioned  exactly  alike  in  every  part,  but  of  diff  sizes)  increase  so 
much  more  rapidly  than  their  clear  spans,  that  although  a  small  one  may  safely  bear 
a  load  of  many  times  its  own  wt,  a  much  larger  one  will  break  down  without  any 
load.  Having  by  experiment  found  the  constant  of  def  for  any  given  material,  the 
def  of  any  similar  beam  of  the  same  material,  whether  larger  or  smaller,  and  loaded 
at  the  center,  may  be  found  thus  : 

wl,,rB,fe  _  sMfli     Ci°  c°"'"°t 

' 


Breadth    v    Cube  of  depth 
limit  in  ins  in  ins     *          in  ins. 

The  limit  of  elasticity  of  a  beam  of  any  particular  form, 
or  material,  is  determined  by  experiment  with  a  similar  beam,  as  in  the  case 
of  constants  for  breakg  loads.  &c.  Thus,  load  a  beam  at  the  center,  by  the  careful 
gradual  addition  of  small  equal  loads  ;  carefully  note  down  the  def  that  takes  place 
within  some  mins  (the  more  the  better)  after  each  load  has  been  applied;  in  order  to 
ascertain  when  the  defs  begin  to  increase  more  rapidly  than  the  loads  ;  for  when  this 
takes  place,  the  load  for  elastic  limit  has  been  reached.  Sec  Remarks,  Art  9  and  29. 


198  STRENGTH   OF   MATERIALS. 

It  Is  not  the  defs  of  the  whole  beam  that  are  to  be  noted,  but  those  of  its 
clear  span  only.  Several  beams  should  be  tried  to  get  an  average 
constant ;  for  even  in  rolled  iron  beams  of  the  same  pattern,  and  same  iron,  there  is 
a  very  appreciable  diff  of  strengths  and  defs. 

Then,  to  get  the  constant,  so  as  to  apply  it  to  similar  beams,  using  the  total  load 
applied  during  the  equal  defs,  including  %  wt  of  beam, 

Constant  for  greatest        ,*«,     X 
center   loads   within  = 


Breadth  v  Square  of  depth 


LX 


in  ins 


Constants  for  greatest  center  loads  within  limits  of  elastici- 
ty, may  be  had  near  enough  for  common  practice  by  taking  one-third  of  the  breaking 
constants  in  the  table  on  p  185 ;  except  those  for  rolled  iron  and  steel.*  It  is  assumed 
always  that  the  load  is  not  subject  to  jars  or  vibrations ;  these  would  increase  the  defs. 
REM.  Within  the  limits  of  elasticity,  a  beam  of  irregular  shape,  such  as  a  T,  or  a 
Hodgkinson  beam,  a  triangle,  Ac,  will  bend  to  the  same  extent,  whether  its  top  or 
its  bottom  be  uppermost. 

To  find  the  greatest  center  load  that  a  given  beam,  sup- 
ported at  both  ends,  can  sustain  without  exceeding  its  elas- 
tic limit,  (beam  rectangular.) 

}/3  of  the 

Greatest  center  load      Breadth  ^  Square  of  depth  ^,  constant  on 
within  elas  limit  of  =    in  ins    ^S  *n  *n8  page  185.* 

beam  in  fts  Span  in  feet 

We  will  remark  that,  in  practice,  it  is  frequently  difficult  to  ascertain  with  pre- 
cision when,  or  under  what  load,  the  defs  actually  do  begin  to  increase  more 
rapidly  than  the  successive  loads.  For  although  theoretically  the  defs  are  equal  for 
equal  loads,  until  the  elastic  limit  is  reached,  yet  in  practice  they  are  only  nearly 
equal,  up  to  that  point  This  is  owing  to  the  fact  that  no  material  composing  a  beam 
is  perfectly  uniform  throughout  in  texture  and  strength;  so  that  instead  of  perfect 
equality  of  defs,  we  shall  have  an  alternation  of  larger  and  smaller  ones.  Therefore, 
some  judgment  is  reqd  to  determine  the  final  point;  in  doing  which,  it  is  better,  in 
case  of  doubt,  to  lean  to  the  side  of  safety. 

The  def  of  a  beam  of  any  form  whatever  of  cross  section, 

if  within  the  limit  of  elasticity,  may  be  found  approximately  thus, 

load    v  Cube  of  span  v       f  . 
Def  in  =     in  ft>8  x     in  inches     ' 
1DS<         mod  of  elas  v  moment  of  inertia 
p  632         A        in  ins,  p  195 

Coef  d.  Beams  supported  at  both  ends  ;  center  load 02083 

"        "    uniform     '•    01302 

"  fixed  at  one  end ;  loaded  at  the  other 33333 

"      "  "      "    ;  load  uniform 12500 

This  formula  gives  the  def  produced  by  the  load  only.  To  find  that  arising  from 
the  weight  of  the  beam  itself,  consider  said  wt  as  a  uniform  load ;  then  find  the  re- 
sulting def  by  the  same  formula,  and  add  it  to  that  of  the  load. 

In  a  rectangular  beam  supported  hor  at  both  ends  and  loaded  at  the 
middle  within  elas  limit,  the  def  in  ins  will  be 

/  Load    ,   .625  wt  of  clear  span\  y,  cube  of  clear  span 
.  \in  fos   '        of  beam  in  fos       /  *  in  ins 

~~  4  X  coef  elas  X  breadth  in  ins  X  cube  of  depth  in  ins 

And  the  center  load  in  Ibs,  (including  .625  wt  of  clear  span  of  beam) 
required  to  produce  any  given  def  in  ins  within  elas  limit  of  such  a  beam  will  be 

4  v  pnpf  P!««  v  breadth  v  cube  of  depth  v  given  def 
b  A    in  ins    A         in  ins        A      in  ins 


cube  of  clear  span  in  ins. 


*  Except  for  wrought  iron  and  steel  ;  for  which  take  the  whole  con- 
stant, 


STRENGTH   OF   MATERIALS.  199 

Table  of  constants  for  the  deflections,  within  the  safe,  or 
elastic  limits,  of  hot  rectangular  beams,  supported  at  both  ends  and  loaded 
at  the  center.    The  timbers  are  supposed  to  be  well  seasoned;  if  not,  the  constant 
should  be  increased. 
'White  oak  ......................  00023*          White  pine  ............................  ] 

Best  southern  pitch  pine,  )    ftnft.77  #          Ordinary  yellow  pine  ...............  | 

and  white  ash  .....  .......  j  -00027  Spruce  ...................................  \  .00032* 

Hickory  ........................  00016*  Good  straight-grained  hemlock. 

Ordinary  oaks  ..........................  J 

Cast  iron  ...........  000018  to  .000036  ....................  ......  ......  Mean  .000027* 

Bar  iron  .............  000012  to  .000024  .................................  Mean  .000018 

Stool,  rolled  .....  000010  to  .000020  .................................  Mean  .000015 

Full  and  reliable  experiments  on  the  strength  and  deflections  of  the  various  steels 
are  much  needed.  See  table  of  safe  loads  and  clefs,  p  191. 

It  is  evident  that  the  stiffer  the  material  is,  the  smaller  will  be  its  constant  for  bend- 
ing. All  these  constants  vary  somewhat  with  the  quality  of  the  metal.  The  defs  also  of 
timber  of  the  same  kind,  vary  so  much  with  the  degree  of  seasoning,  the  age  of  the 
tree,  the  part  it  is  cut  from,  Ac,  that  the  writer  considers  it  mere  affectation  to  pre- 
tend to  assign  constants  for  practical  use,  more  nearly  approximate  than  he  has  here 
done.  They  are  averages  deduced  from  his  own  experiments  on  good  pieces,  well 
seasoned  ;  and  the  loads  were  allowed  to  remain  on  for  months,  instead  of  minutes, 
as  usual.  Every  structure  is  more  or  less  exposed  to  vibrations  and  jars,  which  in 
time  increase  the  deflections.  In  several  instances,  our  experimental  timbers  bore 
their  breakg  loads  for  months  before  they  actually  gave  way.  And  in  all  kinds,  less 
than  •£§  of  the  breakg  load  produced  in  a  few  months  a  permanent  set,  or  def. 

The  following  are  deduced  from  single  experiments  only.    An  allowance  is  made 
ffor  the  weight  of  the  beam. 

Rolled  iron  beams  proportioned  exactly  as  the  7-inch  Phoenix  beam,  A,p  210,  .0000303f 
"        "  "  30  Ibs,  9  inch,  "          "   .  .................  0000321f 

"        "  "  50  "  heavy  9  inch         "  "       B,  p  210,  .0000264 

41%  Ibs,  12  inch  "  "  ................  0000313f 

"        "        "  512J  Ibs,  15  inch  "  "  .................  0000365f 

66%  Ibs,  15  inch  "  "  .................  0000438 

Art.  27.  To  find  the  def  in  inches,  of  a  hor  rectangular 
beam,  supported  at  both  ends,  and  loaded  at  its  center,  with 
any  given  load  within  its  elasticity;  mult  the  weight  of  the  clear 
beam  itself,  in  Ibs,  by  the  decimal  .625.  Add  the  prod  to  the  given  center  load  in  Ibs. 
Call  the  sum  the  total  load.  Mult  together  this  total  load,  the  cube  of  the  span  in 
ft,  and  the  constant  from  the  upper  table.  Also  mult  together  the  breadth  in 
ins,  and  the  cube  of  the  depth  in  ins.  Div  the  first  prod  by  the  last  one. 

Ex.  What  will  be  the  def  of  such  a  beam  of  average  white  pine,  9  ins  broad,  12 
ins  deep,  21  feet  clear  span,  and  weighing  450  Ibs;  with  a  neat  center  load  of  1218.75 
Ibs? 

Here  first,  450  X  .625  =  281.25  Ibs.  And  281.25  -f  1218  75  =  1500  Ibs  total  load. 
Hence, 

1500  X  21»  X  -Const.      1500  X  9261  X  .00032      4445.2 
--  ~  - 


-OT-  ~    9X1728         - 

See  table  of  safe  loads  and  deflections,  p  191. 

REM  1.    When  the  load  is  all  at  one  point  not  at  the  center, 

as  at  o,  Fig  10,  mult  together  the  two  dists  o  a,  o  g,  from  the  load  to  the  points  of 
support.  Mult  the  prod  by  4.  Div  the  result  by  the  clear  span.  Use  the  quot  as  if 
it  were  the  span,  in  the  last  rule.  The  wt  of  the  beam  is  not  here  taken  into  account  ; 
it  will  of  course  somewhat  increase  the  def. 


*  Averages  near  enough  for  ordinary  practice  by  the  writer's  own  trials.  Call- 
ing- the  average  elastic  dot*  of  a  steel  beam,  1,  that  of  a  similar 

average  wrought  one  will  be  1.2  :  and  that  of  a  cast  one  1.8.  If  that  of  an  average  cast  beam  be  1, 
that  of  a  wrought  one  will  be  .67  ;  and  that  of  a  steel  one  .56.  If  that  of  a  wrought  one  be  1,  cast  will 
be  1.5;  and  steel  .83. 

t  We  believe  that  these  four  beams  have  the  same  proportions,  as  nearly  as  the  process  of  making 
them  will  admit  of;  so  that  .000033  may  be  taken  as  a  near  enough  avera'ge  for  all  four.  As  before 
remarked,  extreme  accuracy  must  never  be  expected  in  such  matters.  Two  halves  of  the  same  iden- 
tical beam  will  often  give  differences  greater  than  this. 


200  STRENGTH    OF    MATERIALS. 

HEM.  2.  When  the  neat  load  is  equally  distributed  along" 
the  span,  instead  of  all  being  at  the  center,  then  for  an  equivalent  total  center 
load,  add  together  the  neat  load,  and  the  entire  wt  of  the  clear  span  of  beam  ;  and 
mult  the  sum  by  the  dec  .625.  With  the  resulting  equivalent  center  load,  proceed 
precisely  as  in  the  foregoing  example. 

Ex.    The  def  of  the  foregoing  beam  of  white  pine,  9  ins  broad,  12  ins  deep,  21 
feet  span,  weighing  450  Ibs,  and  bearing  an  equally  distributed  load  of  1218.75  Bbs? 
Here  first  450  +  1218.75  =  1668.75.     And  1668.75  X  -625  =  1042.97  Bbs  =  equivalent 
center  load.    Hence 

104-2.97  X  21"  X  -00032          3090.862 

9X12»  -  *  -T666T    ^  -1987  iD8'  reqd  def' 

REM.  3.  With  an  equally  distributed  load,  including  the  wt  of  the 
beam,  the  def  is  only  %,  or  the  .625  part  as  great  as  it  would  be  if  the  same  total 
load,  including  the  entire  wt  of  the  beam,  were  all  applied  at  the  center. 

RfcM.  4.  If  the  beam  in  any  of  these,  or  the  following  cases,  is  inclined,  as 
in  Fig  11,  use  the  hor  dist  o  y,  instead  of  the  actual  span  o  c. 

Art.  28.  RULE  1.  To  find  the  neat  center  load  which  will  (to- 
g-ether with  the  wt  of  the  beam  itself)  produce  any  given  clef 
within  the  elastic  limit  of  the  beam  ;  find  the  cube  of  the  clear  length 
in  feet;  mult  this  cube  by  the  constant  from  the  table  on  p  199.  Also  mult  the 
breadth  in  ins,  by  the  cube  of  the  depth  in  ins.  Div  the  first  prod  by  the  last  one. 
Div  the  given  def  in  ins,  by  the  quot,  for  the  total  reqd  load  in  Ibs.  Mult  the  wt  of 
the  clear  length  of  the  beam  in  R>s  by  .625,  and  deduct  the  prod  from  the  load  so  ob- 
tained, for  the  mat  load.  By  formula, 

Cube  of  length  .,  Constant 

Total  load,  in  feet          X    in  p  199 

including    =  •ut  iect    -^-  -  =•  -  • 
wt  of  beam         *n  *n8  Breadth    v   Cube  of  depth 

in  ins     A           in  ins. 

Ex.  What  center  load  in  Ibs  will  (together  with  the  wt  of  the  beam  itself)  pro- 
duce a  def  of  .286  of  an  inch,  in  a  beam  of  white  pine,  21  it  span,  9  ins  broad,  12  ins 
deep,  and  which  weighs  450  fibs  ?  See  table,  p  191. 

Cube  of  21.        Const.  Breadth.     Cube  of  12. 

Here      9261     X  .00032  =  2.9635.  And      9      X     1728    =  15552. 


And   --  -  •°001906-  And  -  150°  tt 


For  the  neat  load  we  must  deduct  .625  of  the  wt  of  the  beam  ;  or  450  Ibs  X  -625  = 
281.25  Ibs;  so  that  the  neat  load  is  1500  —  281.25  =  1218.75  ft>s,  as  in  Ex  1,  Art  27. 

If  the  load  is  uniformly  distributed,  use  precisely  the  same  rule  for  get- 
ting the  total  load.  Then  mult  this  load  by  1.6.  Deduct  the  entire  wt  of  the  clear 
length  of  beam. 

Ex.    What  equally  distributed  load  will  deflect  the  foregoing  beam  .1987  ins? 

Here,  proceeding  as  before,  the  only  diff  is  that  instead  of  .286  def,  we  have  .1987 

def  to  be  div  by  .0001906.    And  —  -—  1—  =  1042.5  Ibs,  as  the  equivalent  center  load. 

And  1042.5  X  1.6  =  1668  Ibs  for  the  total  distributed  load,  including  the  entire  wt 
of  the  beam,  or  450  Bbs.  Hence  1668  —  450  =  1218  Ibs,  the  neat  distributed  load  reqd  ; 
agreeing  with  the  preceding  example  within  %  of  a  Ib  ;  the  difif  being  owing  to  a 
neglect  of  small  decimals  in  the  calculation. 

RULE  2.  The  length,  depth,  neat  center  load,  and  def  being: 
given,  to  find  the  breadth. 

Neat  cen  load  ,.  Cube  of  length  v  Constant 
1*  in  feet          X  in  Art  26 


_       ,— 

Cube  of  depth  v    Def  ~    approx. 

in  ins          ^  in  ins 
Or  sufficient  for  the  neat  load  alone. 

Now  calculate  the  wt  of  a  beam  with  the  breadth  already  found.    Mult  this  wt  by 
.625,  then  say,  as 

w     .  Breadth  .625  of  the  Additional 

Neat  center     .         firgt         .    •      ,vei  ht  of     •        breadth 

•        found  the  beam  reqd. 

Add  these  two  breadths  together,  and  their  sum  will  be  the  total  breadth  reqd,  more 
approximately  ;  but  still  somewhat  too  small,  inasmuch  as  it  provides  only  for  the 


STRENGTH    OF    MATERIALS. 


201 


wt  of  the  beam  of  the  breadth  first  found,  and  not  for  that  having  the  additional 
breadth.    This  may  readily  be  calculated  and  added.     See  table,  p  191. 

RULE  3.     The   length,  breadth,  neat   center   load,  and  def, 
being:  given,  to  find  the  depth. 


iii  Ibs 


X 


iu  feet 


1X 


in  Art  26 


Cube  of  depth 
:          in  ins 
approx. 


Breadth  v     Def 
in  ins     *  in  ins 

Take  the  cube  root  of  this  for  the  depth  itself,  approximately.  REM.    This, 

like  the  breadth  given  by  the  preceding  formula,  is  too  small,  inasmuch  as  it  does 
not  allow  for  the  wt  of  the  beam.  Therefore,  when  greater  accuracy  is  required, 
proceed  thus:  Calculate  the  wt  of  a  beam  having  the  depth  just  found.  Mult  this 
wt  by  .625.  Add  the  prod  to  the  neat  center  load.  Consider  the  sum  as  a  new  neat 
center  load;  and  using  it  instead  of  the  neat  center  load  first  given,  go  through  the 
whole  calculation  again,  to  obtain  a  new  cube  of  depth.  The  cube  root  of  this  will 
be  more  nearly  correct ;  but  still  a  trifle  too  small,  for  the  same  reason  as  in  the  fore- 
going case.  See  table,  p  191. 

Art.  29.  Deflection  not  to  exceed  ^  of  an  inch  for  each  foot 
Of  clear  length,  or  j|~g-  part  of  the  clear  length.  To  obtain  constants  for  such 
defs,  at  the  center  of  any  hor  rectangular  beam  supported  at  each  end,  place  any 
load  that  is  entirely  within  its  elastic  limit.  Also,  measure  the  def  in  ins  produced 
by  said  center  load.  Then, 

,ft  v  Breadth  v  Cube  of  depth  v  Deflection 

<    in  ins    >          in  ins          x      in  ins  Constant  for 

def  of 


Center  load 
in  ft>s 


Cube  of  clear  length 
in  feet 


of  the  span. 


*To  be  more  exact,  .625  of  the  wt  of  the  beam  itself  should  be  added  to  the  actual 
neat  load.    See  Rem.  Art  26. 

In  the  same  way,  constants  may  be  found  for  beams  of  any  form  whatever.    If, 
instead  of  ^V»  tne  def  mns*>  n°t  exceed,  ^,  -g-1^,  Ac,  of  an  inch  per  foot,  then  sub- 
stitute 30,  &c,  for  the  40  of  the  formula. 
Table  of  constants  or  coefficients  for  deflections  of  ?V  inch 

per  foot  span,  of  horizontal  rectangular  beams  supported 

at  both  ends,  and  loaded  at  the  center. 


Hickory 0064 

White  Oak 0092 

Lombardy  Poplar 0230 

Teak 0080 

Horse  Chestnut 0170 

Average  Cast  Iron  0011 

Average  Steel 00044 


Best  Southern  Pitch  Pine 0108 

White  Ash " 

White  Pine,  or  Common  Yel- ") 

low 

Spruce,  good  Hemlock I 

Red  and  Black  Oak 

Average  Wrought  Iron .'..  .00066 


.0128 


1.  Rolled  iron  beams,  proportioned  exactly  as  7  inch  Phoenix  beam,  A,  p  210,...  .00121 


2. 
3. 
4. 
5. 
6. 


30  ft,  9  in 
heavy  9  in 
41%  Ibs,  12  in 
51%  Ibs,  15  in 
66%  fts,  15  in 


.00129 
B,    p  210,...  .(10106 

00125 

00131 

00175 


REM.     In  experimenting   for  constants   of 

any  kind,  with  beams  of  irregular  cross-sections,  this,  for  in- 
stance, it  is  quite  immaterial  which  breadths  and  depths  are 
measd;  thus,  for  the  breadth  we  may  take  a  6,  I  in,  cd.  or  oet 
&c;  and  for  the  depth,  either  we,  lr,  md,  bo,  &c.  It  is  only 
necessary  to  state  what  parts  actually  have  been  taken,  so 
that  the  corresponding  ones  may  be  measd  in  any  other  beam 
which  is  to  be  calculated  by  the  constant  derived  from  the 
experiment.  This  remark  applies  to  all  constants  involving 

the  breadth  and  the  depth.  The  constant  itself  will  of  course  vary  according  to 
which  dimensions  are  taken  in  the  experiment ;  but  the  results  derived  from  it  when 
applied  to  other  beams  of  similar  forms,  will  not  be  affected  thereby,  if  the  corre- 
sponding parts  be  measd  in  both  cases  * 

*  We  may  ereu  take  any  single  oblique  measurement,  as  a  b,  Im,  nc.  ad,  &c.  and  call  it  both  the 
breadth  and  the  depth.    This  applies  to  rectangular,  or  to  any  other  shaped  beam*. 


202  STRENGTH   OF   MATERIALS. 

Art.  SO.  RULE  1.  To  find  the  greatest  center  load  which  any 
given  nor  rectangular  beam,  supported  at  both  ends,  can 
sustain  without  bending  more  than  4  J  o  of  its  span. 

Breadth  v  Cube  of  depth 
in  ins    >  in  ins  _  Reqd  load 

*  Square  of  span  v  Constant 
in  feet          X  in  Art  29 
and  generally  near  enough  for  practice.  A 

To  be  more  accurate,  a  deduction  must  be  made  for  the  wt  of  the  beam  itself.  ':o 
get  the  neat  load.  To  do  this,  mult  the  wt  of  the  span  of  the  beam  by  .625,  and  'de- 
duct the  prod  from  the  approx  load  just  found. 

REM.  1.    The  load  may  also  be  found  by  tables  in  pp  204,  205. 
RKM.  2.    If  the  load  is  uniformly  distributed,  use  the  same  rule  or 
formula;  but  mult  the  resulting  load  by  1.6;  and  then,  if  an  allowance  is  reqd  for 
the  wt  of  the  beam,  deduct  the  wt  of  the  entire  span  for  the  neat  load. 

RULE  2.    The  neat  center  load,  length,  and  depth  being  given. 
to  find  the  breadth,  such  that  the  beam  shall   not  deflect 
more  than  ¥|7  of  its  span.     See  tables,  pp  204,  205. 
Square  of  length  ^/  Neat  center  load  .,  Constant 
in  feet  X  in  B>s  X  in  Art  29 

Cube  of  depth  in  ins 
and  generally  near  enough  for  practice. 

Then  say,  as       Centered 


Add  these  two  breadths  together,  for  the  reqd  one  very  approx,  but  a  mere  trifle 
too  small.  See  Remark  after  Rule  3,  Art  28. 

RULE  3  The  neat  center  load,  length,  and  breadth,  being 
given,  to  find  the  depth,  such  that  the  beam  shall  not  deflect 
more  than  ^fa  of  its  clear  length.  See  tables,  pp  204,  205. 

Square  of  length  v  Neat  cen  load  y   Constant  Cube  of  depth 

in  feet  £          in  ft>s  _     ^   in  Art  29    _  in  ing< 

Breadth  in  ins  approx. 

Take  the  cube  root  of  this  for  the  approx  depth  itself,  generally  near  enough  for 
practice.  Then  calculate  the  wt  of  the  entire  clear  span  of  a  beam  having  this 
depth,  mult  it  by  .625,  and  add  the  prod  to  the  neat  center  load.  Consider  the  sum 
as  a  new  neat  center  load;  and  using  it  instead  of  the  one  first  given,  go  through  the 
whole  calculation  again,  for  a  new  cube  of  depth.  The  cube  root  of  this  will  be  the 
reqd  depth  more  approx,  but  a  little  too  small,  for  the  reason  given  at  Rule  3,  p  201. 

If  the  neat  load  is  uniformly  distributed,  first  mult  it  by  .625.  Use  the 
prod  as  a  center  load,  and  by  the  foregoing  formula  find  the  first  approx  depth  Then 
calculate  the  wt  of  the  entire  clear  length  of  a  beam  having  that  depth.  Mult  thie 
wt  by  .6v?o,  and  add  it  to  the  prod  used  as  a  center  load.  Consider  the  sum  as  a  new 
confer  load;  and  using  it  instead  of  the  one  first  used,  go  through  the  whole  calcu- 
lation again,  for  a  ww  cube  of  depth.  The  cube  root  of  this  will  be  the  reqd  depth, 
approx.  but  a  mere  trifle  too  small,  for  the  reason  given  at  Rule  3,  p  201. 

RULE  4.    To  find  the  side  of  a  square  beam,  when  its  center 
load  and  clear  length  are  given,  not  to  deflect  more  than  ^-5  of  its  span. 
Square  of  length  v  Center  load  v  Constant  _      A  certain 
in  feet  in  ft»s        x  in  Art  29  ~  fourth  power. 

Next,  find  the  fourth  root  of  this  fourth  power;  th;it  is,  take  its  sq  rt,  then  take 
the  sq  rt  of  said  sq  rt.  This  last  sq  rt  will  be  one  side  of  a  square  beam,  generally 

*  We  must  now  employ  the  square  of  the  length  ;  not  its  cube,  as  in  the  preceding  Arti. 


STRENGTH   OF   MATERIALS. 


203 


near  enough  for  practice ;  but  which  will  answer  truly  only  for  the  given  center  load, 
minus  .6.5  of  the  wt  of  the  cle.ir  beam;  which  wt  can  now  be  calculated. 

But  if  the  beam  is  to  answer  lor  the  given  center  load,  plus  the  wt  of  the  beam, 
the  calculation  must  now  be  repeated,  thus :  Find  the  wt  of  the  beam  just  calculated ; 
mult  it  by  .625;  add  the  prod  -to  the  given  center  load.  Then,  with  this  increased 
center  load,  employ  the  formula  as  before,  obtaining  a  new  fourth  power.  The  sq  rt 
of  the  sq  rt  of  this  will  be  the  reqd  side  very  approx,  but  a  mere  trifle  too  small. 
See  Kern  after  Rule  3,  Art  28,  p  201 ;  also  table,  p  205. 

RULE  5.  To  find  the  diam  of  a  solid  cylinder,  when  its  cen- 
ter load  and  clear  length  are  given.  Mult  the  load  by  1.7,  and  with 
this  increased  load  proceed,  as  in  the  foregoing  case  of  a  square  beam,  to  find  a  cer- 
tain 4th  power.  Take  the  sq  rt  of  the  sq  rt  of  this,  for  a  diani  sufficient  for  the  given 
center  load,  minus  .625  of  the  wt  of  the  beam.  For  a  larger  diam,  sufficient  for  the 
wt  of  the  beam  also,  first  mult  said  wt  just  found  by  .625.  Add  the  prod  to  the  ori- 
ginal given  center  load.  Mult  the  sum  by  1.7  for  a  new  load;  then,  with  this  new 
load,  repeat  the  formula  as  before,  for  a  new  4th  power.  The  sq  rt  of  the  sq  rt  of 
this  will  be  the  reqd  diam  more  apprax,  but  a  trifle  too  small.  See  Rem,  after  Rule  3, 
Art  28. 

The  stiffness  of  a  cyl  is  to  that  of  a  square  beam,  whose  breadth  and 
depth  are  each  equal  to  the  diam  of  the  cyl,  as  .589  to  1 ;  or  that  of  the  square  one 
is  to  that  of  the  cyl  as  1  to  .589,  or  as  1.698  to  1 ;  in  practice  we  may  use  .6  and  1.7. 
Hence,  the  cylinder  will  bend  1.7  times  as  much  as  a  square  one,  under  the  same 
load. 

When,  in  any  of  the  foregoing  cases,  the  beam  is  inclined,  as  in  Fig  11 
take  the  hor  dist  o  y  for  the  span,  instead  of  o  c. 


STONE  BEAMS, 

Table  of  safe  quiescent  extraneous  loads  for  beams  of  g;ood 
building  granite  one  inch  broad,  supported  at  both  ends,  and  loaded  at  the 
center;  assuming  the  safe  load  to  be  one-tenth  of  the  breaking  one;  and  the  latter 
to  be  100  Ibs  for  a  beam  1  inch  square,  and  1  foot  clear  span.  The  half  weight  of 
the  beams  themselves  is  here  already  deducted  by  the  rule  in  Art  12,  p  186,  at  170 
Ibs  per  cub  ft. 


j 

CLEAR  SPANS  IN  FEET. 

a 

M 

1    1   2 

3   |   4 

5 

6 

7 

8 

10 

12 

15 

20 

Safe  center  load  ;  in  pounds. 

1 

10 

5 

2 

40 

20 

13 

10 

3 

90 

45 

29 

21 

17 

4 

160 

79 

52 

39 

31 

26 

21 

5 

250 

124 

82 

61 

48 

40 

34 

6 

360 

179 

119 

89 

70 

58 

48 

42 

32 

7 

490 

244 

162 

120 

96 

79 

67 

58 

45 

36 

27 

16 

8 

639 

319 

212 

158 

126 

104 

88 

76 

59 

47 

36 

22 

10 

999 

499 

331 

248 

197 

163 

139 

120 

94 

76 

58 

38 

12 

1439 

718 

478 

357 

284 

236 

201 

174 

137 

111 

85 

58 

14 

1959 

978 

650 

487 

388 

322 

274 

238 

188 

153 

118 

81 

16 

2559 

1278 

850 

636 

507 

421 

359 

312 

246 

201 

157 

109 

18 

3239 

1818 

1077 

806 

643 

534 

455 

396 

313 

257 

200 

141 

20 

3999 

1998 

1329 

995 

794 

660 

563 

490 

388 

319 

249 

176 

22 

4839 

2417 

1609 

1205 

961 

800 

682 

594 

470 

387 

303 

216 

24 

5758 

2877 

1916 

1434 

1145 

951 

813 

708 

562 

463 

362 

260 

27 

7288 

3642 

2425 

1815 

1450 

1205 

1030 

898 

713 

588 

462 

332 

30 

8998 

4496 

2995 

2243 

1791 

1489 

1273 

1110 

882 

728 

573 

415 

33 

10888 

5441 

3624 

271* 

2168 

1803 

1542 

1345 

1069 

883 

666 

505 

36 

12958 

6476 

4314 

3231 

2581 

2147 

1836 

1603 

•  1275 

1054 

832 

606 

If  uniformly  distributed  over  the  clear  span,  the  safe  extraneous 
loads  will  be  twice  as  great  as  those  in  the  table. 

For  good  slate  on  bed  the  safe  loads  may  be  taken  at  about  3  times ;  for 
good  sandstone  on  bed  at  about  one-half;  and  lor  g*ood  marble  or 
limestone  on  bed  at  about  the  same  as  those  in  the  table.  See  table,  p  185. 


204 


STRENGTH    OF   MATERIALS. 


STRENGTH    OF    MATERIALS. 


205 


Art.  32.  Table  of  greatest  center  loads  of  square  beams  of 
cast  iron,  supported  at  both  ends,  and  reqd  not  to  bend 
more  than  .tJ0  of  an  inch  per  foot  of  clear  leiig'th,  or  ?J  5 

part  of  the  span.    For  W.  Pine  div  by  12;  or  in  practice  by  18. 

The  loads  are  about  y1^  part  greater  than  would  be  given  by  our  constant  .0011  for 
average  cast  iron,  Art  29.  Wrought  iron  will  bear  about  -|  more  than  cast,  with  the 
same  safe  deflection.  But  .625-(or  %)  of  the  wt  of  the  beam  itself  must  be  deducted 
from  thf-se  center  loads.  If  the  load  is  equally  distributed,  it  will  be  1.6  times  as 
great  as  these  tabular  center  loads  ;  but  in  this  case  the  wt  of  the  entire  clear  length 
of  the  beam  is  to  be  deducted.  These  deductions  are  rarely  reqd  in  practice. 


Greatest 
Centre 
Loads,  in 
Pounds. 

112 
224 
336 

448 
560 
672 
784 
893 
J,008 
"1,1  20 
1,232 
1,344 
1,456 
1,568 
1,680 
1,792 
1,904 
2,016 
2,128 
2,240 
2,800 
3,360 
3,920 
4,480 
5,600 
6,720 
7,K40 
8,960 
10,080 
11,200 
13,440 
15,680 
17,920 
20,160 
22,400 
24,640 
26,880 
29,120 
31,360 
33,«WO 
35,840 
38,080 
40,:i20 
42,560 
44,800 
49,2SO 
53,760 
58,240 

Clear  Spans,  in  Feet.    (From  Tredgold  ) 

4 

01 

A 

In. 
1.2 
1.4 
1.6 
1.7 
It8 
1.8 
1.9 
2.0 
20 
2.1 
2.1 
2.2 
2.2 
2.3 
2.3 
2.4 
2.4 
2.4 
2.5 
2.5 
2.6 
2.8 
2.9 
2.9 
3.1 
3.3 
3.4 
3.5 

6 

8 

.a 

H, 
£ 

In. 
1.7 
2.0 
2.2 
•2.4 
2.5 
2.6 
2.7 
2.8 
2.9 
3.0 
3.0 
3.1 
3.1 
3.2 
3.2 
3.3 
3.4 
3.4 
3.5 
3.5 
3.7 
3.9 
4.0 
4.1 
4.4 
4.6 
4.8 
4.9 
5.1 
5.2 
6.5 
5.7 
5.'.) 
6.0 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

J3 
P, 
& 

In. 
1.4 
1.7 
1.9 
2.0 
2/2 
2.2 
2.3 
2.4 
2.5 
2.6 
2.6 
2.7 
2.7 
2.8 
2.8 
2.9 
2.9 
3.0 
30 
3.0 
3.2 
3.4 
3.5 
3.5 
3.8 
4.0 
41 
4.3 
4.4 
4.5 

€ 

p. 
P 

In. 
1.9 
2.2 
2.4 
2.6 
2.8 
2.» 
3.0 
3.1 
3.2 
3.3 
3.4 
3.5 
3.5 
3.6 
3.6 
3.7 
3.8 
3.8 
3.9 
3.9 
4.1 
43 
4.5 
4.7 
4.9 
5.1 
5.3 
5.5 
5.7 
5.S 
6.1 
6.:j 
6.6 
6.8 
6.9 
1 

JL 

In. 
2.0 
2.4 
2.7 
2.9 
3.0 
3.2 
3.3 
3.4 
3.5 
36 
3.7 
3.8 
3.8 
3.9 
4.0 
4.0 
41 
42 
4.2 
4.3 
4.5 
4.7 
49 
5.1 
5.4 
5.7 
5.8 
6.0 
62 
6.4 
6.7 
69 
7.2 
7.4 
76 
7.8 
7.9 
8.1 
8.3 
8.4 
8.5 
8.7 
8.8 
8.9 
9.0 
9.2 
9.4 
9.6 
9.8 

'  P 

In. 

2.2 
2.6 
2.9 
3.1 
33 
3.4 
3.6 
3.7 
3.8 
3.9 
40 
4.1 
4.2 
4.2 
4.3 
4.4 
44 
4.5 
4.6 
46 
4.9 
5.1 
5.3 
5.5 
5.8 
6.1 
6.3 
65 
6.7 
69 
7.2 
7.5 
•7,8 
8.0 
8.2 
84 
8*6 
88 
8.9 
9.1 
92 
9.4 
9.5 
9.6 
9.7 
10.0 
10.2 
10.4 
10.6 

P« 

P 

In. 
2.4 
2.8 
3.1 
3.3 
3.5 
3.7 
3.8 
39 
4.0 
4.2 
4.3 
4.4 
4.4 
4.5 
4.6 
4.7 
4.7 
4.8 
4.9 
4.9 
5.2 
5.5 
5.7 
59 
6.2 
6.5 
6.7 
7.0 
7.2 
7.4 
7.7 
8.0 
8.3 
8.5 
8.8 
9,Q 
9.2 
9.4 

3 

9.8 
100 
10.1 
10.3 
10.4 
10.7 
10.9 
11.1 
11.4 

Ot 

a> 
P 

In. 
2.5 
3.0 
3.3 
3.5 
3.7 
3.9 
4.1 
4.2 
4.3 
4.4 
4.5 
4.7 
4.7 
4.8 
4.9 
5.0 
5.0 
5.1 
6.2 
5.2 
5.5 
5.8 
6.0 
6.2 
6.6 
6.9 
7.1 
7.4 
7.6 
7.8 
8.2 
8.5 
8.8 
9.0 
9.3 
9.5 
9.7 
9.9 
10.1 
10.3 
10.4 
10.6 
10.8 
10.9 
11.0 
11.3 
11.5 
11.8 
12.0 

P. 
3 

In. 

2.6 
3.1 
3.4 
3.7 
3.9 
41 
4.2 
4.4 
4.5 
4.7 
4.8 
4.9 
4.9 
50 
5.2 
5.2 
5.3 
5.4 
5.4 
55 
5.8 
6.1 
6.3 
6.5 
6.9 
7.3 
7.5 
7.8 
8.0 
8.2 
8.6 
8.9 
9.3 
9.5 
9.8 
10.0 
10.2 
10.4 
10.6 
10.8 
11.0 
11.  '2 
11.3 
11.5 
11.6 
11.9 
12.2 
12.4 
12.7 

p. 

P 

In. 

2.7 
3.3 
3.6 
3.9 
4.1 
43 
4.4 
4.6 
4.7 
4.9 
6.0 
5.1 
5.2 
6.3 
54 
5.5 
5.5 
6.8 
5.7 
5.8 
6.1 
6.4 
6.7 
6.8 
7.3 
7.6 
7.9 
8.2 
8.4 
8.6 
9.0 
9.4 
9.7 
10.0 
10.3 
10.5 
10.8 
11.0 
11.1 
11.4 
11.5 
11.7 
119 
12.2 
12.5 
12.8 
13.0 
13.3 
13.5 

1 

In. 
2.9 

3.4 
3.8 
4.0 
4.3 
4  5 
4.6 
4.8 
4.9 

3 

5.3 
6.4 

5.5 
6.6 
5.7 
5.8 
5.9 

eio 

6.0 
64 
6.7 
6.9 
7.2 
7.6 
7.9 
82 
85 
8.8 
9.< 
9.4 
9.8 
10.1 
10.4 
10.7 
11.0 
11.2 
11.5 
11.7 
ll.f 
12.0 
12.2 
12.4 
12.« 
12.7 
13.0 
13.4 
13.6 
13.9 

A 

1 

In 

30 
3.6 
3.9 
4.2 
4.4 
4.6 
4.S 
5.0 
5,1 
5.3 
5.4 
55 
5.6 
5.7 
5.8 
6.9 
6.0 
6.1 
6.2 
6.3 
6.6 
7.0 
7.2 
7.6 
7.9 
8.3 
8.6 
89 
9.1 
94 
98 
10.2 
10.6 
10.9 
11.2 
11.5 
11.7 
11.9 
12.1 
12.3 
12.5 
12.7 
12.9 
13.1 
13.2 
136 
139 
14.2 
144 

.c 
p. 
P 

In. 

3.1 
3.7 
4.1 
43 
4.6 
4.8 
5.0 
5.2 
6.3 
54 
5.6 
57 
5.9 
60 
6.1 
6.2 
6.v 
6.4 
65 
6.5 
6.9 
7.2 
7.5 
7.7 
8.2 
8.6 
8.9 
92 
95 
97 
102 
106 
10.9 
11.3 

n.e 

11.9 
1-2.1 
12.4 
126 
12> 
13.0 
13/2 
13.4 
13.6 
13.8 
14.1 
14.4 
14.7 
15.0 

i 

P, 

8 

In. 
3.2 

3.8 
42 
45 
4.8 
5.0 
5.2 
5.4 
5.5 
5.7 
5.8 
5.9 
6.0 
6.1 
6.2 
6.4 
65 
6.6 
6.7 
6.8 
7.2 
7.5 
7.7 
8.0 
85 
89 
92 
95 
9.8 
10.1 
10.5 
11.0 
11.3 
11.7 
12.0 
12.3 
12.5 
128 
13.0 
132 
13.5 
13.7 
139 
141 
14.2 
146 
14.9 
1.^.2 
15.* 

.4 
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.7 
.8 
.9 
8.0 
8.1 

1 

62,720 

206 


STRENGTH   OF   MATERIALS. 


207 


at  the  center  of  solid  horizontal  CYLINDRICAL  beams  of 

tie  weight  of  half  the  beam  itself,  which  must  be  deducted  in  order  to  obtain  the 
lly  distributed  along  the  enti(»  clear  length,  it  will  be  twice  as  great  as  at  the  center.  In  this  case, 
rt  iron  cyls  willbear  about  1  fourth  more  than  cast.  For  W.  Pine,  spruce, 
ite  ash,  or  best  Southern  yel  pine  by  4  or  6.  For  the  breaking  load  of  a  square  beam,  whose  breadth 
loads  by  1.7.  (ORIGINAL.; 

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4.  Table  of  breaking  loads  ii 
>n,  supported  at  both  ends;  ii 

g  load.  To  do  this,  use  the  last  column.  If  the 
of  the  entire  clear  length  must  be  dedt 
aks,  divide  the  loads  by  4.5  or  bv  6.8;  and  for  w 
ch  equals  the  diameter  of  the  cylinder,  multiply 

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208 


STRENGTH   OF    MATERIALS. 


FiglS 


Art.  35.     Hodgkinsoii's  beams  have  nearly  1%  times  the  strength  of  a 
beam  of  equal  wt  whose  top  and  bottom  flanges 

0 ,T     are  equal.     Mr.  Hodgkinsori,  having  found  that  on 

an  average,  cast  iron  reqd  about  fii^  times  as  much 
force  to  crush  it  as  it  did  to  pull  it  apart,  contrived 
the  beam  (of  which  Fig  15  is  a  cross-section  at  the 
center)  in  which  the  upper  or  compressed  rib  or 
flange,  ?/,  has  but  Y\  of  the  area  of  the  lower  or 
extended  one,  6.*  The  top  flange  he  therefore  as- 
sumed to  be  safe;  inasmuch  as  its  area  is  some- 
what greater  than  the  proportion  of  1  to  6%:  and 
hence,  breaking  will  take  place  from  the  yielding 
of  the  bottom  flange  by  extension.  As  the  result 
of  his  experiments,  he  gives  the  following  rule, 
when  the  load  is  applied  at  top,  or  equally  on  both 
sides  of  the  beam.  See  Item,  next  page. 
Area  of  bot  flange  v  depth  oo  Constant 
=  in  sq  ins  '  in  ins  '  2.166 

Clear  length  in  feet. 

When  the  lower  flange  is  as  much  as  about  '2*4  inches  thick,  experiments  show 
that  part  of  the  breakg  load  thus  obtained  should  be  deducted;  because  thick  castings 
are  proportionally  weaker  than  thin  ones.  Half  the  wt  of  the  beam  itself  must  be 
deducted,' for  the  neat  breakg  load;  this,  however,  is  necessary  only  when  the  beams 
are  very  lori^;  for  such  as  are  used  for  ordinary  building  purposes,  it  may  be  ne- 
glected. If  the  load  is  equally  distributed,  it  will  be  twice  as  great;  but  the  entire 
wt  of  the  beam  must  then  be  deducted. 

Ex.  The  upper  rib  u  =  3  ins  X  1  inch  =  3  sq  ins  area;  bottom  rib  6  =  1%  ins 
X  12  ins  =  18  sq  ins  area;  total  depth  oo,  15  ins ;  clear  span,  20  ft.     Here, 
18  X  15  X  2.166  584.82 

— 20 —  — 20 —  =  *"•*"  t°ns>  tne  reqd  load,  including  %  the  beam. 

Now  to  find  the  wt  of  half  the  beam,  we  may  proceed  thus:  Mult 
the  entire  area  of  its  cross  section  in  sq  ins,  by  the  clear  span  in  ins.  This  gives  us 
the  cub  ins  of  iron  contained  in  the  beam ;  and  these  div  by  8600,  give  the  wt  of  the 
beam  in  tons ;  because  8600  cub  ins  of  cast  iron  weigh  about  1  ton  ;  or  near  4  cubic 
ins  1  ft).  Thus,  if  the  vert  rib  contains  12  sq  ins,  then  since  the  two  flanges  con- 
tain 21,  the  entire  section  is  33  sq  ins;  and  the  span  being  240  ins,  we  have  33  X  240 

=  7920  cub  ins  of  iron.    And  ~~  =  .92  of  a  ton,  the  wt  of  the  beam.    One-half 

8600 

of  this,  or  .46  ton,  taken  from  the  breakg  load  29.241  tons,  leaves  28.78  tons  as  the  neat 
breakg  load ;  showing  that  in  such  cases  as  this  it  is  scarcely  worth  while  in  practice 
to  make  the  deduction.  These  beams  are  not  always  made  of  the  same  section 
throughout,  (see  Fig  16,)  but  diminish  toward  the  ends;  this  method  is  therefore  not 
always  strictly  correct,  but  no  great  accuracy  is  needed  in  such  cases. 

To  find  the  size  of  a  Hodgkinson  beam,  reqd  to  break  nnder 
a  given  center  load,  having  the  depth.  Mult  the  given  load  in  tons 
by  the  clear  span  in  feet.  Mult  the  constant  2.166  by  the  total  depth,  oo,  in  ins. 
Div  the  first  prod  by  the  last;  the  quot  will  be  the  area  of  the  bottom  rib  in  sq  ins. 
This,  div  by  6,  will  be  the  area  of  the  top  rib.  The  bottom  rib  is  usually  made  from 
6  to  8  times  as  wide  as  it  is  thick;  and  the  top  one  from  3  to  6  times,  The  thickness 
of  the  stem  is  usually  a  little  greater  at  bottom  than  at  top;  the  average  thickness 
being  from  ^  to  -fa  of  the  depth  of  the  beam. 

To  save  iron,  the  width  of  the  bottom  flange,  and  of  the  top  one  also  if  thought 

proper,  may  be  reduced  by  curves 
to  about  %  as  great  at  each  end 
of  the  beam  as  at  its  center;  as 
shown  by  the  middle  sketch  of 
Fig  16,  of  which  the  upper  sketch 
is  a  side  view.  Or,  leaving  the 
dimensions  of  those  flanges  un- 
altered, the  depth  of  the  vertical 
rib  may  be  reduced  toward  the 
ends,  as  shown  by  the  loweet 
sketch.  The  theoretical  curve  is 
here  an  ellipse.  When  the  width 

is  reduced,  the  very  ends  may,  for  stability,  be  widened  out,  as  at  e,  which  is  a  top  view. 
The  vert  rib  is  generally  strengthened  by  casting  brackets 

*  In  practice  %  is  much  better  and  safer  than  %. 


FlglG 


STRENGTH    OF    MATERIALS. 


on  each  side  of  it,  as  in  the  upper  sketch.  These  should  not  extend  entirely  to  tne 
upper  rib,  as  they  then  expose  the  beam  to  crack  as  it  cools.  To  prevent  this  tend- 
"eticy,  they  may  be  attached  alternately  to  the  top  and  bottom  ribs.  The  upper  ones, 
however,  are  rarely  needed. 

In  designing  these  beams,  as  well  as  in  all  other  castings,  it  is  important  to  avoid 
sudden  transitions  from  thin  to  thick  parts  ;  and  to  keep  all  parts  as  nearly  as  possi- 
ble of  the  same  thickness.  Otherwise  the  castings  are  apt  to  warp  and  crack  in 
cooling.  Also,  bear  in  mind  that  the  resistance  or  strength  per  sq  inch  is  considera- 
bly less  in  thick  castings  than  in  thin  ones. 

Item.  The  above  rule  for  breakg  loads  is  safe  when  the  load  is  equally  disposed 
on  top,  or  on  each  side  of  the  vert  web  ;  and  when  said  web  and  the  flanges  are  pro- 
portioned to  each  other  about  the  same  as  those  used  in  Mr.  Hodgkinson's  experi- 
ments. But  subsequent  investigators  have  found  his  beams  to  break  with  but  little 
more  than  half  the  loads  given  by  the  rule,  when  applied  to  only  one  side,  as  bo,  or 
wo,  Fig  15,  of  the  top  or  bottom  flange.  W.  II.  Barlow,  C.  E.,  London,  experimenting 
since  Hodgkinson,  finds  that  when  a  cast-iron  beam  is  liable  to  be  loaded  on  only  one 
side  of  tne  flange,  the  top  flange  should  have  an  area  equal  to  %  that  of  the  entire 
cross-section  of  the  beam  ;  and  for  beams  so  proportioned,  he  gives  the  following  : 


.  Constant 
<     2.333 


in  tons 


Clear  span  in  feet. 


Other  experimenters  recommend  that  even  for  loads  pressing  vertically  through  the 
upright  rib,  tlio  lower  flange  should  have  but  about  3  instead  of  6  times  the  area  of 
the  upper  one.  Cast  beams  should  always  be  tested. 

l^he  average  ultimate  resistance  of  steel  to  compression  being  about  twice  that  to 
extension,  a  Hodgkinson  beam  of  that  metal  should  have  its  lower  flange  of  twice 
the  area  of  its  upper  one.  Much  uncertainty  exists  in  the  whole  matter. 

Art.  36.  For  the  purpose  of  ready  reference,  we  give  a  few  ex- 
perimental results  with  cast-iron  beams  of  various  shapes :  being  the  actual  center 
breakg  loads  in  tons  of  sound  beams.  Some  beams  of  Sterling's  toughened  cast 
iron  gave  results  full  Yz  higher  than  those  of  common  iron. 

Actual  center  breakg;  loads  in  tons,  of  cast-iron  beams.  Clear 
spans  in  feet.  Breadths  and  depths  in  inches. 

1,5x.5^P 

K  co     Span  4U  ft. 
2.5*. 5  I          Br  load  2  tons. 


*.,       The  above  inverted. 
|  0      Br  load  2.3  to  2.9  tons. 


Span  4V  ft 

325  ton 


The  above  inverted. 


£      span  41^  ft. 

]  VO  f 3'7  t°BB 

2 .  27x.  fi^aaLa       Br  load  \      ^ 
I     4.2. 


*      Span  11%  ft  f 
Br  load  20  tons. 


Span  18  ft. 

Br  load  22  to  28  tons. 


00     Span27%ft.t 
|#l  *~    Br  load  2d^  to 


*  As  shown  by  dd,  Fig  15. 

t  "After  bearing  17  tons,  the  beam  was  unloaded,  and  its  elasticity  appeared  to  be  but  little  if  at  all 
Injured."    Def  under  4^  tons.  .15  inch  ;  8%  tons,  .3  inch  ;  17  tons,  l.t  inches. 
;  About  two  hundred  of  these  beams  were  tested  by  center  loads  of  12  tons.     Def  &  U>  ^  inoh. 

14 


210 


STRENGTH   OF   MATERIALS. 


Span  15  ft. 
I  j    ,.      Br  load  12^  tons. 


Br    load    10.5    to 
11.6  tuns. 


"l«       Span  19  ft. 

52        Br  load  50  to  54  tons. 

By  formula,  p  194, 
it  should  have  been 
but  40  tons. 


15  X  2/j 


H* 


Span  30%  ft. 
Br  load  58  tons.* 


m 


In  describing  such  beams,  it  is  better  to  give  the  entire  depth  of  the  beam;  for 
when  the  depth  of  the  wel>  is  given,  doubts  arise  whether  it  is  meant  to  include  the 
thicknesses  of  the  two  flanges,  or  not.  Every  writer,  almost,  that  we  have  seen, 
leaves  us  in  this  doubt. 

REM..  In  beams  either  larg-er  or  smaller  than  these,  but  whose 
cross-sections  are  proportioned  exactly  as  these  are,  and  whose  spans  are  the  same 
that  these  have,  the  center  breakg  loads  will  be  as  the  cubes  of  their  cross-section 

,  %  ,  2,  3,  or  10  times  as  large  every 


lines.    Thus,  in  a  beam  which  is    i 


way,  except  in  span, 

the  breakg  load  will  be  TTToTT  '  T7  '  ^8  >  8  >  27»  or  100°    times  as  great. 

If  the  spans  also  differ,  first  find  the  load  as  above,  as  if  they  were  the 
game;  then  say,  as  the  new  span,  is. to  the  span  given  in  our  Figs,  so  is  the  breakg 
load  thus  found,  to  the  actual  breakg  load  for  the  new  beam.  Thus,  suppose  we  wish 
to  make  a  cast-iron  beam,  4  times  as  large  every  way  as  the  dimensions  given  in  the 
first  of  these  Figs ;  except  its  span,  which  is  to  be,  say  10  ft,  instead  of  4%  ft.  Here 
the  first  breakg  load  is  found  tobe4X4X±  =  64  times  as  great;  or  2  tons  X  64 
«=  128  tons.  Next, 

New  Span.        Span  in  Fig.  First  load.  Actual  load. 

10          :          4.5          : :          128          :          57.6  tons. 

In  such  cases  we  must,  however,  have  regard  to  Rem,  Art  11.  The  foregoing 
process  applies  equally  to  beams  of  any  other  shapes,  such  as  the  following  ones ;  or 
whether  solid  or  hollow,  &c;  and  of  any  other  materials;  so  that  if  we  have  all  the 
dimensions,  and  the  breakg  load  of  any  beam  whatever,  we  may  find  that  for  another 
one  of  the  same  material,  and  of  the  same  proportions  of  cross-section.  It  may 
become  advisable  in  important  cases,  to  even  make  one 
or  more  model  beams  of  some  hitherto  untried  form ; 
and  to  break  them,  in  order  to  find  the  breakg  weight 
of  the  actual  beam  of  the  same  material.  In  doing  this, 
the  defs  should  abo  be  measd,  in  order  to  see.whether 
those  of  the  actual  beam  may  not  be  too  great.  See 
Art  26,  &c ;  and  Art  46. 


Art.  37.    Figs  17  show  some  varieties  of  the 
rolled  I  beam.    They  have  two  equal  flanges, 
and  a  stem  or  web  of  uniform  thickness.    They  are 
3  v     called  6, 9, 15,  &c,  inch  beams  according  to  their  lotal 

GFor  their  strengths  as  beams  see  p  212, 
213;  and  as  pillars  or  struts,  p  638. 


'6.25 


17 


*  This  iron  was  "  Sterling's  toughened,"  haying  about  16  per  cent  of  wrought  scrap  melted  in  it. 
Kaeh  of  the  89  beams  was  tested  bjr  a  cent«r  load  of  20  tons,  which  produced  deft  of  from  %  to  ft  inca. 
Entire  length,  34^  ft. 


STRENGTH    OF    MATERIALS. 


211 


Table  of  safe  quiescent  distributed  loads  in  tons  (224O  Ibs) 
of  channel  bars  as  beams  with  the  web  vertical.  The  safe  loads  are  here 
taken  at  one-third  of  the  ultimate  one  for  iron  of  superior  quality ;  but  for  aver- 
age iron  it  will  be  better  in  practice  to  reduce  them  about  one-sixth  part  before 
deducting  the  wt  in  tons  of  the  span  of  the  beam  itself,  and  which  is  here  in- 
cluded in  the  loads.  'If  liable  to  much  vibration,  as  in  bridges,  deduct  one-third 
to  one-half.  The  beams  are  supposed  to  be  stayed  against  bend- 
ing horizontally.  Three  stays  per  beam  will  suffice  for  the  longest.  For 
a  table  of  these  beams  as  pillars,  see  p  640. 


Spn. 

Hvy 

wt. 

12  in 

Load. 

Def. 

Me 

wt. 

'd  12 

Load. 

in. 

Def. 

Hv 

Wt. 

y  loi 

Load. 

in. 

Def. 

Hf 

wt. 

<1  10 

Load. 

in. 

Def. 

Ft. 

Lbs. 

Tons. 

Ins. 

Lbs. 

Tons. 

Ins. 

Lbs. 

Tons. 

IDS. 

Lbs. 

Tons. 

Ins. 

10 

500 

17.82 

.13 

300 

12.32 

.13 

350 

1089 

.16 

230 

8.30 

.16 

14 

700 

12.50 

.26 

420 

8.75 

.26 

490 

7.77 

.31 

322 

6.00 

.31 

18 

900 

9.64 

.43 

540 

6.84. 

.43 

630 

6.07 

.52 

414 

4.64 

.52 

22 

1100 

7.87 

.65 

660 

5.62 

.65 

770 

4.91 

.78 

506 

3.75 

.78 

26 

1300 

6.66 

.90 

780 

4.73 

.90 

910 

4.10 

1.09 

598 

3.21 

1.09 

30 

1500 

5.80 

1.20 

900 

4.10 

1.20 

1050 

3.57 

1.45 

690 

2.77 

1.45 

II  vy  9  in. 

Med  9  in. 

Hvy  8  in. 

Med  8  in. 

10 

300 

8.29 

.18 

180 

6.07 

.18 

250 

6.25 

.20 

160 

4.73 

20 

14 

420 

5.90 

.35 

253 

4.34 

.35 

350 

4.46 

.39 

224 

3.48 

.39 

18 

540 

4.61 

.58 

324 

3.39 

.58 

450 

3.48 

.65 

288 

268 

.65 

2* 

660 

3.77 

.86 

396  i    277 

.86 

550 

2.86  i    .97 

352 

2.14 

.97 

26 

780 

3.19 

120 

468 

2.33 

1.20 

650 

2.4t   !  1.39 

416 

1.78 

1.39 

30 

900 

2.76 

1.61 

540 

2.03 

1.61 

750 

2.05     1.81 

480 

1.60 

1.81 

Hvy  7  in. 

Med  7  in. 

Hvy  6  in. 

Hvy  5  in. 

10 

200 

4.55 

.23 

140 

3.75 

.23 

110 

2.32 

.27 

100 

1.70 

.32 

14 

280 

3.30 

.45 

196 

2.68 

.45 

154 

1.61 

.58 

140 

1.16 

.63 

18 

360 

2.50 

.75 

252  !    2.14 

.75 

198 

1.34 

.87 

180 

0.89 

1.05 

22 

440 

2.05 

1.11 

308 

1.70 

1.11 

26 

520 

1.78 

1.56 

364 

1.43 

1.56 

Dimensions  of  the  above  channel  bars  in  ins,  from  out  to 
out  each  way. 


Depth. 

wt 

K 

Wdth 
of 

Flge. 

Avg 
Ths 
of 
Flge. 

Ths 
web. 

Area. 

Depth. 

wt 
per 
Ft. 

Wdth 
Flge. 

Avg 
Ths 
of 
Flge. 

Ths 
of 
web. 

Area. 

Ins. 

Lbs. 

Ins. 

Ins. 

Ins. 

Sq  Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Ins. 

Sqln. 

Heavy.  ...12 

50 

3.20 

.78 

.95 

15.01 

Heavy....  7 

20 

2.55 

.53 

.55 

600 

Medium..l2 

30 

2.70 

.78 

.45 

9.00 

Medium..  7 

14 

2.29 

.53 

.30 

4.21 

Light  12 

20% 

3.00 

.41 

.31 

6.25 

Light  7 

10^ 

1.98 

.44 

.22 

3.18 

Heavy  ..  10 

35 

2.97 

.72 

.72 

10.50 

Heavy....  6 

11 

2.00 

.50 

.25 

3.30 

Medium..!  0 

23 

2.61 

.72 

.36 

6.90 

Light  6 

7^ 

1.75 

.34 

.19 

2.25 

Light  10 

15% 

2.50 

.34 

.31 

4.62 

Heavy....  5 

10 

2.00 

.50 

.30 

3.00 

Heavy....  9 

30 

2.83 

.63 

.70 

9.00 

Light  5 

VA 

1.55 

.37 

.18 

1.9o 

Medium..  9 

18 

2.43 

.63 

.30 

5.40 

Heavy....  4 

73| 

1.78 

.41 

.28 

2.33 

Light  9 

14^ 

2.50 

.34 

.34 

4.30 

Light  4 

6 

1.54 

.34 

.23 

1.80 

Heavy....  8 

25 

2.64 

.59 

.64 

7.50 

Heavy....  3 

6 

1.71 

H4 

.40 

1.80 

Medium..  8 

16 

2.30 

.59 

.30 

4.80 

Light  3 

5 

1.50 

.34 

.20 

1.50 

Light  8 

12M 

2.00 

.47 

.26 

3.75 

Rein.  These  tables  are  condensed  from  the  useful  "Tables  and 
Information  on  Wrought  Iron,"  issued  by  Messrs  Carnegie  Brothers 
&  Co,  of  Pittsburgh,  Penna.  The  €o  make  to  order  heavier 

channels  without  extra  charge  per  ft,  by  increasing  the  thick- 
ness of  the  web,  but  not  altering  the  flanges  otherwise  than  that 
the  increased  ths  of  web  adds  to  their  out  to  out  width ;  their  aver- 
age thickness  remaining  unaltered.  They  also  make  Deck  Beams, 
Fig  18,  and  all  varieties  of  I,  L,  -f,  T,  <&c,  bars,  segment  columns 
of  various  kinds,  bridge  work,  roofs,  &c. 


212 


STRENGTH    OF    MATERIALS. 


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STRENGTH    OF    MATERIALS. 


213 


Art.  38.  Rolled  Iron  beams  of  the  New  Jersey  Steel  and 
Iron  Company. 

Made  at  the  Trenton,  N.  J.,  Iron  Works,  Cooper,  Hewitt  &  Co. 
Morris,  Wheeler  &  Co.,  Market  &  Sixteenth  Sts.,  Philadelphia,  agents.* 

Similar  to  the  Phoenix  beams,  Fig.  17,  p  210.  J 
See  "  Cautions  "  below  preceding  table. 


Depth 

Thicks. 

Flange 

Weight 

Load  Coef.    For 

Area  of 

of  beam. 

of  web. 

width. 

per  yd. 

use  see  below. 

section. 

Ins. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Tons. 

sq.  ins. 

loi^ 

.6 

5% 

200 

748000 

334 

20 

16% 

1A 

5 

150 

551000 

246 

15 

12«^ 

.6 

51^ 

170 

511000 

228 

17 

1-/4 

.48 

4.79 

125 

377000 

168 

12.5 

10  v^ 

.47 

5 

135 

360000 

161 

13.5 

10$ 

az 

4.1^ 

105 

286000 

128 

10.5 

9 

.58 

4/2 

125 

268000 

120 

12.5 

9 

% 

4 

85 

189000 

84.4 

8.5 

9 

.3 

'&A 

70 

152000 

67.9 

7.0 

8 

% 

4/2 

80 

168000 

75.0 

8.0 

8 

.3 

4 

65 

135000 

60.3 

6.5 

7 

% 

3/<> 

60 

1020<  X) 

45.5 

6.0 

6 

.3 

3/^ 

50 

76800 

34.3 

6.0 

6 

\/ 

3 

40 

62600 

28.0 

4.0 

5 
5 

\ 

3 

2% 

40 
30 

49100 
38700 

21.9 
17.3 

4.0 
3.0 

4 

A 

3 

37 

36800 

16.4 

3.7 

4 

g 

2% 

30 

30100 

13.4 

3.0 

Channel  iron,  upright,  thns  ]. 

l?6 

.68 

4 

140 

381000 

170 

14.0 

.42 

3 

85 

238000 

106 

8.5 

9  * 

.43 

3i/j 

70 

146000 

65.2 

7.0 

9 

.33 

20 

50 

104000 

46.4 

5.0 

6 

.40 

45 

58300 

26.0 

4.5 

6 

.28 

2y<* 

33 

45700 

20.4 

3.3 

5 

.20 

1^ 

19 

22800 

10.2 

1.9 

4 

.20 

]1^ 

16.5 

15700 

7.0 

1.65 

3 

.20 

1^1 

15 

10500 

4.7 

1.5 

The  above  table,  as  well  as  that  on  p  212,  is  issued  by  the  above  named  Co.  In 
both,  the  sate  quiescent  equally  distributed  load  is  taken  at  l<s  °? 
the  elastic  limit  of  fair  rolled  iron.  Where  there  is  vibration  the  loads  must  be 
reduced.  The  beams  are  supposed  to  be  stayed  ag-ainst  beiid» 
iiijj'  sideways.  Three  stays  per  beam  suffice  for  the  longest. 

To  find  the  safe  distributed  load  in  tbs  of  any  beam  in  the  table: 
Take  out.  the  corresponding  number  in  the  column  of  coefficients.  Divide  it  by 
the  clear  span  in  feet.  Deduct  the  wt  of  the  clear  span  of  beam.  Caution  1  opp. 

To  find  the  deflection  in  inches  which  this  safe  load  will  produce.  Square 
the  span  in  ft.  Divide  this  square  by  70  times  the  depth  of  the  beam  in  ins. 

General  rule  for  strength  of  rolled  I  beam.**  of  any  size. 
When  proportioned  somewhat  like  those  in  common  use,  and  stayed  against 
bending  sideways,  the  safe  (one-third  of  the  elastic  limit)  quiescent  uniformly 
distributed  load  in  tons  (2240  tbs)  may  be  found  by  the  following  approx  Itule. 
Add  together  the  area  in  sq  ins  of  an  entire  one  of  the  two  equal  flanges ;  and  one 
fourth  of  the  area  of  the  entire  web  of  the  full  depth  of  the  beam.  Mult  the  sum 
by  3  times  the  entire  depth  in  ins.  Divide  by  the  span  in  feet.  Deduct  the  wt 
of  the  span  of  the  beam  in  tons.  For  the  def  in  ins  under  the  load  so  found, 
mult  the  sum  of  the  two  areas  above  found  by  180  times  the  square  of  the  depth 
in  ins.  Call  the  product  P.  Then  mult  the  load  in  tons  by  the  cube  of  the  span 
in  ft.  Divide  this  prod  by  P. 

*  One  of  the  most  extensive  and  favorably  known  iron  houses  in  Philadelphia. 
J  Price  in  Philada,  1880,  about  4>£  cts  per  ft). 
In  Belgium  I  beams  are  rolled  21.6  ins  deep;  334  Ibs  per  yd. 


214 


STRENGTH    OF    MATERIALS. 


Art.  39.     Cooper   A    Hewitt's    box  •  beams,  Fig 

IP,  made  by  the  Trenton  Iron  Co.  These  beams  consist  of  two 
channel  irons,  of  6^  X  2%  X  \4  inch  ;  one  at  top,  and  one  at  bot- 
tom ;  and  of  two  vert  sides,  of  ^|  inch  plate-iron,  18  ins  deep.  They 
weigh  69  fbs  per  foot  run.  The  Co  makes  also  larger  and  variously 
modified  beams.  The  following  are  the  results  with  a  beam  like  the 
fig ;  '^'H  ft  J°ng,  19  ft  5  ins  clear  span.  The  ends  unconfined  further 
than  being  steadied  sideways. 


Cen  load.  ft>s. 

Def.  Ins. 

Cen  load.  fts. 

Def.  Ins. 

0 

H 

76862 

1H 

12990 

£ 

Interval  of  26  days. 

19920 

1  6 

81342 

iH 

24230 

¥ 

85524  at  once  a 

IT! 

28744 

crackling        noise 

32284 

72 

commenced.    In 

37844 

9 

10  min, 

2& 

42387 

78 

In  1  hour, 

3c 

46923 

ii 

90302 

3 

51460 

% 

With  a  side  defln  of 

iff 

55985 

1  3 

H 

This      increased 

60553 

« 

until  the  side  plates  gave  way 

65089 

11 

at  their  bottom  edges,  in  an 

69954 

a 

hour. 

Rule  for  strength  of  riveted  box  and  I  beams.  Figs  19  to  25. 
The  greatest  safe,  uniformly  distributed,  quiet  load  in  tons  (2240  Ibs)  for  such 
beams  of  any  size,  well  made,  and  so  proportioned  as  to  be  secure  against  yielding 
either  sideways  or  by  buckling,  may  be  found  near  enough  for  practice  by  using 
the  "General  Rule,"  p  213;  and  taking  only  three-fourths  of  the  resulting  load. 
If  the  top  and  bottom  flanges  diifer  in  area,  use  the  least  one. 

Art.  41.  Fairbairn  plate,  and  box 
beams,  or  girders.  These  are  made  of 
rolled  plates,  and  angle  iron,  riveted  together. 
The  plates  are  usually  from  /%  to  %  inch  thick. 
The  angle-irons  from  2^  by  "Eft  by  %  to  6  by  6 
by  1  inch.  The  rivets  from  %  to  1%  inch  diam ; 
and  driven  from  3  to  6  ins  apart,  from  center  to 
center.  Other  shapes  of  rolled  iron  are  also  fre- 
quently introduced,  as  the  channel-iron,  cc;  T 
iron,  t,  &c,  &c.  Figs  22  and  23  are  common  modes 
of  constructing  the  plate  beam  for  buildings, 
and  for  bridges  of  moderate  spans.  Fig  25  is  an- 
other form.  Frequently  several  thicknesses  of 
plate-iron  are  riveted  together,  to  form  the  top 
and  bottom  flanges,  to  obtain  the  reqd  amount 
of  section.  The  vertical  web,  r,  or  iv,  is  likewise 
frequently  composed  of  two  thicknesses.  In  Fig 
24  of  the  box-beam,  each  of  the  two  flanges  con- 
sists of  two  thicknesses.  This  form  of  beam, 
when  on  a  large  scale,  constitutes  the  tubular 
bridge. 

Mr.  Fairbairn  makes  transverse  area  of  the 
top  flange  equal  to  1%  that  of  the  bottom  one. 

He  states  that  with  this  proportion  no  separate  allowance  need  be  made  for  the  rivet- 
holes,  &c;  that  large  experience  sanctions  it,  and  that  no  gain  of  strength  attends 
making  either  one  of  the  flanges  to  exceed  this  limit.  There  is  roor  *—  J— " 

rimpntsai 

The  angle- 


ngth 

maitiug  t.ituei-  unu  ui  uie  uauges  10  exceeu  uns  iimir.     A  here  is  room  for  doubt 
whether  his  proportion  for  the  flanges  is  the  best.     More  experiments  are  needed  on 
this  point.     He  gives  no  rules  for  the  thickness  of  the  vert  members, 
irons  a  a,  the  channel-irons  c  c,  &c.  are  to  be  con       '    ' 
such  iron 
must  the 


a  a,  the  channel-irons  cc,  <&c.  are  to  be  computed  as  part  of  the  flanges.  All 
rons,  as  also  any  others  used  in  the  bottom  flanges,  are  subject  to  tension;  and 
therefore  be  strongly  spliced  together  at  their  joints. 


re  ue  strongly  spiiceu  logemer  at  uieir  joinis. 

Many  engineers,  however,  make  both  flanges  of  equal  area;  conceiving  that 
portion  to  be  safer  in  consequence  of  the  weakness  of  the  lower  joints.  The  u 
flanges  being  compressed,  are  not  appreciably  weakened  by  their  joints. 


pro- 
upper 


STRENGTH    OF    MATERIALS.  215 


For  the  strength  of  the  single-plate  beam,  Figs  22  and  23,  Fair- 
bairn  gives  as  follows :  Tr  are^  of  bofc 
Cen  breakg  load  in  tons  ")          flange  in  sq  ins  v  Total  depth  of  beam  v  Constant 

including  yz  weight  '  \  =         in  sq  ins        A  in  ins  A         75 

of  the  beam  itself,       J  Clear  span  in  ins. 

For  the  box-beam,  Fig  24,  precisely  the  same,  except  use  a  constant  80  in- 
stead of  75.  Some  very  competent  authorities,  however,  regard  this  as  too  great; 
and  maintain  that  the  constant  should  not  exceed  75  in  either  case. 

Half  the  wt  of  the  beam  must  be  deducted,  to  obtain  the  neat  center  breakg  load; 
or  the  whole  of  it  for  a  uniform  load.  In  beams  for  buildings,  however,  or  in 
bridges  under  40  to  50  ft  span,  this  deduction  will  rarely  be  necessary. 

As  a  general  approximation,  when  these  beams  exceed  a  very  moderate  span  as 
bridges,  about  ^  to  ^  part  must  be  added  to  the  wt  of  wrought  iron  as  deduced  from 
a  neat  transverse  section,  such  as  those  in  the  above  Figs,  to  allow  for  splicing- 
plates,  stifferiers,  chucks,  rivets,  &c. 

The  following  girders  have  for  many  years  sustained  locomotive  traffic: 
1st, like  Fig 23;  single  track;  2  girders,  15  ft  apart  from  center  to  center;  length 
of  each,  28  ft;  clear  span,  24  ft;  total  depth  of  girders,  28  ins;  top  and  bottom  T- 
irons,  t,  horizontally  7%  X  J^;  vert^  flange,  3  Xj^j  the  four  longitudinal  splicing- 

prd- 


plates,  *  s,  each  6  X  %'•>  vert  web  u,  %  thick.  2d,  like  Fig  22 ;  single  track ;  2  gi 


arid  before  the  proper  proportions  were  known.  3d,  like  Fig  23  ;  single  track  ;  2 
girders,  length  of  each  68  ft;  clear  span,  60  ft  ;  total  depth  of  girders,  4  ft  3  ins;  top 
and  bottom  T-irons,  t,  6  X  yV  5  vert  flange,  3  X  K'  tne  4  'longitudinal  splicing-plates 


,  , 

s,Tach  6%  X  /4  ;  vert  web,  w,  %  inch  thick,  luveted  to  the  top  and  bottom  are 
plates  like  the  dotted  ones  in  Fig  22;  each  of  these  is  2  ft  wide,  %  inch  thick  at  the 
center  of  the  girders,  and  %  at  the  ends.  All  have  vertical  stiffeners. 

The  following  are  in  use  on  the  Charing  Cross  Railway,  England:  all  like 
Fig  22.  1st,  for  a  single  track,  41  ft  clear  span,  total  length  46  ft  ;  total  depth  2^  ft  ; 
top  plate,  dotted,  16  X  %:  bottom  plate  15  X  M  ;  vert  web,  w,  %  thick  ;  top  angle- 
irons^  X  4  X  %;  bottom  ones  4  X  4  X  %\  rivets  %  diam,  3  ins  apart  from  center  to 
center  ;  weight  of  1  girder,  4  %tons.2d,for  a  single  track,  51  ft  clear  span,  total  length 
56;  total  depth  3^  ft  ;  top  plate,  dotted,  18  X  %  ;  bottom  plate  16  X  %;  vert  web, 
w,  %  thick;  top  and  bottom  angle-irons,  a,  6,  4  X  4  X  %\  rivets  %  diam,  4  inches 
apart  from  center  to  center;  weight  of  each  girder,  5^  tons.  3d.  For  three  railway 
tracks,  2  girders  only,  39  ft  apart  from  ceu  to  cen;  clear  opan  73;  total  length  82  ; 
total  depth,  7%  ft  at  center  of  span  ;  6  ft  at  ends  ;  top  plate,  dotted,  2  ft  2  ins  wide  ; 
composed  of  4  thicknesses  of  plates,  each  %  thick,  at  center  of  span  ;  and  3  such 
plates  at  the  ends  ;  bottom  plate  2  ft  wide,  one  thickness  of  %;  and  3  of  %  each,  at 
the  center  of  the  span  ;  at  the  ends  one  of  the  %  is  omitted.  Vert  web,  w,  1^  thick 
at  centre  of  span  ;  '%  for  11  ft  at  each  end;  top  and  bottom  angle-irons,  6  X  6  X  M; 
weight  of  each  girder,  25  tons;  which  would  suffice  for  a  single-truck  girder  100  ft 
span.  Web  stiffened  every  3  ft  ;  as  at  i  ?',  Fig  28  ;  by  Ts  of  6  X  3  X  %• 

The  transverse  iron  floor-girders,  when  the  main  beams  are  but  about  15  ft  apart, 
fora  single  track,  and  the  girders  about  5  to  6  ft  apart,  are  like  the  shaded  parts  of 
Fig  22;  that  is,  without  top  and  bottom  plates.  Depth  about  Id  iris  ;  vert  web  ^ 
inch  thick;  4  angle-irons,  each  3^  X  3^  X  Y^\  weight  about  800  JJbs.  In  such  of 
the  Charing  Cross  railway  bridges  as  have  their  main  beams  39  ft  apart,  and  where 
the  transverse  floor-beams,  also  39  ft  long,  support  3  tracks,  the  last,  3  ft  apart,  have 
the  following  dimensions:  Total  depth,  16  ins;  top  and  bottom  plates,  dotted,  15  X 
%  each  ;  vert  web  y±  thick  ;  four  angle-irons  of  5  X  3  X  %>  Transverse  floor-beams 
may  rest  upon  the  top  of  the  main  beams  ;  or  upon  the  inside  portion  of  the  bottom 
flange.  The  last  is  not  as  favorable  to  the  strength  of  the  main  beams  as  the  former  ; 
or  as  when  the  floor-beams  are  placed  beneath  the  bottom  of  the  main  ones.  In  all 
cases  the  two  beams  are  riveted  together. 

Art.  42.  The  construction  of  plate  and  tnbiilai  girders  is 
not  as  simple  as  might  be  supposed  from  the  foregoing  Figs.  They  are  composed  of 
separate  sheets  of  iron,  not  exceeding  about  3  X  12  ft  in  breadth  and  length  in  the 
largest  bridges;  while  in  small  ones,  much  smaller  dimensions  must  be  employed. 
"Whenever  two  plates  come  into  contact,  the  joint,  whether  vert  or  hor,  must  be 
strengthened  by  riveting  upon  both  sides  of  it  at  least  narrow  strips  of  plate  iron 
6  to  8  ins  wide,  called  covering-plates,  or  splicing  plates.  Or,  since  the  girders  re- 
quire vert  stiffeners»the  vert  joints  are  frequently  covered  on  both  sides  of  the  platei 


216 


STRENGTH    OF    MATERIALS. 


Fig  ZJ 


by  vert  T-irons,  as  ww,  Fig  26;  in  which  a  a  represent  two  plates  whose  vert  joint 
is  to  be  thus  strengthened.    When  the  middle  web  w,  of  the  T-irons,  does  not  project 

sufficiently   to   impart    the 
L  ,      reqd    stiffness   to   the   vert 

.  o  ^m>^>  o  web,  tt  or  #,  of  the  beam, 
broad  strips  s,  Fig  27,  of 
plate  iron,  may  be  intro- 
duced instead  of  it,  in  con- 
nection with  4  angle-irons, 
as  bb.  These  last  are  riv- 
eted together  through  the 
vert  plates  gg  of  the  main 
beam ;  and  through  the  stiff- 
eners  ss-  thus  protecting  the  joint  of  the  former,  and  holding  the  latter  firmly  in 
place.  These  stiffening-plates  are  frequently  made  to  project  more  at  bottom  than  at 
top;  and  are  at  times  strengthened  by  angle-iron  riveted  along  their  outer  edge  and 
at  their  base ;  thus  making  them  very  effective  as  braces  also. 

The  transverse  floor-beams  or  gird- 
ers, extending  from  one  main  beam  to  the  other, 
are  so  spaced  as  to  meet  the  stiffeners,  instead  of  rest- 
ing upon  the  weaker  intermediate  parts  of  the  main 
girders.  These  transverse  beams  themselves  generally 
require  to  be  stiffened  in  a  similar  manner,  at  inter- 
vals of  usually  from  3  to  8  It,  as  in  the  main  girders. 
The  stiffeners  near  the  ends  of  the  main  girders,  or 
those  resting  on  the  piers  and  abuts,  are  (especially 
in  large  spans)  placed  nearer  together,  and  made 
stouter  than  the  others,  because  upon  them  rests  the 
wt  of  bridge  and  load.  The  T  and  L  irons,  &c,  used 
to  stiffen  the  sides  of  the  Britannia  tubes,  460  ft  span, 
weigh  half  as  much  as  the  plates  which  compose  the 


The  vert  T-irons  to,  or  angle-irons  &,  generally  have  to  be  bent 
both  at  top  and  bottom,  as  shown  at  i  ?',  Fig  28,  in  order  to  pass  the 
hor  angle-irons  of  the  upper  and  lower  flanges.  The  joints  of  the 
•c  long  narrow  strips  a,  Fig  28,  which  compose  the  top  and  bottom 
plates  of  the  flanges,  must  also  be  connected  together  by  riveted 
splicing-plates.  Those  at  the  lower  flanges  require  especial  care  in 
this  respect,  inasmuch  as  they  undergo  tension.  The  same  applies 
to  the  hor  angle-irons  h  //,  Fig  2v),  which  also  must  be  firmly  con- 
nected at  their  ends.  This  is  done  by  pieces  cc  of  bent  iron  in 
lengths  of  18  to  24  inches. 
Art.  43.  Mr.  William  Fairbairn,  of  England,  gives  the  fol- 
lowing table,  calculated  for  a  double-track  railway,  supported  by  two  box 
beams,  or  by  two  girders  He  adopts  6  tons  per  foot  run  of  the  span,  as  the  center 
breakg  load  of  the  bridge ;  or  12  tons  per  foot  run,  uniformly  distributed,  including 
the  wt  of  the  bridge  itself.  This  12  tons,  he  says,  is  equal  to  about  six  times  the 
maximum  load  that  can  practically  be  brought  upon  the  bridge.  For  spans  up  to  100 
ft,  he  takes  1  ton  per  running  foot  as  the  wt  of  the  bridge  itself,  and  2  tons  as  that 
of  the  rolling  load,  or  two  trains;  and  from  100  to  300  ft  spans,  1^  tons  as  the  wt  of 
the  Bridge;  and  1%  tons  as  that  of  the  load;  the  total  being  3  tons  per  foot  in  both 
cases. 

The  depths  in  the  table  are  equal  to  ^  of  the  span,  for  spans  less  than  150  ft:  for 
those  exceeding  150  ft,  they  are  ^.*  The  breakg  loads  include  the  wt  of  the  bridge 
itself.  The  depths  are  those  at  the  center  o*"  the  span.  Except  in  quite  small  spans, 
some  saving  of  material  is  effected  by  diminishing  the  depth  from  the  center  to  the 
ends.  Single-track  bridges  will  require  but  about  -fs  the  quantity  of  material 
given  in  the  table;  the  depth  of  girder  remaining  the  same.  If  the  depth  is  dimin- 
ished, the  areas  of  the  flanges  must  of  course  be  increased;  or  if  the  depth  is 
increased,  the  flanges  may  be  reduced;  for,  as  is  seen  by  the  foregoing  formula,  Art 
44,  the  strength  varies  as  the  depth,  when  the  flanges  remain  the  same.  The  propor- 
tions of  the  table  will  serve  for  plate  beams,  as  well  as  box;  the  diff  of  strength 
being  but  as  75  to  80. 


*  These  depths  are  much  smaller  than  is  usual  in  truss  bridges  In  the  United  States.    Here  from 
j*£  to  fy  are  the  common  ones ;  and  are  certainly  to  be  preferred  for  practical  reasons  of  econom?. 


STRENGTH   OF   MATERIALS. 


217 


Table  by   Mr.    Fairbairn,   of  the    Proportions   of  Tabular 
Girder  Bridges,  consisting?  of  two  girders,  of  from   3O  to 
3OO  ft  span  ;  and  intended  to  sustain  two  railway  tracks. 

(The  sufficiency  of  these  dimensions  has  been  questioned  by  high  authority.) 

Clear  Span. 

Center  Breaking 
Load  of  Bridge, 
or  of  the  two 
Girders. 

Sectional  Area 
of  bottom  of  one 
Girder. 

Sectional  Area 
of  top  of  one 
Girder. 

Depth  of  a 
Girder,  at 
the  center  of 
the  Span. 

Feet. 

Tons. 

Inches. 

Inches. 

Feet.    In. 

30 

180 

14.63 

17.06 

2         4 

35 

210 

17.06 

19.91 

2        8 

40 

240 

19.50 

22.75 

3        1 

45 

270 

2194 

25.59 

3        6 

50 

300 

24.38 

28.44 

3      10 

55 

330 

26.81 

31.28 

4        3 

60 

360 

29.25 

34.13 

4       7 

65 

390 

31.69 

36.97 

5        0 

70 

420 

34.13 

39.81 

5        5 

75 

450 

36.56 

42.67 

5        9 

80 

480 

39.00 

45.50 

6        2 

85 

510 

41.44 

48.34 

6        7 

90 

540 

43.88 

51.19 

6      11 

95 

570 

46.31 

54.03 

7        4 

*100 

600 

48.75 

56.88 

7        8 

110 

600 

53.63 

6256 

8        6 

120 

720 

58.50 

68.25 

9        3 

130 

780 

63.38 

73.94 

10        0 

140 

840 

68.25 

79.63 

10        9 

150 

900 

73.13 

85.31 

11        6 

160 

960 

90.00 

105.00 

10        8 

170 

1020 

95.63 

11156 

11        4 

180 

1080 

101.25 

118.13 

12        0 

190 

1140 

106.88 

124.69 

12        8 

200 

1200 

112.50 

131.25 

13          4 

210 

1260 

118.13 

137.81 

14        0 

220 

1320 

123.75 

14438 

14        8 

230 

1380 

129.38 

150.94 

15        4 

240 

1440 

135.00 

157.50 

16        0 

250 

1500 

14063 

164.06 

16        g 

260 

1560 

146.25 

170.63 

17        4 

270 

1620 

151.88 

177.19 

18        0 

280 

16SO 

15750 

183.75 

18        8 

290 

1740 

163.13 

190.31 

19        4 

300 

1800 

168.75 

196.88 

20        0 

Art.  44.  Moments  of  Rupture  and  of  Resistance.  By  Moment 
of  Rupture  or  Breaking  Moment  is  meant  the  tendency  of  a  load  (including  the 
wt  of  the  beam  itself  or  not,  as  the  case  may  be)  to  break  a  beam  by  a  lengthwise 
pulling  apart  of  some  of  its  fibres  and  a  crushing  of  others,  by  the  aid  of  leverage 
afforded  by  the  beam  itself.  It  is  often  called  simply  the  moment  of  the  load.  It 
is  also  this  moment  that  tends  to  bend  the  beam  as  a  necessary  consequence  of  the 
stretching  and  crushing  alluded  to;  and  when  considered  with  reference  to  its 
bending  effect  instead  of  (as  now)  its  straining  and  breaking  ones  it  is  called  the 
Bending'  Moment  or  Moment  of  Flexure,  or  of  Deflection,  of  the  load. 
The  load  and  the  wt  of  the  beam  together  tend  also  to  sever  the  beam  transversely 
or  across  its  length  by  a  process  called  shearing.  See  "  Shearing,"  p  642,  but  with 
that  we  have  nothing  to  do  here. 

The  following  rules  both  on  Moments  of  Rupture  and  on  Strains  apply  to 
horizontal  closed  beams  (see  Open  and  Closed  Beams,  p  644)  of  any  form 
whatever  of  cross  section,  whether  rectangular,  circular  or  like  Figs  22  to  24,  p 
214,  &c.  Those  on  moments  apply  also  to  hor  open  beams  like  the  trusses  of 


a  bridge,  &c,  in  which  the  load  is  supposed  to  be  concentrated  at  the  panel-points. 
But  in  such  the  shearing  tendency  vanishes  (see  Rem  2,  Art  5,  p  644);  and  the 
manner  of  resisting  the  moment  of  rupture,  as  also  the  strains  caused  thereby, 
differ  entirely  from  those  of  closed  beams. 


218  STRENGTH   OF   MATERIALS, 

If  the  beam  is  inclined,  the  moment  of  rupture  may  still  be  found  by 
using  the  hor  span  and  segments  instead  of  the  inclined  ones;  but  the  resulting 
longitudinal  strains,  as  well  as  the  shearing  forces  become  changed,  involving 
much  complication.  We  confine  ourselves  therefore  to  hor  beams. 

This  subject  must  not  be  confounded  with  the  principle  of  Equal- 
ity of  UIoiiieiits,  Art  51,  p  476,  frequently  used  for  finding  the  strains  along 
the  members  of  a  truss. 

A  load  causes  no  moment  or  strain  on  any  part  of  a  beam  that  is 
not  between  the  load  and  the  assumed  fulcrum  point.  Thus  in  Fig  29  calling  c 
the  fulcrum  of  the  load  on  c  x,  no  load  between  c  and  e  produces  any  moment 
between  c  and  /,  because  it  has  no  fulcrum  and  hence  no  leverage  there. 

The  weight  of  the  beam  itself  is  not  here  included.  When  required 
to  be  so,  consider  it  as  a  uniform  load,  and  use  Case  3  or  Case  12,  p  220  and  221, 
and  add  the  result  to  that  obtained  for  the  load. 

The  deflection  in  ordinary  cases  may  be  found  by  the  rule  on  page  198. 

Art.  45.  Having  the  moment  of  rupture  of  the  load,  it  is  necessary 
to  know  whether  the  moment  of  Resistance,  or  simply  the  Resistance 
of  the  beam  (which  term  see  Art  5  of  Open  and  Closed  Beams,  p  646)  is 
sufficient  to  withstand  it.  This  Resistance  in  any  hor  solid  or  closed  beam  of  any 
form  whatever  of  cross-section  may  be  found  thus,  subject  however  to  the  first 
paragraph  of  Art  2.%  p  194.  Mult  its  Moment  of  Inertia  in  ins  by  the  Constant 
of  Rupture  (terms  defined  on  p  195),  and  divide  the  prod  by  the  dist  in  ins  of  the 
farthest  fibre  (or  according  to  Prof  Wood,  of  the  farthest  fibre  on  the  side  which 
will  yield  first.)  from  the  Neutral  Axis,  or  cen  of  grav  of  the  section.  Or  as  the 
rule  is  usually  expressed  by  formula,  I  C  -f-  t.  In  a  rectangular  or  circular  beam, 
or  in  one  with  equal  flanges  and  uniform  web,  or  in  any  other  in  which  the  parts 
above  and  below  the  neutral  axis  are  similar  and  equal,  the  dist  of  the  farthest 
fibre  will  in  any  event  be  half  the  depth  of  the  beam.  All  the  dimensions 
must  be  in  the  same  measure,  that  is  all  in  ft,  or  all  in  ins. 

The  Constant  for  Rupture  for  average  rolled  iron  is  about  45000  fcs  or 
say  20  tons  per  sq  inch.  Cast  iron  36000  ft>s  or  16  tons.  Good  straight-grained, 
well-seasoned  white  pine  or  spruce  8100  fos  or  3.6  tons ;  yellow  pine  9000  or  4 ;  good 
oaks  10000  or  nearly  4.5.  But  as  large  beams  are  liable  to  defects  and  imperfect 
seasoning,  not  more  than  about  two-thirds  of  tnese  constants  should  be  used  in 
practice.  See  Table,  p  185.  For  either  square  or  rectangular  beams 
it  will  be  easier  and  as  correct  to  find  the  mom  of  res  thus.  Mult  together  the  sec- 
tional area  in  sq  ins,  the  depth  in  ins,  and  the  constant  for  rupture  per  sq  inch ; 

and  divide  the  prod  by  6.    This  is  usually  expressed  by  formula,  R  =  — - — .   For 

a  solid  flanged  beam  of  equal  flanges  and  uniform  web  it  will  be  suf- 
ficiently close  to  find  its  resistance  thus.  Mult  the  area  of  only  one  entire  flange 
by  the  depth  between  the  centers  of  grav  of  the  two  flanges.  Also  mult  one-sixth 
of  the  area  of  the  web  (in  clear  of  the  flanges)  by  its  depth.  Add  the  two  pro- 
ducts together.  Mult  the  sum  by  the  ult  tensile  or  compressive  strength  per  sq 
inch  (whichever  is  least)  of  the  material.  For  average  wrought  iron  this  may  be 
taken  at  36000  fos  or  16  tons.  For  average  cast  iron  at  18000  ros  or  8  tons  when  the 
flanges  are  equal. 

Rein.  1.  Theoretically  the  webs  of  closed  flanged  beams  need  only 
be  strong  enough  to  bear  safely  the  vert  shearing  forces  of  the  load,  (see  "Shear- 
ing," p  642),  and  in  practice  this  view  may  answer  for  quite  short  beams;  but  in 
long  ones  there  is  a  tendency  to  warp  or  twist  sidewise,  which  must  be  met  by 
stiffening  the  web  either  by  an  increase  of  thickness,  or  by  introducing  vert 
stitfeners  i  ?;,  Fig  28,  p  216,  or  in  some  other  way.  We  cannot  pretend  to  give  rules 
for  either  of  these.  Perhaps  the  best  advice  is  to  observe  the  stiffeners  of  flanged 
girders  of  successful  plate-iron  bridges,  of  which  a  few  examples  will  be  found  on 
p  215. 

The  web  members  of  open  beams  must  be  calculated  like  those  of 
truss  bridges. 

Rem.  2.  The  horizontal  strain  at  any  given  point  of  a  chord  of 
a  hor  open  beam,  is  found  by  dividing  the  moment  of  rupture  at  that  point,  by 
the  depth  of  the  beam  between  the  centers  of  grav  of  the  flanges  at  the  same 
point,  see  Art  9,  p  647.  In  closed  flanged-beams  also  the  same  rule  is  fre- 
quently used  as  being  safe  and  sufficiently  correct  for  practice.  The  span,  depth, 
&c,  must  all  be  in  the  same  dimension,  that  is  all  in  ft,  or  all  in  inches,  &c. 

Art.  46.  In  ordinary  practice  beams  of  wood  or  iron  are  used  either  as  hor 
cantilevers,  as  Fig  29,  firmly  fixed  at  one  end,  or  as  hor  beams  supported  at  each 
end,  -as  Fig  31. 


STRENGTH   OF   MATERIALS. 


219 


I 


General  Rule  for  moments  of  rapture  in  fior  cantilevers,  no 

matter  how  irregularly  the  load  or  loads  may  be  distributed.  Bear  in  mind  that 
only  that  part  of  the  load  which  is  beyond  (towards  the  free  end  from;  any  as- 
sumed point  tends  to  break  the  beam  at  that  point  as  a  fulcrum,  and  that  it  does 
so  with  a  leverage  =  dist  of  the  cen  of  grav  of  that  part  of  the  load  from  the  point. 
The  other  part  of  the  load  has  no  moment  at  that  point.  Thus  the  whole  load 
o  x  tends  to  break  the  beam  at  g  or  I  with  a  leverage  =  a  g  or  a  i  as  the  case  may 
be,  a  being  the  cen  of  grav  of  the  load.  And  so  for  the  moment  at  any  other 
point  c,  Fig  29,  as  a  fulcrum,  find  the  wt  of  all  the  load  c  x  between  c  and  the  free 
end  /  of  the  beam.  Also  find  the  cen  of  grav 
*  of  that  part  of  the  load.  Mult  the  weight 
just  found  by  its  leverage  c  s. 

Example  1.  We  use  a  uniform  load  in 
order  to  illustrate  the  rule  more  readily.  Let 
the  hor  yellow  pine  beam  i 1  be  7  ft  long;  its 
breadth  and  its  depth  i  e  each  6  ins ;  the  whole 
load  o  x  4  tons ;  and  c  the  point  or  fulcrum  at 
which  the  moment  of  the  load  is  reqd.  Then  the  wt  of  the  load  between  c  and  t 
Is  3  tons ;  and  its  cen  of  grav  s  is  1.5  ft  or  18  ins  from  c.  Hence  the  loads  moment 
at  c  =  3  tons  X  18  ins  leverage  =  54  inch-tons.  That  is,  a  load  of  3  tons  tends 
with  a  leverage  of  18  ins  to  rupture  the  beam  at  c.  Now  is  the  resisting  moment 
or  the  strength  of  the  beam  at  c  sufficient  to  withstand  this?  Using  our  first  rule 
we  have  Moment  of  Inertia  in  ins  (p  195)  is  =  (its  area  X  square  of  depth)  -t-  12 
=  (36  X  36)  -=-  12  =  108 ;  the  Constant  of  Rupture  for  average  yellow  pine  is  (p 
185)  500  X  18  =  9000  ft>s  or  4  tons ;  and  the  dist  of  the  farthest  fibre  (on  either 
side)  from  the  neutral  axis  is  3  ins.  Consequently  we  have  the  resistance  of  the 
beani  =  (108  X  4)  -r-  3  =  144  inch-tons,  or  2.67  times  the  moment  of  the  load ;  that 
is  the  beam  has  a  safety  of  2.67.  Or  by  the  formula  R  =  ad  C  -  " 
we  have  the  resistance  =  (36  X  6  X  4)  +•  6  =  144  inch- tons  as  before. 


-  6  (Art  45,) 


Example  2.  Let  Fig  30  be  a  rolled  iron  I  beam  cantilever  of  the 
cross  section  shown  in  ins  at  S,  projecting  hor  10  ft  or  120  ins,  and  bearing  a  con- 
centrated load  of  2  tons  at  its  free  end.  Its 
moment  of  inertia  (p  196)  in  ins  is  184;  Con- 
stant of  Rupture  p  185  =  (2250  X  18)  =  40500 
ibs  =  18.08  tons.  Dist  of  farthest  fibre  from 
neutral  axis  on  either  side  =  half  depth  =  5 
ins.  Hence  its  moment  of  resistance  or  (I 
X  C)  -f-  t  at  any  section,  =  (184  X  18.08  -f-  5)-= 
665.  3  inch-tons.  The  moment  of  the  load  at 
the  section  i  e  is  =  2  X  120  =  240  inch-tons, 
therefore  the  beam  has  a  safety  of  2.77  at  i  e. 

By  the  approx  process,  Art  45.  the  moment  of  the  beam  is  (4  X  9)  -f 
(.667  x  8)  X  16  =  661.4  inch-tons,  or  nearly  as  before. 

Art.  47.  General  Rule  for  M  of  Hup  in  hor  beams  supported 
at  each  end,  no  matter  how  irregularly  the  load  or  loads  may  be  distributed. 
Let  i  n,  Fig  31,  be  such  a  beam  of  yellow  pine 
of  6  ft  or  72  ins  span,  6  ins  square,  and  loaded 
with  3  tons.  First  find  the  cen  of  grav  c  of 
the  whole  load  a  r,  and  what  portion  (1.25 
and  1.75  tons)  of  said  load  rests  on  each  sup- 
port i  and  w,  thus,  as  whole  span  :  whole  load 
: :  either  arm  :  portion  at  other  arm.  Con- 
sider the  upward  reactions  thus  found  (1.25  and  1.75  tons)  of  the  two  supports  to 
be  two  forces  acting  vert  upwards  against  the  ends  of  the  beam  at  f  and  n  as  de- 
noted by  the  arrows.  Let  o  be  any  point  whatever  in  the  beam  at  which  as  a  ful- 


,31. 


»01.  t- 

iqji  e  a  SCO  xnjg 


follow  if  we  use  n  and  the  1.75  tons  reaction,  but  with  the  load  x  o. 

Item.  1.  If  there  is  110  load  between  i  and  the  fulcrum  point,  as  would 
be  the  case  if  the  moment  had  been  reqd  at  any  point  between  i  and  a  instead  of 
at  o,  then  the  above  p  by  itself  is  the  moment.  Thus  e  is  12  ins  from  t,  hence  the 
moment  at  e  of  the  entire  load  a  x  is  1.25  X  12  =  15  inch-tons. 


220  STRENGTH   OF   MATERIALS. 

Rem.  2. 

end  i  or 
it  prod 
p642. 

Rem.  3.  The  resistance  of  the  beam,  Fig  31,  at  any  section,  or  its  a  d 
C  •*-  6  (Art  45),  is  =  (36  X  6  X  4  tons)  -=-  6  =  144  inch-tons.  Therefore  at  o  it  has  a 
safety  of  144  -H  36  =  4. 

Art.  48.    Although  the  foregoing  general  rules  apply  to  all  the  following 
cases,  still  these  last  will  often  expedite  calculations. 

Case  1.    Concentrated  load  at  free  end.  Fig  32. 

Greatest  moment  is  at  o,  and  =  load  Xo  n.  At  any  other  point 
a  it  is  =  load  X  «  n.  Make  o  v  =  greatest  moment,  join  v  n; 
then  a  cis  the  moment  at  any  point  a.  For  the  strain  in 
tons  or  Ibs  attending  the  moment  in  this  case  or  in  any  of 
the  following,  see  Art  45. 

Rem.  The  dotted  line  v  n  of  moments,  and  those  in  some  of  the  other  figs,  do 
not  entirely  control  the  modifications  of  beams,  Figs  5  to  9, 
but  assist  the  shearing  forces  in  doing  so. 

Case  2.    Concentrated  load  at  any  point  a,  Fig 

33.  Greatest  moment  is  at  o,  and  =  load  X  o  a.  At  c  it  is  = 
load  X  c  a.  Make  o  v  =  greatest  moment,  join  v  a.  Then  c  e 
is  the  moment  at  any  point  c.  The  load  has  no  moment  be- 
tween a  and  n. 

Case  3.    Uniform  load  throughout,  Fig 34.  Greatest  moment  is  at  o, 
and  =  whole  load  X  half  o  n.    At  n  it  is  o.    At  any  point  a  it 
Qll          is  =  load  on  a  n  X  half  a  n.     Make  o  v  —  greatest  moment, 
**C  •-*"•       draw  the  dotted  parabola  with  its  vertex  at  n.    Then  a  c  gives 

fv the  moment  at  any  point  a. 

If  the  load  is  not  uniform  the  greatest  moment  is  » 


0 


a    TV 


whole  load  X  dist  from  o  to  its  cen  of  grav. 


d  and  o,  i  "" 
ment  =  (\\-4- 
e  cen  of  "'  4- 


Case  4.     Load  on  one  part,  Fig 

35.  Greatest  moment  is  at  o,  and  =  load  X  dist  from  o  to  cen 
of  grav  c  of  load.  At  any  point  t  between  the  load  and  o, 
moment  =  load  X  tc.  At  any  point  a  in  the  load,  mor 

load  on  a  s  X  dist  a  e  of  th 

grav  of  load  on  o  *  from  a. 

Case  5.  Several  loads,  w  x  y,  Fig  36.  Find 
their  centres  of  grav  c,  a,  s.  Greatest  moment  is  at  o, 
and  =  wXco  +  xXao  +  yXso.  Or  first  find  the 
common  cen  of  grav  of  all  the  loads,  and  mult  its  dist 
from  o  by  the  sum  of  the  three  loads.  Between  the 
loads  the  moment  at  g  =  y  X  g  s; 

Case  6.  One  uniform  load  and  one  local 
one,  Fig  37.  Greatest  moment  is  at  o.  Find  that  of  the 
uniform  one  by  Case  3 ;  and  that  of  the  local  one  by  case 
4,  and  add  them  together.  No  moment  between  a  and  n. 

0  Case  7.    Concentrated 

S  load    at    center,    Fig.   38. 

Greatest  moment  is  at  center,  and  =  half  load  X  half 

\  \  span.    At  the  supports  it  is  o.    Make  c  s  =  moment  at 

f-     r  *\*^  center,  join  s  o,  s  a ;  then  n  t  —  moment  at  any  point  ». 

y       (J     j  \.  Or tlie  moment  at  any  point  n  =  half  load  X  a  n,  n  being 

0*|             p      M — tjir  the  nearest  support. 

CaseS.  Concentrated  g 

load  not  at  center.  Fig.  x~x 

39.    Greatest  moment  is  at  the  load,  and  is  =  (load  X  eo          A  / 
X  «  «)  -T-  o  a.    Make  e  s  =  moment  at  load,  join  s  o,  s  a ;  v  /-^ 

then  at  any  point  c  the  moment  is  c  t.    Or  at  any  point         /{  (  jp 

c.  moment  =  (load  X  a  c  X  a  e)  -+•  span  o  a ;  a  being  always  7\f~f*~ft~ 

the  support  nearest  the  point.    No  moment  at  o  or  a.  Ul  L    " 


Jll 
(ill 


STRENGTH   OF   MATERIALS. 


220* 


Case  9.  Several  concentrated  loads  x  y  z,  Fig  40.  By  Case  8  find 
the  greatest  moment  of  each  load  separately, 
and  for  each  of  them  draw  its  dotted  vertical 
and  two  inclined  lines  as  in  this  fig.  Then  for 

hxx"  ^L  the  moment  at  any  point  whatever  ase,  measure 

/'  \  the  vert  dists  (in  this  case  e  o,  e  a,  e  c)  to  the 

sloping  lines,  and  add  them  together.    For  it 
is  plain  that  at  e  we  have  e  o  for  the  moment 
of  the  load  x  at  that  point ;  e  a  for  that  of  the 
load  y ;  and  e  c  for  that  of  the  load  z ;  and  so  at 
any  other  point.    Or  make  en  =  eo  +  ea  + 
6  c ;  also  make  A-  i  and  m  h  respectively  equal 
to  the  three  dists  aboye  *  and  m,  and  join  j  hi 
n  k.    Then  at  any  point  along  the  beam  j  k  the 
vert  dist  to  these  upper  lines  gives  the  moment. 
Case  1O.    Uniform  load  from  end  to  end,  Fig  41.    The  greatest 
moment  is  at  the  center  c,  and  is  =  half  load  X  quarter 
span.    At  any  other  point  e  moment  =  half  load  on  e  o 
X  e  a ;  or  to  half  load  on  ea  X  e  o.    Make  c s  =  half  load 
X  quarter  span,  and  draw  a  parabola  o  s  a,  then  at  any 
point  e  the  moment  is  =  e  t. 

The  shearing  or  vertical  strain  at  the  center 
is  zero  or  nothing.    See  Art  6,  p  644. 

Rem.  1.  The  weight  of  the  beam  itself  is  usually 
suchya  load,  but  is  frequently  so  small  compared  with  the  load  that  in  this  and 
other  cases  it  may  be  neglected. 

Rem.  2.  The  greatest  moment  of  rupture  that  can  occur  at  any  given  point  on  the 
span  is  when  the  load  covers  the  span  from  end  to  end ;  and  in  beams  or  trusses 
of  uniform  depth  the  hor  strains  at,  any  given  section  are  then  also  greater  than 
under  any  partial  load ;  so  that  if  the  chords  are  then  strong  enough  in  every  part, 
they  will  be  strong  enough  for  any  partial  load :  which  is  not  the  case  with  web 
members;  any  one  of  which  is  most  strained  when  the  longest  segment  reaching 
to  it  is  loaded.  See  Rem  3,  Art  6,  p  644. 

Case  11.  Uniform  load  from  a  support  to  part  way  across, 
Fig  42.  Find  the  cen  of  grav  g  of  the  load,  and  by  Art  47  what  portion  of  it  rests 
on  each  support  o  and  x.  Then  by  General  Rule,  Art  47, 
the  moment  at  o  or  x  —  o.  At  n  or  at  any  point  a  be- 
tween n  and  x  it  is  =  portion  or  reaction  at  xX  a;  n  (or 
x  a  as  the  case  may  be).  At  any  point  c  between  n  and 
o  moment  is  =  reaction  at  x  X  x  c  —  (load  on  c  n  X 
half  c  n)  or  to  reaction  at  o  X  o  c  —  (load  on  o  cX  half 
o  c).  This  plainly  applies  to  unequal  loads  also,  if  instead  of  //<///  c  n 
or  halfo  c,  &c,  we  use  the  dist  of  the  given  point  from  the  cen  of  grav  of  the  load. 
To  find  the  place  of  greatest  moment  t  if  the  load  is  uni- 
form say,  as  twice  xoinoiinoint.  When  the  load  covers 
the  whole  beam  it  becomes  Case  10. 

Case  12.    Uniform  load  reaching:  to  neither 
support,  Fig.  43.    From  either  support  proceed  as  from 
x  in  Fig  42,  except  as  to  greatest  moment,  which  find  by  trial.* 

*  On  this  subject  see  "  Humber's  Strains  in  Girders,"  to  which  the  writer  is  chiefly  indebted  for  the 
foregoing. 


43. 


STRENGTH   OF    IRON   PILLARS.  221 

STEENGTH  OF  IKON  PILLARS, 


Caution.     See  Remarks  on  this  edition,  p.  ix. 

Our  practical  knowledge  of  the  strength  of  pillars  or  columns,  of  different  mate- 
rials, and  of  different  forms  of  cross-section,  is  to  this  time  very  imperfect.  The  best 
authorities  on  iron  pillars  are  Eaton  Hodgkinson  and  Lewis  Gordon,  both  of  England. 
Their  results,  however,  differ  considerably,  although  the  rules  of  both  are  based  in  a 
great  measure  upon  the  same  experiments  tried  by  Mr.  Hodgkinson.  Mr.  Gordon's 
rules,  however,  are  much  the  most  simple,  and  inasmuch  as  the  principle  upon  which 
they  are  founded  seems  to  be  equally  as  reliable  as  that  adopted  by  Hodgkinson,  we 
shall  confine  ourselves  to  them.  See  REMARK,  p.  240  and 


HOLLOW  CYLINDRICAL  IRON  PILLARS, 

Gordon's  rnle  for  the  breaking?  load  in  Ibs.  of  hollow  cylin- 
drical iron  pillars,  with  flat  ends,  firmly  fixed,  and  with  the 
loi*d  pressing1  upon  them  equally  throughout.  The  thickness  of 
the  ring,  or  single  thickness  of  metal,  not  to  exceed  about  ys  of  the  outer  diam. 

Find  the  area  in  square  ins  of  solid  metal  contained  in  the  ring,  or  transverse 
section  of  the  pillar.  Square  the  length  of  the  pillar  in  ins.  Also  square  the  outer 
diam  in^ns.  Then, 

RULE  1.    For  Hollow  Cast  Iron  Cylindrical  Pillars.* 
Breakg  load  =  Metal  area  in  sq  ins  X  80000  1  _ 

in  Ibs.  sq  of  length  in  ins  -4-  sq  of  outer  diam  in  ins 

600 

Or  in  words,  mult  the  transverse  metal  area  in  sq  ins.  by  80000,  and  call  the  product 
P.  Div  the  square  of  the  length  in  ins.  by  the  square  of  the  diam  in  ins.  Div  the 
quot  by  600.  To  this  last  quot  add  1.  Div  P  by  the  sum.  The  result  will  be  the 
reqd  load  in  Ibs. 

For  the  breaks*  load  in  tons  use  35.71  instead  of  80000. 

Ex.  What  is  the  breakg  load  in  Ibs,  of  a  round  hollow  cast  iron  pillar,  6  ins  outer 
diam  ;  10  ft  or  120  ins  long;  and  having  a  transverse  metal  area  of  11  sq  ins? 

Here  the  sq  of  the  length  is  1202  —  14400  ins  ;  and  the  diam2  =  62=  36  ins.   Hence 


~~600~~ 
Which  is  equal  to  21.42  tons  per  sq.  inch  of  metal  area. 

RULE  2.    For  Hollow  Wrought  Iron  Cylindrical  Pillars.* 

Breakg  load  ^  _  Metal  area  in  sq  ins  X  36000  f 

in  Ibs  sq  of  length  in  ins  -j-  sq  of  outer  diam  in  ins 

4500 

For  the  breakg  load  in  tons,  use  16.07  instead  of  36000. 
Ex.     A  wrought  pillar  of  the  same  dimensions  as  the  foregoing.     Here, 
Breakg  load  =     H  X  36000_   =    396000         396000  =  f 

in  fos  1.  14400  -s-  36       1  -f  .0889       1.0889 

4500 
162.4  (tons) 

—  :-  =  14.76  tons  per  sq  inch  of  metal  area. 
11  (area) 

*  Wrought  iron   columns  shorten  at  the  average  rate  of  about  % 

inch  in  30  feet,  under  loads  of  4  tons  per  sq  inch  of  metal  cross-section  ;  and  cast  iron  ones  average 
about  twice  as  much. 

t  This  80000  is  Gordon's  assumed  crushing  load  in  fts  (35.71  tons)  per  sq  inch  (a  rather  low  and  safe 
one)  for  average  cast  iron,  in  pieces  about  two  diams  hieh  ;  the  36000  fts  (16.07  tons)  is  the  corre- 
sponding crushing  load  for  wrought  iron  ;  both  as  assumed  by  Gordon.  Therefore,  to  find  the  load 
at  once  in  tons,  we  may  use  35.71  instead  of  80000  in  Rule  1  ;  "and  16.07  instead  of  the  36000  in  Rule 
2.  By  doing  this,  and  at  the  same  time  omitting  the  metal  area,  we  at  ouce  get  the  breakg  load  in 
tons  per  sq  inch. 

Item..  For  good  American  Iron  we  may  uae  40000  instead  of  36000;  but  for  average  iron 
36000  is  the  beet. 


222  STRENGTH   OF   IRON   PILLARS. 

HOLLOW  SQUARE  IRON  PILLARS. 


Breaking  loads  of  hollow  square  iron  pillars,  with  flat 
ends,  firmly  fixed,  and  with  the  load  pressing  upon  them 
equally  throughout.  The  single  thickness  of  the  metal  nut  to  exceed  about 
%  of  the  width  of  a  side. 

Find  the  area  of  solid  metal,  in  sq  ins,  contained  in  the  square  ring  or  transverse 
section  of  the  pillar.  Square  the  length  of  the  pillar  in  ins;  also  square  one  outer 
bide  in  ins.  Then 

RULE  3.    For  hollow  square  cast  iron  pillars. 
Breakg  load  _  Metal  area  in  sq  ins  X  80000 

in  fi)8  1  4-  8(1  °^  length  in  ins  -f-  sq  of  one  side  in  ins 

800 

Or  in  words,  mult  the  metal  area  in  sq  ins  by  80000,  and  call  the  prod  P.    Div  the 
square  of  the  length  in  ins  by  the  sq  of  one  outer  side  in  ins.      Div  the  quot  by  800. 
To  this  last  quot  add  1.    Div  P  by  the  sum. 
For  the  breakg  load  in  tons  use  35.71  instead  of  80000. 
Ex.     A  cast-iron  hollow  square  pillar,  %  inch  thick,  6  ins  on  each  outer  side;  10 
ft,  or  120  ins,  or  20  sides  long;  and  having  a  transverse  metal  area  of  11  sq  ins. 

Here  the  square  of  the  length,  or  1202  =  14400;  and  the  sq  of  an  outer  side,  or  62 
=  36.  Hence  \ 

Breakgload  _  -11X80000  880000  _  880000  _ 

in  ft>s  j  ,    14400  -i-  36       1  -f  .5         1.5 

800 
7^-  =  23.8  tons  per  sq  inch;  as  per  Table  page  232,  for  a  square  hollow 

cast  pillar  20  sides  high.  A  round  pillar  of  the  same  area,  and  20  diams  high, 
was  found  by  Rule  1  to  have  a  breakg  load  of  235.7  tons.  Therefore,  when  both  are 
20  diams  or  sides  high,  the 

square  one  is  •   .'    =  1.111  times,  or  1J,  as  strong  as  the  round  one.    The  same  pro- 

portion, however,  does  not  hold  good  at  other  equal  heights,  as  may  be  found  by 
comparing  the  round  and  square  ones  of  Table,  p.  232. 

RULE  4.  For  hollow  square  wrought  iron  pillars,  the  same  as  for 
cast  iron,  except  that  36000  is  to  be  used  instead  of  80000  and  6000  instead  of  800; 
that  i8'  Breakg  load  =  Metal  area  in  sq  ins  X  36000 

in  fi>s  sq  of  length  in  ins  -5-  sq  of  one  side  in  ins 

6000 

For  the  breakg  load  in  tons  use  16.07  instead  of  36000. 
Ex.    A  square  hollow  pillar  of  wrought  iron,  like  the  preceding. 


6000 

hence,  165-'3ftops)  =  15.07  tons  per  sq  inch,  as  per  Table  p.  232. 
'  11  (ins  area) 

The  breakg  load  of  a  given  hollow  square  wrought  iron 
pillar  may  be  found  by  Table  p.  232. 


SOLID  CYLINDRICAL  IRON  PILLARS. 


Breakg  loads  of  solid  cylindrical  iron  pillars,  with  flat 
ends,  firmly  fixed,  and  with  the  loads  pressing-  upon  them 
equally  throughout. 

Find  the  area  of  metal  in  sq  ins  contained  in  the  transverse  section  of  the  pillar. 
Square  the  length  of  the  pillar  in  ins;  also  square  its  diam  in  ins.  Then, 


STRENGTH   OF   IRON   PILLARS.  223 

RULE  5.    For  solid  cylindrical  cast  iron  pillars. 

Break-  load  _  Metal  area  in  sq  ins  X  ^0000 

in  fes  sq  of  length  in  ins  -r-  sq  of  diam  in  ins 

300 

For  the  breakg  load  in  tons  use  35.71  instead  of  80000. 
Ex.    A  solid  cylindrical  cast-iron  pillar  6  ins  diam,  and  10  ft,  or  120  ins  long. 
Here  the  metal  area  in  sq  ins  of  the  transverse  section,  by  Table  of  Circles,  p.  18, 
is  2a274.    The  sq  of  the  length  in  ins,  or  1202,  is  14400;  and  the  sq  of  the  diam  6  is 

^Breakg  load  _  28.274  X  80000  ,2261920  __  2261920  _ 
in  fl>s  I""  14400  -=-  36  ~~  1  +  1.33  ~     2.33     " 

300 
Or  to  ™™_  433.4  tons. 

RULE  6.    For  solid  cylindrical  wrought  iron  pillars. 

Break"1  load  _  Metal  area  in  sq  ins  X  36000 

in  S)s  -       sq  of  length  in  ins  -j-  sq  of  diam  in  ins 

2250 

For  the  breakg  load  in  tons  use  16.07  instead  of  36000. 
Ex.   A  solid  cylindrical  wrought  iron  pillar  6  ins  diam ;  and  10  ft,  or  120  ins  long. 
Here,  as  in  the  preceding  case,  the  metal  area  is  28.274  sq  ins ;  the  sq  of  the  length 
in  ins  =^14400;  and  the  sq  of  the  diam  =  36  ins.     Hence, 

Break*  load  _  28-2^  X  36000  _  1017864  _  1017864 
in  fa        ~  I       14MO+?6       f+~T8        TIT   ' 

2250 

This  pillar  and  the  foregoing  cast  iron  one  are  of  the  same 
dimensions;   and  each  of  them  is  20  diams  long.      Hence  at  20  diams  long  a 

solid  wrought  iron  cylindrical  pillar  is  •-•  ~       =  0.89  times  as  strong  as  a  cast  one. 
See  Table  5,  page  230,  of  such  breaking  loads. 


SOLID  SQUARE  IRON  PILLARS. 

Breakg  loads  of  solid  sqnare  iron  pillars,  with  flat  ends, 
firmly  fixed,  and  with  the  loads  pressing  upon  them  equally 
throughout.  Find  the  area  of  metal  in  sq  ins  contained  in  the  transverse 
section  of  the  pillar.  Sq  the  length  of  the  pillar  in  ins ;  also  sq  one  side  of  the 
pillar  in  ins.  If  rectangular,  square  only  the  least  side.  Then, 

RULE  7.    For  solid  square  cast  iron  pillars. 

Breakg  load  =  Metal  area  in  sq  ins  X  80000 

in  tt»s  ^gq  of  length  in  ins  -f  sq  of  one  side  in  ins 

~loo 
For  the  breakg  load  in  tons  use  35.71  instead  of  80000. 

Ex.  A  solid  square  cast  iron  pillar,  6  ins  sqnare,  and  10  ft,  or  120  ins  long.  Here  the  metal  area 
of  transverse  section  is  36  sq  ins.  The  sqof  the  length  in  ins,  or  1  2  O2  —  14400;  and  the  sq  of  one 
side  in  ins,  fi2  =  36.  Hence, 

Breakg  load  =      *  X  80000      _  2880000      2880000  _ 
in  Its  ,  ^  14400 -r  36  ~    1  +  1    ~~        2 

400 

RULE  8.    For  solid  sqnare  wrought  iron  pillars. 

Breakg  load  -  Metal  area  in  sq  ins  X  36000 

in  Bbs  sq  of  length  iu  ins  -f-  sq  of  one  side  in  ins 

3000 

For  the  breakg  load  in  tons  use  16.07  instead  of  36000. 

Ex.  A  solid  square  wrought  iron  pillar,  6  ins  square.  10  ft,  or  120  ins  long.  Here  the  metal  area 
of  transverse  section  is  36  sq  ins.  The  sq  of  the  length  in  ins,  or  120*-  14400,  and  the  sqof 
one  side  iu  ins,  or  62  -  36.  Hence, 

Breakg  load  -    36  X  M000_          1296000        1296000  _ 
infts  1  ^  14400  -36"         l-f.133          1.133    ~ 

3000 
See  Table  6,  page  231,  of  such  breaking  loads. 


224 


STRENGTH   OP   IRON   PILLARS. 


Table  1.     HOLLOW   CYLIND  CAST  IRON  PILLARS. 

Breaking  loads,  flat  ends,  perfectly  true, and  firmly  fixed; 
and  the  loads  pressing  equally  on  every  part  of  the  top. 

By  Gordon's  formula. 
For  diams  or  lengths  intermediate  of  those  in  the  table,  the 

loads  may  be  found  near  enough  by  simple  proportion. 

For  thicknesses  less  than  those  in  the  table,  the  breaking  loads 

may  safely  be  assumed  to  diminish  in  the  same  proportion  aa  the  thickness,  while  the  outer  diam 
remains  the  same.  But  for  greater  thicknesses  than  those  in  the  table,  the  loads  do  not  increase 
as  rapidly  as  the  new  thickness.  Still,  in  practice,  they  may  be  assumed  to  do  so  approximately, 

if  the  new  thickness  does  not  exceed  about  ya  part  of  the 
outer  diam. 


•S 

CAST  IRON.    THICKNESS  &  INCH.    (Original.) 

a 

§"* 

Outer  Diameter  in  inches. 

ll 

(3 

2      j 

2& 

2^ 

w 

3 

2X 

4       . 

4& 

5      |     5% 

6 

3 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

i 

45.2 

52.5 

59.8 

66.8 

74.1 

88.5 

103.0 

117.1 

131.3 

145.5 

159.8 

1 

2 

36.2 

43.4 

51.0 

58.6 

66.5 

81.5 

96.7 

111.3 

125.9 

140.5 

155.2 

2 

3 

27.2 

34.1 

41.4 

48.8 

56.7 

71.9 

87.6 

102.7 

117.8 

132.9 

148.1 

3 

4 

20.2 

26.3 

32.9 

39.7 

47.0 

61.9 

77.5 

92.9 

108.3 

123.7 

139.1 

4 

5 

15.1 

20.2 

25.9 

32.0 

38.6 

62.6 

67.4 

82.6 

97.9 

113.4 

129.1 

5 

6 

11.6 

15.8 

20.7 

25.9 

31.6 

44.3 

58.2 

72.7 

87.7 

103.0 

118.7 

6 

7 

9.1 

12.5 

16.6 

21.0 

26.1 

37.4 

50.1 

63.7 

78.1 

92.9 

108.3 

7 

8 

7.3 

10.1 

13.5 

17.3 

21.7 

31.6 

43.2 

55.8 

69.3 

83.4 

98.4 

8 

9 

5.9 

8.2 

11.1 

14.3 

18.2 

26.9 

37.3 

48.8 

61.5 

74.9 

89.2 

9 

10 

4.9 

6.9 

9.3 

12.0 

15.4 

23.1 

32.4 

42.9 

54.6 

67.1 

80.7 

10 

11 

4.1 

5.8 

7.9 

10.3 

13.2 

20.0 

28.3 

37.8 

48.6 

60.3 

73.0 

11 

12 

3.5 

5.0 

6.8 

8.9 

11.4 

17.4 

24.9 

33.5 

43.3 

54.2 

66.1 

12 

13 

3.0 

4.2 

5.8 

7.6 

9.9 

152 

21.9 

29.7 

38.8 

48.8 

60.0 

13 

14 

2.6 

3.6 

5.1 

6.7 

8.7 

13.5 

19.5 

26.6 

34.9 

44.1 

54.5 

14 

15 

2.3 

3.2 

4.5 

6.0 

7.7 

11.9 

17.4 

23.9 

31.4 

39.9 

49.7 

15 

16 

2.0 

2.8 

4.0 

5.3 

6.9 

107 

15.6 

21.5 

28.4 

36.4 

45.6 

16 

18 

1.6 

2.3 

3.2 

4.2 

5.5 

8.6 

12.7 

17.5 

23.4 

30.2 

38.0 

18 

20 

1.3 

1.8 

2.6 

3.4 

4.5 

7.1 

10.5 

14.7 

19.7 

25.6 

32.3 

20 

25 

1.7 

2.3 

8.0 

4.7 

7.0 

9.8 

13.3 

17.6 

22.5 

25 

30 

1.2 

1.6 

2.1 

3.2 

5-0 

7.0 

9.4 

12.6 

16.1 

30 

35 

2.4 

3.7 

5.2 

7.1 

M 

12.2 

35 

40 

1/2 

1.9 

2.8 

4.0 

5.4 

7.2 

9.5 

40 

\ 

4.31  |     4.91 

Veight 

5.53 

Of  1  f 

6.13 

50t  Of 

6.75 

Length  of  pill 
7.97  |      9.22 

ar  in  pounds. 

10.4     |    11.7     |    12.9 

14.1 

Area  of  ring  of  solid  metal  in  square  inches. 
1.38  |      1.57  |      1.77  |    1.96    |      2.16  |      2.55  \      2.95  |      3.34  |      3.73  |      4.12  \      4.52 


a^ 

J 

CAST  IRON.     THICKNESS  X  INCH.    (Original.) 

a 

ft 

3 

Outer  Diameter  in  inches. 

5 

5^ 

6 

W 

7 

1* 

8 

8}* 

9 

10 

11 

12 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

2 

239 

268 

297 

325 

354 

383 

412 

440 

469 

526 

583 

640 

7 

4 

205 

235 

266 

296 

326 

356 

387 

416 

445 

504 

563 

622 

4 

6 

166 

196 

227 

257 

288 

319 

350 

380 

410 

471 

532 

593 

6 

8 

131 

159 

188 

218 

248 

279 

310 

340 

371 

432 

494 

557 

8 

10 

103 

127 

154 

182 

210 

240 

270 

300 

330 

391 

454 

517 

10 

12 

82 

103 

126 

151 

177 

205 

233 

262 

292 

351 

413 

475 

12 

14 

66 

84 

104 

126 

149 

174 

200 

227 

255 

313 

372 

434 

14 

16 

54 

69 

87 

106 

126 

149 

173 

198 

224 

277 

334 

394 

16 

18 

45 

58 

73 

90 

108 

128 

149 

172 

196 

246 

300 

357 

18 

20 

37 

48 

62 

77 

93 

111 

130 

?5l 

173 

219 

270 

323 

20 

22 

32 

41 

53 

66 

80 

96 

113 

132 

151 

194 

241 

292 

22 

25 

25 

34 

43 

54 

65 

79 

93 

109 

126 

164 

206 

252 

25 

30 

18 

24 

81 

89 

48 

59 

70 

83 

96 

126 

160 

199 

80 

35 

13 

18 

24 

30 

36 

44 

53 

63 

73 

98 

126 

159 

35 

40 

10 

14 

18 

23 

28 

35 

42 

50 

59 

78 

102 

129 

40 

45 

8 

11 

15 

19 

23 

28 

34 

41 

48 

64 

84 

107 

45 

50 

6 

9 

12 

15 

19 

23 

28 

33 

39 

53 

70 

89 

50 

60 

4 

6 

9 

11 

14 

17 

20 

24 

28 

38 

50 

65 

60 

70 

3 

4 

6 

8 

10 

12 

15 

18 

21 

29 

38 

50 

70 

80 

3 

4 

5 

6 

8 

9 

11 

13 

16 

22 

30 

38 

80 

'Weight  of  1  foot  of  length  of  pillar,  in  pounds. 
22.1  |    24.5   |   27.0    |   29.4    |    31.9   |    34.4   |    36.9    |    39.4  |   41.9  I   46.6  |  51.6  |    56.6 


7.07  |    7.85   |    8.64 


Area  of  ring  of  solid  metal,  in  square  inches. 

.43  f  10.2   |    11.0   |   11.8   I  12.6   |   18.4   |   14.9   I   16.5  |    18.1 


STRENGTH   OF   IRON   PILLARS. 


225 


Table  1.    HOLLOW  CYLIXD  CAST  IROX  PILLARS. 

BREAKING  LOADS.— (Continued.)  BY  GORDON'S  RULE. 


Length  in 
feet. 

CAST  IRON.  THICKNESS  1  INCH.  (Original.) 

1  Length  in 
feet. 

Outer  Diameter  in  Inches. 

12    13 

14 

15 

16 

18 

20 

22 

24 

27 

SO 

86 

4 

Tons. 
1188 

1301 

1415 

1530 

1645 

Tons. 
1874 

Tons. 
2103 

Tons. 
2330 

Tons. 
2557 

Tons. 
2*96 

Tons. 
3236 

Tons. 
3915 

4 

6 

1138 

1253 

1368 

1484 

1601 

1833 

2066 

2295 

2525 

2866 

3208 

3890 

6 

8 

1065 

1184 

1303 

1423 

1543 

1779 

2015 

2247 

2479 

2S24 

3170 

3860 

8 

10 

989 

1110 

1231 

1355 

1475 

1716 

1957 

2193 

2430 

2780 

3128 

3823 

10 

12 

909 

1030 

1152 

1275 

1399 

1644 

1889 

2129 

2369 

2723 

8076 

3778 

12 

14 

829 

949 

1071 

1195 

1320 

1566 

1813 

2056 

2300 

2659 

3016 

3726 

14 

16 

756 

873 

992 

1114 

1237 

1484 

1733 

1979 

2226 

2589 

2951 

3668 

16 

18 

683 

796 

913 

1034 

1155 

1401 

1651 

1899 

2147 

2515 

2879 

3604 

18 

20 

618 

727 

840 

958 

1077 

1320 

1568 

1817 

206(5 

2437 

2805 

3536 

20 

22 

559 

663 

772 

887 

1002 

1241 

1486 

1734 

1982 

2356 

2726 

3464 

22 

24 

508 

606 

709 

818 

929 

1163 

1404 

1650 

1»99 

2272 

2644 

3387 

24 

26 

459 

553 

651 

756 

803 

1090 

1326 

1570 

1816 

2188 

2560 

3308 

26 

28 

418 

506 

598 

697 

800 

1020 

1250 

1489 

1733 

2103 

2475 

3226 

28 

30 

380 

462 

549 

644 

743 

954 

1178 

1412 

1653 

2020 

2392 

3143 

30 

32 

347 

424 

506 

595 

689 

893 

1110 

1338 

1575 

1938 

2308 

3059 

32 

34 

318 

390 

467 

552 

641 

836 

1046 

1268 

1500 

1857 

2225 

2974 

34 

36 

592 

359 

432 

511 

596 

783 

984 

1199 

1427 

1779 

2143 

2889 

36 

38 

SB 

331 

400 

475 

556 

734 

928 

1136 

1361 

1704 

2063 

2804 

38 

40 

247 

305 

370 

441 

518 

687 

874 

1076 

1296 

1630 

984 

2720 

40 

42 

229 

283 

344 

411 

484 

645 

825 

1019 

1232 

1560 

909 

2636 

42 

44 

212 

263 

321 

383 

452 

605 

776 

963 

1169 

1491 

834 

2.552 

44 

46 

197 

246 

299 

358 

423 

569 

734 

915 

1114 

1428 

765 

2474 

46 

48 

183 

229 

280 

335 

397 

536 

694 

869 

1060 

1367 

696 

2396 

48 

50 

170 

213 

261 

314 

373 

505 

656 

824 

1008 

1306 

1627 

2319 

50 

55 

144 

181 

222 

269 

320 

438 

573 

725 

893 

1172 

1474 

2135 

55 

60 

124 

157 

192 

233 

278 

383 

503 

641 

795 

1054 

1333 

1964 

60 

65 

105 

134 

165 

202 

242 

335 

443 

568 

709 

951 

1211 

J808 

65 

70 

98 

118 

146 

178 

213 

297 

394 

507 

636 

853 

1099 

1664 

70 

80 

73 

92 

113 

139 

168 

235 

315 

408 

516 

705 

914 

1414 

80 

90 

58 

73 

91 

112 

136 

191 

257 

335 

426 

586 

768 

1209 

90 

100 

48 

60 

75 

92 

112 

157 

213 

279 

356 

491 

651 

1040 

100 

Weight  of  one  foot  of  length  of  pillar,  in  pounds. 
108  |    118    |    128     |    138     |    147     |    167    j    187    j    206    |    226    |    255    |    285    |    344 

Area  of  ring  of  solid  metal,  in  square  inches. 
34.6  |     37.7  I     40.8   |     44.0   |     47.1   |    53.4   |    59.7  |     66.0  |   72.2   |    81.7   |    91.1  |  110.0 


Si 

OAOJ.  •Lxiuj*  .  j.  xii^/jvox  Xioo  o  xi.x  uxLj^o.  (.unginai.j 

.2 
•S« 

in 

Outer  Diameter  in  Feet. 

s 

3 

31* 

4 

*1A 

5 

5>* 

6 

7 

8 

10 

12 

3 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

10 

10806 

12878 

14908 

16967 

18986 

21038 

23052 

27143 

31196 

39254 

47375 

10 

15 

10453 

12564 

14628 

16712 

18754 

20825 

22855 

26972 

31045 

39133 

47273 

15 

20 

9996 

12150 

14250 

16369 

18440 

20533 

22585 

26737 

80837 

38962 

47131 

20 

30 

8884 

11102 

13275 

15459 

17593 

19743 

21847 

26084 

30257 

38486 

46729 

30 

40 

7688 

9907 

12113 

14344 

J6531 

18735 

20891 

25224 

29476 

87838 

46175 

40 

50 

6554 

H702 

10888 

13126 

15341 

17580 

19780 

24107 

28532 

37037 

45484 

50 

60 

5553 

7575 

9690 

11892 

14100 

1K349 

18570 

23049 

27460 

86103 

44666 

60 

70 

4704 

6570 

8576 

10702 

12870 

15097 

17319 

21824 

26287 

35058 

43737 

70 

80 

3998 

5699 

7570 

9595 

11692 

13873 

16070 

20566 

25048 

83924 

42712 

80 

90 

3417 

4953 

6683 

8588 

10594 

12706 

14856 

19304 

23791 

32725 

41670 

90 

100 

2940 

4321 

5909 

7665 

9588 

11614 

13700 

18065 

22521 

31481 

40438 

100 

110 

2547 

3788 

5238 

6888 

8677 

10606 

12613 

16869 

21270 

30213 

39220 

110 

125 

1950 

2954 

4400 

5864 

7483 

9257 

11132 

14650 

19448 

27664 

36692 

125 

150 

1532 

2350 

3353 

4547 

5900 

7417 

9058 

12701 

16668 

25182 

34127 

150 

Weight  of  one  foot  of  length  of  pillar,  in  pounds. 
972    |    1150    |    1325    |    1503    |    1678    |    1856    |    2031    |    2388   |    2740     |    8444     |    4153 

Coefficients  of  safety  for  hollow  cast  iron  pillars. 

Mr.  Jam^s  B.  Francis,  of  Lowell,  Mass.,  a  high  authority,  in  his  "  Tables  of  Cast  Iron  Pillars." 
recommends  that  in  order  to  allow  for  unequal  loading,  imperfect  casting,  bad  end  bearings,  side 
blows.  &c.,  we  should  not  take  the  safe  load  at  more  than  one-fifteenth  of  Hodgkinson's  breaking 
load,  if  the  pillars  are  roughly  cast,  and  the  ends  not  perfectly  pinned  aud  adjusted;  and  one-fifth 
when  they  are  so.  and  the  loads  about  equally  distributed. 

It  will  be  seen  by  the  last  table  on  p  242,  how  our  Gordon's  loads  differ  from  Hodgkinson's ;  but 
we  think  that  the  same  proportions  of  ours  may  be  taken  as  safe ;  depending  on  the  above  con- 
iitions. 


226 


STRENGTH   OF   IRON   PILLARS. 


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STRENGTH  OF   IRON   PILLARS. 


227 


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228 


STRENGTH   OF   IRON   PILLARS. 


HOLLOW  CYLINDRICAL,  WROUGHT  IRON  PILLARS. 

Table  4,  of  breaking  loads  in  tons  of  hollow  cylindrical 
wrought  iron  pillars,  with  flat  ends,  perfectly  true,  and 
firmly  fixed,  and  the  loads  pressing  equally  on  every  part 
Of  the  top.  Calculated  by  Gordon's  formula.  No  pains  have  been  taken  to  have 
the  last  figure  of  the  loads  perfectly  correct  in  every  case. 
(Original.) 


Length  in 
feet. 

WROUGHT  IRON.    THICKNESS  H  INCH. 

Length  in 
feet.  | 

Outer  diameter  in  inches. 

H 

1      1     IX     \     1* 

Wi      \       2        |      2X      |      2*      1    2«      |       3 

BREAKING    LOAD. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

1 

3.64 

5.27 

6.88 

8.50 

10.1 

11.7 

13.2 

14.8 

16.4 

18.0 

1 

2 

2.94 

4.64 

6.32 

8.00 

9.6 

11.2 

12.8 

14.5 

16.1 

17.8 

2 

3 

2.30 

3.86 

5.57 

7.28 

8.9 

10.6 

12.2 

13.9 

15.6 

17.3 

3 

4 

1.77 

3.13 

4.74 

6.36 

8.1 

9.9 

11.6 

13.3 

15.0 

16.7 

4 

5 

1.36 

2.51 

4.07 

5.66 

7.3 

9.1 

10.8 

12.5 

14.2 

16.0 

5 

6 

1.04 

2.03 

3.46 

4.91 

6.6 

8.3 

9.9 

11.6 

13.4 

15.2 

6 

7 

.81 

1.65 

2.91 

4.24 

5.7 

7.4 

9.1 

10.8 

12.6 

14.4 

7 

8 

.61 

1.36 

2.46 

867 

5.1 

6.7 

8.3 

9.9 

11.7 

13.5 

8 

9 

.50 

1.05 

2.03 

3.18 

4.5 

6.0 

7.5 

9.1 

10.8 

12.6 

9 

10 

.41 

.95 

1.75 

2.77 

4.0 

5.4 

69 

84 

10.1 

11.8 

10 

11 

.34 

.81 

1.52 

2.41 

3.6 

4.8 

6.2 

7.7 

9.3 

11.0 

11 

12 

.29 

.70 

1.34 

2.14 

3.2 

4.3 

5.6 

7.0 

8.6 

10.2 

12 

13 

.24 

.60 

1.16 

1.88 

2.8 

3.9 

5.2 

6.5 

8.0 

9.5 

13 

14 

.21 

.53 

1.03 

1.69 

2.5 

3.5 

4.7 

6.0 

7.4 

8.9 

14 

15 

.19 

.47 

.91 

1.50 

2.3 

3.2 

4.3 

5.5 

6.9 

8.3 

15 

16 

.18 

.42 

.84 

1.38 

2.1 

2.9 

4.0 

5.1 

6.4 

7.7 

16 

18 

.14 

.33 

.67 

1.11 

1.7 

2.4 

3.4 

4.4 

5.6 

6.8 

18 

20 

.27 

.55 

.91 

1.4 

2.0 

2.8 

3.7 

4.7 

5.8 

20 

25 

.9 

1.4 

2.0 

2.6 

3.4 

4.2 

25 

Weight  of  one  foot  of  length  of  pillar,  in  pounds. 
.820      |     1.15     |     1.47     I     1.80     |    2.13     |     2.45    |     2.78     |     3.11     |    3.43     1 3.77 

Area  of  ring  of  solid  metal,  in  square  inches. 
.246      |    .344     J     .442     |     .540    |     .638     |     .736    |    .835     |     .933     |    1.03    |    1.13 


5 

WROUGHT   IRON.    THICKNESS  y±  INCH. 

.2 

fl 

Outer  diameter  in  inches. 

II 

a 

2      |    2J4       |      V&    |      2%    |        3      |      33^    |       4      |    4^      |      5        |      5^    |      6 

$ 

BREAKING    LOAD. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

i 

21.9 

25.4 

28.3 

31.4 

34.5 

40 

47 

53 

60 

66 

72 

1 

2 

21.1 

24.3 

27.6 

30.7 

33.9 

40 

47 

53 

60 

66 

72 

2 

3 

19.9 

23.1 

26.4 

29.7 

33.0 

39 

46 

52 

59 

65 

71 

3 

4 

18.6 

21.8 

25.3 

28.5 

31.9 

38 

45 

51 

58 

64 

71 

4 

5 

17.0 

20.4 

23.5 

27.3 

30.7 

37 

44 

50 

57 

63 

70 

5 

6 

15.4 

18.8 

22.1 

25.7 

29.2 

36 

43 

49 

56 

62 

69 

6 

7 

13.9 

17.3 

20.5 

23.8 

27.8 

34 

41 

47 

54 

61 

68 

7 

8 

12.5 

15.6 

19.1 

22.3 

25.9 

32 

40 

46 

53 

60 

67 

8 

9 

11.2 

14.2 

17.5 

20.6 

24.3 

30 

38 

44 

51 

58 

65 

9 

10 

10.0 

13.0 

16.1 

19.1 

22.7 

29 

37 

43 

50 

57 

64 

10 

11 

9.0 

10.7 

15.7 

17.6 

21.1 

27 

35 

41 

48 

55 

62 

11 

12 

8.1 

10.6 

13.5 

16.4 

19.6 

26 

33 

40 

46 

54 

61 

12 

13 

7.3 

9.6 

12.4 

15.1 

18.2 

24 

31 

38 

44 

52 

59 

13 

14 

6.6 

8.8 

11.3 

14.0 

17.0 

23 

30 

36 

43 

51 

57 

14 

15 

6.0 

8.0 

10.4 

12.9 

15.8 

21 

28 

34 

41 

49 

55 

15 

16 

5.5 

7.3 

9.5 

12.0 

14.6 

20 

27 

33 

40 

47 

54 

16 

18 

4.5 

6.0 

8.0 

10.3 

12.7 

18 

24 

30 

37 

43 

50 

18 

20 

3.8 

5.1 

6.8 

8.7 

11.0 

16 

21 

27 

34 

40 

47 

20 

25 

7.9 

12 

16 

21 

27 

33 

39 

25 

30 

13 

17 

22 

27 

32 

30 

35 

10 

14 

18 

22 

27 

35 

40 

14 

18 

23 

40 

45 

11 

15 

19 

45 

50 

8 

12 

16 

50 

Weight  of  one  foot  of  length  of  pillar,  in  pounds. 
5.23      |    5.90     |    6.53    |    7.20    |    8.50     |    9.83    |    11.1     |    12.4    |    13.7    |    15.0 

Area  of  ring  of  solid  metal,  in  square  inches. 
1.57      j    1.77     j    1.96    |    2.16    |    2.55     |    2.95    |    3.34    i    3.73    |    4.12    (    4.51 


STRENGTH    OF   IRON   PILLARS. 


229 


HOLLOW  CYLINDRICAL  WROUGHT  IRON   PILLARS. 


Table  4,    (Continued.)    (Original.) 


Length  in 
feet. 

WROUGHT  IRON.     THICKNESS  *  INCH. 

Length  in 
feet. 

Outer  diameter  in  inches. 

5      |    5&    |      6      |    6^    I      7      I    7^    |      8      |    8^     1      9      |     10 

11     |     12 

BREAKING  LOAD. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

2 

112 

125 

139 

152 

166 

177 

189 

201 

214 

238 

263 

290 

2 

4 

110 

123 

136 

149 

163 

174 

186 

199 

212 

237 

262 

289 

4 

6 

106 

119 

132 

145 

158 

171 

184 

197 

210 

235 

261 

288 

6 

8 

101 

114 

127 

140 

154 

167 

181 

194 

207 

232 

258 

284 

8 

10 

95 

108 

123 

136 

149 

162 

176 

189 

203 

228 

254 

280 

10 

12 

89 

102 

116 

129 

143 

157 

171 

185 

199 

224 

250 

276 

12 

14 

8? 

95 

108 

122 

137 

151 

165 

179 

194 

219 

245 

272 

14 

16 

76 

89 

103 

117 

131 

145 

160 

173 

187 

213 

240 

268 

16 

18 

70 

83 

97 

110 

124 

138 

153 

166 

180 

207 

235 

263 

18 

20 

64 

77 

91 

104 

117 

131 

145 

159 

173 

201 

227 

257 

20 

22 

58 

70 

83 

96 

109 

123 

138 

151 

165 

192 

220 

250 

22 

25 

52 

64 

76 

89 

102 

115 

129 

143 

157 

183 

212 

241 

25 

30 

42 

52 

63 

74 

87 

100 

113 

127 

141 

167 

195 

224 

30 

35 

34 

43 

53 

64 

75 

87 

99 

112 

125 

151 

178 

207 

35 

40 

27 

35 

44 

53 

64 

75 

86 

98 

110 

135 

163 

190 

40 

45 

23 

30 

38 

46 

55 

65 

76 

87 

98 

123 

148 

174 

45 

50 

19 

24 

32 

38 

47 

56 

66 

76 

87 

109 

133 

158 

50 

60 

15 

19 

24 

29 

36 

43 

51 

60 

69 

88 

109 

132 

60 

70 

11 

14 

18 

23 

28 

34 

40 

48 

56 

73 

91 

111 

70 

80 

9 

11 

14 

18 

22 

27 

32 

37 

44 

57 

74 

93 

80 

90 

7 

9 

11 

14 

18 

22 

26 

31 

36 

49 

63 

78 

90 

100 

18 

22 

26 

80 

41 

53 

66 

100 

Weight  of  one  foot  of  length  of  pillar,  in  pounds. 
26.2  I    28.8  |    31.4  |    34.0  |    36.6  |    39.3    |    42.0    |    44.7    |    49.7    |    55.0   | 

Area  of  ring  of  solid  metal,  in  square  inches. 
7.85  |    8.64  |    9.43  |    10.2  |    11.0  |    11.8   |    12.6    |    13.4   |    14.9        16.5   | 


Table  4,  (Continued.)    (Original.) 


a 

WROUGHT  IRON.    THICKNESS  1  INCH. 

5 

S*5 

Outer  diameter  in  inches. 

tig 

JT 

13      |      14    |      15    |      16    |      17    |      18    |      20    |      22    |      24 

26    |      28    |    30 

_r 

BREAKING  LOAD. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

i 

603 

653 

704 

753 

805 

854 

955 

1056 

1157 

1257 

1357 

1458 

i 

10 

588 

638 

691 

742 

795 

846 

949 

1049 

1149 

1248 

1354 

1457 

10 

20 

543 

595 

651 

702 

759 

810 

913 

1016 

1120 

1223 

1327 

1430 

20 

30 

479 

538 

594 

645 

699 

758 

866 

973 

1077 

1186 

1289 

1394 

30 

40 

415 

470 

528 

584 

636 

691 

806 

912 

1027 

1130 

1237 

1348 

40 

50 

355 

405 

462 

516 

670 

627 

740 

848 

961 

1067 

1179 

1294 

50 

60 

300 

348 

400 

452 

605 

659 

669 

781 

891 

1005 

1115 

1228 

60 

70 

256 

300 

348 

398 

448 

499 

606 

715 

824 

936 

1046 

1160 

70 

80 

215 

255 

298 

344 

392 

440 

543 

649 

757 

868 

978 

1092 

80 

90 

185 

222 

261 

303 

347 

392 

489 

590 

694 

800 

910 

1023 

90 

100 

"157 

190 

225 

262 

303 

345 

436 

532 

631 

735 

843 

955 

100 

110 

134 

162 

193 

227 

264 

302 

386 

474 

568 

666 

770 

877 

110 

125 

111 

135 

162 

192 

225 

259 

336 

416 

605 

598 

697 

799 

125 

150 

82 

101 

122 

145 

171 

198 

262 

328 

405 

485 

674 

666 

150 

175 

62 

78 

95 

112 

133 

155 

208 

266 

331 

<00 

478 

560 

175 

200 

49 

60 

74 

89 

106 

124 

168 

216 

269 

328 

395 

467 

200 

Weight  of  one  foot  of  length  of  pillar,  in  pounds. 
136  I     147   |    157     |    168    |    178     |  199     |    220    |    241     |    262     |    283    {   304 

Area  of  ring  of  solid  metal,  in  square  inches. 
40.8  |    44.0  |    47.1   |    50.3  |    53.4   |  59.7   |    66.0  |    72.3  |    78.5   [   84.8  |    91.1 


The  breaking  loads  for  less  thicknesses  may  safely  be  assumed  t* 

diminish  at  the  same  rate  as  the  thickness. 


230 


STRENGTH   OF   IRON   PILLARS. 


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^   (MS<J'MC<l<MC4rHrHrHr 


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--- 


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-'  CO  <N  i-H  rn'  rH 


§  S2C 


SSSSS5S 


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ft  CO 


§3 
2 
ii 


STRENGTH   OF   IRON   PILLARS. 


231 


§ 

53 


a 

s 


I 


S   -3 


ft  d 

8- 

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r-\       §COlN-<*OSOiOCOCO<NC5fHr-lOO      I 

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232 


STRENGTH    OF   IRON    PILLARS. 


TABLE  OF   BREAKING  LOADS  OF  IRON  PILLARS, 

in  tons  per  square  inch  of  metal  area.  Deduced  from  Gordon.  The  ends  are 
supposed  to  be  planed  to  form  perfectly  true  bearings;  arid  all  parts  to  be  equally 
pressed.  The  last  is  rarely  the  case  in  practice.  If  the  pillar  is  rectailg'U- 
lar  instead  of  square,  use  the  least  side  for  a  measure  of  length.  (Original.) 


Hollow 

Hollow 

Solid 

Solid 

Round.* 

Square.* 

Round. 

Square. 

Length 

Length 

measd 

measd 

Breakg  oads  per 

in  sides 

Breakg  loads  per 

Breakg  loads  per 

n  sides 

Breakg  loads  per 

sq  inch  of  metal 

or 

sq  inch  of  metal 

sq  inch  of  metal 

or 

sq  inch  of  metal 

area  of 

diams. 

area  of 

area  of 

diams. 

area  of 

transverse  section. 

transverse  section. 

transverse  section. 

transverse  section. 

Cast. 

Wrt. 

Cast. 

Wrt. 

Cast. 

Wrt. 

Cast. 

Wrt. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

35.7 

16.1 

1 

35.7 

16.0 

35.6 

16.1 

1 

35.6 

16.1 

35.5 

16.1 

2 

35.5 

16.0 

35.2 

16.1 

2 

35.4 

16.0 

35.2 

16.1 

3 

35.3 

16.0 

34.6 

16.0 

3 

34.9 

16.0 

34.8 

16.0 

4 

35.0 

16.0 

33.9 

15.9 

4 

34.3 

16.0 

34.3 

16.0 

5 

34.6 

16.0 

33.0 

15.9 

5 

33.6 

16.0 

33.7 

lfi.0 

6 

34.2 

16.0 

31.9 

15.8 

6 

32.8 

15.9 

33.0 

15.9 

7 

33.7 

16.0 

30.6 

15.7 

7 

31.8 

15.8 

32.3 

15.8 

8 

33.1 

15.9 

29.3 

15.6 

8 

30.8 

15.7 

31.5 

15.8 

9 

32.4 

15.9 

28.0 

15.5 

9 

29.7 

15.7 

30.6 

15.7 

10 

31.7 

15.8 

26.7 

15.4 

10 

28.6 

15.6 

29.7 

15.6 

11 

31.0 

15.8 

25.4 

15.2 

11 

27.4 

15.5 

28.8 

15.5 

12 

30.3 

15.7 

24.1 

15.1 

12 

26.3 

15.3 

27.9 

15.5 

13 

29.5 

15.7 

22.8 

14.9 

13 

25.1 

15.2 

27.0 

15.4 

14 

28.7 

15.6 

21.6 

14.8 

14 

24.0 

15.1 

26.0 

15.3 

15 

27.9 

15.5 

20.5 

14.6 

15 

22.9 

15.0 

25.1 

15.2 

16 

27.1 

15.5 

19.4 

14.4 

16 

21.8 

14.8 

24.2 

15.1 

17 

26.3 

15.4 

18.3 

14.2 

17 

20.7 

14.7 

23.3 

15.0 

18 

25.4 

15.3 

17.2 

14.0 

18 

19.7 

14.5 

22.3 

14.9 

19 

24.6 

15.2 

16.2 

13.8 

19 

18.8 

14.3 

21.4 

14.8 

20 

23.8 

15.1 

15.3 

13.6 

20 

17.9 

14.2 

20.6 

14.7 

21 

23.0 

15.0 

14.5 

13.4 

21 

17.0 

14.0 

19.8 

14.6 

22 

22.2 

14.9 

13.7 

13.2 

22 

16.2 

13.8 

19.0 

14.4 

23 

21.5 

14.8 

12.9 

13.0 

23 

15.4 

13.6 

18.3 

14.3 

24 

20.8 

14.7 

12.2 

12.8 

24 

14.6 

13.5 

17.5 

14.2 

25 

20.1 

14.6 

11.6 

12.6 

25 

13.9 

13.3 

16.8 

14.0 

26 

19.4 

14.5 

11.0 

12.4 

26 

13.3 

13.1 

16.2 

13.9 

27 

18.7 

14.4 

10.4 

12.1 

27 

12.7 

12.9 

15.5 

13.7 

28 

18.0 

14.3 

9.90 

11.9 

.28 

12.1 

12.7 

14.9 

13.5 

29 

17.4 

14.2 

9.41 

11.7 

29 

11.5 

12.6 

14.3 

13.4 

30 

16.8 

14.0 

8.93 

11.5 

30 

11.0 

12.4 

13.8 

13.2 

31 

16.3 

13.9 

8.50 

11.3 

31 

10.5 

12.2 

13.2 

13.1 

32 

15.7 

13.8 

8.10 

11.1 

32 

10.0 

12.0 

12.7 

12.9 

33 

15.2 

13.6 

7.70 

10.8 

33 

9.59 

11.8 

12.3 

12.8 

34 

14.6 

13.5 

7.36 

10.6 

34 

9.18 

11.6 

11.7 

12.6 

35 

14.1 

13.3 

7.04 

10.4 

35 

8.79 

11.4 

11.3 

12.5 

36 

13.6 

13.2 

6.71 

10.2 

36 

8.42 

11.2 

10.9 

12.3 

37 

13.2 

13.1 

6.41 

10.0 

37 

8.07 

11.0 

10.5 

12.2 

38 

12.7 

13.0 

6.11 

9.79 

38 

7.75 

10.8 

10.1 

12.0 

39 

12.3 

12.9 

5.85 

9.59 

39 

7.44 

10.7 

9.75 

11.9 

40 

11.9 

12.7 

5.62 

9.39 

40 

7.14 

10.5 

9.40 

11.7 

41 

11.5 

12.6 

5.40 

9.20 

41 

6.86 

10.3 

9.07 

11.6 

42 

11.1 

12.4 

5.19 

9.00 

42 

6.60 

10.1 

8.76 

11.4 

43 

10.8 

12.3 

4.99 

8.81 

43 

6.35 

9.95 

8.46 

11.3 

44 

10.4 

12.2 

4.79 

8.64 

44 

6.11 

9.77 

8.16 

11.1 

45 

10.1 

12.0 

4.59 

8.46 

45 

5.89 

9.59 

7.88 

10.9 

46 

9.80 

11.9 

4.42 

8.27 

46 

5.68 

9.42 

7.61 

10.8 

47 

9.50 

11.7 

4.26 

8.10 

47 

5.48 

9.25 

7.36 

10.6 

48 

9.20 

11.6 

4.11 

7.94 

48 

5.28 

9.09 

7.13 

10.4 

49 

8.94 

11.4 

3.97 

7.76 

49 

5.10 

8.92 

6.91 

10.3 

50 

8.66 

11.3 

3.84 

7.60 

50 

4.92 

8.77 

5.90 

9.61 

55 

7.47 

10.7 

3.22 

6.86 

55 

4.16 

8.00 

5.10 

8.92 

60 

6.49 

10.0 

2.75 

6.18 

60 

3.57 

7.30 

4.44 

8.29 

65 

5.68 

9.43 

2.37 

5.59 

65 

3.09 

6.67 

3.90 

7.69 

70 

5.01 

8.84 

2.06 

5.06 

70 

2.70 

6-10 

3.44 

7.14 

75 

4.44 

8.30 

1.80 

4.59 

75 

2.37 

5.59 

3.08 

6.64 

80 

3.97 

7.77 

1.59 

4.18 

80 

2.10 

5.13 

2.46 

5.73 

90 

3.21 

6.88 

1.27 

3.49 

90 

1.68 

4.34 

2.02 

4.99 

100 

2.65 

6.02 

1.04 

2.95 

100 

1.37 

3.71 

No  care  has  been  tafeen  to  have  the  last  figure  of  a  load  perfectly  correct. 

*  See  Coefficients  of  safety  near  foot  of  p.  225,  and  "  Remarks  on  this  edition,"  p.  ix. 


STRENGTH   OF   IRON   PILLARS. 


233 


Pillars  with  rounded  ends,  as  in  Fig  1,  are  usually  assumed  to  have 
only  y%  the  strength  ot  those  with  flat  ends,  firmly  fixed;  or  %,  if  only  one  end 
(either  top  or  bottom)  is  rounded.  In  iron  bridges  and  roofs,  the  ends  of  pillars  and 
of  oblique  struts  frequently  have  not  full  fair  bearings  for  their  ends ; 
but  are  sustained  by  means  of  pins  or  bolts  driven  through  (across) 
them,  at  either  one  or  both  ends,  as  at  p.  These  are  about  interme- 
diate in  strength  between  those  with  flat  and  those  with  rounded 
ends ;  as  the  writer  is  informed  by  Mr.  C.  Shaler  Smith,  as  the  results 
of  experiments  by  him.  These  remarks  apply  equally  to  hollow 
pillars;  and  to  both  wrought  and  cast  iron,  and  wood;  but  there  is 
a  good  deal  of  uncertainty  about  this  and  all  such  matters.  The 
terms  hinged,  jointed,  or  pinned  are  not  applied  to  pillars 
whose  ends  are  merely  rounded,  as  in  Fig  1,  but  only  to  those,  whether  f 
flat  or  rounded,  which  have  pins. 

Ultimate  crippling  strengths  in  Ibs  per  sq  inch  of  metal 
section  of  the  four  wrought  iron  pillars  below.  These  formulas 
are  deduced  by  Chs.  Shaler  Smith,  from  many  tests  by  G.  Bouscaren,  C.  E.,  of 
large  pillars  of  good  American  iron.  The  lower  Table  is  an  abridgment  of  the 
full  ones  by  C.  L.  Gates,  C.  E.,  in  the  Trans.  Am.  Soc.  C.  E.,  Oct.,  1880. 
length  between  end  bearings  ^ 

least  diameter  d 
For  safety  take  from  } 


H 


-  both  in  the  same  measure ;  and  is  to  be  squared. 
,  according  to  circumstances. 


Flat  ends 

One  pin  end- 
Two  pin  ends. 


1200 


Ultimate  and  safe  loads  in  ft>s  per  sq  inch,  of  the  above  four  pillars,  with 
flat  ends,  and  equally  loaded.  Coef  of  Safety  =  4  -4-  .05  H.  By  C.  L.  Gates,  C.  E. 


H. 

A.  Square  Col. 

B.  Phoanix  Col. 

C.  Amer 

can  Col. 

D.  Common  Col. 

Ult. 

Safe. 

Ult. 

Safe. 

Ult. 

Safe. 

Ult. 

Safe. 

15 

37067 

7822 

40476 

8521 

34434 

7249 

33693 

7093 

16 

86876 

7683 

40212 

8377 

34167 

7118 

33339 

6946 

18 

36470 

7443 

39645 

8091 

33597 

6856 

32589 

6651 

20 

36024 

7205 

39030 

7806 

32982 

659(5 

31790  I 

6358 

22 

35544 

6970 

38373 

7514 

32327 

6338 

30952 

6069 

25 

34767 

6622 

37317 

7110 

31285 

5959 

29639 

5646 

30 

3:3344 

6063 

35424 

6440 

29435 

5352 

27375 

4977 

35 

31806 

5531 

33406 

1  5810 

27512 

4789 

25108 

4367 

40 

30198 

5033 

31352 

5226 

25584 

4264 

22919 

3820 

45 

28562 

4570 

29310 

4690 

23701 

3792 

20857 

3337 

50 

26932 

4143 

27321 

4203 

21900 

3369 

18952 

2916 

55 

25333 

3728 

25415 

3765 

20203 

3004 

17214 

2550 

60 

23787 

3398 

23611 

3373 

18621 

2660 

15643 

2235 

234 


STRENGTH   OF   IRON   PILLARS. 


Table  of  rolled  segment-columns  of  the  Phoenix  Iron4 
Company,  Fig  B,  p  233.    See  p  233,  for  the  strength. 


Four-segment  Columns. 

The  least  outer  diams  are  those  of  the  circles  formed  by  the  segmental  pieces;  the 
greatest  are  from  out  to  out  of  flats. 


Four  Segments. 

Four  Flats.  f 

Total. 

External  Diam. 

Thick- 
ness. 
Inch. 

Area. 
Sq.Ins. 
of  4. 

Weight. 
Lbs.pr  ft. 
of  4. 

30.25 
37.10 
44.00 
51.00 

Size. 
Ins. 
1    ofl. 

Area. 
Sq.Ins. 
of  4. 

Weight. 
Lbs.  pr  ft. 
of  4. 

Area. 
Sq.Ins. 

Weight. 
Lbs.  pr  ft. 

Least. 
Ins. 

Greatest. 
Ins. 

5-16 

8.71 
10.79 
12.84 
14.85 

3X9  216 

4.5 
6. 
6.75 
10. 

13.94 
20.28 
22.S1 
33.79 

13.21 
16.79 
19.59 
24.85 

4419 
67.38 
66.81 
84.79 

8 
8/t 

Sfj 

Six-segment  Columns. 


Six  Segments. 

Six  Flats.f 

Total. 

External  Diam. 

Thick- 
ness. 
Inch. 

Area. 
Sq.Ins. 
of  6. 

Weight. 
Lbs.  pr  ft. 
of  6. 

Size. 
Ins. 
ofl. 

Area. 
Sq.Ins. 
of  6. 

Weight. 
Lbs.  pr  ft. 
of  6. 

Area. 
Sq.Ins. 

Weight. 
Lbs.  pr  ft. 

Least. 

lus. 

Greatest. 
Ins. 

~^A" 

7-16 

23.02 

76.56 

4X% 

15. 

50.68 

38.02 

127.24 

13  A 

Eight-segment  Columns. 


Eight  Segments. 

Eight  Flats.f 

Total. 

External  Diam. 

Thick- 
ness. 
Inch. 

Area. 
Sq.Ins. 
of  8. 

Weight. 
Lbs.  pr  ft. 
of  8. 

Size. 
Ins. 
ofl. 

Area. 

Sq.Ins. 
of  8. 

Weight. 
Lbs.  pr  ft. 
of  8. 

Area. 
Sq.Ins. 

Weight. 
Lbs.  pr  ft. 

Least. 
Ins. 

Greatest. 
Ins. 

7-16 

30.64 

102 

5X% 

24.96 

83.2 

55.6 

185.2 

16^ 

22% 

RULE  9.  For  such  forms  of  transverse  section  as  the  follow 


•K  Price,  Philada,  1880,  5  to  5^  cts  per  ft. 

4-  The  Fliits  (.strengthening  strips  between  the  segments')  are  not  now  used. 


STRENGTH   OP   IRON   PILLARS. 


RULE  10.    In  such  forms  of  cross-section  as  Fig's  5  to  14,  Ac,  in 

order  to  attain  the  greatest  accuracy  according  to  present  theory,  we  must  introduce 
in  the  formulas  the  square  of  the  least  radius  of  yy  ration  (see  Glos- 
sary) of  the  cross-section.  Gordon's  Rules  1  to  8,  pp.  221  to  223,  would  then  assume 
the  following  shapes.  This  however  would  not  alter  the  results  of  any  of  them,  in- 
asmuch as  they  are  based  upon  this  principle. 


For  Cast  Iron.                              For  Wrought  Iron. 

toacUn      Area  °f  metal  in  8q  inS  X  8000°  'Bre^«      Area  of  metal  in  sq  ins  X  3600° 

ft3m          1+    ( 

When  modified  b 
mulas  for  several  sh 

Solid  rec-  ^m 
tangle      "• 
(or  square)  Q 

Thin    hollow 
square,     _- 

of  uniform     U 
thickness.* 

Thin    hollow 
rec  tan-    a 

uniform    •""" 
thickness.* 

Solid  cyl-  ^ 
inder.      W 

Thin  hollow 
cylinder,  y^ 

of   uniform    ^J 
thickness.* 

Cross     of    J 
equal  JU 
arms.         I 
O 

Angle  iron,  of 
ribs   of  * 
eq  u  al    1 
lengths  ~0 

and  thick-  *    ° 

nesses.f 

Length2  in  ins  \             !>•             i    i     (     Length2  in  ins     ^ 

4800  X  Rad  gyr.2/ 
y  inserting  the  actual  square  o 
apes  would  read  thus,  all  the  di 

CAST  IROX. 

Metal  area  X  80000 

V36000  X  Rad  gyr.2/ 
"  the  radius  of  gyration,  the  for- 
mensions  being  iu  inches  : 

WROUGHT  IRON. 

Metal  area  X  36000 

Length  2  in  ins         ^ 

Length  9  in  ins        . 

1+1           v  least  side  2  in  ins  1 

1  +  /%«nnnx/  least  side3  in  ins  J 

Metal  area  X  80000 

V           X             12.            / 

Metal  area  X  36000 

t         Length  2  in  ins         x 

Length  a  in  ins        ^ 

1  "*"  (  1800  X  °ne  8id°  2  iD  iil8  ) 

1+(36000-°Ue8ideainin8) 

*                            6.              ' 
Metal  area  X  80000 

yODUUU  X                   g                    / 

Metal  area  X  36000 

Length  2  in  ins 

Length  a  in  ins 

1  4.  1            fc2vC~*"3M     ) 

l+(QCftAAfca    ^c-fSa^J 

V           \12c  +  a/' 

Metal  area  X  80000 

>          ^12          c-fa^ 
Metal  area  X  36000 

Length  2  in  ins 

/        Length  2  in  ins        x 

I  -f-  I                  Diam  2  in  ins     1 

1  +  (                  Diam  2  in  ins  ) 

16.             ' 

Metal  area  X  80000 

\  36000  X             -.  g  / 
Metal  area  X  36000 

Length  2  in  ins 

Length  2  in  ins        x 

1  ~*~  1  4800  V    I)iam  2  in  ins     1 

1  +  (                   Diam  2  in  ins  ) 

Metal  area  X  80000 

\36000  X    •          g             / 
Metal  area  X  36000 

t         Length  2  in  ins         s 

Length  2  in  ins        . 

1  +  (  1800  X      a  °  2  iu  in8      ) 

1  +  (                   a  o  2  in  ins     ) 

>  ^                      24.             1 
Metal  area  X  80000 

\36000  X             24.          ' 
Metal  area  X  36000 

t         Length  2  in  ins         ^ 

Length  a  in  ins 

1  +  (  4800  Y      n  °  2  in  ins      ) 

1  -|-  (                   n  o  a  in  ins     ) 

y  4»uu  X            ^ 

\3600                    24            J 

&  By  thin,  wo  believe,  is  meant  that  the  single  thickness  of  metal  shall  not 
exceed  about  ^  part  of  the  diam,  or  side.  When  the  thickness  is  greater  than  ^  diam,  or  side,  the 
loads  given  by  the  formulas  will  be  appreciably  too  great  for  practical  use.  See  ".Remarks  on 
this  edition,"  p.  ix. 

i  This  and  the  next  answer  also  for  T  iron. 


236 


STRENGTH   OF   IKON   PILLARS. 


Angle  iron  of 
ribs      of    • 
unequal  01 
lengths,     bn 

but      equal        C 
thicknesses. 

H    iron.  £t     C 

Call    the    I      |£ 
area  of  the  H& 
web    W,    1     Iv 
and  that  of  0     11 
the  two  flanges  a  o, 
c  n,  E.* 
-f 

CAST  IROX. 

Metal  area  X  80000 

WROUGHT  IROX. 

Metal  area  X  36000 

Length  a  in  ins 

/        Length  3  in  ins        v 

1+  r                 /  o*  X    c3  \    ] 
\180Q  X    I                       )  / 

1  +  (  36000    y/°aXc2l\j 

Metal  area  X  80000 

Metal  area  X  36000 

/        Length  a  in  ins        x 

,        Length  a  in  ins          . 

1  +  UoOX  (m2XareaF)j 

1+  (  36000  V  (  m*  v  afea  *  )  ) 

\i5uu  x   \i2     F4-W'' 

*"|ir            Channel  in 
the  outer  end  d,  o 

E§$sl^           p  *  nf  t.hft  t.hiplrne 

>n.    Let  d  e  be  the  depth  from 
f  a  flange  /  o,  to  the  middle  line 
3s  of  the  web  w  w  o  o.    Find  the 
that  of   the    two  flanges  /  o, 
are  inches.    Then,  for  cast 

%4I 

J 

IPO    total   area;    also 

w 

T^rfK                     iron, 

Brkg 
load  = 
in  Ibs 


Metal  area  X  80000 


Length  2  in  inches 


1  -f  (4800X«*«»X 
T      *ou  ^ 


*"'ea  flX  area, 


webNX 
area  /  / 


N  \12  X  total  area         4  X  sq  of  tot  alc«  /  , 

For  wrought  iron,  instead  of  80000,  and  4800,  use  36000  in  both  cases. 

Table  of  breaking  loads  in  tons  (224O  Ibs)  per  square  Inch 
of  metal  area,  of  pillars  or  struts  of  H  section  of  uniform 
thickness,  by  the  above  formula.f  The  heights  or  lengths  of  the 

pillars  are  in  flanges  a  o,  or  c  n.    See  "  Remarks  "  below.  Original. 


Hts 

Cast. 

Wrt. 

Hts 

Cast. 

Wrt. 

Hts 

Cast. 

Wrt. 

Hts 

Cast. 

Wrt. 

in 

Figs. 

Tons. 

Tons. 

in 

Figs. 

Tons. 

Tons. 

in 
Figs. 

Tons. 

Tons. 

in 

Figs. 

Tons. 

Tons. 

1 

35.5 

16.1 

12 

20.2 

15.0 

23 

9.38 

12.9 

38 

4.12 

9.56 

2 

34.9 

6.0 

13 

18.8 

14.9 

24 

8.79 

12.7 

40 

3.76 

9.15 

3 

34.0 

6.0 

14 

17.5 

14.7 

25 

8.26 

12.4 

42 

3.45 

8.76 

4 

32.9 

6.0 

15 

16.3 

14.5 

26 

7.77 

12.1 

45 

3.04 

8.21 

5 

31.5 

5.9 

16 

15.1 

14.3 

27 

7.33 

11.9 

50 

2.51 

7.37 

6 

30.0 

5.8 

17 

14.1 

14.0 

28 

6.89 

11.7 

55 

2.09 

6.62 

7 

28.3 

5.6 

18 

13.1 

13.8 

29 

6.53 

11.5 

60 

1.77 

5.95 

8 

26.6 

5.5 

19 

12.2 

13.7 

30 

6.18 

11.3 

70 

1.32 

4.85 

9 

24.9 

5.4 

20 

11.4 

13.5 

32 

5.66 

10.9 

80 

1.02 

4.00 

10 

23.3 

15.3 

21 

10.6 

13.3 

34 

5.02 

10.4 

90 

.81 

3.33 

11 

21.6 

15.1 

22 

10.0 

13.1 

36 

4.54 

10.0 

100 

.66 

2.81 

The  H  is  stronger  than  the  L,  T,  -f ,  or  LJ  section,  the  diff  increasing 
with  the  height.  The  table  may  be  used  for  riveted  columns  with 
I  webs  and  LJ  flanges  if  properly  made. 

Remarks.    This  table  was  calculated  for  a  square  H,  with  a  uniform 

thickness  of  one-twelfth  of  its  width,  but  it  will  answer  near  enough  in  practice  if  the  H  is  longer  in 
either  direction  than  in  the  other,  not  exceeding  one-fourth  part ;  and  tor  any  uniform  thickness  from 

one-twenty-fourth  to  one-fifth  of  the  least  out  to  out  width  of  the  H.  If  wider 
along  the  flanges  than  across  them  (within  the  above  limits)  the  strength 

per  sq  inch  becomes  somewhat  greater  than  the  tabular  one,  and  vice  versa,  varying  with  the  pro- 
portions of  the  section,  the  height,  and  whether  cast  or  rolled.  Still  an  allowance  of  6  per  cent  will 

suffice  for  either  extreme.    If  the  web  is  thinner  than  the  flanges, 

the  strength  per  square  inch  of  metal  area  of  the  entire  H  increases  somewhat,  and  vice  versa. 

If  the  uniform  thickness  of  the  H  should  be  as  great  as  one-fifth  of  its 

least  out  to  out  widt.h,  the  increase  of  strength  per  sq  inch  will  not  exceed  about  6  per  ct  in  any  pillar 
or  strut  either  wrought  or  cast;  and  the  diminution  of  strength  in  a  uniform  thickness  of  only  one- 

twenty-fourth  said  least  width  would  be  at  about  the  same  rate.  If  the  flanges 
taper  toward  their  ends,  as  is  usually  the  case,  it  is  well  for  safety  to  take  their 
metal  area  at  only  what  it  would  be  if  they  were  uniformly  of  about  their  end  thickness. 

*  The  writer  doubts  whether  this  formula  applies  after  a  o  exceeds  about  twice  a  c. 

f  For  tables  of  pillars  of  L,  T,  I,  +,  and  LJ  sections,  see  pages  637  to  640. 


STRENGTH    OF   IRON   PILLARS. 


237 


In  arches  of  cast  iron  for  bridges,  &c,  it  is  usual  among  English 
engineers  not  to  allow  more  than  2^  tons  (5600  ft>s)  of  compression,  or  thrust,  per  sq 
inch.  Brunei  never  subjected  cast-iron  pillars  to  more  than  1%  tons  (3360  fos)  per 
sq  inch.  C.  Shaler  Smith  *  employs  as  maximum  working  strains,  J.  of  the  calcu- 
lated breaking  strain  for  such  hollow  chords  and  posts  of  bridges  as  are  1  inch  or 
more  in  thickness,  and  not  more  than  15  diams  long.  For  posts,  only  ^ ;  when  not 
less  than  %  inch  thick,  nor  more  than  25  diams  long ;  or  from  y1^  to  ^y,  when  % 
thick  or  less,  and  more  than  25  diams  long. 

The  young  engineer  must  bear  in  mind  that  the  breakg  and  the 
safe  loads  per  sq  inch,  of  pillars  of  any  given  material,  are  not  constant  quantities ; 
but  diminish  as  the  piece  becomes  longer  in  proportion  to  its  diam.  If  a  very  long 
piece  or  pillar  be  so  braced  at  intervals  as  to  prevent  its  bending  at  those  points, 
then  its  length  becomes  virtually  diminished,  and  its  strength  increased.  Thus,  if  a 
pillar  100  ft  long  be  sufficiently  braced  at  intervals  of  20  ft,  then  the  load  sustained 
may  be  that  due  to  a  pillar  only  20  ft  long.  Therefore,  very  long  pillars  used  for 
bridge  piers,  &c,  are  thus  braced;  as  are  also  long  horizontal  or  inclined  pieces, 
exposed  to  compression  in  the  form  of  upper  chords  of  bridges;  or  as  struts  of  any 
kind  in  bridges,  roofs,  or  other  structures. 

Mistakes  are  sometimes  made  by  assuming,  say  5  or  6  tons  per  sq  inch,  as  the  safe 
compressing  load  for  cast  iron ;  4  tons  for  wrought ;  1000  pounds  for  timber ;  without 
any  regard  to  the  length  of  the  piece. 

But  although  the  final  crushing  loads,  as  given  in  tables  of  strengths  of  materials, 
are  usually  those  for  pieces  not  more  than  about  2  diams  high,  they  will  not  be  much 
less  for  pieces  not  exceeding  4  or  5  diams. 

Cautions.  Remember  a  heavily  loaded  cast-iron  pillar  is  easily  broken  by  a 
side  blow.  Cast-iron  ones  are  subject  to  hidden  voids.  All  are  subject  to  jars  and 
vibrations  from  moving  loads.  It  very  rarely  happens  that  the  pres  is  equally  dis- 
tributed over  the  whole  area  of  the  pillar;  or  that  the  top  and  bottom  ends  have  per- 
fect bearing  at  every  part,  as  they  had  in  the  experimental  pillars.f  Cast  pillars  are 
seldom  perfectly  straight,  and  hence  are  weakened. 

Hollow  pillars  intended  to  bear  heavy  loads  should  not  be  cast 
with  such  mouldings  as  aa;  or  with  very 
projecting  bases  or  caps  such  as  g,  Fig  19. 
It  is  plain  that  these  are  weak,  and  would 
break  off  under  a  much  less  load  than 
would  injure  the  shaft  of  the  pillar.  When 
such  projecting  ornaments  are  required, 
they  should  be  cast  separately,  and  be  at- 
tached to  a  prolongation  of  the  shaft,  as 
co",  by  iron  pins  or  rivets. 

Ordinarily,  it  is  better  to  adopt  a  more 
simple  style  of  base  and  cap,  which,  as  at 
&,  can  be  cast  in  one  piece  with  the  pillar, 
without  weakening  it. 


Hodginson  states  that  while  the  quantity  of  material  is  the  same 
in  both  pillars,  no  strength  is  gained  in  hollow  ones  by  making 
the  diams  greater  at  the  middle  than  at  the 
ends  ;  but  that  in  solid  ones,  with  rounded  ends,  there  is  a  gain 
of  about  i.th  part ;  and  in  those  with  flat  ends,  of  about  Jth  or 
^th  part,  by  making  the  diam  at  the  middle  about  \%  or  2  times 
that  at  the  ends.  Also  that  a  uniform  round  pillar  has  the  same 
strength  as  a  moderately  tapering  one  whose  diam  at  half-way  up 
is  equal  to  the  uniform  diam  of  the  cylindrical  one. 

Also,  that  when  a  flat-ended  pillar,  Fig.  2,  is  so  irregularly 
fixed,  that  the  pressure  upon  it  passes  along  its  diagonal  a  a,  it 
loses  two-thirds  of  its  strength.  Hence  the  necessity  for  equalizing,  as  far  as  possi- 
ble, the  pressure  over  every  part  of  the  top  and  bottom  of  the  pillar;  a  point  very 
difficult  to  secure  in  practice. 

*  Of  the  very  skilful,  experienced,  and  intelligent  firm  of  Smith  &  Latrobe,  civ  engs,  and  bridge- 
builders,  Baltimore,  Md.,  The  Baltimore  Bridge  Co. 

f  In  important  cases,  both  ends  should  be  planed  perfectly  true; 

as  is  done  ill  irou  bridges,  &c. 


238 


STRENGTH   OF  WOODEN   PILLARS. 


Steel  pillars.  Mr.  Kirkaldy  experimented  with  a  small  steel  pillar  or  tube 
of  Shortridge,  Howell  &  Go's  homogeneous  metal.  Its  length  was  4  ft,  or  25.6  diams ; 
outer  diam  1%  inch:  inner  diam  1^;  thickness  %  inch.  Area  of  cross-section  1% 
sq  ins.  Flat  ends.  Under  67300  Ibs,  or  30  tons  total  pressure,  or  17.14  tons  per  sq 
inch  of  solid  metal  section,  it  bent  very  slightly.  On  increasing  the  pressure  con- 
siderably, the  pillar  sprang  out  from  under  the  load.  Our  preceding  table  gives  22% 
tons  total,  or  13  tons  per  sq  inch,  as  the  ultimate  load  for  a  wrought-iron  tube  of  the 
same  size. 

Mr.  M.  G.  Love,  Paris,  as  the  result  of  a  trial  with  small  steel  rods,  about  .4 
inch  diam,  and  which  had  a  tensile  strength  of  48  tons  per  sq  inch,  suggests  that  the 
comparative  strength  of  pillars  of  wrought  iron,  cast-iron,  and  steel,  are  probably 
about  as  follows :  At  from  1^£  to  5  diarns  in  length,  steel  and  cast-iron  ones  have 
equal  strengths  ;  and  either  of  them  is  about  twice  as  strong  as  wrought  iron.  At  10 
diams,  steel  is  1%  times  as  strong  as  cast ;  and  2.2  times  as  strong  as  wrought  iron. 
At  40  diams,  steel  is  4  times  as  strong  as  cast;  and  2.7  as  strong  as  wrought.  But 
this  needs  confirmation.  Now  that  powerful  and  accurate  testing  machines  are 
coming  more  into  use,  we  may  hope  that  the  doubts  at  present  existing  on  such 
subjects  will  be  set  at  rest. 

Mr  Stoney  advises  that  until  then  steel  pillars  should  not  be  trusted 
with  more  than  1.5  the  loads  of  wrought  iron  ones. 


WOODEN  PILLASS, 


The  strengths  of  pillars,  as  well  as  of  beams  of  timber,  depend  much  on  their  de- 
gree of  seasoning.  Hodgkinson  found  that  perfectly  seasoned  blocks,  2  diams 
long,  required,  in  many  cases,  twice  as  great  a  load  to  crush  them  as  when  only 
moderately  dry.  This  should  be  borne  in  mind  when  building  with  green  timber. 

In  important  practice,  timber  should  not  be  trusted  with  more  than  ^  to  %of  ita 
calculated  crushing  load ;  and  for  temporary  purposes,  not  more  than  %  to  %. 

Mr.  Charles  Shaler  Smith.  €.  E.,  of  St.  Louis,  prepared  the 
following  formula  for  the  breaking  loads  of  either  square  or  rectangular 
pillars  or  posts,  of  moderately  seasoned  white,  and  common  yellow  pine,  with  flat 
ends,  firmly  fixed,  and  equally  loaded,  based  upon  experiments  by  himself. 

It  is  Gordon's  formula  adapted  to  those  woods;  and  gives  results  considerably 
smaller  than  Hodgkinson's,  as  is  shown  on  p  242.  It  is  therefore  safer. 

Call  either  side  of  the  square,  or  the  least  side  of  the  rectangle,  the  breadth.    Then, 

Breakg  load  in  Ibs,  per  1        . 5000t 

Rule.      sq  inch  of  area,  of  a     >  ==  /sq  of  length  in  ins         v     \ 

pillar  of  W  or  Y  pine   J       1  +  (sq  of  breadthip  ini|  X  .004  J 

Or  in  words,  square  the  length  in  ins ;  square  the  breadth  in  ins ;  div  the  first  square 
by  the  second  one;  mult  the  quot  by  .004  ;  to  the  prod  add  1 ;  div  5000  by  the  sum. 

Ex.  Breakg  load  per  sq  inch,  of  a  white  pine  pillar  12  ins  square,  and  30  ft,  or  360 
ins  long.  Here  the  sq  of  length  in  ins  is  3602  =  129600.  The  square  of  the  breadth  is 

122  =  144 .  and  -^-  =  900 ;  and  900  X  -004  =  3.6;  and  3.6  +  1  =  4.6.  Finally,  — 

=  1087  Ibs,  the  reqd  breakg  load  per  sq  in.  As  the  area  of  the  pillar  is  144  sq  ins, 
the  entire  breakg  load  is  10H7  X  1^-4  =  156528  tt>s,  or  69.9  tons.  See  table,  p  242. 

Recent  experiments  on  wooden  pillars  20  ft  long,  and  13  ins  square,  by  Mr. 
Kirkaldy,  of  England,  confirm  the  far  greater  reliability  of  Mr.  Smith's  formula. 
Hence  we  present  the  following  new  set  of  original  tables  based  upon  it. 

For  solid  pillars  of  cast  iron  and  of  pine,  whose  heights  range 
from  5  to  tiO  times  their  side  or  diam,  we  may  say,  near  enough  for  practice,  that  a 
cast  iron  one  is  about  1G%  times  as  strong  as  a  pine  one;  but  no  such  approximate 
ratio  holds  good  between  wrought  iron  and  pine,  or  between  cast  and  wrought  iron. 

*  Each  projects  1  Inch  in  a  4  inch  column  ;  \%  in  a  6  inch  ;  1%  in  all  larger, 
t  The  breaking  load  in  Ibs  per  sq  iuch  iu  short  blocks,  by  Mr.  Smith. 


STRENGTH    OF   WOODEN    PILLARS. 


239 


Table  of  breaking:  loads  in  tons  of  square  pillars  of  half 
seasoned  white  or  common  yellow  pine  firmly  fixed  and 
equally  loaded.  By  C.  Shaler  Smith's  formula.  (Original.) 


is 

Side  of  square  pine  pillar,  in  inches. 

II 

§1 

1       I    IK     1    IX    1    IK    1      2      |    2J4    |    2tf         UJi    |      3          3^    |    3^         3%     |      4 

^f_ 

BREAKING  LOAD. 

i 

1.42 

2.54 

3.99 

5.73 

7.80 

10.1 

12.8 

15.7 

18.9 

22.3 

26.1 

30.1 

34.5 

1 

M 

1.17 

2.22 

3.59 

5.26 

7.25 

9.6 

12.2 

15.1 

18.3 

21.7 

25.4 

29.2 

33.7 

Ji 

.97 

1.93 

3.19 

4.80 

6.74 

9.0 

11.6 

14.5 

17.6 

21.0 

24.7 

28.6 

33.0 

}^j 

% 

.81 

1.66 

2.81 

4.35 

6.19 

8.4 

10.9 

13.7 

16.8 

20.2 

23.9 

27.8 

32.1 

K 

2 

.68 

1.44 

2.48 

3.92 

5.66 

7.8 

L0.2 

12.9 

15.9 

19.3 

23.0 

26.9 

31.2 

2 

J4 

.57 

1-24 

2.19 

3.53 

5.17 

7.2 

9.6 

12.3 

15.2 

1&.5 

22.0 

25.8 

30.1 

B 

H 

.49 

1.07 

1.93 

3.16 

4.70 

6.7 

8.9 

11.5 

14.3 

17.6 

21.1 

24.9 

29.1 

N 

.42 

.93 

1.71 

2.85 

4.29 

6.2 

8.3 

10.8 

13.5 

16.7 

20.1 

23.8 

28.0 

H 

3 

.36 

.82 

1.52 

2.55 

3.89 

5.6 

7.6 

10.0 

12.7 

15.8 

19.2 

22.9 

27.0 

3 

.28 

.63 

1.21 

2.07 

3.23 

4.8 

6.6 

8.8 

11.3 

14.2 

17.4 

20.9 

24.8 

4      . 

.22 

.50 

.98 

1.70 

2.70 

4.0 

5.7 

7.7 

9.9 

12.7 

15.7 

19.0 

22.7 

4 

J6 

.18 

•40 

.81 

1.42 

2.28 

3.4 

4.9 

6.7 

8.8 

11.4 

14.1 

17.2 

20.7 

/4 

5 

.15 

.34 

.68 

1.19 

1.94 

3.0 

4.2 

5.8 

7.7 

10.0 

12.6 

15.5 

18.8 

5 

.12 

.28 

.57 

1.02 

1.67 

2.6 

3.7 

5.1 

6.8 

8.9 

11.3 

14.0 

17.1 

X 

6 

.10 

.24 

.49 

.86 

1.44 

2.3 

3.3 

4.6 

6.1 

8.0 

10.2 

12.7 

15.6 

6 

.09 

.21 

.43 

.74 

1.26 

2.0 

2.9 

4.1 

5.4 

7.2 

9.2 

11.6 

14.2 

7 

.08 

.18 

.37 

.66 

1.11 

1.8 

26 

3.6 

4.9 

6.5 

8.3 

10.5 

12.9 

7 

.07 

.16 

.33 

.59 

.98 

1.6 

2.3 

3.3 

4.4 

5.9 

7.6 

9.6 

11.8 

K 

8 

.06 

.14 

.29 

.52 

.87 

1.4 

2.0 

2.9 

3.9 

5.2 

6.8 

8.7 

10.8 

8 

/4 

.05 

.12 

.26 

.47 

.78 

1.2 

1.8 

2.6 

3.5 

4.8 

6.2 

7.9 

9.9 

% 

9 

.05 

.11 

.23 

.42 

.71 

1.1 

1.6 

2.3 

3.2 

4.3 

5.6 

7.2 

9.1 

9 

i^ 

.10 

.21 

.37 

.64 

1.0 

1.5 

2.1 

2.9 

3.9 

5.1 

6.6 

84 

$6 

10 

.09 

.19 

.34 

.58 

.93 

1.4 

,  2.0 

2.7 

3.6 

4.7 

6.1 

7.8 

10 

.17 

.31 

.53 

.86 

1.3 

1.8 

2.5 

3.4 

4.4 

5.7 

7.2 

11 

.16 

.28 

.48 

.79 

1.2 

1.7 

2.3 

3.1 

4.1 

5.3 

6.7 

11 

.14 

.26 

.44 

.72 

1.1 

1.5 

2.1 

2.9 

3.8 

4.9 

6.2 

12 

.13 

.24 

.41 

.65 

1.0 

1.4 

2.0 

2.7 

3.4 

4.5 

5.8 

12 

3 

.21 

.35 

.55 

.84 

1.2 

1.7 

2.3 

3.1 

4.0 

5.0 

13 

4 

.18 

.31 

.46 

.70 

1.0 

1.4 

2.0 

2.7 

3.5 

4.4 

14 

5 

.27 

.41 

.63 

.91 

1.2 

1.7 

2.4 

3.1 

3.9 

15 

6 

.24 

.37 

.57 

.78 

1.1 

1.5 

2.1 

2.7 

3.5 

16 

7 

.50 

.70 

1.0 

1.4 

1.9 

2.4 

3.1 

17 

8 

.45 

.66 

.92 

1  3 

1.7 

2.2 

2.8 

18 

20 

.76 

1.0 

1.4 

1.8 

2.3 

20 

ll 

Side  of  square  pine  pillar,  in  inches. 

II 

K.S 

*X   1    *%    1    *%    1     5      |    534    |    5}^    |    5%          6      |    6J4        6}^    |    6%    |      7   |    7}£ 

K.2 

BREAKING  LOAD. 

Tons.   Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

2 

35.8 

40.6 

45.7 

51.1 

56.8 

62.8 

69.0 

75.5 

82.3 

89.4 

96.8 

104.5 

112.4 

2 

3 

31.4 

36.1 

41.1 

46.2 

51.8 

57.7 

63.8 

70.2 

76.9 

84.0 

91.3 

98.9 

106.7 

3 

4 

268 

31.2 

35.9 

,  40.8 

46.1 

51.7 

57.6 

63.8 

70.4 

77.3 

84.5 

92.1 

99.9 

4 

5 

22.6 

26.5 

30.8 

35.4 

40.5 

45.8 

51.5 

57.4 

63.7 

70.3 

77.2 

84.5 

92.1 

5 

6 

18.9 

22.5 

26.4 

30.5 

35.2 

40.1 

45.4 

51.0 

57.0 

63.3 

69.9 

76.8 

84.1 

6 

7 

15.8 

19.0 

22.6 

26.2 

30.5 

35.0 

39.9 

45.0 

50.5 

56.5 

62.8 

69.4 

76.3 

7 

8 

13.3 

16.1 

19.2 

22.5 

26.3 

30.4 

34.9 

39.7 

45.9 

50.4 

56.2 

62.4 

69.0 

8 

9 

11.3 

13.7 

16.5 

19.5 

22.9 

26.6 

30.7 

35.0 

39.9 

44.8 

50.2 

56.0 

62.1 

9 

10 

9.7 

11.8 

14.2 

16.9 

19.9 

23.2 

26.9 

30.9 

35.2 

39.9 

44.9 

50.3 

56.0 

10 

11 

8.3 

10.2 

12.4 

14.8 

17.5 

20.4 

23.8 

27.4 

31.3 

35.6 

40.2 

45.1 

50.4 

11 

12 

7.2 

8.8 

10.7 

12.9 

15.4 

18.0 

21.1 

24.3 

27.9 

31.8 

36.0 

40.6 

45.5 

12 

13 

6.2 

7.7 

9.4 

11.4 

13.6 

16.0 

18.8 

21.7 

24.9 

28.5 

32.4 

36.6 

41.1 

13 

14 

5.5 

6.8 

8.3 

10.1 

12.1 

14.2 

16.7 

19.4 

22.4 

25.7 

29.2 

33.1 

37.3 

14 

15 

4.8 

6.0 

7.4 

9.0 

10.8 

12.7 

15.0 

17.5 

20.2 

23.2 

26.4 

30.0 

33.9 

15 

16 

4.4 

5.4 

6.7 

8.1 

9?8 

11.5 

13.6 

15.8 

18.3 

21.0 

24.0 

27.3 

30.8 

16 

18 

3.6 

4.4 

5.5 

gj 

8^0 

9^4 

12.3 
11.2 

14.3 
13.0 

16.6 
15.1 

19.1 
17.4 

19.9 

24.9 
22.7 

25^8 

18 

19 

3.3 

4.0 

5.0 

6^0 

7.3 

8.6 

102 

11.9 

13.8 

16.0 

18.3 

20.9 

23.7 

19 

20 

3.0 

3.7 

4.6 

5.5 

6.6 

7.8 

9.3 

10.9 

12.6 

14.6 

16.8 

19.2 

21.8 

20 

22 

2.5 

3.0 

3.8 

4.6 

5.6 

6.6 

7.9 

9.2 

10.7 

12.4 

14.3 

16.3 

186 

22 

24 

.  2.1 

2.6 

3.2 

3?9 

4.7 

5.6 

6.7 

7.9 

9.1 

10.6 

12.2 

14.1 

16.0 

24 

26 

1.8 

2.2 

2.8 

3.4 

4.1 

4.9 

5.8 

6.8 

7.9 

9.2 

10.6 

12.2 

13.9 

26 

28 

1.5 

1.9 

2.4 

2.9 

3.5 

4.2 

5.1 

5.9 

6.9 

8.0 

9..? 

10.7 

12.2 

28 

30 

13 

1.7 

2.1 

2.6 

3.1 

3.7 

4.4 

5.2 

6.1 

7.1 

8.2 

9.4 

10.8 

30 

32 

1.2 

1.5 

1.9 

2.3 

2.7 

3.2 

3.9 

4.6 

5.4 

6.3 

7.3 

8.4 

9.6 

32 

34 

1.1 

1.3 

1.7 

2.0 

2.4 

2.9 

3.5 

4.1 

4.8 

5.6 

6.6 

7.5 

8.6 

34 

36 

1.0 

1.2 

1.5 

1.8 

2.2 

2.6 

3.1 

3.7 

4.3 

5.0 

5.8 

6.7 

7.7 

36 

38 

.9 

1.1 

1.3 

1.6 

2.0 

2.4 

2.8 

3.3 

3.9 

4.5 

5.3 

6.1 

7.0 

38 

40 

.8 

1.0 

1.2 

1.5 

1.8 

2.1 

2.6 

8.0 

3.5 

4.1 

4.8 

5.5 

6.3 

40 

Continued  on  next  page. 

16 


240 


STRENGTH   OF   WOODEN   PILLARS. 


Table  of  breaking:  loads  in  tons  of  square  pine  pillars,  with 
flat  ends  firmly  fixed,  and  equally  loaded.    (Continued.) 

Original. 


.11 

Side  of  square  pine  pillar,  in  inches. 

II 

S.2 

7H 

1%    \     8     |    8J4    |    8^    |    8%    |    9      I    9H    i    9H    |    9%    |     10     |   10}*  |   10^ 

K.3 

BREAKING  LOAD. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons 

Tons. 

Tons. 

2 

120.6 

129.1 

137.9 

147.0 

156.3 

165.9 

175.8 

186-0 

196.4 

207.2 

218.2 

229.5 

241.0 

2 

4 

107.9     11H-3 

125.0 

133.9 

143.0 

152.6 

162.4 

172.5 

182.8 

193.4 

204.2 

215.3 

2?6.6 

4 

6 

91.7  j    99.7 

108.0 

116.5 

125.3 

134.$ 

J43.-5 

153.0 

162.8 

173.5 

184.4 

195.6 

207.0 

6 

8 

75.9      83.2 

90.8 

98.7 

106.8 

115.5  !  124.4 

133.6 

143.0 

153.0 

1W3.2 

173.7 

184.4 

8 

10 

62.0 

68.5 

75.3 

82.4 

89.7 

97.7 

105.8 

114.3 

123.0 

132.3     141.8     151.6    161.6 

10 

12 

50.7 

56.5 

62.5 

68.7 

75.1 

82.2 

89.6 

97.2 

105.0 

1135     122.2    131.2    140.5 

12 

14 

41.8 

46.7 

51.9 

57.3 

62.9 

69.2 

75.7 

82.4 

89.4 

97.1  1  105.0 

113.2     121.6 

14 

16 

34.7 

38.9 

43.4 

48.1 

53.0 

58.5 

64.3 

70.3 

76.5 

83.3 

90.4 

97.7     105.3 

16 

18 

29.1 

32.7 

3(5.6 

40.7 

45.0 

49.8 

549 

60.1 

65.6 

71.7 

78.0 

84.6  j    91.4 

18 

20 

24  6 

•>>7  7 

31  1 

34  7 

38  5 

42  7 

47  2 

51  8 

56  7 

6'>  0 

67  6 

7'i  5 

79  6 

20 

23 

19.6       22.1 

24.8 

27.7 

30.9 

34.4 

38.0 

41.9 

46.0 

50.5 

55.2 

60.* 

65.4 

23 

26 

15.8 

17.8 

20.1 

22.5 

25.2 

28.1 

31.1 

34.4 

379 

41.6 

45.6 

49.8 

54.3 

26 

29 

13.1 

14.8 

16.7 

18.7 

20.9 

23.4 

25.9 

28.6 

31.6 

34.9 

38.2 

41.9 

45.6 

29 

32 

10.9 

12.3 

13.9 

15.7 

17.6 

19.7 

21.8 

24.2 

26.7 

29.5 

32.3 

35.5 

38.7 

32 

35 

9.3 

10.6 

11.9 

13.4 

15.0 

16.8 

18.7 

20.7 

22.8 

25.2 

27.7 

30.5 

33.3 

35 

38 

8.0 

9.1 

10.2 

11.5 

12.9 

14.6 

16.3 

18.0 

19.7 

21.8 

23.9 

26.3 

28  8 

38 

41 

6.9 

7.9 

8.9 

10.0 

11.2 

12.6 

14.1 

15.6 

17.2 

19.1 

21.0 

23.1 

25.2 

41 

44 

6.0 

6.9 

7.8 

8.8 

9.8 

11.0 

12.3 

13.6 

15.0 

16.7 

18.4 

20.2 

22.1 

44 

50 

4.7 

5.4 

6.1 

6.9 

7.7 

8.8 

10.0 

11.3 

12.6 

13.7 

14.9 

16.2 

17.5 

50 

f| 

10%  | 

Side  of  square  pine  pillar,  in  inches. 

II 

W.2 

11    |  iiy^  \  \\K  \  u%  |    12    | 

|    10%  |     11    |  lltf   |  11&  |  11%    |    12 

BREAKING  LOAD. 

BREAKING  LOAD. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

4 

238.9 

251.0 

263.2 

275.9 

288.8 

302.1 

26.5 

28.6 

31.1 

33.8 

36.7 

39.9 

42 

6 

218.8 

230.6 

242.7 

255.2 

267.9 

281.0 

24.2 

26.4 

28.7 

31.2 

33.9 

36.8 

44 

8 

195.5 

207.0 

218.5 

230.4 

242.5 

255.1 

22.4 

24.3 

26.4 

28.7 

31.2 

33.9 

46 

10 

172.2 

183.0 

194.1 

205.6 

217.3 

229.5 

20.7 

22.6 

24.5 

26.7 

28.9 

31.4 

48 

12 

150.3 

160.3 

170.7 

181.5 

192.5 

204.0 

19.2 

20.9 

22.7 

24.7 

26.8 

29.2 

50 

14 

130.5 

139.8 

149  4 

159.4 

169.6 

180.2 

17.8 

19.5 

21.1 

23.0 

25.0 

27.2 

52 

16 

113.2 

121.7 

1304 

139.6 

149.0 

158.8 

16.6 

8.2 

19.7 

21.5 

23.3 

25.4 

54 

18 

98.7 

106.2 

114.1 

122.5 

13KO 

140.0 

15.5 

7.0 

18  4 

20.1 

21.8 

23.7 

56 

20 

86.2 

92.9 

100.0 

107.6 

115.4 

123.6 

14.5 

5.9 

172 

.18.8 

20.4 

22.2 

58 

22 

75.6 

81.7 

88.1 

95.0 

102.0 

109.5 

13.6 

4.9 

16.2 

*17.7 

19.2 

20.9 

60 

24 

66.7 

72.2 

77.9 

84.1 

90.5 

97.3 

11.8 

2.9 

13.9 

15.2 

165 

18.0 

65 

26 

59.1 

64.0 

69.2 

74.9 

80  6 

86.8 

10.1 

11.1 

12.0 

13.2 

14.3 

15.6 

70 

23 

52.6 

57.1 

61.8 

66.9 

72.1 

77.7 

8.9 

9.8 

10.6 

11.6 

12.5 

13.7 

75 

30 

47.0 

51.1 

55.3 

60.0 

64.7 

69.9 

78 

8.6 

9.4 

10.2 

11.1 

12.1 

80 

32 

42.1 

46.0 

49.9 

54.0 

58.4 

63.0 

7.0 

7.7 

8.4 

9.1 

9.9 

10.7 

85 

34 

38.2 

41.5 

45.0 

48.8 

52.8 

57.1 

6.2 

6.8 

T.4 

8.1 

8.8 

9.6 

90 

36 

34.6 

37.7 

40.9 

44.3 

48.0 

51.8 

5.6 

6.1 

6.7 

7.3 

8.0 

8.7 

95 

38 

31.5 

34.2 

37.2 

40.4 

43.9 

47.4 

5.1        5.6 

6.1 

6.6 

7.2 

7.8 

100 

40 

28.8 

31.3 

34.0 

37.0 

40.1 

43.5 

3.5  '     3.9 

4.2 

4.6 

5.0 

5.5 

120 

Continued  on  next  page. 

Remarks.    Mr  Kirkaldy  found  for  Riga  and  Bantzic  firs, 

20  ft  long,  and  13  ins  square,  (or  18^  sides  high,)  148  and  138  tons  total ;  or  .876 
and  .817  tons,  (1963  and  1829  ft>s,)  per  sq  inch.  Mr  Smith's  rule  gives  for  common 
pine,  160  tons  total ;  or  .947  ton,  or  2121  R>s,  per  sq  inch.  Hodgkinson  would  give 
for  Riga  about  297  tons  total. 

Each  of  Mr  Kirkaldy's  20-ft  pillars  shortened  about  .6  of  an  inch  total;  or  .03 
inch  per  ft;  or  Vg  °f  an  inch  in  4  ft  2  ins,  under  a  mean  of  1900  ft»s  per  sq  inch. 

The  writer  has  known  8  unbraced  pillars  of  hemlock, .tolerably  seasoned, 
12  ins  square,  and  42  ft  high,  to  be  gradually  loaded  each  with  32  tons,  or  71680 
ibs  total ;  (or  .2222  ton,  or  498  ft>s  per  sq  inch)  without  appreciable  yielding.  As- 
suming their  strength  and  stift'ness  to  be  about  as  for  Mr  Smith's  pine,  (as  in  all 
our  tables,)  they  should  by  him  yield  at  39.9  tons  total.  With  these  same  data, 
but  with  Hodgkinson's  formula,  they  should  yield  at  69.3  tons;  and  with 
Hodgkinson's  own  data,  for  seasoned  red  deal,  at  91.6  tons.  See  Remarks,  p  242. 


STRENGTH    OF   WOODEN   PILLARS. 


241 


Table  of  breaking  loads  in  tons  of  square  pillars  of  half- 
seasoned  white  or  common  yellow  pine,  with  flat  ends 
firmly  fixed,  and  equally  loaded.  By  C.  Shaler  Smith's  formula. 
(Continued.) 

As  this  table  was  partly  made  by  interpolation,  the  last  figure  is  not  always  pre- 
cisely correct. 

Original. 


!•!' 

s.s 

Side  of  square  pine  pillar  in  inches. 

|l 

ffl.2 

13  j  14   15  |  16  |  17  |  18  |  19  |  20   21  |  22  |  23  |  24 

BREAKING  LOAD. 

Tons.  |  Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

4 

358 

418 

482 

552 

625 

703 

786 

872 

964 

1060 

1161 

1265 

4 

6 

335 

394 

456 

526 

599 

676 

760 

847 

938 

1033 

1134 

1236 

6 

8 

308 

367 

429 

500 

572 

649 

732 

818 

910 

1005 

1106 

1208 

8 

10 

281 

339 

400 

466 

537 

612 

694 

780 

870 

964 

1064 

1166 

10 

12 

252 

307 

365 

432 

502 

576 

656 

740 

829 

922 

1022 

1124 

12 

14 

225 

277 

333 

397 

464 

536 

614 

696 

784 

876 

973 

1074 

14 

16 

201 

250 

303 

363 

428 

497 

573 

652 

739 

829 

925 

1024 

16 

18 

179 

224 

274 

331 

392 

458 

531 

608 

692 

780 

873 

972 

18 

20 

160 

201 

248 

301 

359 

422 

492 

566 

647 

732 

822 

919 

20 

22 

143 

182 

224 

274 

329 

388 

455 

526 

604 

686 

773 

866 

22 

24 

127 

163 

203 

249 

301 

357 

421 

488 

563 

642 

726 

816 

24 

26 

115 

148 

184 

226 

275 

328 

389 

453 

523 

599 

680 

767 

26 

28 

103 

133 

167 

206 

252 

302 

359 

420 

490 

5SO 

638 

721 

28 

30 

93 

121 

152 

189 

231 

278 

332 

389 

453 

522 

597 

677 

30 

32 

84 

109 

138 

173 

212 

256 

307 

361 

421 

487 

558 

635 

32 

34 

76 

99 

126 

159 

196 

237 

284 

335 

392 

455 

523 

597 

34 

36 

69 

91 

116 

146 

180 

219 

264 

312 

366 

426 

490 

560 

36 

38 

63 

84 

107 

134 

166 

203 

245 

290 

341 

397 

458 

525 

33 

40 

58 

77 

99 

124 

154 

188 

227 

270 

318 

372 

429 

494 

40 

42 

54 

71 

91 

115 

143 

175 

212 

253 

298 

349 

403 

465 

42 

44 

50 

66 

84 

107 

133 

163 

198 

236 

280 

328 

380 

438 

44 

46 

46 

61 

78 

99 

123 

152 

185 

221 

263 

308 

358 

413 

46 

48 

43 

57 

73 

92 

115 

142 

173 

207 

247 

290 

337 

389 

48 

50 

40 

53 

68 

86 

107 

133 

162 

194 

231 

272 

317 

367 

50 

52 

37 

50 

64 

81 

101 

124 

152 

182 

217 

256 

300 

347 

52 

54 

35 

47 

60 

76 

95 

117 

144 

172 

205 

242 

283 

328 

54 

56 

33 

44 

56 

71 

89 

110 

135 

162 

193 

228 

267 

310 

56 

58 

31 

41 

52 

67 

84 

103 

127 

153 

182 

215 

253 

294 

58 

60 

29 

38 

49 

63 

79 

98 

120 

144 

172 

204 

240 

280 

60 

65 

25 

33 

43 

55 

69 

86 

105 

126 

151 

179 

211 

246 

65 

70 

22 

29 

37 

48 

60 

74 

92 

111 

134 

159 

187 

218 

70 

75 

19 

25 

33 

42 

53 

66 

82 

98 

118 

141 

166 

195 

75 

80 

16 

22 

29 

37 

46 

58 

72 

87 

105 

125 

148 

174 

80 

85 

14 

19 

26 

33 

41 

52 

65 

78 

94 

112 

132 

156 

85 

90 

13 

17 

23 

30 

37 

46 

58 

70 

85 

102 

120 

141 

90 

95 

12 

16 

21 

27 

33 

42 

53 

64 

77 

93 

108 

127 

95 

100 

11 

14 

19 

24 

30 

38 

48 

58 

70 

84 

99 

117 

100 

110 

10 

12 

16 

20 

26 

33 

40 

48 

58 

70 

82 

97 

110 

120 

9 

il 

14 

17 

22 

28 

34 

41 

49 

60 

71 

83 

120 

130 

7 

9 

12 

14 

18 

23 

29 

36 

43 

52 

61 

72 

130 

140 

6 

8 

10 

12 

16 

20 

25 

31 

37 

44 

53 

62 

140 

150 

5 

7 

9 

11 

14 

18 

22 

27 

32 

38 

46 

54 

150 

160 

5 

6 

8 

10 

13 

16 

20 

24 

29 

34 

41 

48 

160 

170 

4 

5 

7 

9 

11 

14 

17 

21 

25 

30 

36 

43 

170 

180 

4 

5 

6 

8 

10 

12 

15 

19 

22 

27 

32 

38 

180 

190 

3 

4 

5 

7 

9 

11 

14 

17 

20 

24 

29 

34 

190 

200 

3 

4 

5 

6 

8 

10 

12 

15 

18 

22 

26 

31 

200 

242 


STRENGTH   OF   WOODEN   PILLARS. 


Table  of  breaking:  loads  in  tons,  or  in  Ibs  per  sq  inch,  of 
cross  section  of  half  seasoned  square  pine  pillars,  whose 
heights  are  measured  by  one  of  their  sides.  By  formula,  p.  238. 


4*  00 

•&-= 

II 

II 

£  8 

tt'O 

§s 

B.fl 

S.S 

SQ  n». 

Tons. 

Lbs. 

Tons. 

Lbs. 

Tons. 

Lbs. 

Tons. 

Lbs. 

i 

2.2232 

4980 

26 

.6027 

1350 

51 

.1960 

439 

76 

.0924 

207 

2 

2.1969 

4921 

27 

.5697 

1276 

52 

.1888 

423 

77 

.0902 

202 

3 

2.1544 

4826 

28 

.5398 

1209 

53 

.1826 

409 

78 

.0879 

197 

4 

2.097S 

4699 

29 

.5116 

1146 

54 

.1763 

395 

79 

.0862 

193 

5 

2.0290 

4545 

30 

.4853 

1087 

55 

.1705 

382 

80 

.0839 

188 

6 

1.9513 

4371 

31 

.4607 

1032 

56 

.1647 

369 

81 

.0821 

184 

7 

1.8665 

4181 

32 

.4379 

981 

57 

.1598 

358 

82 

.0799 

179 

8 

1.7772 

3981 

33 

.4165 

933 

58 

.1545 

346 

83 

.0781 

175 

9 

1.6857 

3776 

1  34 

.3969 

889 

59 

.1496 

335 

84 

.0763 

171 

10 

1.5942 

3571 

35 

.3781 

847 

60 

.1451 

325 

85 

.0746 

167 

11 

1.5040 

3369 

36 

.3599 

809 

61 

.1406 

315 

86 

.0728 

163 

12 

1.4165 

3173 

37 

.3447 

772 

62 

.1362 

305 

87 

.0714 

160 

13 

1.3317 

2983 

38 

.3295 

738 

63 

.1321 

296 

88 

.0696 

156 

14 

1.2513 

2803 

39 

.3152 

706 

64 

.1286 

288 

89 

.0683 

153 

15 

1.1745 

2681 

40 

.3018 

676 

65 

.1250 

280 

90 

.0670 

150 

16 

1.1027 

2470  • 

41 

.2889 

647 

66 

.1210 

271 

91 

.0656 

147 

17 

1.0353 

2319 

42 

.2772 

621 

67 

.1179 

264 

92 

.0638 

143 

18 

.9723 

2178 

43 

.26ol 

696 

68 

.1147 

257 

93 

.0625 

140 

19 

.9134 

2046 

44 

.2554 

572 

69 

.1112 

249 

94 

.0616 

138 

20 

.8585 

1923 

45 

.2455 

550 

70 

.1085 

243 

95 

.0603 

135 

21 

.8076 

1809 

46 

.2357 

528 

71 

.1054 

236 

96 

.0589 

132 

22 

.7603 

1703 

47 

.2268 

508 

72 

.1027 

230 

97 

.0576 

129 

23 

.7165 

1605 

48 

.2183 

489 

73 

.1000 

224 

98 

.0567 

127 

24 

.6755 

1513 

49 

.2100 

472 

74 

.0973 

218 

99 

.0554 

124 

25 

.6380 

1429 

50 

.2031 

455 

75 

.0951 

213 

100 

.0545 

122 

Remark.— Gordon  and  Hodgkinson  compared.  The  difference 
between  them  is  greater  in  wooden  pillars  than  in  hollow  cast  iron  ones.  More- 
over, in  the  latter,  Gordon  is  sometimes  greater,  sometimes  less,  than  Hodgkin- 
son,  as  seen  per  lower  table.  Mr.  Smith's  assumed  strength  and  stiffness  of  pine 
may  safely  be  taken  at  about  one-fourth  less  than  Hodgkinson's  for  his  red  deal ; 
and  with  this  assumption,  Hodgkinson's  rule  would  make  the  strength  of  Smith's 
pine  (in  all  our  tables  for  wooden  pillars)  greater,  to  the  extent  shown 
by  the  multipliers  in  the  following'  table.  The  truth  is  probably  between 
the  two.  See  Remark,  p.  240. 


Htin 
sides. 

Multr. 

Htin 
sides. 

Multr. 

Ht  in 
sides. 

Multr. 

Htin 
sides. 

Multr. 

Htin 
sides. 

Multr. 

5 

1.04 

12.5 

1.23 

20    !     1.44 

35 

1.72 

50 

1.67 

7.5 

1.09 

15 

1.30 

25     1      1.54 

40 

1.76 

60 

1.63 

10 

1.15     ' 

17.5 

1.37 

30     !      1.64 

45 

1.71 

80 

159 

Gordon's  and  Hodgkinson's  hollow  cylindrical  cast  iron 
pillars  compared. 

The  thickness  is  usually  from  ^  to  Y8  of  the  outer  diameter ;  and  for  these 
limits,  the  column  G,  (Gordon),  and  H,  (Hodgkinson),  show  the  proportions  of 
the  breaking  loads.  See  "  Remarks  on  this  edition,"  p.  ix. 


Thickness  =  ^  of  the  outer  diameter. 

Htin 
Diams. 

G. 

H. 

1  Htin 
Diams. 

G. 

1 
1 

H. 

Htin 
Diams. 

G. 

H. 

Htin 
Diams. 

G. 
1 

H. 

Htin 
Diams. 

G. 

1 

.90 
.97 

5 
8 

1 
1 

1.25 
1.12 

10 
15 

1.11 
1.02 

20 
30 

1 
1 

.97 
.941 

40 

50 

.90 

.88 

70 
100 

Thickness  =  ya  of  the  outer  diameter. 

5 
8 

1 
1 

1.23  il     10 
1.10  ||     15 

1 
1 

1.08  1  1     20 
.98  1  1     30 

\ 

.92  1  1     40 
.88  1|     50 

1 
1 

.8111        70     Jl 
.80  1|      100    |l 

.82 
.88 

TRUSSES.  243 

TKUSSES, 


Art.  1.  When  the  span  of  a  bridge,  roof,  &c,  becomes  so  .Treat  that  single  solid 
beams,  supported  at  their  ends,  cannot  be  employed,  we  resort  to  compound  beams, 
called  trusses,  composed  of  several  pieces  so  arranged  and  united  as  to  furnish 
.  the  reqd  strength.  The  designing,  construction,  and  erection  of  trusses  of  great 
span,  especially  when  of  iron,  involve  such  a  multiplicity  of  important  detail,  that, 
like  the  building  of  locomotives,  cars,  &c,  they  have  become  a  specialty,  or  a  dis- 
tinct branch  of  business,  to  which  persons  confine  themselves  to  the  exclusion  more 
or  less  of  other  departments ;  and  thus  attain  a  degree  of  skill  beyond  the  reach  of 
the  general  engineer.*  The  latter,  however,  should  possess  a  knowledge  of  the  sub- 
ject sufficient  at  least  to  enable  him  to  form  a  well-grounded  opinion  of  the  general 
merits  of  a  design ;  and  to  guard  him  against  the  adoption  of  one  involving  serious 
imperfections.  In  a  volume  like  this  we  can  aim  at  nothing  more  than  an  attempt 
to  illustrate  some  few  general  principles.  We  shall  confine  ourselves  to  such  trusses 
as  are  in  common  use;  showing  first  the  effects  of  uniform  stationary  loads,  as  in 
the  case  of  roofs ;  and  then  those  of  moving  loads,  such  as  an  engine  and  train  on  a 
bridge. 

Art.  2.  Most  of  the  bridge  trusses  in  common  use  have  two  long,  straight,  par- 
allel upper  and  lower  members  It,  ap;  and  I  t,  ap,  Figs  10, 11,  p.  254,  called  the 
chords ;  or  in  England,  the  booms.  Vertical  pieces  placed  between,  and  con- 
necting the  upper  and  lower  chords,  are  called  posts,  when  they  sustain  compres- 
sion; and  vertical  ties,  or  suspension  rods,  &c,  when  they  sustain  tension 
or  pull.  The  oblique  pieces  seen  in  these  figs  are  called  braces,  strut-braces, 
main-braces,  &c,  when  resisting  pres  or  thrust;  or  tie-braces,  tension- 
braces,  main  oblique  ties,  oblique  suspension-rods.  &c,  when 
resisting  pulls.  Sometimes  the  same  piece  is  adapted  to  bear  both  tension  and  com- 
pression alternately;  and  may  then  be  called  a  tie-strut  or  a  strut-tie.  The 
oblique  members  alluded  to  are  sometimes  called  main-braces,  whether  they  are 
struts  or  ties ;  to  distinguish  them  from  counter-braces,  or  counters.  These 
last  are  not  shown  in  Figs  10  and  11,  but  are  seen  in  Figs  28  and  31,  crossing  the 
main  braces  diagonally.  These  posts,  braces,  counters,  ties,  &c,  serve  not  only  to 
keep  the  two  chords  asunder,  and  to  prevent  them  from  bending;  but  to  transform 
the  transverse  strains  produced  by  the  wt  of  the  truss  and  its  load,  into  other  strains, 
acting  longitudinally,  or  lengthwise,  along  the  diff  members ;  and  to  conduct  said 
strains  along  the  truss,  to  the  firm  supports  of  the  abuts.  A  load  placed  at  any  one 
of  these  members  is,  of  course,  partly  supported  by  each  abut;  one  part  of  it  travels 
up  and  down  alternately  between  the  chords,  and  along  the  successive  members, 
until  it  reaches  one  abut ;  and  the  other  part,  in  like  manner,  goes  to  the  other  abut. 
These  members,  therefore,  perform  the  duty  of  the  vert  web  of  the  Hodgkinson 
beam ;  or  of  the  I  rolled  beams,  or  of  the  tubular  girder;  and  on  this  account  are  col- 
lectively called  the  web  members,  in  contradistinction  from  the  chords.  Each 
portion  of  any  load,  while  being  transferred  by  the  web  members,  from  the  spot  at 
which  it  is  placed  on  the  truss,  to  its  final  point  of  support  on  the  abut,  produces  a 
strain  equal  to  itself  upon  every  vert  web  member  along  which  it  travels  between  the 
parallel  chords  ;  while  upon  each  oblique  member  encountered  on  its  way,  it  produces 
a  strain  greater  than  itself,  in  the  same  proportion  that  the  oblique  member  is  longer 
than  a  vert  one. 

Whether  the  web  members  are  strained  by  compression,  or  by  tension;  or,  in 
other  words,  whether  they  act  as  struts,  or  as  ties,  the  amount  of  strain  will  be  the 
same.  In  either  case  the  straining  agent  is  the  same  identical  force,  namely  the  wt, 
or  vert  force  of  gravity  of  the  truss  itself,  and  of  its  load ;  and  (Art  18,  of  Force  in 
Rigid  Bodies)  whether  this  force  exhibits  itself  as  a  pusli,  or  as  a, pull,  neither  its 
amount,  nor  its  direction  undergoes  any  change.  So  far,  therefore,  as  regards  the 
broad  principle  involved  in  the  duty  performed  by  the  web  members,  they  might  be 
divided  simply  into  verticals,  and  obliques.  We  shall  frequently  so  desig- 
nate them. 

Whatever  amount  of  strain  the  upper  end  of  an  oblique  produces  in  one  direction 
against  the  upper  chord,  that  same  amount  will  its  lower  end  produce  against  the 
parallel  lower  chord  ;  but  in  the  opposite  direction.  That  is,  if  the  top  or  head  of 
any  oblique,  pushes  the  upper  chord  toward  the  right  hand  ;  its  foot  will  pull  the 
lower  chord  to  the  same  extent  toward  the  left  hand.  This,  however,  is  notpre- 

*  The  first  writer  to  whom  we  are  indebted  for  a  knowledge  of  correct  principles  on  this  subject  is 
S.  WHIPPLK,  C.  E.,  the  first  edition  of  whose  book  (beyond  all  doubt  the  pioneer  one)  bears  date, 
Utica,  N.  York,  1847.  He  was  followed  by  Bow.  of  England,  and  Haupt,  of  this  country,  both  in  1851. 
The  Murphy-Whipp^e  bridge  (of  which  Mr.  John  W.  Murphy,  C.  E.,  has  built  several  of  the  besU 
owes  its  name  to  these  two  gentlemen. 


244 


TRUSSES. 


cisfly  correct,  inasmuch  as  when  the  oblique  is  a  strut,  the  pres  at  its  foot  is  some- 
what greater  than  at  its  head,  because  the  foot  supports  also  the  wt  of  the  strut 
itself;  or  if  the  oblique  is  a  tie,  with  its  head  attached  to  the  upper  chord,  then  the 
strain  is  a  little  greater  at  the  head  than  at  the  foot ;  because  then  the  head  upholds 
the  wt  of  the  oblique,  and  the  foot  sustains  none  of  it.  This  remark  applies,  of 
course,  to  verts  also.  Another  exception  is.  when  the  ends  of  two  obliques  meet 
each  other;  as  those  at  the  center  of  the  trusses,  in  Figs  1U  and  11.  If,  in  such  cases, 
the  ends  of  the  obliques  abut  aynnsteacli  oilier,  instead  of  being  separately  attached 
to  the  chord,  they  will  at  that  point  exert  their  strains  against  each  other,  instead  of 
against  the  chord. 

In  any  oblique,  as  c  d,  Fig  1,  the  vert  dist  a  c  between  its  ends;  and  the  hor  dist 
a  d  between  the  same,  are  called  its  vert  and  hor  spreads,  or  stretches,  or 
readies. 

Art.  3*  There  is  a  great  din0  in  princinle  between  two  classes  of  trusses  in 
common  use.  In  some  of  them,  two  chords  are  absolutely  essential,  as  in  the 
Howe  truss,  p  283;  the  Pratt,  p  284;  the  Lattice,  p  285;  the  Warren,  p  280;  and 
their  various  modifications,  known  as  the  Murphy-W  hippie,  the  Linville,  the  Latrop, 
Ac,  <fec,  which  differ  only  in  certain  unessential  details.  In  the  Howe  and  Pratt 
trusses  there  is  no  diff  whatever  of  broad  principle.  The  distinction  between  them 
consisting  chiefly  in  the  fact  that  in  Howe's  the  verts  are  ties,  or  suspension  rods  ; 
and  the  obliques,  struts ;  while  in  Pratt's,  the  verts  are  posts;  and  the  obliques,  ties. 
In  all  these  the  strains  on  the  verts  and  main obliques  (not  on  the  counters;  are  least 
at  the  center  of  the  truss;  and  increase  gradually  toward  the  end  of  it;  while  those 
on  the  chords  fas  in  an  ordinary  wooden  beam)  are  greatest  at  the  center,  and  least 
at  the  ends.  Hence,  also,  such  are  called  beam  trusses.  The  strains  on  the 
counters  are  also  greatest  at  the  center. 

But  there  is  another  class,  called  suspension  trusses,  of  which  the  Fink, 
Figs  26  and  27  ;  and  the  Bollman,  Figs  22,  24,  25,  are  the  principal  representatives. 
In  these  but  one  chord  is  essential  for  a  perfect  truss.  See  Figs  25,  27.  From  this 
chord  the  web  members  are  suspended :  and  to  it  alone  do  they  all  transfer  their 
strains ;  and  the  -strain  upon  this  chord  is  uniform  from  end  to  end.  Figs  25,  27, 
show  perfect  bridges,  with  but  one  chord  each.  In  Figs  24,  26,  n  vi,  n  n,  appear  to 
be  chords:  but  strictly  speaking  they  are  not;  they  are  merely  longitudinal  pieces 
for  upholding  the  cross-beams  of  the  flooring,  when  the  roadway  is  placed  at  the 
bottom  of  the  truss.  They  have  not  to  resist  tension,  as  in  beam  trusses. 

In  all  the  foretneutioned  trusses  the  roadway  may  be  placed  on  either  the  top  or  the  bottom  chord; 
constituting  in  the  lirsr,  ca*»-  :•  top  road,  or  a  deck  bridge:  and  in  the  second,  a  bottom 
road,  or  a  through  bridge. 

Art.  4.  That  part  of  a  truss,  such  as  Figs  10,  28,  31,  &e,  that  is  comprised  be- 
tween two  adjacent  verts,  is  called  a  panel ;  thus,  in  Fig  10,  eij  d,  djkc,  &c;  and 
in  Fig  31,  of  Pratt,  tynw,  is  a  panel.  The  Triangular  or  Warren  truss,  Fig  11,  or 
Fi<?  23  g,  has  no  verts,  as  essential  parts  of  it ;  and  its  subdivisions  are  called  simply 
triangles;  and  a  panel  is  a  length  of  truss  equal  to  the  width  of  a  triangle.  Verts 
are  sometimes  added  to  it  when  the  spaces  a  b,  b  c,  c  d.  Fig  11,  become  too  long  for 
safely  supporting  the  roadway  without  them;  thus  dividing  the  truss  into  half 
panels.  It  is  not  a  matter  of  practical  importance  as  regards  strength,  whether 
the  number  of  panels  in  a  truss  be  odd  or  even  ;  but  it  is  usually  even,  with  a  vert 
at  the  center  of  a  truss.  In  Fig  1,  a  b  c  o,  b  d  o  w,  are  half  panels. 

A  panel-point,  as  a,  6,  d,  c,  o,  or  n,  Fig  1,  is  one  at  which  web-meinbers 
meet  a  chord,  or  a  rafter  in  a  bridge  or  roof,  as  Fig  14,  b,  c,  k,  &c,  p  260. 

The  length  of  a  panel  is  its  horizontal  measurement,  The 
best  inclination  of  obliques,  as  regards  economy  of  material  in  the 
web,  is  when  their  least  angle  (t p  o,  or  s  o  n,  Ac,  Fig  10)  with  the  chord  is  45°. 
This  applies  also  to  the  admirable  Warren  truss ;  in  which  the  triangles  are,  however, 
usually  made  equilateral.  When  the  span  is 
great,  and  the  height  of  truss  correspondingly  so, 
if  the  panels  be  made  square,  or  nearly  so,  with 
a  view  to  secure  this  inclination  of  about  45°  for 
the  obliques,  the  verts  (as  to,  sn,  Ac,  Fig  10) 
will  become  so  far  apart,  that  the  stretches  or 
dists/>  o,  on,  &c,  become  too  long  to  be  safe  for 
upholding  their  loads  of  engines,  cars,  &c,  with- 
out additional  precautions.  When,  therefore, 
the  expense  or  inconvenience  resulting  from  this 
would  be  too  great,  the  verts  may,  as  in  Kig  1,  be 
placed  so  near  together  as  to  make  half  panel* 


TRUSSES. 


245 


much  higher  than  they  are  long;  and  the  obliques  (both  the  main  ones,  and  the 
dotted  counters)  then  run  across  one  vert,  as  in  the  Fig:  or  across  two,  if  neces- 
sary. In  the  Warren  girder,  the  expedient  is  to  introduce  verts ;  or  else  a  second  set 
of  triangles,  as  in  Fig  1,  omitting  the  verts.  From  8  to  12  or  15  ft  apart  are  ordinary 
'  dists  for  verts  in  bridges  of  moderately  large  spans.  Frequently  panels  are  made 
considerably  higher  than  long;  disregarding  the  economical  angle  of  4u°.  In  large 
bridges,  the  main  obliques,  instead  of  being  each  in  one  piece,  are  usually  made  of 
two  or  more  parallel  pieces,  disposed  in  such  a  manner  as  to  let  the  counters  pass 
between  them  diagonally,  without  mutual  interference.  Each  lower  chord  (and  fre- 
quently the  upper  one  also)  in  large  spans,  is  usually  made  up  of  several  parallel 
beams  of  wood,  or  bars  of  iron,  side  by  side. 

In  the  Newark  Dyke  Bridge,  England,  a  transverse  section  of  each  lower  chord  shows  14  iron  bars. 
This  enables  us  to  employ  smaller  beams  and  bars  ;  and,  moreover,  secures  greater  widtk  of  truss; 
thereby  diminishing  the  tendency  to  lateral  or  sideways  motion.  It  in  no  way  affects  the  amount  of 
the  strains,  or  the  mode  of  calculating  them  ;  but  is  a  mere  matter  of  mechanical  expediency,  or  of 
economy. 

When  the  truss  is  very  high,  the  posts  are  sometimes  made  as  in 
Fig  2,  where  c  and  c  are  the  upper  and  lower  chords ;  and  pp  a  post 
consisting  of  two  hollow  cast-iron  pillars,  bolted  together  by  their 
broad  flanges  at  s  s.  At  ss  is  also  placed  a  hor  +  shaped  casting,  of 
4  arms ;  the  outer  extremities  of  which  servo  to  keep  in  place  the 
iron  bars  oo,  for  stiffening  the  post  sideways. 

Various  other  ways  are  in  use  for  building  or  composing  large  posts,  as  well  as 
oblique  struts,  bv  means  of  T,-f-,  U,  H,  O,  or  other  shaped  irons,  riveted  together. 
The  rolled  iron  hollow  column  of  the  Phoenix  Iron  Co.  of  Philadelphia,  (p  233, 
"Strength  of  Iron  Pillars, ")  is  coming  into  very  common  use  for  such  purposes. 
Lower  chords  of  iron  may  be  made  of  round  or  square  bars  ;  or  of  flat  ones  joined 
at  their  ends  by  means  of  splicing-plates,  as  at  B,  Figure  3,  p  654,  or  by  bolts  or 
pins  passing  through  eyes  at  the  ends  of  the  bars;  see  Figure  41.  Figures  37 
to  40  show  modes  of  joining  the  beams  for  lower  chords  of  wood  so  as  to  resist 
tension.  Upper  chords  sustain  compression  only  ;  and  need  but  a  simple  butt 
joint. 

To  allow  free  expansion  and  contraction  from 
changes  of  temperature,  when  tbe  span  of  an  iron  bridge 
truss  exceeds  about  75  or  100  ft,  one  end  of  it  should  rest  upon  roll- 
ers ;  or  some  other  device  (see  p  295)  be  substituted  for  the  same 
purpose.  If  this  be  neglected,  the  abuts  will  be  exposed  to  displace- 
ment; or  if  these  be  sufficient  to  resist  the  expansive  force,  the 
truss  itself  will  be  likely  to  buckle,  if  of  wrought  iron  ;  or  to  be  frac- 
tured, if  of  cast  iron,  or  having  a  cast-iron  upper  chord. 

If  in  Fig  10  wTe  imagine  lines  crossing  the  panels  diag,  as  the  main  obliques  shown 
in  the  Fig  do,  but  in  the  opposite  direction,  as  shown  in  Figs  28  arid  31,  they  will  rep- 
resent counter-braces,  or  counters.  These,  like  the  main  obliques,  are  in 
some  cases  struts,  and  in  others  ties.  Although  important  members,  they  are  less  so 
than  the  main  obliques.  They  are  unnecessary  when  the  load  is  uniform  and  station- 
ary, as  is  usually  assumed  to  be  the  case  in  roofs ;  )ind  are  required  only  when  the 
load  is  unequal,  or  a  moving  one,  as  in  a  train  crossing  a  bridge.  In  this  last  case  they 
act  chiefly  while  the  span  is  but  partially  loaded.  If  the  train  at  any  moment  covers 
the  entire  span,  and  is  of  uniform  wt,  their  action  ceases  for  that  time.  Their  office 
is  solely  to  counteract  the  deranging  tendency  of  the  unequal  loading  of  difl"  parts 
of  the  truss,  as  shown  in  Figs  9%,  p  253.  In  Fig  96,  an  excess  of  load  along  ao 
would  tend  to  derange  the  main  braces  ho  and  ta;  and  this  would  be  counteracted 
by  counters  across  co  and  ts.  The  same  thing  may  be  effected  by  arranging  the  main 
braces,  ho,  ta,  so  as  to  bear  tension  as  well  as  compression.  The  bad  effects  of  une- 
qual loads  must  plainly  become  greater  in  proportion  as  the  load  is  heavier  than  the 
truss  itself;  and  when  the  bridge  becomes  very  heavy,  so  that  the  load  must  extend 
over  several  panels  before  its  effects  become  serious,  but  little  couriterbracing  is 
needed;  and  that  at  arid  near  the  center  only;  whereas,  in  a  very  light  bridge,  the 
counters  should  extend  from  the  center,  where  they  are  most  strained;  to  near  the 
ends,  where  the  strain  upon  them  is  least.  Inasmuch  as  we  shall  first  speak  of  uni- 
formly loaded  trusses,  we  shall  not  here  say  more  respecting  counters.  See  Remark, 
Art  10,  p  252. 

It  would  at  first  sight  appear  that  the  several  parts  of  abridge  truss  must  be  most 
strained  when  covered  from  end  to  end  with  its  maximum  load  ;  but  this  is  true  only 
of  the  chords;  and  of  the  mam  obliques  and  verts,  as  la,  tp,  Fig  10,  at  the  ends  of 
the  truss.  The  other  web  members  are  more  strained  by  a  part  of  the  load  as  it  passes 
along  the  truss  ;  so  that  if  they  be  correctly  proportioned  for  a  full  load,  they  will 


246 


TRUSSES. 


be  too  weak  for  a  partial  one.  If  all  be  made  as  strong  as  the  end  ones,  they  will,  it 
is  true,  be  safe  fora  passing  load;  but  this  would  require  an  expense  of  material  that 
would  be  justified  only  in  the  case  of  moderate  spans,  especially  of  wood ;  in  which 
the  additional  trouble  and  expense  of  getting  out  and  fitting  together  pieces  of 
many  diff  sizes,  may  more  than  counterbalance  the  saving  in  material. 

Art.  5.  Trusses  with  moving-  loads  require  calculation  diff  from 
that  for  uniform  loads.  We  shall  first  treat  of  the  latter  only  ;  and  in  so  doing  shall 
not  employ  the  shortest  methods,  but  such  as  will  render  the  general  principles  clear 
to  any  one  acquainted  with  the  simple  elements  of  "Composition  and  Resolution  of 
Forces."  The  strains  on  trusses  may  be  found  with  all  the  accuracy  needed  for  prac- 
tical purposes,  by  means  of  diagrams  drawn  to  a  scale.  The  same  division  of  the  scale 
that  answers  for  a  foot  of  length,  may  also  represent  a  ton,  1000  ibs,  or  any  other 
convenient  wt,  load,  or  strain,  and  may  thus  be  used  for  measuring  the  lines  which 
represent  such. 

The  chords,  verts,  and  obliques  heretofore  mentioned,  constitute  all  the  essential 
elements  of  a  complete  truss:  but  other  pieces  are  necessary  for  a  complete 
bridge:  such  as  roof  and  floor  beams;  transverse  bracing  for  connecting  two  par- 
allel trusses  with  one  another,  so  as  better  to  resist  lateral  or  sidewise  motion  from 
winds  or  lurchings  of  trains ;  bars  for  tying  the  truss  to  the  piers  and  abutments  in 
gome  cases,  &c.  The  same  may  be  said  of  the  extension  frequently  made  at  the  ends 
of  either  an  upper  or  a  lower  chord  of  a  bridge,  as  shown  at  n  n  in  the  bottom  chords 
of  Figs  24  and  26.  Here  the  trusses  are  perfect  without  the  extensions  ;  but  the  bridge 
requires  them,  to  allow  the  load  to  reach  and  to  leave  it.  They  may  be  needed  for 
the  same  purpose  in  an  upper  chord  of  a  top-road  bridge ;  or  for  extending  a  roof 
over  an  entire  span,  &c.  The  end  vert  posts  ss,  of  the  same  Figs,  are  not  parts  of 
the  truss,  but  supports  for  upholding  it ;  also,  the  posts  p  and  d,  Fig  28,  are  not  ea 
sential  to  the  truss. 

RKM.  Besides  the  forms  of  truss  already  mentioned,  there  are  many  others,  in  some  of  which 
arches  are  introduced  either  as  principal  members,  or  merely  aa  auxiliaries ;  as  Town's  Lattice* 
Fig  33  ;  the  Bow  and  String,  Fig  35;  and  the  Burr,  Fig  36,  all  of  much  merit.  The  Lattice 
and  the  Burr  have  both  fallen  into  undeserved  disrepute,  from  the  fact  that  being  the  first  trusses 
that  were  extensively  introduced  upon  the  railroads  in  this  country,  they  were  built  too  weak  for  the 
heavy  engines  and  trains  of  the  present  day,  and  consequently  failed. 

Art.  6.     Chords.     When  a  beam  a  b,  Fig  3,  supported  at  both  ends,  breaks 

either  und?r  its  own  wt,  or  under  the 
action  of  a  load  placed  on  top  of  it,  or 
suspended  from  it  below,  it  does  so  be- 
cause the  lower  fibres,  near  its  center 
Z,  are  pulled  asunder  ;  and  its  upper 
ones  at  u,  crushed  together  to  such  aa 
extent  as  to  offer  no  effective  resist- 
ance. The  fig  shows  this  in  a  some- 
what exaggerated  manner.  The  ex- 
treme upper  particles  at  M,  and  the  ex- 
treme lower  ones  at  /,  being  the  most 

strained,  give  way  first ;  and  the  strength  of  the  beam  being  thereby  diminished,  the 
adjacent  ones  give  way  in  rapid  succession.  The  compressed  particles  of  the  beam 
are  all  atmve  a  certain  point  n ;  while  the  extended  ones  are  below  it.  If  we  imagine 
an  infinitely  fine  needle  to  be  held  perp  to  this  page,  and  in  that  position  to  be  stuck 
through  the  point  w,  passing  entirely  through  the  beam,  or  page,  then  the  infinitely 
fine  hole  thus  made  will  pass  along  what  is  called  the  neutral  axis  of  the  beam. 
It  is  so  named  because  the  fibres  situated  in  that  line,  and  which  were  cut  in  two  by 
the  needle,  are  neither  compressed  nor  extended,  until  the  strain  becomes  so  great 
that  on  its  removal  the  beam  will  not  entirely  recover  itself ;  or.  in  other  words, 
until  the  strain  exceeds  the  elastic  limit  of  the  beam.  Within  the  limits  of  elasticity, 
the  neutral  axis  maybe  assumed  to  pass  through  the  cen  of  grav  of  the  cross-section 

of  the  beam.  Thus,  if  the  cross-section  be  of 
any  of  the  forms  shown  in  Fig  4,  then  so  long 
as  the  beam  is  safe,  or  the  load  within  the 
elastic  limits,  the  line  na  will  pass  along  its 

•»,.  f-w^L—  v$s$$^§$^  cen  °f  grav ;  which  is  at  the  same  time  its 

Eo'4-  ™    ^^  ^^  neutral  axis.    But  the  chords  of  a  truss 

S  differ  essentially  in  condition  from  the 

fibres  of  the  beam,  inasmuch  as  the  truss  has  no  neutral  axis  about 
which  as  a  center  its  fibres  are  extended  or  compressed  to  different  degrees 
depending  on  their  dists  from  it.  On  the  contrary  the  fibres  o'f  the  two  chords 
react  upon  each  other  with  nearly  uniform  intensity,  with  leverages  of  so  nearly 
uniform  length  that  their  mean  or  the  depth  of  truss  between  the  centers  of  grav 


TRUSSES. 


247 


of  the  chords  may  be  taken  as  their  one  length.  See  "  Open  Beams,"  p  647.  There- 
fore the  same  quantity  of  material,  that  composes  the  beam  a  6,  Fig  3,  will  present 
farmore  resistance  to  bending  or  breaking,  if  it  be  cut  in  two  lengthwise,  and  con- 
verted into  top  and  bottom  chords  of  a  truss ;  for  the  reason  that  the  two  chords  are 
then  so  far  removed  from  each  other  that  all  their  particles  are  strained  to  nearly  the 
same  extent  at  the  same  time ;  so  that  all  the  fibres  in  the  upper  one  must  be  crushed, 
and  all  those  in  the  lower  one  be  pulled  apart,  at  the  same  instant,  rJefore  the  truss  can 
give  way  ;  whereas,  in  the  single  beam  a  />,  the  extreme  upper  and  lower  fibres  break 
first:  then  those  next  to  them,  and  so  on,  one  after  the  other.  They  do  not  all  act 
unitedly,  as  they  do  in  the  chords.  It  is  this  principle  that  gives  so  much  strength 
to  I  iron ;  to  the  Hodgkinson  beam,  &c,  (aided  by  increased  leverage.) 

Art.  7.  In  the  designing  of  trusses,  especially  such  as  may  have  to 
bear  unequal  loads  at  different  parts,  as  in  a  bridge,  the 
point  chiefly  to  be  aimed  at  is  to  dispose  its  various 
parts  so  as  to  form  a  series  of  properly  connected  tri- 
angles,  because  in  that  shape  they  present  more 
resistance  to  derangement  of  form,  than  in  figs  of  a 
greater  number  of  sides.  Thus,  in  the  three  beams  at 
a,  Figs  4^,  with  a  bolt  at  each  junction  or  joint,  the 
triangular'torm  evidently  cannot  be  changed  by  any  but  a  force  sufficient  to  either 
bend  or  break  either  the  beams  or  the  bolts.  But  in  the  4-sided  fig  t>,  the  form  may 
readily  be  changed  to  that  at  c,  by  a  force  at  n  entirely  too  small  to  injure  either 
the  beams  or  the  holts.  In  «  the  bolts  assist  to  prevent  change  of  form;  but  in  b 
they  are  merely  pivots,  around  which  great  changes  may  easily  take  place. 

Before  the  strains  can  be  calculated,  and  the  truss  propor- 
tioned to  those  strains,  ITS  WEIGHT  MUST  BE  KNOWN;  for  this  tends  to  break  it,  as  well  as  the  extrane- 
ous load.  But,  on  the  other  hand,  we  cannot  learn  its  wt  until  we  know  the  size  of  its  iliff  members. 
In  this  dilemma  we  must  assume  for  it  an  approximate  wt,  based  upon  our  knowledge  of  somewhat 
similar  trusses  already  built.  This  becomes  the  more  necessary  as  the  truss  increases  in  size,  so  that 
its  own  wt  becomes  greater  in  proportion  to  th sit  of  the  load.  The  table,  p  296,  gives  safe  assumed 
wts  for  bridge  trusses;  and  p  300  will  aid  in  the  case  of  roofs.  In  very  small  spans,  especially  of 
bridges,  the  load  is  generally  so  much  greater  than  the  wt  of  the  truss,  that  the  latter  might  almost 
be  neglected  entirely. 

'Rem.  For  finding  the  strains  on  a  paneled  truss  by  means  of  a  drawing,  it 
is  best  to  represent  each  member  by  a  single  line,  as  in  Figs  1, 10, 14,  23,  &c.  Such 
is  called  a  skeleton  drawing;  or  diagram  of  the  truss.  Each  of  the  parts 
into  which  the  panel-points  divide  either  chord,  or  a  rafter,  is  to  be  regarded  as 
a  separate  member. 

As  will  be  shown  farther  on,  a  load  consisting  of  some  portion  of  the  wt  of  the 
truss  and  its  load,  is  assumed  to  be  supported  at  each  panel-point.  All  the  forces 
which  meet  at  any  panel-point  (namely,  the  aforesaid  partial  load,  and  the  forces 
acting  lengthwise  of  the  members  which  meet  there)  hold  each  other  in  equili- 
brium, or  would  theoretically  keep  each  other  in  position  without  any  aid  from, 
pins,  rivets,  or  other  fastenings. 

The  forces  acting1  upon  a  truss  (omitting  wind)  are  the  downward 
one  of  the  wt  of  itself  and  load ;  and  the  upward  one  of  the  reaction  of  the  abut- 
ments ;  and  these  two  forces  are  equal.  They  produce  all  the  strains  along  the 
members. 

The  figs  on  pages  292,  294,  show  some  details  of  modes  of  uniting  the  mem- 
bers of  a  truss  to  each  other  at  the  panel-points,  and  elsewhere. 

Fig-  5  is  the  most  simple  form  of  a  roof  truss.     It  consists  of  two 

equal  rafters  o  a,  o  b; 
and  a  nor  tie-beam  a  b. 
Here,  as  in  roofs  gen- 
erally, the  entire  weight 
of  the  truss,  and  of  its 
load  of  roof  -  covering, 
•now,  wind,  &c,  may  be 
assumed  to  be  uniformly 
distributed  across  the 
whole  span.  A  roof  con- 
sists of  several  trusses, 
placed  usually  from  8  to 
12  ft  apart;  but  some- 
times much  less,  and  at 
others  much  more.  The 

nal  timbers,  p,  p,  called 
wall -plates,  stretching 
along  the  top  of  the  wall ; 
and  serving  to  distribute 
the  wt  of  the  truss  and 
its  load  over  a  greater 
area.  On  the  rafters,  and  at  interval*  of  a  few  ft,  are  fixed  pieoes  of  timber  called  purling,  of 


248 


TRUSSES. 


small  scantling,  running  across  from  truss  to  truss ;  to  which  the  laths  or  boards  are  nailed  which 
support  the  shingles,  tin,  or  slate,  &c,  which  forms  the  roof-covering. 

A  truss  plainly  supports  all  the  purlins,  roof-covering,  snow.  &c,  &c,  which  occupy  the  space  half- 
way on  each  side  of  it  to  the  next  truss.  Thus,  suppose  a  span  of  30  ft ;  and  each  ra'fter  to  be  16.8  ft 
long ;  and  if  the  trusses  are  say  12  ft  apart  from  center  to  center,  and  if  we  assume  (as  it  is  generally 
well  to  do,)  that  the  wt  of  the  truss,  covering,  snow,  &c,  may  amount  to  40  fts  for  every  sq  ft  of  area 
of  roof;  each  truss  has  to  sustain  33.6  X  12  X  40  =  16128  fts,  including  its  own  weight.  Strictly  the 
wt  of  the  tie-beam  shoura  be  omitted ;  because  in  Fig  5  no  part  of  it  is  upheld  by  the  rafters.  It  is 
very  trifling  however  in  comparison  with  the  load. 

To    find  the  strains  upon  the  different  parts  of  a  truss, 

Fig"  5.f  Jb'irst  calculate  in  the  manner  just  shown,  the  entire  wt  in  K>s  of  a  truss 
and  its  load.  Through  the  center  U  of  either  rafter  draw  a  vert  line  Hr.  From  o  draw  a  hor  line 

0  H.     Join  H  a.     Now  on  the  vert  line  H  r,  lay  off  H I  by  any  convenient  scale  to  represent  the  entire 
uniformly  distributed  wt  of  one  rafter  and  its  load;  and  draw  the  hor  line  IE.     Then  will  IE  give 
by  the  same  scale  the  hor  force  at  the  head  of  the  rafter ;  and  H  E  the  amount  and  direction  of  'the 
oblique  force  *  which  presses  the  foot  of  the  rafter ;  but  which  diminishes  gradually  to  nothing  at  its 
head.1I     The  hor  force  at  the  foot  of  the  rafter  will  be  equal  to  that  at  its  head  ;  t  and  equal  also  to 
the  hor  pull  along  the  whole  length  of  the  tie-beam.t 

There  is  no  force  acting  in  the  direction  o  a  of  the  actual  length  of  the  rafter  in  such  a  truss  as  Fig  5. 

But  the  rafters  have  to  be  considered  in  another  point  of  view  no  less  important  than  their  ability 
to  sustain  this  pres  at  their  feet.  Each  rafter  is  an  inclined  b'eam  supported  at  both  ends ;  and 
bearing  a  heavy  load  equally  distributed  along  it ;  and  we  must  find  the  dimensions  reqd  for  this 
purpose  also;  and  add  them  to  those  req'l  for  other  purposes. 

These  dimensions  may  be  found  by  the  rules,  p  18i*,  or  by  tables,  pages  191,  or  204,  according  to 
the  case. 

The  sizes  in  Fig  .1  may  be  found  in  the  following-  manner. 

Take,  for  example,  a  truss  of  white  pine,  of  30  t'c  span,  and  1%  ft  rise.  The  wt  of  the  entire  roof, 
ano-r,  &<J.  &c,  40  fts  per  sq  ft  of  roof  area.  Trusses  12  ft  apart  from  center  to  center ;  so  that  each 
truss  will  have  to  sustain  a  total  load  (including  its  own  wt,)  of  33.6  X  *0  X  12  =  16128  fts,  which  we 
may  call  16000,  or  each  rafter  8000  fts.  We  will  calculate  each  part  with  a  safety  of  3  ;  which  we 
think  is  abundantly  sufficient,  with  the  assumption  of  40  fts  per  sq  ft.  First  prepare  a  diagram 
of  the  truss,  on  a  scale  of  say  %  inch  to  a  ft.  This  diagram  will  consist  of  but  three  lines.  We  will 
use  the  same  scale  of  %  inch  to  represent  1000  fts  of  either  wt  or  strain.  Make  H I  by  scale  equal  to 

1  inch  ;  that  is  to  the  8000  fts  uniformly  distributed  wtof  one  rafter  and  its  load.   Also  draw  I  E,  and 
measure  it.    It  will  be  equal  in  this  case  (accidentally)  to  H  I,  or  8000  ft»s  ;  and  this  is  the  amount  of 
pull  along  the  tie-beam.   Now  we  see  by  table,  page  177,  that  average  white  pine  breaks  under  a  pull 

of  10000  fts  per  sq  inch ;  so  that  for  a  safety  of  3,  we  must  not  subject  it  to  more  than — —  —  3333 

fts  pull  per  sq  inch.  The  weakest  part  of  the  tie-beam  is  where  it  is  cut  into,  near  the  ends,  for  foot- 
ing the  rafters ;  and  even  what  is  there  left  by  the  cut,  is  usually  still  farther  reduced  by  the  holes 
of  the  bolts  or  spikes  driven  into  it  through  the  feet  of  the  rafters.  Therefore,  allowing  for  these 
things,  we  must  give  to  the  tie-beam  at  that  point  a  transverse  section,  of  Solid  wood,  equal  at  least  to 

—TTT-  —  2.4  sq  ins.  This  would  no  doubt  be  sufficient  to  resist  the  pull ;  but  there  are  other  considera- 
tions, such  as  danger  of  sagging  or  breaking  down  if  persons  should  get  on  it ;  or  if  a  moderate  load 
should  chance  to  be  laid  upon  it,  &c,  whicli  cause  the  tie-beam  (even  when  unloaded  even  by  the  wt  of  a 
plastered  ceiling  below, II  as  is  here  supposed  to  be  the  case,)  to  be  made  about  as  large  as  a  rafter. 
If,  instead  of  a  beam,  we  had  used  an  iron  rod  to  resist  the  8000  fts  pull,  we  should  have  reqd  OIIK. 
with  a  breaking  strength  of  8000  X  3  =  24000  fts;  and  by  the  table  of  bolts,  page  376,  we  see  that  a 
diam  of  full  1-4-  inch  would  suffice  if  upset ;  or  of  full  1.04  inch  if  not  upset. 

Now,  as  to  the  rafters,  each  of  them  is  an  inclined  beam,  supported  at  both 

*  As  this  strain  H  E  arises  from  the  wt  of  the  rafter  and  load,  it  is  greatest  at  the  foot  of  the 
rafter  which  bears  all  said  wt.  At  U,  half-way  up,  it  is  only  half  as  great;  and  at  the  head  of  tha 
rafter,  which  has  no  wt  of  truss  above  it,  it  is  nothing;  and  therefore,  so  far  as  said  strain  alone  is 
concerned,  the  rafter  might  terminate  in  a  point  at  top.  But  it  requires  some  thickness  there  to  pre- 
vent being  snapped  off  by  the  hor  pres  caused  by  the  rafters  leaning  against  each  other;  and  still 
more  to  enable  it  to  bear  the  transverse  strain  arising  from  its  own  wt,  and  from  the  load  which  13 
spread  uniformly  over  it;  as  will  be  seen.  The  increase  reqd  for  this  last  purpose,  is  sufficiently  great 
to  provide  in  itself  against  the  top  hor  pres;  which,  therefore,  requires  no  special  provision. 

t  The  foot  of  each  rafter  tends  to  slide  or  push  outward  horizontally  in  the  direction  of  the  arrows 
t  and  v ;  each  with  a  force  equal  to  I  E.  But  the  tie-boara  prevents  them  from  so  doing,  and  thus  con- 
verts their  pushes  into  pulls  against  each  other;  and  thereby  into  a  pulling  strain  along  the  whole 
tie-beam  itself,  to  an  amount  equal  to  one  of  the  forces;  as  two  men  pulling  against  each  other  at  the 
two  ends  of  a  rope,  each  with  a  force  of  10  fts,  only  strain  the  rope  10  fts.  In  other  words,  it  requires 
two  equal  opposing  forces  of  10  fts  each,  to  produce  one  strain  of  10  fts.  See  Arts  13,  IS,  33,  of  Force 
in  Rigid  Bodies. 

+  And  there  is  a  hor  strain  to  the  same  degree  generated  at  every  point  along  the  length  of  each 
rafter. 

H  The  weight  of  an  ordinary  lathed  and  plastered  ceiling  is 

about  10  fts  per  sq  ft ;  and  that  of  an  ordinary  floor  of  1^  inch  boards,  to- 
gether with  the  usual  3  by  12  inch  joists,  15  ins  apart  from  center  to  center,  is  from  10  to  12  fts  per  sq 
ft.  In  preliminay  calculations  it  is  well  to  take  the  two  together  at  25  fts  per  sq  ft. 

^[  The  principle  upon  which  the  lines  IE,  HE,  become  the  measures  of  the 
strains  at  a  upon  the  tie-beam  and  upon  the  foot  of  the  rafter,  is  explained  at  Rem  2,  p  168. 


TRUSSES. 


249 


«n/l«,  and  uniformly  loaded  with  8000  fts  ;  which  is  equal  to  a  center  load  of  4000  Ibs.  But  for  a  safety 
of  3  against  4000,  we  will  find  its  dimensions  for  a  breaking  center  load  of  12000.  Being  inclined* 
its  length  must  here  be  taken  as  if  measd  horizontally  between  its  end  supports,  or  15  ft.  See  Art  15, 
p  188.  We  will  assume  for  it  some  probable  approx  depth  ;  say  9  ins.  Then  by  Art  20,  p  189,  we  find 
that  for  a  breaking  center  load  of  12000  fts,  its  breadth  will  be  4.94.  say  5  ins.  Therefore,  to  be  safe 
with  4000  tts  center  load,  each  rafter  must  at  it*  head  be  5  ins  broad,  by  9  ins  deep.*  But  it  must 
be  larger  at  its  foot,  because  we  have  not  yet  provided  for  the  crushing  pres  at  its  foot,  indicated  by 
the  line  H  E,  which  by  scale  measures,  say  10500  Ibs.  Now,  we  find  by  table,  page  174.  that  average 
white  pine  or  spruce  absolutely  crushes  under  a  pres  of  say  6000  B>s  per  sq  inch  ;  therefore  it  will  have 


a  safety  of  3  under  2000  tts  per  sq  inch  ;  so  that  we  must  provide  for  the  foot  of  each  rafter 

sq  ins  of  area  for  resisting  this  pressure  alone.     These  5^  sq  ins  may  be  ndded  either  in  the  breadth 


— 

.  may  b 

of  our  5  X  9  rafter,  thus  making  it  say  5.6  by  9  ;  or  to  its  depth,  making  it  full  5  by  10  at  foot.     We 


y 


Fit,  6 


,  . 

may  diminish  the  rafter  regularly  from  the  bottom,  until  it  becomes  equal  to  the  top:  but  in  practice 
it  is  not  worth  while  to  do  this,  unless  timbers  with  the  proper  taper  happen  to  be  at  hand.  Make 
the  tie-beam  about  the  same  size  as  a  rafter. 

In  the  next  three  trusses  we  shall  not  enter  into  this  detail 
of  calculation;  as  we  conceive  that  this  example  suffices  to 
elucidate  its  principle. 

Art.  8.  Next  to  Fig  5,  in  point  of  simplicity,  is  Fig  6  ;  which  represents  a  truss 
for  either  a  bridge  or  a  roof  of  mode- 
rate span.  It  has  two  equal  rafters, 
and  a  hor  tie-beam  a  6  as  before; 
but  with  the  addition  of  a  king- 
post, king-rod,  or  suspension-rod 
o  n.  Either  the  tie-beam,  or  the 
rafters,  or  both,  may  be  uniformly 
loaded.  It  is  immaterial  whether 
the  load  on  the  former  be  equal  to 
that  on  the  latter  or  not.  We  shall 
here  consider  the  truss  only  as  that 
of  a  roof.  Let  y  y  be  points  half- 
way between  the  king-rod  and  the 
abutments.  Then  will  the  king-rod 
sustain  all  the  weight  of  the  portion 
y  y  of  the  tie-beam  and  its  load. 
The  portions  of  the  tie-beam  and  its 
load  between  y,  y,  and  the  walls 
a;,  w,  are  sustained  directly  by  the 

walls.  The  entire  wt  of  the  truss  and  its  load,  it  is  plain,  is  sustained  ultimately  by  the  ahuts.  or 
walls  x  w  ;  but  the  wt  of  y  y  and  its  load  does  not  reach  the  walls  until  after  having,  as  it  were,  first 
travelled  up  the  king-rod  to  o,  and  from  there  down  the  rafters  to  a  and  b  •  or,  indirectly,  by  a  cir- 
cuitous route.  That  the  king-rod  sustains  all  between  y  and  y,  will  be  evident  when  we  reflect  that 
a  beam  a  b,  when  firmly  suspended  at  its  center  n,  may  be  regarded  as  two  separate  beams  n  b,  n  a. 
One-half,  of  the  beam  n  b.  and  its  load  would,  in  that  case,  manifestly  be  borne  by  the  wall  x,  and 
the  other  half  by  n;  and  so  with  n  a.  Therefore,  n  upholds  one-half  of  the  beam  a  I  and  its  load  ; 
or,  in  other  words,  all  between  y  and  y.  The  king-rod  transfer.--  the  wt  of  and  on  y  y,  to  the  heads 
of  the  rafters  at  o.  This  wt  may,  therefore,  be  considered  precisely  in  the  light  of  one  resting  upon 
o;  and  we  may  proceed  to  find  the  strains  which  it  produces  upon  the  rafters  and  tie  beam,  by  Art 
33,  p;ige  461,  of  Force  in  Rigid  Bodies.  Namely,  on  o  n  make  o  t,  by  scale,  equal  to  said  total  wt  of 
yy  and  its  load,  and  the  wt  of  the  king-rod  itself.  Complete  the  parallelogram  of  forces  omtd; 
and  draw  its  hor  diag  m  d.  Then  will  o  m.  o  d  measure  the  strains  produced  by  said  total  weight 
only,  along  their  respective  rafters;  and  cm,  c  d  the  pulling  forces  produced  by  the  same  wt  only 
al'ing  the  tie-beam  a  b  ;  causing  strain  all  along  it  equal  to  one  of  them.  Art  13,  p  449,  of  Force  in 
Rigid  Bodies. 

It  will  be  observed  that  in  this  truss,  therefore,  unlike  Fig  5,  there  z's  a  strain  o  m,  or  o  d,  running 
lengthwise  through  the  rafters  from  head  to  foot.  To  resist  this  Strain,  f  It  <>  rafter 
nitlSt  be  regarded  as  a  pillar,  with  a  load  on  its  top  equal  to  the  strain  ;  and  the 
safe  area  reqd  for  upholding  it  may  be  found  by  means  of  the  table  of  strength  of  wooden  pillars,  on 
p  ige  239.  Call  this  area  when  found,  a.  Next,  consider  a  rafter  as  a  beam  supported  at  both  ends. 
and  sustaining  a  center  load  equal  to  half  its  own  wt  and  actual  load;  and  in  that  respect  proceed 
precisely  as  in  Fig  5.  to  find  its  safe  breadth  and  depth  by  rules,  page  189.  To  the  area  of  section 
which  results,  add  area  a  :  and  the  sum  will  be  the  final  area  reqd  for  the  head  o  of  the  rafter.  Call 
it  !,.  Then  measure  H  K,  as  iu  Fig  5,  for  the  pres  caused  at  the  foot  only  of  the  rafter  by  the  weight 
of  the  rafter  and  its  load.  Divide  this  by  the  safe  amount  of  2000  Ibs,  as  was  done  in  Fig  5;  or  bv 
a  less  number  if  a  safety  greater  than  3  is  reqd.  The  quot  will  be  the  area  needed  for  that  purpose 
alone.  Add  it  to  area  6;  and  the  sum  will  be  the  total  area  reqd  at  the  foot.  This  area  may  now 
be  regularly  diminished  upward,  bv  reducing  either  the  breadth  or  the  depth,  until  at  the  head  it  is 
equal  to  area  b  :  but  this  is  rarely  done  even  in  iron  roofs  ;  the  area  at  foot  being  continued  to  the  head. 

We  must  not  diminish  both  the  breadth  and  the  depth,  because  we  should  thereby  reduce  the  pro- 
portion  of  area  required  at  every  point  of  the  length,  to  resist  this  pressure,  which,  although  greatest 
»t  the  foot,  does  not  disappear  entirely  until  at  the  very  head  o. 

The   pull   on   the   tie-beam  will  be  I  E  added  to  cm  or  cd.     Find  the 

*  This  is  not  a  bad  proportion  of  breadth  to  depth.  If  we  had  assumed  say  15  ins  for  the  depth, 
we  should  have  got  a  rafter  so  thin  as  to  be  laterally  weak.  Frequently,  two  or  three  assumptions 
and  calculations  may  have  to  be  made  before  we  hit  'upon  a  satisfactory  "proportion- 


250 


TRUSSES. 


safe  area  by  dividing  their  sum  by  3333,  which  is  the  number  of  Ibs  per  sq  inch,  giving  a  safety  of  3. 
Then  regarding  half  the  length  of  the  tie-beam  supported  at  both  euds,  aud  loaded  at  its  center  with 

or  by  table,  page  191.  The  resulting  area,  added  to  the  safe  area  for  the  pull  just  found,  will  be  the 
eutire  section  of  the  tie-beam,  uuless  some  addition  be  made  to  the  depth,  to  allow  tor  what  is  cut 
away  for  the  feet  of  the  rafters.*  See  Rem,  p  19'2,  also  Rem,  p  487. 

As  to  the  vertical  king-rod,  n  o,  it  must  be  strong  enough  to  bear  safely 

a  pull  equal  to  its  own  weight,  added  to  the  weight  of  aud  upou  y  y.  Jf  the  rod  is  of  good  bar  iron, 
it  should  have  one  square  inch  for  a  safety  of  3,  of  cross-section  for  each  20000  tbs  of  said  weight; 
or  see  table,  page  376.  If  of  wood,  it  must,  for  a  safety  of  3,  have  at  least  one  sq  inch  for  about  each 
3333  fts  of  said  weight.  A  safety  of  3  will  be  enough  if  the  bar  is  not  liable  to  vibration. 

When  the  king-rod  is  of  wood,  it  is  improperly  termed  a  king-post.  Since  a  post  is  intended  to  sus- 
tain a  load  on  its  top,  the  term  might  lead  to  the  inference  that  the  upper  ends  of  the  rafters  rested 
upon,  or  were  upheld  by  the  king  post;  whereas,  as  we  have  seen,  they  actually  uphold  it. 

We  add  the  calculated  approximate  dimensions  for  a  truss, 
Fig1  6,  of  30  ft  span ;  and  7V£  ft  rise.  Trusses  12  ft  apart  cen  to  cen.  Wt  of 

rafters  and  load  on  top  of  them,  40  Ibs  per  sq  ft  of  area  of  roof,  Wt  of  and  on  the  tie-beam,  includ- 
ing floor,  ceiling,  load,  and  momentum,  100  Ibs  per  sq  ft.  Timber  white  pine.  Safety  of  each  piece  3. 
Rafters  8>£  ins  broad,  by  11  deep,  at  foot,  and  7%  by  11  at  head.  Tie-beam  8^  broad,  by  11  deep, 
without  any  allowance  for  cutting  at  feet  of  rafters.  King-rod  l^j  inch  diam  if  upset;  or  full  1%  if 
not  upset.-|-  See  note,  p  263. 

With  no  floor  or  loading:  on  the  tie-beam,  except  its  own  wt,  say 

1000  Ibs,  we  have,  approximately  enough,  rafters  6^  ins  broad  by  9  ins  deep,  at  foot;  6  by  9  at  top. 
Tie-beam,  say  same  as  rafter,  or  6%  X  9.  King-rod,  ^  inch  diam  if  upset ;  %  inch  if  not  upaet;  but 
it  would  be  expedient  to  make  it  rather  more.  Trusses  12  ft  apart  center  to  center. 

Art.  9.  In  Fig  7  we  have  a  truss  consisting  of  two  rafters,  a  6,  a  d;  a  tid-beam, 
6  d;  a  king-rod,  a  c;  and  two  struts  or  braces,  e  c,  h  c.  Either  the  rafters  or  the  tie-beam,  or  both, 
may  be  supposed  to  be  uniformly  loaded. 


Here, 


R 


as  in  Fig  6,  the  king-rod  a  c,  upholds  the  weight  of  the  portion  y  y 
.  of  the  tie-beam,  and  of  any  load  of 

' _JH  flour,  ceiling,  people,  &c,  that  may  be 

'*"    "  placed  upon  that  portion  ;  together 

with  its  own  weight.  But  it  also  sus- 
tains, in  addition  to  these,  the  weight 
of  the  two  struts  e  c,  he;  part  of  the 
weight  of  the  portions  z  r,  aud  x  it, 
of  the  rafters  ;  and  part  of  the  weight 
of  the  roof-covering,  snow,  &c,  that 
may  rest  on  said  portions.  That  it  up- 
holds itself,  y  y,  and  the  struts,  is  al- 
most self-evident;  but  that  it  upolds 
part  of  z  r,  aud  xu,  and  their  loads,  is 
not  at  first  sight  so  apparent.  Such 

struts  are  introduced 

illtO  trusses  when  the  rafters 
become  so  long  as  to  be  in  danger  of 

•*  bending  too  much,  or  of  breaking  un- 

der their  loads;  or  else  requiring  the 

use  of  inconveniently  large  timbers  to  make  them  of.  They  act  like  posts  in  aff  irding  partial  support 
to  the  rafters.  They  carry  a  part  of  the  strain  upon  the  rafters  down  to  the  foot  c.  of  the  king-rod; 
and  the  king-rod  carries  it  from  there  up  to  the  tops,  a,  of  the  rafters.  From  a  it  passes  down  through 
the  entire  length  of  the  rafters  to  their  feet.  Thus,  it  is  seen  that  the  actiou  of  the  struts  consists  in 
relieving  the  rafters  from  a  transverse,  or  cross-strain  which  would  endanger  their  safety;  and  in 
converting  it  into  a  longitudinal  strain  in  the  direction  of  their  length,  in  which  they  can  resist  it 
with  less  danger.  As  we  proceed  with  the  subject  of  trusses  for  bridges  as  well  as  roofs,  it  will  be 
seen  that  this  is  the  grand  duty  of  such  struts  and  obliques  generally.  In  roofs  they  thus  assist  the 
rafters  ;  and  in  bridges,  the  chords.  See  Rem,  p  375. 

To  find  the  actual  amount  of  strain  in  Ibs  which  the  struts 
thus  convey  from  the  rafters  to  the  foot  of  the  king-rod,  draw  e  o  and  h  n  verti- 
cally ;  and  make  each  of  them,  by  any  convenient  scale,  equal  to  the  weight  in  fts  of  either  zr  or  xu, 
and  its  load.  From  o  and  n  draw  the  dotted  lines,  o  i.  n  w,  parallel  to  the  struts ;  and  o  fc.  n  v,  par- 
allel to  the  rafters  ;  thus  completing  the  parallelograms  of  forces,  ekoi,  and  hwnv.  Draw  the  hori- 
zontal diagonals  i  k.  and  v  w. 

Then  by  Composition  and  Resolution  of  Forces,  either  ek  or  h  v,  measured  by  the  same  scale  as  be- 
fore, will  give  the  longitudinal  strain  in  Ibs  upon  each  one  of  the  struts.  This  strain  presses  the  struts 
lengthwise  from  head  to  foot.  Their  feet  are  pressed  in  addition  by  the  weight  of  the  struts  them- 
selves; and  Mis  pressure  diminishes  regularly  toward  their  heads  or 'tops,  where  it  is  nothing;  as  in 
the  case  of  the  rafters  in  Fig  5.  In  practice,  the  wt  of  the  struts  is  so  trifling  in  comparison  with  that 
of  the  roof  portions  which  they  sustain,  that  it  may  be  neglected,  and  its  safe  dimensions  may  be  found 

by  pa*es2:58, 239.  Therefore,  each  strut  may  be  regarded  as  if  a  vert 

pillar,  bearing  a  load  equal  to^fe  or  h  v.  NOW,  the  strain  e  k,  along  the  strut 


:ompounded  or  composed  of  the  vert  strain  e  s,  (which  is  equal  to  half  c 
wt  of  and  on  zr;)  and  of  the  hor  strain  a  k.    And  the  strain  h  v  along  the  st 


o.  or  one-half  of  the 
•ut  h  c,  is  compounded 


*In  cases  where  no  appearance  of  sagging  would  be  admissible,  it  is  not  always  enough  that  the 
rafters  and  tie  beam  be  safe ;  for  they  mav  be  perfectly  safe,  and  yet  sag  too  much  for  some  purposes. 
When  such  is  the  case,  refer  to  table,  page  204. 

tWe  have  known  country  road  bridges,  Fig  6,  of  30  ft  span,  and  7^  ft  rise,  of  two  trusses  18  ft 
apart,  in  wnicn  neither  the  timbers,  nor  the  probable  loads,  were  larger  than  in  this  example. 


TRUSSES. 


251 


ef  the  vert  strain  h  t,  (which  is  equal  to  half  of  h  n,  or  one-half  of  the  wt  of  and  on  x  « ;*  and  of  the 
nor  strain  t  v.  These  two  hor  strains  s  k  and  tv  neutralize  or  counteract  each  other,  by  pressing 
against  each  other  at  the  feet  of  the  struts ;  and  therefore  only  the  vert  ones  e  s  and  h  t  pull  upon 
the  king-rod  ;  and  they  pull  it  to  an  extent  equal  to  half  the  weights  of  and  on  z  r  and  xu.* 

The  kiii$?-rod,  therefore,  upholds  in  all,  1st,  the  weight  of  the  two 

struts  ;  2d,  the  wt  of  and  on  y  y  ;  3d,  half  the  wt  of  and  on  z  r  and  x  u  ;*  and  4th,  ite  own  wt.  It 
must,  therefore,  have  sufficient  sectional  area  to  safely  sustain  a  pull  equal  to  the  sum  of  these  lour. 
This  area  may  be  found  by  means  of  the  table  of  bolts  on  p  376. 

Make  a g  by  scale  equal  to  the  sum  of  these  four  wts.  Draw  am,  gl  parallel  to  the  rafters;  and 
Imhor. 

For  the  dimensions  of  the  rafters,  ab,ad,  commencing  with  what 

they  require  as  beams,  supported  at  the  ends,  bear  in  mind  that  the  introduction  of  the  struts  ec,  h  c 
converts  each  rafter,  as  a  b,  into  two  shorter  ones,  a  e,  e  b ;  each  of  which  sustains,  in  the  present 
case,  only  one-half  the  load  of  and  on  the  whole  rafter ;  or  only  %  of  it  as  a  center  load.  Find  the  safe 
dimensions  for  the  short  beam,  with  its  smaller  center  load,  by  rules,  p  189,  or  by  table,  p  191. 

Next,  regard  the  rafter  as  if  a  vert  pillar,  which  has  to  support  on  its  top  the  pressure  indicated  by 
either  o  m  or  a  L  But  in  doing  this,  remember  that  the  struts  virtually  reduce  each  rafter,  as  a  6,  to 
two  short  pillars  a  e,  e  b;  which  require  much  less  area  than  the  whole  rafter  would.  This  area 
may  be  found  by  the  rule  on  pag«  238 ;  or  by  the  table,  page  239 ;  and  added  to  that  required  as  a 
beam.  The  sum  will  be  the  total  area  at  the  head  of  the  rafter. 

Next  find  by  means  of  a  line,  H  E,  drawn  on  the  same  principle  as  in  Figs  6,  7,  and  9,  (HI  being 
the  entire  wt  of  a  rafter  and  load,)  the  amount  of  pres  which  the/ooe  of  the  rafter  sustains  from  its 
own  wt  and  load  resting  on  it.  Divide  it  by  2000,  (or  by  whatever  other  number  of  tt>s  may  be  con- 
sidered the  safe  crushing  strength  of  timber.)  The  quot  will  be  the  safe  area  in  sq  ins  reqd  at  the 
foot  for  that  purpose ;  and  added  to  the  areas  previously  found  for  a  beam,  and  for  a  pillar,  it  gives 
the  entire  area  for  the  foot. 

We  may  diminish  either  the  breadth  or  the  depth,  (not  both,)  of  the  rafters  regularly  from  foot  to 
head,  to  accord  with  the  total  areas  found  for  those  points  respectively  ;  or,  which  is  generally  better, 
may  give  to  them  throughout  the  same  area  they  have  at  foot. 

The  tie-beam.  Here  I E  added  to/ra,  or  to  fl,  will  give  by  scale  all  the  pull 
on  the  tie-beam.  Divide  this  pull  by  3333,  the  safe  pull  in  ft>9  per  sq  inch.  The  quot  will  be  the  safe 
area  reqd  for  that  strain.  Then  consider  one-half  of  the  tie-beam  to  be  a  uniformly  loaded  beam  sup- 
ported at  each  end;  and  find  the  safe  dimensions  by  rujes,  p  189,  or  by  table,  p  191.  To  these  dimen- 
sions add  the  area  just  found  for  the  hor  pull ;  the  sum  is  the  entire  area  reqd  for  the  tie-beam,  un- 
less some  addition  be  made  to  compensate  for  the  cutting  away  at  the  feet  of  the  rafters.t 
See  Rem,  p  192. 

Below  are  the  calculated  dimensions  for  two  trusses,  Fig:  7, 
of  4O  ft  span  ;  10  ft  rise ;  and  12  ft  apart  from  center  to  center.  In  the  first  of 
these  the  tie-beam  with  its  floor,  ceiling,  and  other  load,  are  assumed  at  the  rate  of  100  Ibs  per  sq  ft 
of  floor;  while,  in  the  second,  no  specific  load  is  assumed  for  that  member,  for  reasons  before  given. 
In  both,  the  wt  of  the  rafters,  with  their  roof-covering  and  load  of  snow,  and  wind,  is  taken  at  40  Sbs 
per  sq  ft  of  roof  surface  between  the  centers  of  two  trusses.  The  safety  of  each  separate  part  is  taken 
at  3;  except  that  the  unloaded  tie-beam  is  fixed  by  rule  of  thumb.  Timbers  white  pine.  The  great- 
est dimension  in  each  case  is  the  depth.  Dimensions  in  inches.  1st.  Rafters  8  X  10  at  foot;  and 
8  X  9  at  head.  Tie-beam  8  X  15.  Each  strut  43^  X  4>£.  King-rod  \%  diam  if  upset ;  or  2  ins  if  not 
upset.  In  practice,  it  is  better  to  make  the  struts  as  broad  as  the  rafters.  2d.  Rafters  6  X  8  at 
foot;  and  5  by  8  at  head.  The  tie-beam  requires,  theoretically,  only  16  sq  ins  area;  we  will  make 
it  6  X  8,  like  the  rafters.  Each  strut  4>^  X  4^ ;  (the  same  as  in  the  other.)  King-rod  %  diam  if 
upset ;  or  scant  1  inch  if  not  upset.  See  Note,  p  263. 

If  a  tie-rod  were  used  instead  of  a  tie-beam,  its  diam  would  be  \y±  inch  if  upset ;  or  1.6  if  not. 

Art.  1O.  Fig  9  is  a  truss  with  a  tie-beam  a  6;  two  rafters  w  a,  z  b ;  two  queen- 
rods,  J  or  queens,  w  t,  z  t,  and  a  hor 
straining  beam  d.  It  may  repre- 
sent a  roof  uniformly  loaded  along 
the  rafters  and  straining  beam  ;  and 
having  a  uniform  load  along  the  tie- 
beam.  Or  only  one  of  these  loads 
may  be  supposed  to  exist,  as  in  a 
bridge  with  a  load  along  a  b ;  or  a 
roof  with  its  load  along  a  w  z  b.  The 
queen  w  t  supports,  besides  its  own 
weight,  all  the  weight  of  and  on  the 
part  «  y  of  the  tie-beam;  and  the 
other  one  z  t,  that  of  and  on  u  y;  » 
and  r,  each  being  halfway  between 
a  queen  and  an  abut.  These  are  the 

their  proper  diams  can  be  found  by 
table  of  bolts,  p  376.     The  parts  of 

*  Each  strut  will  thus  bear  half  of  the  wt  of  and  on  z  r,  or  li  u,  only  when,  as  in  Fig  7,  the  incli- 
nation of  the  strut  is  the  same  as  that  of  the  rafter.  If  the  strut  is  steeper  than  the  rafter,  it  will 
bear  more  than  half;  but  if  it  is  less  steep  than  the  rafter,  it  will  bear  less  than  half;  the  remainder 
being  in  every  case  borne  by  the  rafter.  The  parallelogram  of  forces  will  of  course  show  all  this. 
When  the  inclinations  of  a  rafter  and  strut  are  not  equal,  we  cannot  draw  hor  diags  ik,  vw;  but 
from  the  points  i,  k,  v,  w,  we  must  draw  hor  lines  to  the  vert  diags  e  o.  and  h  n. 

f  The  introduction  of  the  struts  in  Fig  7,  renders  our  process  for  that  form  of  truss  in  some  meas 
ure  empirical.  It  is  however  safe. 

WHEN  A  TIE-BEAM  is  so  LONG  THAT  IT  MUST  BE  SPLICED,  allowance  must  be  made  for  the  weakening 
effect  of  the  splice.  For  Splices,  see  p  292  ;  and  for  other  joints,  p  294. 

1  The  queens  are  frequently  made  of  wood. 


252 


TRUSSES. 


the  tie-beam  from  «  and  tt  to  the  abuts,  or  walls,  as  well  as  whatever  loads  those  parts  may  bear,  ar« 
sustained  directly  by  the  abuts. 

The  queens  transfer,  as  it  were,  the  weights  of  themselves  and  of  «  y  and  u  y,  with  their  loads,  di- 
rectly to  w  and  z.  To  find  the  strains  on  the  various  parts  of  the  truss,  first  from  the  center  U  of  a 
rafter  aw,  draw  a  vert  line  UH;  and  from  w  draw  a  hor  line  toH  to  meet  it.  Join  Ho.  Make  HI 
by  scale  equal  to  the  wt  of  only  one  rafter  and  its  uniformly  distributed  load.  Also  draw  og  vert, 
and  equal,  by  the  same  scale,  to  the  wt  upheld  by  the  queen-rod  wt,  added  to  one-half  the  wt  of  the 
straining-beam  d,  and  its  load  ;  for  it  also  presses  vert  at  o.  Draw  g  m  hor,  or  parallel  to  the  strain- 
ing-beam ;  and  g  c  parallel  to  the  rafter;  thus  completing  the  parallelogram  ocgmof  forces. 

The  strains  on  the  straining-beam  d.    The  hor  line  IE  and  oc  to- 

gether, give  all  the  hor  pres  against  the  end  w  of  the  straining-beam  d  ;  and  it  is  plain  that  a  similar 
process  on  the  other  side  of  the  truss,  would  give  an  equal  pres  against  the  end  z.  These  two  equal 
pressures  reacting  against  each  other,  produce  a  strain,  equal  to  one  of  them,  throughout  the  entire 
length  of  the  straining-beam  ;  and  therefore,  the  beam  must  be  regarded  as  a  pillar  with  a  load  equal 
to  this  strain,  on  its  top  ;  and  the  dimensions  and  area  of  section,  for  safely  supporting  it,  may  be 
found'by  the  rule,  p  238;  or  table,  p  239. 

But  beside  this,  the  straining-beam,  if  loaded,  must  be  regarded  also  as  a  beam  supported  at  both 
ends  ;  and  the  area  necessary  for  this,  as  found  by  tables,  page  191,  or  204,  must  be  added  to  that 
already  found. 

The  strains  on  the  rafters.    First,  consider  a  rafter  to  a  as  an  inclined 

loaded  beam  supported  at  both  ends  ;  and  find  the  proper  dimensions  and  area,  by  the  rules  on  page 
188  ;  or  by  the  tables,  p  191,  or  204.  Second,  consider  it  as  a  pillar  supporting  a  load  equal  to  o  TO; 
and  by  p  238  or  239,  find  its  safe  area.  Add  this  area  to  the  one  already  found  ;  and  their  sum  will  be 
the  area  of  the  rafter  at  its  head.  Third,  measure  by  scale  the  number  of  Its  pres  indicated  by  H  E. 
Divide  it  by  2000,  (the  crushing  pressure  in  fts  which  ordinary  average  building  timber  can  bear  in 
short  blocks  with  a  safety  of  3.)  The  quot  will  be  the  area  in  sq  ins  reqd  at  the  foot  of  the  rafter,  ex- 
presslv  for  resisting  the  crushing  effect  at  that  point.  Add  this  area  to  the  two  preceding  ones  ;  and 
the  sum  will  be  the  total  area  for  the  foot.  We  may  diminish  to  the  top  ;  or  may  make  it  of  uniform 
section  throughout  its  length  :  the  last  is  generally'most  convenient. 

The  tie-beam.  The  hor  strain,  or  pull  on  the  tie-beam,  will  be  equal  to  the 
push  on  the  straining-beam  ;  and  is  represented  by  I  E  and  o  c  together.  Find  the  safe  area  by  table, 
page  177  :  or  by  dividing  the  hor  strain  by  33»3,  which  is  the  pull  in  fts  per  sq  inch  that  ordinary 
building  timbeV  will  bear  with  a  safety  of  3.  See  Rem,  p  192,  also  Rem,  p  487. 

Then,  since  in  this  truss  the  queens  divide  the  tie-beam  into  three  lengths,  each  of  these  must  be  con- 
sidered as  a  separate  beam,  (loaded  or  unloaded,  as  the  case  may  be,)  supported  at  each  end.  Its  safe 
dimensions  being  found  by  rules,  p  189;  or  tables,  pages  191,  or  204.  as  may  be  reqd.  add  the  area  just 
found  for  resisting  the  pull.  Add,  if  reqd,  an  allowance  for  the  cutting  away  at  the  feet  of  the  rafters. 

Below  are  the  calculated  approximate  dimensions  for  two 
trusses,  Figr  9,  of  sixty  ft  span  ;  15  ft  rise;  and  12  ft  apart  from  center  to 
center.  All  the  conditions  the  same  as  for  the  preceding  example  of  Fig  7.  1st.  Rafters  12  ins  broad, 
by  14  ins  deep  at  the  head  ;  and  12  X  HV>  at  foot.  Straining-  beam  12  broad,  by  12  deep.  Tie-beam 
12  broad,  by  12  deep.  Each  queen  rod  1ft.  ins  diarn  if  upset  ;  IJf  ifnot.t 

2d.  Rafters  10  X  11%  at  the  head;  and  10  X  12}^  at  foot.  Straining-beam  10  X  11.  Tie-beam,  say 
10  X  12.  Each  queen-rod  y5g  inch  diain  if  upset  ;  %  if  not.  Unloaded  tie-rod,  1  -=-3_  or  1}$. 

The  proper  size  for  each  piece,  so  tJiat  they  sJiall  all  be  suitable 

for  tlif  truss,  cannot  be  determined  at  once.  We  must  find  any  dimensions  that 
will  answer  for  each  piece  by  itself;  and  afterwards  adjust  them  by  recalculation,  perhaps  3  or  4 
times.  Great  accuracy  is  not  necessary  in  doing  this.  See  Note,  p  263. 

For  more  on  roof  trusses,  see  pp  257  to  205  ;  294  ;  and  298  to  802. 

REM.  The  truss  in  Figs  9  and  9%  affords  a  good  opportunity  for  alluding,  in  a  general  way,  to 
the  principle  Of  COUnterbracing",  and  to  the"  necessity  (as  stated  in  Art  7)  of  ad- 

hering to  a  triangular  arrangement  of  the  parts  of  a  truss.  So  long  as  this  truss  is  uniformly  loaded 
throughout  its  length,  it  is  well  arranged  for  sustaining  the  resultiug  strains;  because  the  strains  on 
each  side  of  the  center  are  equal,  and  balance  each  other.  But  if  a  heavy  load  be  placed  alo 

ment  sho 


. 
only,  its  tendency  to  depress  that  portion  of  the  truss  will  produce  the  de 


in  Fig  9%  ; 


*  A  strut  or  tie  cannot  be  strained  along  the  direction  of  its  length 

by  a  force  acting  at  one  end,  unless  there  is  at  the  other  end  an  equal  force  acting  in  the  same  straight 
line  but  in  the  opposite  direction,  and  which  may  be  either  one 
single  force,  or  the  resultant  of  two  or  more  forces  neither  of 
which  acts  in  that  direction.  See  Strain,  p  444.  Hence  if  in 
Fig  9  we  place  a  load  at  Z  only,  a  parallelogram  v  e  gn  of  forces 
will  not  give  the  hor  strain  v  e  along  the  beam  Z  W,  because 
there  is  then  no  equal  reacting  hor  force  at  the  other  enrt  in  the 
direction  from  H  towards  W.  In  that  case  a  load  at  Z  only, 
«f  /^%x'/  ^x\.n  (represented  by  z  c  in  Fig  X)  produces  at  z  the  two  strains  z  n, 

I)/  ^£  ^cAI     z  e;  which  last  pressing  towards  a  tends  to  make  z  b  revolve 

around  b  as  a  center,  thus  forcing  z  down  wards,  and  the  joint  w 
upwards,  thereby  causing  the  distortions  seen  in  Figs  9^.  $%. 
The  force  z  e  therefore  evidently  tends  to  break  the  joint  w , 
and  with  a  moment  equal  to  the  force  z  e.  (in  tons  or  Ibs,  &c) 

mult  by  its  leverage  w  o  perp  to  z  a.  See  Art  46,  p  473,  and  Moments,  p  217.  If  the  moment  of 
resistance  of  the  joint  can  withstand  this  the  truss  will  remain  unchanged  ;  btu  a  simple  strut  from 
z  to  a  would  remove  all  danger,  by  sustaining  the  whole  of  the  force  z  e  effectively,  and  thus  relieving 
the  joiut  w  entirely.  See  top  of  p  462. 


--,.          -«y 
I1  Ifl      \« 


TKUSSES. 


253 


because  the  hor  pressure  from  s  toward  t,  Fig  9J^,  will  then  become  greater  than  that  from  t  toward  ». 
The  two  triangular  portions  will  still  retain  their  original  fig;  but  owing  to  the  ease  with  which  the 
4-sided  portion,  n  m  c  «,  has  been  deranged,  and  changed  to  8  t  c  e,  their  position  becomes  altered  to 
tfce  dangerous  one  in  Fig  9^.  The  diag  cm  s  «. 

has  been  lengthened  to  ct;  while  the  diag  "     *•  TI     -m  c     5" 

e  n  has  been  shortened  to  es.     Now,  if  there 


points,  it  would  have  divided  the  whole  truss 
into  triangles  ;  and  then  the  diag  c  m  could 
not  have  become  lengthened  to  c  t  by  any 
strain  less  than  one  sufficient  to  break  this 
iron  bar  by  pulling  it  apart;  therefore  the  truss  would  have  remained  safe,  and  unchanged  in  figure ; 
for  the  bar,  while  preventing  cm  from  lengthening  to  ct,  would,  as  a  consequence,  prevent  en  from 
shortening  to  es.  Or,  omitting  the  iron  bar  at  cm,  suppose  a  stiff,  unbending  inclined  post  to 
be  inserted  between  e  and  n.  This  also  will  divide  the  whole  truss  into  triangles  ;  and  it  is  then 
plain  that  en  could  not  be  shortened  to  es  by  any  strain  less  than  one  sufficient  to  break  the  post  by 
crushing  it.  Therefore,  in  this  case  also,  the  truss  would  have  remained  safe,  and  unchanged  in 
figure  ;  for  the  post,  while  preventing  en  from  shortening  to  e  s,  would,  as  a  consequence,  prevent  en* 
from  lengthening  to  ct.  Either  the  bar  or  the  post  would  be  a  counterbrace  against  the  effect  of  un- 
equal loading.  With  a  uniform  load  it  is  not  needed.  Neither  are  additional  counterbraciug  pieces 
Deeded  in  bridge  trusses  of  the  forms  Figs  10, 11,  12,  13,  provided  each  web  member  is  so  constructed 
as  to  bear  alternately  compression  and  extension.  See  Rem  1,  foot  of  p  306. 


The  next  Fig  9  6,  shows  the  bad  effect  produced  in  a 
truss  longer  than  Fig  9^,  when  the  web  members  are 
not  so  constructed.  In  the  Burr  bridge,  Fig  36,  and  in, 
some  others,  although  the  truss  is  divided  into  trian- 
gles, yet  the  inclined  braces,  ic,  Ac,  are  often  impro- 
perly adapted  to  bear  compression  only ;  their  ends  not 
being  firmly  attached  to  the  chords.  Consequently, 
with  a  heavy  load  at  a,  the  derangement  shown  in  Fig 
9  b  (analogous  to  that  in  Fig  9^)  takes  place.  To  pre- 
vent it,  counterbracing  must  be  resorted  to,  either  by 
inserting  struts  or  ties  along  the  dotted  diagonals;  or 
by  making  the  braces  capable  of  resisting  tension  as 
well  as  compression.  The  last  method  shows  that  counterbracing  can  be  performed  without  the  ad- 
dition of  pieces  specially  called  counterbraces,  an  1  denoted  by  the  dotted  diagonals.  All  that  is  re- 
quired in  the  principle  of  counter  bracing,  is  to  so  arrange  and  connect  the  several  web  members, 
that  the  strain  produced  by  unequal  loading  at  any  point,  as  a,  between  the  abuts  ;  or  along  any 
portion,  of  that  distance,  shall  be  properly  transferred  by  them  to  both  abutments. 

Art.  11.    The  strains  in  snch  trnsses  as  Fisp-i  1O  and  11,  p  254, 

may  be  found  by  three  very  simple  processes  when  the  truss  and  its  load  are  uniform 
from  the  center  each  way.*  When  this  is  the  case  it  is  usual  and  safe  to  assume  that 
the  half  load  e  p.  Fig  10  or  11,  on  the  right  hand  of  the  center  e,  rests  on  the  right 
hand  support^  ;  and  that  the  half  load  e.  a  on  the  left  hand  of  the  center  e,  rests  on 
the  left  hand  support  a.f  It  is  often  assumed  also,  for  simplifying  the  calculations, 
that  the  entire  weight  of  the  truss  and  its  load  is  distributed  along  one  chord  only. 
This  is  plainly  incorrect ;  but  inasmuch  as  the  exfranenus  load  (such  as  the  covering 
of  slate,  snow,  etc.,  on  a  roof,  and  the  travelling  load  on  a  bridge)  in  many  cases  ac- 
tually does  rest  on  one  chord  only,  and  is  great  in  comparison  with  the  weight  of  the 
truss  alone,  the  error  arising  from  the  assumption  in  such  cases  is  not  of  practical 
importance. 

But  in  bridges  of  great  span  the  weight  of  the  truss  may  bear  a  large  proportion 
to  that  of  its  load  ;  or  there  may  be  an  upper  and  a  lower  roadway,  one  resting  on 
each  chord;  and  a  roof  truss  may  have  to  bear  not  only  the  covering,  snow,  &c.,  on 
its  upper  chord  or  rafters:  but  a  floor  with  a  plastered  ceiling  beneath  it,  and  all 
the  load  incident  to  any  ordinary  room,  on  its  lower  chord.  In  such  cases  the  entire 
weight  of  the  truss  and  load  must  be  properly  distributed  along  both  chords  before 
we  can  correctly  find  the  strains.  But  this  will  in  no  way  affect  the  principle  of  the 
three  processes  which  we  are  about  to  explain,  and  as  we  proceed  we  shall  give  di- 
rections for  both  cases. 

*  It  is  not  necessary  that  the  entire  load  should  in  itself  be  uniform ;  but  merely 
uniform  each  way  from  the  center.  Thus  at  e  may  be  say  1  ton  ;  at  a  and  m  each  say 
5  tons ;  at  c  and  n  each  2  tons,  &c. 

t  This  assumption  is  untrue,  and  opposed  to  the  unvarying  law  that 
every  individual  portion  of  the  entire  weight  rests  partly  on  each  support.  Thus, 
one  portion  of  the  load  at  o  rests  partly  on  p  and  partly  on  a;  and  so  with  every 
other  portion  ;  and  on  this  fact  depends  the  difference  in  the  methods  of  calculating 
the  strains  from  uniform,  and  ununiform  or  moving  loads.  When,  however,  the 
weight  of  the  truss  and  load  is  uniform  each  way  from  the  center  we  obtain  correct 
results,  and  more  readily  by  adopting  the  erroneous  assumption.  When  not  thus 
uniform,  the  parallelogram  of  forces  is  not  applicable  to  a  simple  uncounterbraced 
truss. 


254 


TRUSSES. 


TRUSSES.  255 

Beginning  then  with  uniformly  distributed  weights  of 
truss  and  load,  and  assuming  all  of  said  weights  to  rest  on  the  long  chord  a  p, 
prepare  a  correct  skeleton  diagram  of  the  truss  (or  at  least  of  one-half  of  it),  such  as 
Figs  10  and  11,  in  which  the  height  or  depth  e  t,  Fig  10,  is  the  vert  distance  between 
the  centers  of  the  depths  of  the  two  horizontal  chords.  A  scale  of  from  /^  to  J^  of 
an  inch  to  a  foot  will  generally  be  large  enough. 

Then  the  first  process  is  the  very  easy  one  of  ascertaining  how  much  of  the 
total  uniform  weight  is  to  be  considered  as  sustained  at  each  point  of  support  along 
either  the  top  or  the  bottom  chord,  as  the  case  may  be;  remembering  that  one  half 
of  each  end  panel  is  sustained  directly  by  the  abut  nearest  to  it,  as  in  the  preceding 
eases. 

In  order  more  fully  to  illustrate  the  following  Articles,  we  shall  assume  each  of 
the  trusses,  to  Fig  22  inclusive,  to  be  64  ft  long,  and  16  ft  high.  Each  truss  is  as- 
sumed to  be  divided  into  8  equal  panels.  Total  uniform  wt  of  one  truss  and  its  load, 
32  tons ;  or  4  tons  to  a  panel.  Consequently  there  will  be  9  points  of  support  to  each 
truss.  Thus,  in  Figs  14, 15,  and  16,  in  which  the  load  is  supposed  to  rest  on  top  of 
the  truss,  and  in  Figs  10  and  11,  in  which  it  rests  upon  the  bottom,  the  points  of 
support  are  at  a,  b,  c,  d,  e,  m,  n,  o,  p.  Some  of  these  are  not  shown  in  the  first  three 
Figs.  If  both  chords  are  loaded,  there  will  be  points  of  support  in  the  short  one 
also.  Thus,  in  Fig  10  there  will  be  7,  and  in  Fig  11  there  will  be  8  of  them.  Now, 
in  Figs  10  and  11,  w,  x,  y,  etc.,  being  midway  between  the  points  of  support,  it  is 
plain  that  (assuming  all  the  weight  to  be  on  the  lower  chord)  the  point  o  must  sus- 
tain that  portion  of  it  comprised  between  w  and  x;  nail  between  y  and  x ;  while 
the  abut  p  sustains  directly  the  portion  from  w  top.  The  same  principle  applies  to 
all  the  other  trusses ;  and  equally  so  whether  the  panels  be  of  the  same  width  or 
not ;  each  point  of  support  is  assumed  to  sustain  all  the  uniform  wt  of  truss  and  load 
between  itself  and  the  two  points  midway  to  the  adjacent  points  of  support,  how- 
ever unequal  the  two  distances  may  be.  In  our  Figs  10  to  16,  the  strong  dotted  lines 
of  the  web  members  represent  ties ;  the  full  lines,  struts.  The  dots  intimate  that 
chains  may  serve  as  ties.  When  the  panels  are  of  equal  length,  p  o,  o  w,  etc.,  the  dis- 
tance from  p  to  w  will  be  but  half  a  panel ;  consequently  then  each  abut  will  sustain 
but  half  as  much  wt  as  each  other  point.  Therefore,  to  find  the  amount  of  wt  sus- 
tained at  each  of  the  nine  points  of  support,  we  have  only  to  div  the  total  wt  (32 

00 

tons)  by  a  number  less  by  1  than  the  number  of  points.    The  quot  —  =  4  tons,  will 

8 

be  a  full  panel-load,  to  be  at  each  point,  except  the  two  end  ones,  a  and  p,  at  the 
abuts ;  at  each  of  which  it  will  be  but  half  of  one  of  the  full  panel-loads,  or  two 
tons.*  The  amounts  of  these  panel  and  half-panel  loads  should  at  once  be  figured  on 
the  sketch  at  their  proper  points,  as  is  done  in  our  Figs ;  a  2  being  placed  at  each  end 
of  the  truss ;  and  a  4  at  the  other  points.  Each  of  these  panel-loads  of  course  causes 
a  vert  strain  equal  to  itself  where  it  rests.  As  the  strains  on  one  half  of  the  truss 
are  the  same  as  those  on  the  other  half,  the  numbers  need  only  be  written  on  one  of 
them ;  indeed,  the  sketch,  as  a  general  rule,  need  show  but  one-half  of  the  truss. 

If  there  is  a  load  on  the  other  chord  also,  it  must  be  in  the  same 
way  divided  among  the  points  of  support  of  that  chord,  and  be  figured  as  before. 

The  second  process.  All  the  panel-loads  are  of  course  eventually  trans- 
mitted through  the  truss  to  the  abuts ;  as  is  manifest  from  the  fact  that  each  abut 
sustains  half  the  total  load.  But  each  panel-load,  while  travelling,  as  it  were,  up 
and  down  alternate  web  members  from  its  original  point  of  support,  to  the  nearest 
abut,  places,  so  to  speak,  an  additional  load,  or  more  correctly  produces  an  addi- 
tional vert  strain  equal  to  itself,  at  every  intervening  point  of  support  in  each 
chord.f  Our  second  process  consists  in  finding  the  amount  of  this  additional  vert 
ttrain  at  each  point  of  support. 

*  This  of  course  is  only  when  the  end  panels  are  of  the  same  length  as  the  others. 
When  not  so,  the  loads  at  the  points  of  support  and  on  the  abutments  will  plainly 
vary  from  the  above. 

f  The  routes  taken  by  the  panel-load  strains  in  Figs  10  and  11, 
lead  at  once  toward  the  nearest  abut ;  but  in  Figs  14, 15,  and  16,  they  first  go  forward 
to  the  center  e  of  the  truss ;  and  from  thence  are  transmitted  backward  to  the  abuts, 
along  the  inclined  upper  member  e  a,  Figs  14  and  16 ;  or  the  inclined  lower  one  i  a, 
of  Fig  15.  In  Fig  16,  a  peculiar  force  is  generated  by  the  raising  of  the  center  of 
the  tie-bar  a  in;  this  case  will  be  considered  after  the  others.  It  is  very  easy  to  de- 
termine, at  a  glance,  whether  the  panel-load  strains  travel  at  once  toward  the  abuts, 
or  toward  the  center  of  the  truss;  and  it  is  essential  that  this  be  first  done.  We 
have  only  to  assume  or  suppose,  for  the  moment,  that  a  tie  can,  like  a  chain,  sustain 

17 


256  TBUSSES. 

lu  Figs  10  and  11,  with  parotid  horizontal  upper  and  lower  chords,  the  vert  strains 
are  very  easily  found,  thus  :  Remembering  that  only  half  of  the  center  panel-load 
strain  at  e  goes  to  each  abut,  begin  with  the  4  tons  at  e. 

In  Fig.  10  these  4  tons  first  go  up  the  tie  ei  to  i,  where  they  produce  a  vert  strain 
of  4  tons,  which  figure  as  in  the  diagram.  But  at  i  these  4  tons  separate  ;  2  of  them 
going  to  the  abut  p,  and  the  other  2  to  the  abut  a.  The  last  2  first  pass  down  i  d  to 
5,  where  also  they  produce  a  vert  strain  of  2  tons,  which  also  figure,  as  must  be  done 
with  all  that  follow.  At  d  these  2  tons  unite  with,  or  as  it  were  take  up  and  carry 
along  with  them  the  4  tons  already  there  ;  and  the  entire  6  tons  go  up  the  tie  d  j  toj>', 
where  they  produce  a  vert  strain  of  6  tons.  From  j  these  6  tons  go  down  the  strut 
jc  to  c,  where  they  also  produce  a  vert  strain  of  6  tons.  At  c  these  6  tons  take  up 
the  4  tons  already  there,  and  the  entire  10  tons  go  up  ck  to  k;  and  thus  the  process 
continues  until  14  tons  find  their  way  to  the  abut  a,  where  they  meet  the  2  tons  of 
load  already  there  ;  thus  making  16  tons,  or  one-half  the  wt  of  the  truss  and  its  load; 
which  is  a  proof  that  our  work  is  correct  so  far. 

In  Fig  11,  the  4  tons  at  the  center  e  separate  there  ;  2  of  them  going  up  e  i  to  t',  and 
thence  to  the  abut  a  as  before. 

In  either  Fig  if  there  is  a  uniform  load  on  each  chord  there  is  no 
difference  in  the  second  process  ;  for  after  having  by  the  "  first  process  "  divided  each 
load  among  the  points  of  support  of  its  own  chord,  the  portion  at  each  point  must 
be  taken  up  as  it  occurs,  and  carried  on  with  the  others  to  the  abutment  as  before. 

The  third  and  last  process  consists  in  completing  our  sketch,  or  diagram, 
in  such  a  manner  as  to  enable  us  to  measure  by  scale  the  strains  produced  along 
every  member  of  the  truss,  by  these  vert  strains  thus  accumulated  at  the  difif  points 
of  support,  a,  6,  c,  I,  kt  etc. 

To  do  this  in  Fiy  1O  (that  is,  whenever  the  web  members  are  alternately 
vertical  and  oblique)  from  each  point  of  support  of  one  chord  only,  beginning  at  the 
center  apex  i,  draw  a  vert  line  as  ivtjv,  etc.,  to  represent  by  any  convenient  scale, 
the  vert  strain  figured  at  said  point;  except  at  the  center  one  t,  where  the  vert  line 
must  represent  only  half  the  vert  strain,  inasmuch  as  that  is  all  that  goes  to  each 
abut.  Draw  also  the  hor  lines  vu,vu,  etc.  Then  will  each  oblique  line  I'M,./  it,  etc., 
give  by  the  same  scale  the  strain  (2.2,  6.7,  11.2,  15.7)  along  its  own  oblique  web  mem- 
ber, as  figured.  The  hor  Hues  will  give  the  hor  strains  on  the  chords  at  both  ends 
of  each  oblique.  We  have  figured  all  these,  strains  (7,  5,  3,  1)  at  the  head  and  foot  of 
each  oblique.  Each  of  these  hor  strains  extends  from  the  ends  of  the  oblique,  to 
the  center  of  the  chord;  therefore  the  end  stretches  of  the  chords  bear  7  tons  hor 
strain  ;  the  next  ones  7  +  5  =  12  tons  ;  the  next  ones  7  +  5  +  3  =  15  tons  ;  and  the 
center  one  7-f5-f-3-fl  =  16  tons;  all  of  which  are  figured  along  the  chords.* 

We  have  said  that  the  vert,  hor,  and  oblique  sides  of  the  triangles  give  the  strains, 
but  it  would  be  more  correct  to  say  that  each  of  them  gives  a  force,  which  being 
balanced  by  the  other  two.  thereby  causes  a  strain  equal  to  itself,  instead  of  mo- 
tion. See  "  Strains,"  p  444. 


The  hor  strains  at  the  centers  of  the  two  chords  trill  be 
equal  in  both  Figs  10  and  11,  whether  one  or  both  chords  be  uniformly  loaded; 
or  if  the  truss  be  inverted  ;  with  only  the  exception  in  the  foot  note.* 

no  pressure;  and  a  strut  no  pull  ;  then,  by  looking  at  any  point  of  support,  as  at  c, 
Fig  10,  we  see  at  once  that  its  strain,  being  a  pull,  cannot  travel  toward  the  center, 
along  the  strut  cj,  but  must  go  toward  the  abut  a,  through  the  tie  c  k.  Where- 
as in  Fig  14  the  pressing  strain  at  c  cannot  go  toward  the  abut  through  the  tie  c  A-,  but 
only  toward  the  center,  through  the  strut  cj.  When,  as  in  Figs  10  and  11,  the  strains 
travel  at  once  toward  the  nearest  abut,  the  web  members  are  slightest  at  the  center 
of  the  truss,  and  stoutest  near  the  abuts  ;  while  the  chords  are  stoutest  at  the  center, 
and  slightest  at  the  abuts.  But  when  the  strains  first  move  toward  the  center  of 
the  truss,  all  this  is  reversed  ;  as  indicated  by  the  diff  thicknesses  of  the  lines  in 
the  Figs. 

*  When  at  the  central  apex  t,  Fig  10,  the  two  ends  of  the  obliques  td,  ira,  which 
meet  there,  are  so  arranged  as  to  butt  tight  against  each  other,  then  the  center  hor 
strain  of  1  ton  at  that  point  is  not  borne  by  the  chords,  but  by  the  obliques  them- 
selves; so  that  there  will  then  be  that  much  less  strain  at  the  center  point  of  that 
chord  than  along  the  center  stretch  d  m  of  the  other  chord.  But  if  instead  of  this, 
they  abut  against  the  chord,  at  some  little  distance  from  each  other,  then  the  chord 
also  receives  the  strain;  so  that  the  hor  strains  at  the  centers  of  the  two  chords 
become  equal,  as  we  assume  to  be  the  case  in  all  our  Figs  of  uniform  trusses  uni- 
formly loaded  each  way  from  the  center.  The  same  remark  applies  to  the  hor  strain 
of  .5  of  a  tou  at  the  center  apex  e  of  Fig  11. 


TRUSSES.  257 

The  strain  along  the  center  vert  tie  e  i  of  Fig  10,  will  be  equal  to  the  4  tons  at  e ; 
and  when  the  entire  wt  is  assumed  to  be  on  the  Jong  chord,  the  vert  lines  at  the  other 
points  of  support  will  give  the  vert  pulling  strains  on  the  other  verts,  as  6,  10,  14. 

But  with  loads  upon  both  chords  this  last  will  not  be  the  case;  but 
the  strain  on  each  vert  tie  will  then  be  equal  to  the  vert  strain  at  its  foot.* 

If  the  loaded  truss  is  inverted,  the  verts  become  struts  or  posts,  and 
the  obliques  ties  ;  also  the  strain  on  each  vert  is  then  the  one  figured  at  its  top  ;  but 
the  amount  of  strain  on  each  part  of  the  entire  truss  will  remain  as  before. 

In  Fig  10  all  the  uniform  wt  is  on  the  long  chord,  and  the  resulting  strains  are 
all  figured  on  the  diagram.  We  add  the  strains  that  would  occur  in  case  there  were 
an  additional  uniform  load  of  6  tons  on  the  upper  chord  from  I  to  t.  This  would  give 
1  ton  at  each  point  of  support  along  that  chord,  except  the  two  end  ones  I  and  f,  at 
each  of  which  it  would  be  but  .5  of  a  ton.  All  these  must  be  figured  on  the  short 
chord  of  the  diagram,  as  were  4,  4,  Ac,  on  the  long  one.  The  student  may  then 
work  out  the  case  for  himself.  We  repeat  that  uniform  trusses  and  loads  require  no 
counter-bracing. 

For  Fig:  1O,  but  for  a  load  on  each  chord. 


e  i  =   4.    tons. 
dj  =   6.5    " 
c  k  =  11.5    " 
b  I  =  16.5     " 


i  d  =    2.8    tons. 
jc  =    8.39 

k  b  =  13.98 
I  a  =  19.01 


a  6  or    I  k  =    8.5    tons. 
6  cor   fc.;  =  14.75    " 
c  d  or   j  i  =  18.50    " 
d  e  or  at  i  =  19.75    " 


For  the  '*  third  process"  in  Fig  11  (or  when  all  the  web  members  are 
oblique,  whether  equally  so  or  not)  after  having  found  and  figured  the  loads  and  vert 
strains  at  each  point  of  support  precisely  as  directed  for  Fig  10,  then  from  every 
such  point  in  both  chords  draw  a  vert  line  as  ev,  i  u,  d  r,  &c  ;  and  on  it  lay  off  sepa- 
rately by  scale  both  the  vert  strain  that  comes  to  that  point  through  a  web  member 
from  towards  the  center  of  the  truss ;  and  the  one  that  yoesfrom  it  through  another 
web  member  towards  the  abut ;  except  that  at  the  very  starting  point  e,  Fig  11,  there 
is  but  one  vert  strain  (the  one  of  2  tons  going  from  it) ;  and  at  the  very  end  a  also 
there  is  but  one,  namely  that  of  14  tons  coming  to  it  along  the  oblique  /  a  ;  for  the 
2  tons  at  a  are  not  to  be  included,  because  they  do  not  roach  a  by  means  of  a  web 
member.  Therefore  both  at  e  and  at  a  only  a  single  vert  strain  is  to  be  laid  off.f 

*  It  is  so  in  both  cases,  for  in  any  of  these  trusses  under  stationary  loads 
the  strain  along  a  web  tie  whether  vert  or  oblique  may  be  considered  to  commence 
at  its  lower  end,  that  being  the  end  at  which  the  panel  loads  first  act  on  their  route 
to  the  abut, and  up  which  they  as  it  were  work  their  way.  But  under  moving  loads 
the  same  member  may  have  to  act  both  as  a  tie  and  a  strut ;  hence  the  remark 
will  not  apply  to  such.  Referring  to  what  is  said  above,  when  the  entire  wt  is 
on  only  one  chord,  tiie  vert  strains  on  the  two  chords  are  equal  at  any  tie  in  Fig 
10.  Hence  a  line  drawn  to  represent  the  upper  one,  may  be  assumed  to  repre- 
sent also  the  lower  one.  But  when  both  chords  are  loaded,  the  vert  strains 
figured  at  the  two  ends  of  any  tie  are  unequal,  and  we  must  then  have  regard 
to  the  true  principle.  If  the  verts  should  be  struts  or  posts  (as 
if  Fig  10  should  be  inverted)  then  any  strain  along  them  must  be  received  from 
their  tops,  or  the  reverse  of  the  case  with  ties.  It  will  aid  the  student  very 
much  in  what  follows  to  familiarize  himself  with  the  idea  that  strains  pass 
only  down  the  struts,  and  up  the  ties.  Also  that  vert  web 
members  cause  no  nor  strain  along;  either  hor  chord ; 
whereas  the  two  ends  of  any  one  oblique  web  member  always 
cause  equal  strains  on  each  of  two  parallel  chords;  and  one  of 
these  strains  is  a  push  in  one  direction,  while  the  other  is  a  pull  in  the  opposite  di- 
rection. If  the  chords  are  not  hor,  the  strains  on  them  from  both  verts  and  obliques 
will  all  be  more  or  less  oblique. 

f  When  all  the  wt  of  truss  and  load  is  assumed  to  be  on  the  long  chord  as  in  our 
Fig  11,  then  the  vert  strain  that  comes  to  any  point  in  the  short  chord  by  one  web 
member  is  plainly  the  same  in  amount  as  that  which  goes  from  it  by  the  other  web 
member;  and  hence  only  one  vert  measurement  need  be  laid  off  for  it,  as  is  seen  at 
i  v,j  v,  k  v,  I  v,  in  the  Fig.  But  at  the  long  chord  (or  at  both  chords  when  there  is 
a  load  on  both)  the  vert  strain  that  conies  to  any  point  is  less  than  the  one  that  goes 
from  it  towards  the  abut,  and  is  evidently  the  one  last  figured  at  that  point,  as  the 
2,  6,  10,  &c,  tons  at  d,  c,  b,  &c,  in  Fig  11 ;  while  the  one  that  goes  from  that  point 
towards  the  abut  is  as  evidently  equal  to  the  sum  of  the  two  strains  figured  at  that 
point,  as  the  6, 10, 14,  Ac,  tons  at  d,  c,  6,  &c.  When  both  chords  are  loaded  there  will 
be  two  vert  strains  to  be  figured  at  each  point  of  support  in  both  chords,  except  at 
the  very  starting  point,  where  will  be  but  one  in  any  case. 


258  TRUSSES. 

Draw  also  the  hor  lines,  thus  forming  a  series  of  triangles  (as  tvu,  ivu^jvu^jvu^ 
&c,  of  the  upper  chord ;  and  a  v  u,  b  v  M,  6  z  w,  <fcc,  of  the  lower  chord),  each  with 
one  vert,  one  hor,  and  one  oblique  side.  Tbeii  the  hor  strain  on  either 
Chord  at  any  point  of  support  will  be  measured  by  the  two  hor  lines  directly 
opposite  to  said  point,  except  at  the  center  e,  and  at  the  end  a,  at  each  of  which 
it  will  be  measured  by  the  single  hor  line  v  u,  or  v  w,  opposite  each.*  These  hor 
strains  (7,  5,  3,  1  tons  on  the  upper  chord ;  and  3.5,  6, 4,  2,  .6  on  the  lower  chord)  are 
figured  close  to  the  points  of  support  at  which  they  occur;  and  the  fotaZhor  strains 
on  the  several  stretches  of  the  chords  are  figured  midway  of  said  stretches.f 

The  strain  along:  any  oblique  asj  c,  will  be  measured  by  the  oblique 
side.;'  w,  or  c  w,  of  either  one  of  the  two  triangles  on  either  side  of  it;  and  this  will 
be  the  case  whether  one  or  both  chords  be  loaded;  or  if  the  truss  and  load  be  in- 
verted. All  the  strains  in  Fig  11  (loaded  on  the  long  chord  only)  being  figured  on 
the  diagram,  we  give  below  the  strains  that  would  occur  in  case  there  were  an  ad- 
ditional uniform  load  of  7  tons  on  the  short  chord  from  I  to  t ;  which  would  give  1 
ton  at  each  point  of  support  along  that  chord,  except  the  two  end  ones  I  and  £,  at 
each  of  which  it  would  be  but  .5  of  a  ton.  All  these  must  first  be  figured  on  the 
short  chord  of  the  diagram,  as  were  4, 4,  &c,  on  the  long  one.  The  student  may 
then  work  out  the  case  for  himself. 


For  Fig:  11,  but  for  a  load  on  each  chord. 


«  t  =  2.06  tons. 
t  d  =  3.09    " 


c  k  =  12.36  tons. 
k  b  =  13.40    " 
b  I  =  17.53    " 
I  a  =  18.04    " 


ab  =    4.38  tons. 
bc  =  11.88    " 
c  d  =  16.88    " 
d  e  =  19.38     " 
at  e  =  19.88    " 


I  k  —    8.63  tons. 
kj  =  14.88    " 
j  i  =  18.63    " 
t  to  center  =  19.88    " 


Art  12.  The  strains  in  Figs  1O  and  11  may  readily  be  calcu- 
lated (after  having  by  the  "first  and  second  processes"  found  the  vert  strains  at 
all  the  points  of  support)  whether  one  or  both  chords  be  loaded,  or  if  the  truss  and 
load  be  inverted.  Thus,  divide  the  hor  stretch  of  an  oblique  by  its  vert  stretch ;  the 
quotient  will  be  the  natural  tangent  (.5  for  Fig  10,  and  .25  for  Fig  11)  of  the 
angle  (26°  34'  in  Fig  10,  and  14°  2'  in  Fig  11)  which  the  oblique  forms  with  a  vert 
line.  Divide  the  actual  length  of  an  oblique  by  its  vert  stretch;  the  quotient  will 
be  the  nat  secant  (1.12  for  Fig  10,  and  1.03  for  Fig  11)  of  the  same  angle.  Then 
the  strain  along;  any  oblique  in  Fig  10  or  11,  is  found  by  multiplying  the 
vert  strain  that  travels  towards  the  abutment  along  said  oblique,  by  the  nat  secant. 

The  hor  strain  on  either  chord,  caused  by  either  end  of  any  oblique, 
is  in  Fig  10  equal  to  the  vert  strain  that  travels  along  said  oblique  towards  the  abut- 
ment, mult  by  nat  tangent.  And  in  Fig  11,  it  is  equal  to  the  vert  strain  that  comes 
to,  added  to  that  which  goes  from  any  end  of  an  oblique,  mult  by  nat  tang;  except 
at  the  center  and  end,  where  the  single  vert  strain  must  be  mult  by  nat  tang. 

All  this  is  simply  because  that  if  we  assume  the  vert  side  of  any  one  of  the  tri- 
angles to  be  radius  or  1,  then  the  hor  side  becomes  by  that  same  scale  the  nat  tang 
of  the  angle  which  it  subtends ;  while  the  oblique  side  becomes  the  nat  secant  of 
the  same  angle. 

The  principle  of  the  mode  of  finding:  the  strains  in  Figs  10  and 
11  is  this.  We  know  that  if  three  forces  are  in  equilibrium  with  each  other  at  any 
point,  the  lines  which  represent  them  will  form  a  triangle.  Now  at  every  point  of 

*  Each  ef  the  upper  hor  lines  u  «,  u  M,  &c,  in  Fig  11  is  to  be  considered  as  com- 
posed of  two  separate  ones  v  u,  v  M,  &c  ;  the  right  hand  one  of  which  measures  the 
hor  strain  caused  by  the  vert  strain  that  comes  to  i,./,  Ac;  while  the  left  hand  one 
measures  the  hor  strain  caused  by  the  vert  one  that  goes  from  -ij,  &c,  towards  the 
abutment.  Such  lines  as  u  tt,  u  M,  &c,  occur  only  when  all  the  wt  is  on  one  chord; 
for  when  both  chords  are  loaded,  the  vert  strain  that  comes  to,  and  that  which  goes 
from  any  point  of  support  differ,  therefore  requiring  two  unequal  vert  measurements, 
and  two  unequal  hor  lines  at  each  point  of  support  of  both  chords,  except  at  the 
center,  and  at  the  ends ;  at  each  of  which  will  be  but  one. 

_    Total  uniform  wt  X  span 
t  The  common  rule        s  ,...„.,,,„.,. ,.,.,,.,        =  b"  <*™->  *  center 

of  either  hor  chord,  is  not  strictly  correct  except  when  both  chords  extend  the  full 
length  of  the  span,  and  are  both  loaded  throughout  their  entire  length ;  or  in  the 
impossible  case  of  the  entire  wt  being  on  the  long  chord.  Still  in  ordinary  cases  it 
is  a  sufficiently  close  approximation.  On  this  subject  see  Art  19.  p  272. 


TRUSSES. 


259 


support  in  Fig  10  we  have  one  set  of  three  such  forces  ;  and  in  Pig  11,  two  sets.  In 
Fig  10  it  was  not  necessary  to  show  those  at  the  long  chord.  Now  each  set,  or  each 
triangle  represents  a  vert  force,  a  bor  one,  and  an  oblique  one,  keeping  each  other 
in  equilibrium  at  the  point  of  support.  It  is  true  that  there  are  other  forces  acting 
at  the  same  point,  but  as  they  hold  each  other  in  equilibrium,  they  do  not  interfere 
with  the  first  ones.  Thus,  both  the  7  and  the  12  tons  hor  forces  along  the  chord  at 
k  are  balanced  or  held  in  equilibrium  by  the  equal  ones  from  t  and  s,  on  the  other 
half  of  the  truss;  without  disturbing  the  forces  represented  by  the  sides  of  the  tri- 
angles. Hence  by  measuring  those  sides  we  obtain  the  forces  and  strains  themselves. 


The  same  principle  either  by  diagram  or  by  calculation 
applies  to  Figrs  12  to  13%,  when 
uniform  and  uniformly  loaded.  In  those 
Figs  all  the  weight  is  here  assumed  to  be 
on  the  long  chord;  (but  after  what  we 
have  said,  no  difficulty  can  arise  when 
placing  loads  on  the  other  chord  also.) 
All  said  wt  is  first  to  be  properly  dis- 
tributed among  the  points  of  support  on 
said  long  chord,  and  there  figured,  as 
shown  by  the  upper  4s  and  2s  along  that 
chord  in  the  Figs.  This  being  done,  we 
have  figured  all  the  other  vert  strains, 
thus  providing  the  data  for  drawing  the 
vert  sides  of  the  triangles ;  and  these  in 
turn  give  us  the  hor  and  oblique  sides 
which  measure  the  corresponding  strains, 
and  all  of  which  are  drawn  on  the  Figs. 

All  these  trusses  being  uniform  and 
uniformly  loaded  from  the  center  each 
way  require  no  counter  bracing.  Bear  in 
mind  that  the  vert  strains  that  accumu- 
late at  an  abut  must  equal  half  the  wt  of 
the  truss  and  its  load. 

In  Fig:  12,  with  no  oblique  at  the 
center,  the  4  tons  at  a  having  no  oblique 
in  contact  on  either  side  of  them,  go  to  b  ; 
and  on  their  way  to  b  strain  ab  4  tons. 
From  b  all  4  go  along  the  web  members 
to  the  nearest  abut  e  as  figured. 

In  Figs  12%  and  13  the  web  mem- 
bers of  each  are  to  be  regarded  in  some 
dogree  as  belonging  to  two  separate 
trusses,  namely  abode  and  m  n  op  r  in 
Fig  12%;  and  a  b  c  d  e  and  m  n  o  d  e  in 
Fig  13;  and  the  vert  strains  at  their  ends 
are  to  be  found  on  that  assumption,  as 
figured.  In  Fig  13,  o  d  is  a  vertical  tie. 

In  Fig:  13*4  there  being  at  none  of 
the  4  ton  loads  on  the  long  chord  an  ob- 
lique in  contact  with  them  on  either  side, 
they  (like  that  at  e  Fig  10,  or  at  a  Fig  12) 
pass  each  by  itself  vertically  to  the  upper 
chord,  where  figure  them.  Of  those  at  6, 
2  go  to  the  abut  ?.  by  way  of  the  oblique 
b  e  ;  but  the  other  two  4s  nil  go  to  e,  each 
by  its  own  oblique.  Each  of  the  three 
hor  lines  gives  the  strain  produced  along 
the  upper  chord  by  an  oblique ;  but  the 
hor  strain  along  the  lower  chord  is  uni- 
form from  end  to  end,  because  all  the 
forces  that  produce  it  act  at  its  ends  only, 
hor  lines. 


C  4r       <2>4 
14  6 

It  is  equal  to  the  sum  of  the  three 


260 


TRUSSES. 


In  Fig:  13%.  ftt  the 4  ton  loads  at  c  and  e,  there  is  no  oblique  In  contact  with 
them  on  either  side;  therefore  they  pass  at  once  vertically  to  t  and  */,  where  figure 
them.  Then  begin  with  '2  of  the  4  tons  at  a. 

All  loads  that  have  no  oblique  in  contact  oil  either  side,  whether  sustained 
by  vert  ties  or  by  vert  posts,  are  to  be  thus  transferred  at  Olice  to  the  opposite 
chord  in  order  to  meet  an  oblique  along  which  to  travel  to  the  nearest  abut;  and 
said  vert  ties  or  posts  will  be  strained  to  the  amounts  of  said  loads. 

REM.  We  call  attention  to  the  fact,  not  sufficiently  known,  that  this  paral- 
lelogram of  forces  does  not  always  indicate  the  true  strains. 

Thus  in  Fig  9,  p  251,  with  a  load  represented  by  vg,  at  z  only,  and  none  at  w  or  else- 
where, v  e  and  v  n  would  not  represent  the  strains  along  z  w  and  z  6.  Nor  in  Figs  10 
and  11,  if  loaded  on  only  half  the  truss,  would  the  lines  w?>,  uv.  iu,  ju,  etc.,  do  so. 
They  do  so  however  in  \miform  trusses  uniformly  loaded  from  the  center  each  way  ; 
also  in  cases  like  the  Figs  on  pages  461  to  465  where  the  strains  pass  directly  from 
the  loads  to  supports  which  are  in  themselves  rigid  and  fixed.  See  foot  note  p  252. 

Art.  13,  In  Figs  14,  15,  and  16,  (in  which,  as  before,  the  span  is  64  ft,  the  rise 
16  ft,  and  the  load  to  a  full  panel  4  tons,)  the  second  process  (or  that  of  finding 


y    t    y   K       J         -i 

bow  much  additional  vert  strain  is  produced  at  the  several  points  of  support  by  each 
panel-load,  on  its  way  to  the  nearest  abut)  differs  somewhat  from  that  pursued  with 
Figs  10  and  11,  which  have  parallel  upper  and  lower  members.  The  principle  in- 
volved, however,  is  precisely  the  same.*  In  Fig  14,  beginning  at  the.  panel-load  b, 
nearest  tin-  abut,  lay  off  by  scale  the  vert  b  r  equal  to  that  panel-load.  4  tons.  Draw 
r  It'"  parallel  to  the  rafter ;  and  h'"  s"'  hor.  Measure  b  s"',  (2  tons,;  and  carry  forward 
that  amount  to  c,  writing  it  down  over  the  panel-load  4 ;  thus  making  the  vert  strain 
ate  6  tons.  (See  Remark,  Art  15.)  Make  cr  equal  to  this  vert  strain;  draw  rh" 
parallel  to  the  rafter  ;  and  h"  s"  hor,  as  before.  Measure  cs",  (4  tons, )  and  carry  it 
forward  to  d,  writing  it  down,  and  thus  making  the  vert  strain  at  d  8  tons.  Make  d  r 
equal  to  this  vert  strain;  draw  r  h'  parallel  to  the  ratter;  and  //  sr  hor,  as  before. 
Measure  ds',  (6  tons,)  and  carry  it  forward  to  the  center  e,  writing  it  down.  Now  it 
is  plain  that  the  same  process,  on  the  other  half  of  the  truss,  would  bring  another  6 
tons  to  e.  Write  this  down  also,  as  in  Fig  14.  Thus  we  get  for  the  vert  strain  at  e., 
6  -f  4  +  6  =  16  tons,  or  one-half  of  the  wt  of  the  entire  truss  and  its  load.  As  this 
must  always  be  the  case  in  such  trusses,  it  proves  our  work  to  be  correct  so  far. 


Third  process 


ase  in  such  trusses,  it  proves  our  work  to  be  correct  so  lar. 
t.    From  e  draw  a  vert  line  e  r,  by  the  same  scale  as  before, 


*  When  wishing  merely  to  know  the  amount,  without  caring  to  trace  the  progress  of  these  vert 
strains  at  the  points  of  support,  in  order  to  calculate  the  other  strains,  we  may  omit  part  of  the  fol- 
lowing, and  proceed  thus:  At  the  peak  e  of  the  truss,  the  vert  strain  will  always  in  such  trusses  he 
equal  to  half  the  entire  weight  of  the  truss  and  its  load,  (for  which,  per  sq  ft.  see  Table  4,  p  301,) 
and  may  therefore  be  written  down  at  once  at  e.  Next  count  the  entire  number  of  points  of  support 
a,  6,  c,  d,  e.f,  &c,  of  the  truss.  From  this  number,  whatever  it  may  be,  subtrnct  1.  Div  the 
entire  weight  of  the  truss  and  its  load  by  the  rera.  One  half  of  the  quot  will  be  the  vert  strain  at  the 
foot  a  of  the  truss,  which  write  down  at  a.  At  b  the  strain  will  be  twice  that  at  a  ;  ot  c  three  times; 
at  d  4  times  ;  and  so  with  any  number  of  points  of  support  along  a  rafter,  except  at  the  center  one  e, 
where  the  strain  will  be  twice  as  great  as  at  the  preceding  one  d.  But  it  has  been  found  already. 
When  this  has  been  done,  the  vert  lines  b  r,  cr,  d  r,  er,  &c,  may  at  once  be  drawn  equal  by  scale 
to  the  several  vert  strains ;  then  rh,rh,rh,  &c,  parallel  to  the  rafter ;  and  t  h,  a  h,  Ac,  parallel  to 
the  tie.  Then  begin  at  "  Third  Process." 


TRUSSES. 


261 


equal  to  the  16  tons  of  total  load  at  «  ;  draw  r  h  parallel  to  the  rafter  ;  and  k  s  hor. 
The  fig  is  now  ready  for  giving  by  scale  the  strains  along  all  the  ditt'  members  of  the 
truss.  It  is  by  mere  chance  that  the  two  vert  lines  b  I  and  e  r,  representing  the  loads 
at  6  and  e,  happen  just  to  extend  to  the  hor  tie  a  i  in  our  fig.  As  with  the  preceding 
figs,  we  add  the  strains  in  this  case  also.  When  as  usual,  the  points  of  support  a,  6, 
c,  <fcc,  are  equidistant,  the  continuous  additions  for  the  strains  along  the  verts,  or 
along  the  several  divisions  of  a  rafter  or  main  tie,  will  be  equal  to  each  other  in  such 
roofs  ;  but  those  along  the  obliques  will  be  unequal,  on  account  of  their  varying 
obliquities. 


Strains  along:  the  obliques. 

Along  6  k  =  b  "h  —  4.47  tons. 
"     cj=e  K  =  5.66. 


At  i  =  s  h  —  16  tons  :  also  = 


Strains  along:  the  verticals. 

Along  b  I  =  nothing,  except  weight  of  tie-bar 
from  y  to  y. 

"    ck,  —  b  8  =  2  tons. 

"    d.;',:=c  «  =  4.    See  Bern,  p  375. 

«f=2rfs~2X6=:12;  for  while  each  other  vertical  tie  bears  only  the  vert  strain  brought  upon  it  by 
the  oblique  strut  next  nearer  the  abut;  the  center  tie  e  i,  of  coarse  bears  those  from  2  obliques; 
one  on  each  side  of  it. 

Strains  along:  the  horizontal  tie-bar  a  i. 

£  wt  of  truss  and  load  X  %  span 
height  of  truss. 

From  j  to  i  =  s  h  -f-  /  h  —  16  -f  4  =  20. 

«     k  toj  =  s  h  +  s  /i  +  7  A'=  16  -f  4  -f  4  =  24. 
"     atofc  =  s^-f-sA-fs    h  +  s     A=16-f4-|-4-f-4=«28. 
Strains  along:  the  rafter  e  a. 

From  e  to  d  =  hr  =  17.9  tons. 

"     d  to  c  =  A  r  -f  A'r  =  17.9  4-  4.47  =  22.4. 

"     c  to  b  =  h  r  -f  hr  +  7*'r  =  17.9  -f  4.47  -f-  4.47  =  26.8. 

«•     b  to  a  =  h  r  +  ft  r  -f  '/T  r  ='A 'r  =  17.9  +  4.47  +  4.47  +  4.47  =  31.3. 
It  will  be  observed  that  the  hor  components  h  s,  except  the  center  one,  have  equal 
lengths;  also  those  marked  h  r,  parallel  to  the  rafter;  while  the  oblique  ones  have 
not. 

For  a  span  of  1OO  feet,  rise  20  ft,  or  ^  of  the  span ;  trusses  10  feet  apart 
from  center  to  center;  loaded  on  top  only;  the  following  dimensions  will  be  amply 
kmfficient  for  a  covering  of  slate.  Rafters  and  tie-beam,  each  10"  X  12"  deep.  Th« 
rafters  may,  if  preferred,  be  reduced  to  .9  X  12  at  top.  The  verts  of  round  bar-iron 
of  good  quality,  if  not  upset  at  the  screw  ends,  ^£,  %,  1,  and  1%  ins  diam.  The 
obliques  or  braces,  5  X 10,  6  X 10, 8  X  10 ;  thus  making  the  truss-thickness  uniformly 
10".  See  Table  2,  p  299.  For  shorter  spans,  see  NOTE,  p  263. 


Art.  14.  In  Fig  15,  the  process  is  the  same  as  in  Fig  14,  except  that  the  vert  liner 
representing  the  strains  at  the  points  of  support  a,  6,  c,  d,  e,  are  to  be  drawn  upward 
from  I,  k,  j,  t;  and  the  strains  I  s"f,  k  s",  j  s',  are  to  be  carried  forward  to  the  next 


Fig.  15 


262 


TRUSSES. 


e  ons  a  must  stran  te  vert  post  to  tat  same  amount,  since  they 
ried  by  the  post  to  Z,  where  they  are  transferred  to  I  a  and  I  c.  But  in  Fig  14, 
ons  at  6,  rest  directly  upon  6  a  and  b  k,  and  produce  no  «train  whatever  upon 
rt  tie  6  /. 


Span  64  ft. 
Depth  e  t,  10  ft 


Art.  15.  In  Fig  16,  the  process  is  the  same  as  in  Fig  14,  except  that  the  lines  h  s, 
Ac,  must  be  drawn  and  measured  parallel  to  the  inclined  tie  a  i;  instead  of  being  hor. 
Also  &.s'"  must  be  carried  forward  to  c;  c  a"  to  d;  d  s*  to  e.  In  this  manner,  we  find, 
as  before,  the  vert  strain  of  6  4-  4  4-  6  =  16  tons  at  e.  But  we  must  now  add  to 
these  16  tons,  another  strain  generated  by  the  obliquity  of  the  tie-rod  a  i.  This  strain 
is  found  by  mult  the  one  at  e,  (16  tons,  or  half  the  wt  of  the  entire  truss  and  its 
load,)  by  the  vert  dist  n  iy  (6  ft,)  which  the  center  i  of  the  tie-rod  is  raised  above 
the  horizontal  a  u,  (see  Remark  1,  p  266 ;)  and  div  the  prod  by  fche  dist  t  e,  (10  ft,) 

That  is,  — — —  =  9.6  tons;  which  also  write  down  as  in  the  Fig;  making  the  total 

vert  strain  at  e  25.6  tons,  instead  of  the  16  tons  of  Fig  14.* 

ISIow,  make  e  r  by  scale,  =  25.6  tons  ;  draw  r  h  parallel  to  the  rafter  e  a,  and  meet- 
ing the  other  rafter ;  also  draw  h  s,  parallel  to  the  raised  tie-bar  i  a.  Then  the  strains 
along  the  members  of  the  truss  will  be  as  follows,  taken  from  a  Fig  on  a  larger  scale. 

Strains  along  the  verticals.        Strains  along  the  obliques. 

Along  the  one  at  6  =  nothing,  except  weight  of 
tie- bar  for  the  width  of 
one  panel  (8  ft.) 
Along  co  =  bt'"  =  2  tons. 
"      dz  —  cs"   =  4    " 
"      ei  =  2  d«' 4- 9.6  =  21.6. 


Along  60  —  bh'"  =6.43. 
•'  cz  —  ch"  —6.96. 
"  di  =  dh  =8.04. 


Strains  along  the  raised  tie-bar  a  i. 


From  z  to  i  —  h 


£  =  fts  =  26  tons. 


to  i  rr  ha  4-  h's'  =  26  4-  6.5  -  32.5. 

:oz  ~  ha  4-  ft'«'  4-  h"  a"  ~  26  4-  6.5  4-  6.5  =  39. 

too  rr  ha  -f-  A'*'  -f  A"«"  4-  A'"  «'"  =  '26  4-  6.5  -f  ( 


'26  4-  6.5  4-  6.5  4-  6.5  =  48.5. 


Strains  along  the  rafter  e  a. 

From  d  to  f,  —  hr  =  28.5  tons. 


"  c  t  >  d  =  hr  4-  h'  r  =  28.5  4-  7.13  =  35.63. 
"  bto  c  =  hr  4-  h'r  4-  h"  r  -  28.54-7.13 
"  ato&  =  Itr  +  A'r  +  /»"»*+ A'"  r  =  28.5 


4-7.13  =  42.76. 

4-  7.13  4-  7.13  4-  7.13  =  49.9. 


*  It  is  probable  that  the  tie-rod  is  sometimes  raised  in  this  manner  bj  persons  ignorant  of  the  fact 
that  they  thereby  greatly  increase  the  strains  on  the  rafters,  &c. 

All  the  strains  in  Figs  14,  15.  and  16  may  also  be  found  by  pre- 
cisely the  same  process  as  that  for  bowstring  and  crescent  trasses,  in  Art  19,  p  272. 


TRUSSES. 


268 


IV 17 


Firf/8 


RSM.  The  reason  for  measuring  only  parts  of  the  vert  lines  which,  in  Figs  14, 16, 
16,  represent  the  whole  panel-loads,  is  that  the  rafter  a«,  Figs  14  and  16;  or  the  tie 
"at  of  Fig  15,  being  inclined,  also  bear  a  part  of  each  panel-load  ;  and  since  that  part 
does  not  go  forward  to  the  next  point  of  support,  but  goes  backward,  along  said  in- 
clined member,  to  the  abut  at 
a,  it  must  be  omitted  in  the 
second  process.  Thus,  in  Fig 
17,  if  b  a  be  an  inclined  rafter 
resting  on  an  abut  a;  bg  a 
strut ;  and  br  a  vert  line  repre- 
senting the  load  sustained  at 
6  by  b  a  and  bg;  if  we  com- 
plete the  parallelogram  bmrn 
of  forces,  then  will  6m  give 
by  scale  the  strain  along  the 
JN  strut ;  and  b  n  that  along  the 
~  rafter.  The  strain  along  the 
2  strut  is  made  up  or  composed 
of  the  portion  6  s  of  vert  force ; 
and  the  hor  force  sm.  The 
vert  portion  bs  alone  goes  to  the  next  point  of  support;  while  s  m  strains  the  tie  ag 
hor.  So  also  the  strain  bn  is  made  up  of  the  other  portion  (bo  or  sr)  of  the  vert 
force  br;  and  of  the  hor  force  on;  which,  when  the  strain  bn  reaches  a,  become 
again  resolved  into  two;  one  of  which,  bo,  presses  vert  upon  the  abut;  or,  in  other 
words,  transfers  to  the  abut  the  portion  bo  of  the  load  resting  on  6;  while  the  por- 
tion on,  which  is  equal  to  sw,  strains  the  tie  ag  hor. 

But  in  Fig  18,  where  b  r  also  represents  a  load  resting  on  b,  and  supported  by  a 
strut  bg,  and  by  a  hor  chord  b  a,  if  we  complete  the  parallelogram  6  m  rn,  we  have 
the  strain  b  m  along  the  strut,  composed  of  all  the  vert  force  b  r,  and  the  hor  force 

NOTE. 

The  following  may  at  times  save  trouble  in  designing  roof 
trusses.  After  the  dimensions  of  all  the  members  of  a  roof  truss  of  any  span 
have  been  calculated,  then  those  of  any  smaller  span  arranged  in  the  same  manner,  and  having  the 
same  rise  in  proportion  to  its  span  :  but  with  the  trusses  at  the  same  distance  apart  as  in  the  large 
one;  may  be  found  tafely;  and  often  near  enough  for  practice,  thus  : 

Find  the  are*  of  cross  section  of  each  member  of  the  large  truss,  in  sq  ins.  Then  make  the  areas 
of  cross  section  of  the  corresponding  members  of  the  small  truss  in  the  same  proportion  as  its  span 
is  smaller  than  that  of  the  large  one.  The  small  truss  thus  obtained  will  in  fact  be  stronger  for  its 
span  than  the  large  one.  If  the  total  loads  sustained  by  trusses  of  different  spans,  including  the  weight* 
of  the  trusses  themselves,  were  in  proportion  to  the  spans,  then  this  method  would  be  correct.  But, 
with  the  trusses  at  the  same  distance  apart  in  both  oases,  only  the  extraneous  total  load  borne  by  them 
is  in  proportion  to  the  span  ;  while  the  total  weight  of  the  trusses  themselves  is  as  the  squares  of  the 
spans.  In  our  table  of  wts  of  iron  roofs  on  p  300,  it  will  be  seen  that  it  is  based  upon  total  loads  of 
40  Ibs  per  sq  ft  of  ground  covered,  including  the  wts  of  the  trusses  themselves.  Also  that  the  wt  of 
the  truss  itself  of  175  ft  span  is  8.0a  Ibs  per  sq  ft  of  ground  covered ;  so  that  the  greatest  extraneous 
load  of  purlins,  slate,  snow,  &c,  tor  this  span  is  40  —  8.05  =  31.95  Ibs.  If  now  we  proportion  a  roof 
of  35  ft  span  by  the  above  mode,  or  by  the  table  (which  is  based  on  that  mode),  we  find  that  inasmuch 
as  35  is  but  i  of  175,  the  short  truss  will  weigh  but  396  Ibs,  or  ivV  part  of  9379  Ibs,  the  tabular  wt  of 
the  175  ft  span.  We  also  see  that  the  35  ft  truss  will  weigh  but  1.61  Ibs  per  sq  ft  of  ground  covered; 
while  the  175  ft  one  weighs  8.05  Ibs.  or  6.44  Ibs  more.  Therefore  the  small  truss  will  be  as  safe  under 
an  extraneous  load  of  40  — 1.61  =  38.39  Ibs  per  sq  ft,  as  the  large  one  is  under  40—8.05=31.95  Ibs.  Or 
In  other  words,  if  the  large  one  is  strong  enough,  the  small  one  will  be  about  -jt  part  stronger  than 
necessary.  Reductions  however  will  rarely  be  made  to  as  small  as  -^  of  the  original ;  and  where  they 
«o  not  exceed  J^,  the  method  will  answer  very  well  in  practice. 

For  examples  of  reducing,  see  p  301. 

With  the  same  total  load  per  sq  ft,  including  the  wt  of 
the  truss  itself,  and  with  trusses  at  the  same  distance  apart  in  all  cases,  the 
strains  on  the  several  members  of  similar  trusses,  (that  is,  of  trusses  precisely  alike 
except  in  the  size  of  themselves  and  their  parts)  will  be  in  the  same  proportion  as 
the  spans ;  as  will  also  the  areas  of  cross  section,  and  the  wts  per  foot  run  of 
each  individual  member  ;  but  the  total  Wts  of  the  trusses  will  be 
as  the  squares  of  the  spans. 


264 


TRUSSES. 


r  TO.    The  whole  of  6  r  is  transferred  to  the  next  point  of  support ;  while  r  m  and  b  n 
produce  only  hor  strains  along  b  a  and  g  y. 

Art.  16.  The  roof  truss  shown  by  Figs  19,  20,  and  21,  consists  of  two  complete 
J'ink  trusses,  a  <'  y  and  « ?/  p.  Fig  10.  It  is  supposed  to  be  of  the  same  sp;m  and  height 
as  the  others ;  and  to  have  the  same  number  ^9)  of  points  of  support  for  the  weight 


Rise  16  ft. 


TRUSSES.  265 

(32  tons,)  supposed  to  be  uniformly  distributed  along  its  top.  Consequently  from 
our  first  process  there  will. 'as  before,  be  2  tons  of  panel-load  at  each  of  the 
end  supports ;  and  4  tons  at  each  of  the  others.  Write  these  down  as  in  Fig 
20.  Now,  at  either  strut,  as  dg,  draw  a  vertical  line  dv  in  pencil,  by  any  convenient 
scale,  to  represent  a  whole  panel- load,  (-i  tons;)  and  draw  vo  parallel  to  the  rafter//*'; 
and  v  r  parallel  to  the  strut.  Measure  d  r  or  v  o  by  the  scale,  and  write  down  the 
result  (1.77  tons)  near  every  strut,  as  in  the  Fig.  The  object  of  this  will  be  shown 
hereafter.*  The  linns  d  v,  v  o,  v  r,  may  then  be  rubbed  out.  Now  the  strains  from 
the  several  panel-loads,  in  passing  to  their  final  points  of  support  at  the  abuts,  travel 
by  a  route  diif  from  that  in  either  of  the  preceding  cases.  The  part  truss  ex  a  may 
be  regarded,  to  some  extent,  as  being  composed  of  three  separate  trusses ;  namely, 
ex  a,  egc,  a  cm;  as  will  appear  more  plainly  from  en  a,  eic,  and  a  en,  in  Fig  21. 
These  may  be  called  first  and  second  secondary  trusses.  In  Fig  19,  the  half  cp  y  ex- 
hibits a  truss  on  the  same  principle,  but  having  a  greater  number  of  points  of  sup- 
port for  the  uniform  wt.  That  half  truss  consists  of  first,  second,  and  third  second- 
aries, as  shown  by  ey  p,  tgi,  and  es  h.  However  far  this  subdivision  may  be  carried, 
if  the  struts  occur  at  equal  dists,  and  if  the  wt  or  panel-load  supported  by  each  strut 
is  the  same,  then  each  panel-load  resting  on  a  short  strut  will  travel  (one-half  of  it 
each  way)  either  to  the  panel-load  of  the  next  longer  strut ;  or  else  to  one  end  of  the 
rafter.  Thus,  with  our  second  process  at  one  of  the  shortest  struts,  as  6 
m,  Fig  20,  2  of  its  4  tons  go  to  c  at  the  next  longer  strut,  car;  arid  2  of  them  to  the 
end  a  of  the  rafter,  as  written  on  the  Fig.  Then,  at  the  other  of  the  shortest  struts, 
d g,  2  tons  go  to  c ;  and  2  to  the  end  e  of  the  rafter.  We  thus  have  4  tons  at  b  ;  4  at  d ; 
and  8  at  c.  But  of  these  8  tons  at  c,  4  travel  down  the  strut  c  a;,  and  along  the  tie  x  a, 
to  the  end  a  of  the  rafter ;  and  4  along  ex  and  xeto  its  other  end  e  ;  both  of  these  4 
tons  are  therefore  set  down  as  at  a  and  e.  When  there  are  more  points  of  support, 
as  along  the  rafter  ep,  Fig  19,  the  process  is  precisely  the  same :  we  first  adjust  the 
strains  of  the  four  third  secondaries,  e  s  h,  h  r  i,  i  v  k,  kzp\  placing  them  at  e,  //,  i,  k, 
and  p:  then  we  transfer  those  thus  accumulated  at  h  and  k,  to  et  i,  and  p ;  and  finally 
transfer  them  from  i  to  e  and  p,  at  the  ends  of  the  rafter.  Now,  returning  to  Fig  20, 
we  see  that  in  addition  to  the  original  panel-load  of  4  tons  at  e,  we  have  accumulated 
8  tons  of  vert  strain  from  the  other  panel-loads ;  and  it  is  plain  that  the  same  pro- 
cess, performed  along  the  other  half  of  the  truss,  would  bring  2  -f-  4  =  6  tons  more 
to  e,  as  written  in  the  Fig.  Thus  it  appears  that  we  have  16  tons  in  all  at  e ;  and  this 
is  precisely  half  the  weight  (32  tons)  of  the  entire  truss  and  its  load;  and  as  this 
will  always  be  the  case  in  trusses  on  this  principle,  it  proves  our  work  to  be  correct 
thus  far.  In  like  manner,  the  total  strains  accumulated  at  the  other  end  a  of  the 
rafter;  as  well  as  those  at  its  center  c,  must  always  each  be  equal  to  one  quarter  of 
the  weight  of  the  entire  truss  and  load.  Also  the  total  strains  thus  accumulated 
at  any  longer  strut,  will  be  twice  as  great  as  that  at  any  next  shorter  one.  See  Art  18. 

Having  thus  finished  our  second  process  of  finding  the  additional  strains  at  tho 
several  points  of  support  produced  by  the  panel-loads  on  their  way  to  their  final 
points  of  support,  we  have  only  by  our  third  process,  to  complete  the  draw- 
ing, so  that  we  may  measure  by  scale  the  strains  along  all  the  members  of  the 
truss.  To  do  this,  from  the  tops  of  the  struts  draw  vert  lines  b  v,  c  v,  d  v  to  repre- 
sent the  total  vert  strains  accumulated  at  those  respective  points;  namely,  4  tons  at 
6,  4  at  </,  8  at  c.  Braw  v  o,  v  «,  v  »  parallel  to  the  rafter  a  e.  Then  6  o,  c  r>,  d  o  will 
give  the  strains  along  the  struts ;  3.6  tons  on  b  m  or  d  g ;  and  7.2  tons  on  c  x.  Lay  off 
m  *',  x  i,  and  g  i  respectively  equal  to  '6  o,  c  o,  and  d  o;  draw  ij,  ij,  ij\  arid  i  y,  i  y, 
i  y,  parallel  to  the  ties ;  thus  completing  the  parallelograms  of  forces  ij  in  y,  ij  x  y, 
and  ijg  y.  Also  draw  the  diags  yj,  yj,  yj;  and  tne  vert  lines  i  w,  i  w,  i  ?r,  m  u, 
x  M.and  #  u.  Also  lay  off  the  vert  dist  e/equal  to  the  total  vert  strain  (1C  tons) 
ate;  draw  fh  (Fig  21)  parallel  to  the  rafter  a  e,  and  terminating  at  /tin  the  other  rafter 
e  1]  and  h  z  hor  or  parallel  to  the  tie  a  I.  Or,  which  amounts  to  the  same  thin.ir,  is. 
Fig  20  make  ef  equal  to  the  strain  (16  tons)  at  e\  make  e  z  =  to  half  of  tf;  draw 
z  h  hor;  and  hf.  This  saves  the  necessity  of  drawing  more  than  half  the  tru>s. 

Now  m  y  and  mj  give  the  strains  (4  tons  each)  along  the  ties  m  a,  m  c,  caused  by 
the  4  tons  at  6;  which  strains  extend  from  m  to  a  and  c.f  In  like  manner,  g  y  and 


*  When  merely  wishing  to  ascertain  the  strains  along  the  members  of  such  a  truss,  without  caring 
to  tra«e  their  progress,  Ave  may  omit  part  of  the  following ;  and,  after  having  made  a  correct  diagram 
of  the  truss  ;  and  found  and  noted  down  the  force  dr  or  vo  mentioned  above,  we  may  at  once  write 
down  the  vert  strains  at  the  points  of  support,  thus:  At  e  (the  apex  or  peak  of  the  truss)  write  one 
half  of  the  entire  wt  of  a  truss  and  its  load,  (for  which,  per  sqft,  see  Table  4,  p  301,)  at  the  center 
strut  c,  one  fourth ;  at  the  foot  a  of  the  rafter,  also  one  fourth  ;  at  6  and  d,  one  eighth  at  each  ;  and 
when  there  are  four  intermediate  subdivisions  of  the  same  kind,  a-»  along  the  rafter  ep,  Fig  19, 
one  sixteenth  of  said  entire  weight  at  each  of  such  additional  points,  &c.  Then  begin  at  "  Having 
thus  finished  our  second  process." 

t  These  strains  along  the  ties  will  b«  equal  to  those  at  the  points  of  support,  only  where  the  height 
of  the  truss  is  equal  to  y±  of  its  span  ;  as  in  the  case  before  us.  When  the  height  is  less  than  V£,  the 
•trains  oa  the  tie*  will  be  greater  than  those  at  the  points  of  support, ;  and  vice  versa. 


266  TRUSSES. 

gj  give  the  strains  (4  tons  each)  extending  from  g  to  c  and  e.  In  Pig  21,  the  short 
ties,  o  a,  o  c,  i  c,  i  e,  show  this  more  distinctly. 

Next,  x  y  and  xj,  Fig  20,  give  the  strains  (8  tons  each*)  produced  along  x  a  and  x  e 
by  the  8  tons  at  c.*  This  also  is  shown  more  plainly  in  Fig  21,  by  the  ties  n  a  and 
n  e.  Again,  the  nor  line  It  z  will  give  the  strain  (16  tons)  produced  along  the  entire 
nor  tie  a  /,  Fig  21,  by  the  16  tons  at  «.  Fig  20  may  be  considered  one  half  of  Fig  21. 

In  practice,  the  ties  ao,an,  &c,  Fig  21,  of  the  secondaries,  are  not  always  made 
distinct  from  that  (a  I)  of  the  primary  truss  a  e  I;  but  they  are  so  represented  in  Fig 
21,  merely  to  show  more  plainly  that  the  central  portion  a;  a;  of  the  primary  tie  a  I 
needs  only  such  dimensions  as  will  enable  it  to  sustain  the  thrust  produced  by  the 
16  ton  strain  at  e:  whereas,  along  its  portions  x  m,  x  m  it  must  be  stout  enough  to 
bear,  in  addition,  the  thrust  along  the  first  secondary  ties  na,kl\  while  at  its  ends 
m  a,  m  I  it  must  resist  not  only  the  two  preceding  thrusts,  but  also  those  along  the 
second  secondary  ties  o  a,  o  I.  Likewise,  it  is  plain  that  the  portion  g  e  of  the  first 
secondary  tie  n  e,  must  be  stouter  than  the  portion  n  g\  because  \vhen  n  eis  formed 
of  one  bar,  its  portion  g  e  has  to  bear  also  the  thrust  along  the  second  secondary  tie 
i  e.  In  Fig  20,  those  portions  of  the  ties  which  are  most  strained  are  shown  by 
stouter  lines. 

We  have  for  the  total  strains  on  the  ties  as  follows :    See  Rem,  p  375. 

Along  c  m  and  c  g,  strain  —  mj  or  g  y  —  4  tons. 

A  long  x  e,  from  x  to  g.  strain  ~xj—S  tons. 

Along  x  e,  from  g  to  e,  strain  —  x  j  +  ffj  ~  8  +  4  =  12  tons. 

Along  t  a,  from  t  to  x,  strain  =  h  z  =  16  tons. 

Along  t  a,  from  x  to  m,  strain  —  7t^4-zy— 16  +  8  =  24  tons. 

Along  t  a,  from  m  to  a,  strain  =  Az  +  £y  +  7rty:=16  +  8  +  4  =  28  tons. 
The  strain  along  a  rafter  a  e  would  be  equal  throughout,  were  it  not  for  the  small 
strains,  of  1.77  tons  each,  at  />,  c,  and  d  first  found.f  This  will  be  seen  thus  :  The  8 
tons  at  c,  Fig  20,  produco/ortv.s  at  a  and  e  of  5.4  tons  each ;  which  are  found  by  measur- 
ing uj  and  w  ?/,  of  the  middle  parallelogram  of  forces  ij  x  y.  Consequently  these  two 
forces  acting  against  each  other  at  the  opposite  ends  of  the  rafter,  produce  a  uniform 
pressing  strain  of  6.4  tons  throughout  its  entire  length.  Again,  the  4  tons  at  b  produce 
forces  at  a  and  c  of  2,7  tons  each,  found  by  measuring  uj  and  w  y  of  the  parallelogram 
ij  m  y.  Consequently  these  two  forces  acting  at  the  opposite  ends  of  the  half  rafter 
a  c,  produce  a  strain  of  2.7  tons  along  said  half.  But  the  4  tons  at  d  produce  in  like 
manner  the  same  amount  of  strain  along  the  other  half,  e  c.  Finally,  the  10  tons  at 
e,  produce  two  forces,  each  equal  to  e  h (17.9  tons;)  one  of  which  presses  the  entire 
length  of  each  rafter.  Consequently  we  have,  as  pressing  its  entire  length,  the  forces 
e  liy  uj  of  the  middle  parallelogram ;  and  uj  of  the  parallelogram  ij  g  y  ;  or  17.9  + 
6.4  +  2.7  =  26  tons.  Nothing  but  these  26  tons  press  it  from  e  to  d\  but  from  d  to  c 
we  must  add  1.77  tons ;  from  c  to  &  1.77  more ;  and  from  b  to  a  1.77  more  ;  so  that 
we  have  at  last,  for  tne  total  strains  along1  a  rafter, 

From  e  to  d,  strain  =  «  h  +  uy  (of  the  parallelogram  ijgy),  and 

v.j  (of  the  parallelogram  ijxy)  =  17.9  +  6.44-2.7=26  tons. 
"      dtoc,     "      -264-1-77  =  27.77. 
44      ctofc,     "      =26  4- 1.77  4- 1.77  =  29.54. 
"      6  to  a,     "      =26+ 1.77 -fl-77 +  1.77  =  31.31. 

The  center  vert  e  t  may  be  omitted  in  short  spans  ;  since  it  sustains  noth- 
ing but  the  wt  of  the  half  (yy)  of  the  central  spread  xx  of  the  hor  tie  a  I.  Thus  we 
have  the  strains  along  every  member  of  the  truss.  See  Art  29,  p  298. 

REM.  1.  If  the  main  or  primary  tie  is  raised  either  at  its  center,  as 
p  n,  Fig  19  ;  or  if  it  is  raised  only  as  far  as  y,  and  is  then  continued  hor,  as  y  o,  (as  is 
frequently  done,)  in  either  case  we  must  proceed  as  at  Fig  16,  p  262  ;  and,  after  hav- 
ing found  the  vert  strains  at  all  the  points  of  support,  as  before,  we  must  add  to  that 
(16  tons)  at  e,  the  amount  arising  from  mult  said  16  tons  by  the  vert  height  tn  (Fig 
19)  to  which  the  tie  zn  is  actually  raised  above  the  hor  atp,  (or  to  which  it  would  be 
raised  if  the  inclination  of  the  tie  p  y  were  continued  to  n,  instead  of  being  hor  like 
3/0 ;)  and  from  div  the  prod  by  the  remaining  height  ne  of  the  truss.  Then,  as  in 
Art  15,  we  must  lay  off  the  vert  ef,  Fig  20,  equal  to  the  total  vert  strain  at  e,  thus 
found;  and,  after  drawing  fh  parallel  to  the  rafter,  must  draw  hz  parallel  to  the 
inclined  tie,  instead  of  hor. 

REM  2.  We  will  explain  the  reason  for  using  the  portions  uj  and  wy,  of  the  diags 
j  y,  Fig  20,  for  measuring  strains  alon  •*  the  rafter  a  ?.  Take,  for  instance,  the  strut  d  g. 
Here  it  is  evident,  that  since  ig  represents  by  scalo  the  strain  along  the  strut,  the 
two  sides  g  y  and  gj  of  the  parallelogram,  give  the  resulting  strains  along  the  two 

*  See  second  note  at  foot  of  preceding  page. 

t  These  small  strains  become  proportionally  greater  as  the  rise  of  the  truss  increases ;  so  that  when 
the  rise  is  as  great  as  J£  of  the  span,  they  cause  the  pressure  at  the  foot  of  the  rafter  to  be  about  l.i 
times  that  at  its  bead ;  while  at  ^  rise,  it  is  but  about  1.04  times  as  great. 


TRUSSES.  267 

ties.  Now  if  we  take  one  of  these  strains,  say  gj,  and  on  it  as  a  diag  dravr  the  par- 
allelogram of  forces  ujng,  with  two  of  its  sides  vert,  and  two  parallel  to  the  rafter, 
then.;  n  will  give  the  vert  strain  which  gj  produces  at  « ;  and  ju,  the  strain  which 
it  produces  at  e,  in  the  direction  of  the  rafter.  Then,  if  we  measure  the  vert  strain 
j  w,  or  g  u,  we  shall  find  it  to  consist  of  the  2  tons  which  we  originally  transferred 
from  d  to  e,  by  our  second  process;  and,  in  like  manner,  iw  represents  the  2  tons 
transferred  from  d  to  c.  Since,  then,  the  strain  gj  is  made  up  of  uj  and  jn,  and 
since ,/n  was  transferred  to  e  by  calculation  during  our  second  process,  only  j  u  re- 
mained to  be  determined  by  measurement  in  the  third  process. 

REM  3.  It  is  not  necessary  actually  to  draw  all  three  of  the  parallelograms,  as  in  Fit  20.  The  large 
or  center  one  alone  will  suffice ;  for  we  need  only  div  the  several  strains  measured  aloug  the  strut  en, 
Fig  21 ;  and  along  the  ties  na,  ne,  by  2,  to  get  those  along  the  struts  4o  and  4f ;  and  along  the  ties 
ao.  co,  ci,  ei.  And  these,  in  turn,  div  by  2,  will  give  those  along  the  smaller  subdivisions  shown 
between  e  andp,  Pig  19,  if  there  are  such  ;  and  so  on  with  any  number  of  still  smaller  ones. 

REM.  4.  The  student  need  now  have  no  difficulty  in  finding  the  strains  produced 
by  a  uniform  load  on  a  hor  Fink  Truss,  Figs  26,  27 ;  the  process  is  more 
simple  than  in  the  roof;  for  the  chord  being  hor,  and  the  struts  vert,  there  will  be  no 
force  like  the  1.77  tons  at  6,  c,  and  d;  and  the  strain  along  the  chord  will  therefore 
be  uniform  from  end  to  end.  In  Fig  26,  nn  is  not,  properly  speaking,  a  truss  chord, 
but  merely  a  beam  added  only  for  supporting  the  roadway.  If  the  truss  were  in- 
tended for  a  hor  roof,  n  n  would  be  omitted.  The  vert  strains  at  the  top  of  each 
strut  may  be  at  once  written  down,  without  tracing  their  progress  to  those  points. 
Thus,  at  the  half-way,  or  center  strut,  d  c,  write  one  half  the  entire  wt  of  the  truss 
and  its  load:  at  each  quarter-way  strut  mm,  write  one-quarter  of  the  same  entire 
weight ;  at  each  eighth-way  strut,  hhhh,  one-eighth  of  it,  &c.  Then,  from  the  feet 
of  the  struts,  set  up  vert  dists  along  the  struts,  by  scale,  to  represent  the  vert  strains 
just  written  at  the  top  of  each.  From  the  upper  ends  of  these  dists  draw  lines 
downward  parallel  to  the  two  ties  at  the  foot  of  the  strut,  and  ending  in  said  ties  ; 
thus  completing  a  parallelogram  of  forces  at  the  foot  of  each  strut,  as  in  Fig  20. 
Draw  the  hor  diag  of  each  of  these  parallelograms.  Measure  and  add  together  one- 
half  of  the  half-way  or  central  diag;  one-half  of  one  quarter-way  one;  one  half  of 
one  eighth-way  one,  Ac.  The  sum  will  be  the  uniform  hor  strain  along  the  entire 
chord.*  The  strain  along  each  inclined  tie  will  be  found  by  measuring  that  side  of« 
the  parallelogram  which  is  on  said  tie.  For  Fink  bridge  trusses,  see  p  305. 

REM.  5.  Figs  21%  show  a  few  of  the  many  forms  of  the  details 
of  iron  roofs.  Every  maker  has  his  own  modifications  of  them.  Most  of  the 
figs  explain  themselves.  They  will  serve  as  hints.  For  more  on  iron  roofs,  p  298. 

R  and  P  stand  for  rafter  and  purlin.  In  small  roofs,  with  the  trusses  only  3  or  4 
ft  apart,  the  purlins  may,  as  at  6,  be  simple  %  inch  or  ^  round  rods,  about  9  ins 
apart;  and  the  slates  may  rest  immediately  on  them,  being  tied  to  them  by  iron 
wire.  They  may  be  bent  down  at  their  ends,  and  riveted  to  the  rafters.  As  the  dist 
between  the  trusses  increases,  these  purlins  may  be  made  of  flat  iron,  from  1  to  3  ins 
deep,  and  %  inch  thick;  or  of  light  T  iron,  &c;  and  may  be  trussed,  as  at  7,  so  as  to 
admit  of  being  placed  several  feet  apart.  When,  however,  they  have  to  bear  great 
weight,  the  mode  at  c,  Fig  7,  of  confining  their  ends  to  the  rafters,  will  be  too  weak. 
Sometimes  they  may  be  arranged  as  at  y.  Or  the  purlins,  of  either  iron  or  wood, 
may  rest  on  top  of  the  rafters,  as  at  1  and  5 ;  or  their  ends  may  rest  in  a  kind  of 
stirrup,  as  at  t,  Fig  2;  and  at  P,  Fig  4;  in  castings  placed  at  the  "points  of  sup- 
port "  of  the  truss ;  or  they  may  be  confined  to  the  sides  of  the  rafters  by  two  angle- 
irons,  as  at  P,  Fig  9.  Purlins  should,  when  practicable,  be  sup- 
ported only  at  or  near  the  "points  of  support"  of  the  truss :  and 
as  a  general  rule,  it  will  be  expedient  to  arrange  the  number  of  these  points  with 
reference  to  this  particular.  The  rafters  are  then  relieved  from  transverse  strains  ; 
and  may  be  proportioned  with  regard  only  to  the  compressive  strain  in  the  direction 
of  their  length.  Too  little  attention  is  sometimes  given  to  this  point,  and  the  trans- 
verse strain  is  overlooked,  to  the  serious  injury  of  the  roof.  It  is  well,  however,  to 
bear  in  mind  that  thin  deep  rafters  are  liable  to  yield  by  buckling  sideways;  and 
that  this  tendency  is  diminished  by  purlins  well  secured  to  them  between  the  "  points 
of  support."  Sometimes  castings  similar  to  2,  are  used  at  the  heads;  and  3,  at  the 
feet,  of  the  struts  and  vert  ties ;  which  last  have  their  ends  cut  into  right  and  left 
hand  screws,  for  insertion  into  corresponding  female  screws  cut  in  the  castings. 
At  3, 1 1  is  the  main  tie  passing  loosely  through  the  lower  opening  through  the  cast- 
ing. Below  it,  is  seen  the  head  of  a  small  set-screw,  for  tightening  together  the 
casting  and  the  tie ;  to  prevent  the  former  from  slipping  out  of  place.  There  must 
be  different  patterns  of  these  castings,  tu  suit  the  obliquities  of  the  several  obliques; 
or,  in  small  roofs,  the  parts  a  a  may  be  made  with  hinges,  for  the  same  purpose. 

*  If  the  struts  are  equidistant,  and  of  equal  length,  each  succeeding  one  of  these  half  diags  will  be 
%  part  as  long  as  the  oue  that  precedes  it.  The  strains  along  the  oblique  ties  will  not  follow  the 
M*tne  proportion. 


268 


TRUSSES. 


At  4  ami  5  are  cast-iron  shoes  for  supporting  the  ends  of  the  trusses 
upon  the  walls.  With  the  exception,  perhaps,  of  these  shoes,  it  is  better  that  the 
details  generally  should  be  of  wrought  iron. 

At  8  is  a  mode  of  confining-  thin  metal  roof-covering  i,  to 
the  purlins  P,  by  means  of  short  (about  an  inch)  U-shaped  pieces  (c  c  1 1  is  one 
of  them)  of  the  same  metal;  to  which  i  is  riveted  by  an  %  inch  rivet  through  each 
flange  1 1.  This  may  be  adopted  with  corrugated  iron  covering,  which,  by  its  strength, 
allows  the  purlins  to  be  placed  several  ieet  apart.  See  Corrugated  Iron.  Flat 
sheets  require  boards  beneath  them. 

At  10  is  a  mode  of  confining  a  wooden  purlin  P  on  top  of  an  iron 
one  _p,  by  means  of  a  crooked  spike  s  s  s;  which,  after  being  driven  from  below,  is 
clinched  or  bent  on  top.  Wooden  purlins  are  sometimes  thus  required,  for  nailing 
elates  or  plain  sheet  metal.  At  11,  c,  is  a  stick  of  timber  inserted  between  an  iron 


10 


purlin  P  and  the  corrugated  roof-covering  a  a.  To  such  sticks  plastering-laths  may 
be  nailed,  when  the  roof  is  to  be  plastered  beneath,  to  avoid  condensed  moisture. 
There  is  room  for  much  ingenuity  in  all  these  details.  Fig  12  is  a  rafter  made  of  two 
channel-bars  riveted  together ;  with  a  web  member  c  c  between  them.  Two  angle- 
bars  are  often  thus  riveted  together  for  a  rafter.  For  more  on  iron  roofs,  p  298. 

Fig:  18  shows  a  mode  of  tightening  two  lengths  of  a  tie-bar 
by  a  swivel  or  turn  buckle,  t  b.  The  end  of  one  length,  n.  is  cut  into  a  screw ; 
and  the  corresponding  end,  ft,  of  the  turnbuckle  is  a  female  screw,  into  which  n  fits. 
The  end  r  of  the  other  length  may  either  be  made  in  the  same  manner,  or,  as  in  the 
fig,  may  be  plain,  and  be  furnished  with  a  head  c.  The  turnbuckle  revolves  around 
both  rods,  and  of  course  can  thus  tighten  them.  Sometimes  t  b  is  made  of  a  solid 
round  bar,  called  a  double  nut;  with  a  female  screw  tapped  a  few  inches  into 
each  end,  right  and  left.  The  end  of  each  rod,  c  and  2,  is  then  cut  to  a  screw. 

Fig  14  is  a  mode  of  tightening  four  lengths  of  tie-bars  < 


Fig  14  is  a  mode  of  tightening  four  lengths  of  tie-bars  crossing 
each  other,  by  means  of  a  ring.  The  ends  inside  of  the  ring  are  cut  into  screws,  and 
provided  with  tightening  nuts,  as  in  the  fig.  The  rings  are  usually  %  to  l1^  inch 
thick ;  3  to  5  deep ;  and  7  to  10  diam. 

In  Art  *29  are  remarks  on  the  comparative  merits  of  the  foregoing  plans  of  roofs, 
inasmuch  as  they  are  the  kinds  usually  employed. 

In  proportioning  the  sizes  of  struts  for  roofs  or  bridges,  bear  in  mind  Hem 
''The  young  engineer,"  &c,  p  237 ;  and  Bern,  p  375,  for  ties. 


TRUSSES. 


269 


a 

I! 


J 


*- 


Art.  17.    Fig-  22  represents  a  suspension  truss  on  the  Boll- 
man  plan ;  *  the  whole  weight  supposed  to  be  along  the  top  a  p. 

In  this,  the  strain  from  each 
panel-load,  as  for  instance  that 
at  d,  passes  down  to  the  foot  of 
its  supporting  post  dj ;  and  from 
there  is  transferred  to  the  two 
ends  a  and  p  of  the  chord,  by- 
means  of  two  ties,  as  j  a, .;  p, 
upon  which  the  post  stands.  In 
this  manner  the  vert  strain  from 
each  panel -load  is  separately 
sustained;  and  transferred  di- 
rectly as  a  hor  strain  to  the  ends 
of  the  chords,  by  its  own  post 
and  pair  of  ties ;  without  pro- 
ducing, as  in  the  foregoing  cases, 
an  additional  vert  strain  at 
the  points  of  support  of  the 
other  panel-loads.  So  omit 
our  2nd  process ;  and 
having  divided  the  uniform  wt 
of  the  trues  and  its  load,  among 
the  several  points  of  support  a, 
I),  c,  d,  e,  &c,  as  before,  we  pro- 
ceed at  once  to  draw  the  parallel- 
ograms of  forces  v  u  I  g,  v  u  If  p, 
Ac,  for  measuring  the  strains. 
To  do  this  we  have  only  to  set 
up  the  equal  vert  dists  I  r,  k  v, 
j  v,  &c,  each  to  represent  by 
scale  the  4  ton  panel-loads  on 
top  of  the  respective  postfc ;  then 
complete  each  parallelogram  by 
drawing  v  u,  v  g  parallel  to  the 
two  ties  which  support  each 
post.  Then  the  lines  I  u,  I  g ; 
k  it,  k  g,  &c.  give  by  scale  the 
strains  along  the  respective  ties. 
The  end  a  of  the  hor  chord  is 
pressed  hor  by  the  seven  hor 
forces  u  o,  u  on  u  o,,,  u  «„„  &c., 
equal  to  1.75  -f  3  +  3.75  -f  4  -f 
3.75  +  3  -f  1.75  =  21  tons  ;  and 
the  other  end  p  is  in  like  man- 
ner pressed  by  the  seven  corres- 
ponding forces  not  shown  ;  and 
these  two  sets  of  equal  op- 
posing forces  produce  a  strain 
equal  to  one  of  them  ;  or  to 
21  tons,  uniform  throughout  the 
entire  chord.  The  tie  la  car- 
ries to  a  so  much  of  the  weight 
of  the  4  tons  at  6  as  is  rep- 
resented by  I  o,  or  3.5  tons ; 
k  a  carries  to  a  a  weight  equal 
to  k  o,,  or  3  tons ;  j  a  carries  j  o/y, 
=  2.5  tons ;  i  a  carries  i  oin  =  2 
tons ;  w  a,  w  o  =  1.5  ;  x  a,  x  o  = 
1;  andt/a,  yo  =  .5  ton.  All 
these  amount  to  14  tons ;  which, 
with  the  2  tons  of  half  panel- 
load  at  a,  give  16  tons  ;  or  half 
the  entire  weight  (32  tons)  of 
the  truss  and  its  load.  This  is  a 
proof  that  the  strains  have  been  drawn  and  measured  correctly.  The  other  half 
weight  of  truss  and  load  is  carried  in  the  same  way  to  the  other  end  jp,  by  means  of 
the  ties  yp,  xp,  lp,  &c.  These  wts  cause  the  following  strains : 


2- 


-I      co 


*  Invented  bj  Mr.  Wendcl  Bellman,  C.  K. 


270  TRUSSES. 


The  strain  I  u  =  3.91  tons. 


The  strain  I  g  =  1.82  tons. 


Ar0r  =  3.17 
.70  =  4.05     " 
t  g  =  4.47    " 

Each  post  or  vert  is  of  course  strained  to  the  amount  of  a  full  panel-load,  when 
tlie  whole  wt  is  supposed  to  be  on  top  of  the  truss.  If  the  load  is  at  the  bottom  of 
the  truss,  see  "  In  the  Boll  man  truss,-9  near  the  end  of  Art  20%,  p  282. 

BOWSTRING,    AND     CRESCENT     TRUSSES,     UNIFORMLY 
1,OA1>EI>. 

Art.  18.  Before  attempting  to  find  the  strains  on  either  a  uniformly  loaded 
bowstring  or  a  crescent  truss,  Figs  23,  23  b,  23  c,  by  means  of  a  diagram,  the  student  should  familiar- 
ize himself  with  the  following  remarks  : 

RKM.  1.  The  basis  of  the  entire  process  is  that  at  every  point  of  sup- 
port, beginning  at  an  abut  as  the  first  one,  we  have  acting  one  or  more  known  forces,  balanced  or 
held  in  equilibrium  by  either  one  or  two  unknown  ones  ;  and  the  object  at  each  point  is  first,  by 
means  of  the  parallelogram  of  forces,  to  find  the  resultant  of  the  known  ones  ;  and  second,  by  the 
same  principle,  to  resolve  this  resultant  into  two  components  in  the  directions  of  the  unknown  ones. 

This  is  all  that  is  required  in  either  the  bowstring  or  the 
crescent  truss.* 
REM.  2.    While  more  than   two  unknown   forces  exist  at  any 

point  of  support,  their  amounts  cannot  be  found.  If  one  force  is  known, 
and  two  unknown,  the  three  balancing  each  other,  draw  a  line  by  scale  to 
represent  the  amount  and  direction  of  the  known  one  ;  and,  considering  it  as  one  side  of  a  triangle, 
from  its  two  ends  draw  lines  parallel  to  the  two  unknown  ones,  to  meet  each  other,  thus  completing 
the  triangle.  Then  these  last  two  will  by  the  scale  give  the  two  unknown  ones  ;  because  when  three 
forces  meeting  at  one  point,  balance  each  other,  three  lines  representing  them  both  in  amount  and 
in  direction,  will  form  a  triangle.  See  Rem  2,  p  468.t 

If  there  are  two  or  more  known  forces,  first  find  the  single  re- 
sultant of  them  all  (Art  36,  p  466),  and,  taking  it  as  one  side  of  the  triangle,  find  the  other  two  sides 
(that  is,  the  two  unknown  forces)  as  before.  After  a  little  practice  the  student  will  find  it  unneces- 
sary to  draw  more  than  half  the  sides  of  the  parallelogram  of  forces. 

REM.  3.  The  bow  is  to  be  considered  straight  from  apex  to 
apex.  If  actually  curved  it  will  be  much  weakened. 

RKM.  4.  At  each  point  of  support,  or  apex,  consider  every  member  that  meets  there,  to  be  a  force 
either  pushing  towards  said  point,  if  along  a  strut;  or  pulling  from 

it,  if  along  a  tie.    This  is  shown  by  the  arrows  in  Figs  23,  23  a,  <fcc. ;  thus  in 
Fig  23,  at  o,  the  forces  along  the  struts  co,  so,  and  qo,  push  towards  o;  qo  also 
pushes  towards  q  ;  while  the  force  along  the  tie  r  o  pulls  from  o,  as  also  from  r.  All 
loads  on  the  bow  are  forces  pushing  vert  downwards. 
REM.  5.    If  the  known  forces  are  not  all  alike  at  any  point,  (that 

is,  neither  all  pulls  nor  all  pushes,)  then  while  constructing  the  parallelograms  of  forces,  one  or  more 
of  the  forces  must  be  changed,  and  be  regarded  as  acting  at  the  opposite  side  of  said  point,  but  in  the 
same  direction  as  before :  so  as  to  make  them  all  alike  ;  otherwise  the  parallelogram  will  not  give  the 
correct  resultant.  See  Rem  1,  p  458.  For  instance,  in  Fig  23  a,  we  have  (as  will  be  seen  hereafter) 
three  known  forces  acting  at  c,  namely  e  c  pushing  towards  c :  a  load  (not  shown)  on  the  bow  at  c, 
pushing  vert  downwards  towards  c  ;  andp  c  pulling  from  c.  In  this  case  we  must,  while  drawing 
the  sides  of  the  parallelograms,  either  consider  the  pull  p  c  to  be  changed  to  a  push  in  the  direction 

*  The  same  process  applies  equally  to  all  our  fig-s  from  1O 
to  16  also  ;  whether  uniformly  loaded  on  one  chord  or  on  both ;  also  to  the  in~ 
verted  bowstring,  and  to  Fig  23(2.  In  this  last,  as  in  the  others,  the  lower  member  (nnn)  is  a  tie 
which  prevents  the  truss  from  spreading,  and  thus  causes  it  to  exert  only  vert  pres  against  the  abuts. 
This  is  a  distinctive  character  of  all  so-called  truss  girders,  including  all  our 
figs  back  to  Fig  5.  But  when  nnn  in  converted  into  an  arch  for  sustaining 
compression,  it  ceases  to  be  a  tie,  and  the  truss  then  exerts  hor  as  well  as  vert 
force  against  the  abuts ;  and  becomes  what  is  called  a  braced  arch.  The  pro- 
cess requires  a  slight  modification  before  it  can  be  used  for  such,  as  shown  in  Art  20,  p  274.  Al- 
though strictly  speaking  the  process  does  not  apply  to  the  Fink  truss,  Figs  19.  20,  21,  because  we 
there  encounter  three  unknown  forces  at  any  point  where  three  web  members  as  cm,  ex,  eg,  Fig  20, 
meet  at  a  rafter  as  at  c,  still  as  we  can  readily  determine  the  strain  on  one  of  these,  ex,  by  means  of 
the  entire  vert  strain  at  such  point,  (which  strain  can  be  found  in  a  Fink  truss  by  a  mere  mental 
calculation)  we  thus  reduce  the  unknown  forces  to  two,  and  therefore  may  employ  the  process  even 
for  such  trusses.  The  student  would  do  well  to  test  Figs  10  to  16  by  this  process. 

f  Any  one  of  the  three  forces  is  then  the  anti-resultant,  or  bal- 
^        ancer,  or  opposer,  of  the  other  two;  and  if  three  arrow-heads 

^>  X  showing  their  directions  be  added  to  the  sides  of  the  triangle  as  in  this  Fig,  they  will, 
^  ^  as  it  were,  chase  each  other  around  the  triangle  ;  that  is,  the  head 
H  "  '  ^  of  any  one  of  them  will  touch  the  butt  end  of  the  one  next  to  it ;  or  no  two  arrow-heads 
•will  meet.  But  this  is  not  so  when  three  sides  of  a  triangle  represent  two 
forces  and  their  resultant,  or  equivalent  in  effect ;  for  the  arrow-head  of  the  re- 
sultant will  then  meet  that  of  one  of  the  other  forces.  See  Kent.  2,  p  468. 


TRUSSES. 


271 


J8 


272 


TRUSSES. 


from  t  towards  e ;  or  the  other  two  to-  b«  similarly  changed  to  pulls.  The  first  of  course  is  the  easier. 
According  as  the  known  forces  (after  one  or  more  are  so  changed,  if  necessary)  are  pulls  or  pushes, 
the  diagonal  or  resultant  will  be  the  same. 

IU.M".  6.    To  decide  whether  an   unknown   force  is  a  pull  or 

a  push  ;  that  is  whether  the  member  along  which  said  force  acts  is  a  tie  or  a 
strut.  Having  found  the  resultant  of  the  known  forces,  add  to  it  an  arrow-head  to  show  its  direc- 
tion. Then  having  on  this  resultant  completed  the  triangle  by  means  of  the  two  unknown  forces, 
arid  arrowheads  to  them  also,  placing  them  so  that  the  three  arrows  shall  chase  each  other  around 
tlie  triangle.  Then  imagine  each  arrow  of  the  unknown  forces  to  be  placed  one  at  a  time  without 
changing  its  direction,  with  its  head  at  the  apex  or  point  of  support  under  consideration,  as  if  it  were 
pushing  against  said  point.  It.  in  this  position,  it  coincides  witl),  or 
covers,  the  corresponding  member,  the  member  is  a  strut.  But  if  the 
arrow-head  then  touches  that  side  of  the  apex  which  is  opposite  to  the  member, 
the  member  is  a  tie.  This  simple  rule  will  save  much  perplexity,  and 

should  be  thoroughly  learned. 

When  there  is  hut  one  unknown  force,  and  it  is  found  to  form  a 

straight  line  with  the  resultant  of  the  known  ones,  then  its  arrow  must  point  in  the  opposite  direc- 
tion from  that  of  the  resultant.  We  may  add  that  of  the  two  unknown  forces  at  any  apex,  one  is  al- 
ways along  either  the  bow  or  the  string ;  and  must  plainly  be  a  strut  in  th.e  first  case,  and  a  lie  in  th« 
last  one.  For  the  present  we  will  call  it  the  chord-force.  The  other  unknown  force  will  be  along  a 
web  member.  Now  it  can  always  be  seen  at  once  that  the  resultant  and  the  chord  force  together  tend 
to  displace  the  apex  at  which  they  act,  by  moving  it  either  outwards  or  inwards,  from  or  towards  the 
truss  ;  and  if  we  simply  consider  that  it  is  the  duty  of  the  web-member  to  counteract  this  displacing 
tendency,  we  shall  have  no  trouble  in  deciding  whether  it  must  for  that  purpose  pull  or  push  at  the 
apex,  or  in  other  words  be  a  tie  or  a  strut. 

RKM.  7.  After  having  found  the  resultant  of  the  known  forces,  the  forces  themselves  may  be  con- 
sidered as  tio  longer  existing. 

REM.  8.    To  find  the  strains  correctly  requires  g-reat  care  and 

attention.  Plain  as  the  foregoing  remarks  are,  the  student  will  in  his  first  attempts  probably  com- 
mit many  errors.  A  little  practice  however 'will  rectify  this. 

A  good  metallic  parallel  ruler  on  rollers,  and  about  18  inches  long, 

is  almost  indispensable.  Paper  ruled  in  squares  facilitates  the  work.  The  lead-pencil  must  be  kept 
sharp,  and  the  lines  should  be  drawn  lightly'.  .  A  scale  of  from  %  to  %  of  an  inch  to  a  foot  or  ton  will 

generally  be  found  convenient.  It  will  be  difficult  to  find  all  the 
strains  to  within  the  nearest  .1  or  .2  of  a  ton,  or  even  more;  be- 
cause, as  the  work  progresses  errors  which  are  inappreciable  at  the  start  may  insensibly  enlarge 
themselves.  It  will  be  seen  from  our  table  of  bowstrings  of  80  ft  span,  p  274,  that  the  strains  on  some 
of  the  web  members  are  less  than  .1  of  a  toil  ;  so  that  the  diagram  may  even  with  great  care  mislead 
us  to  the  extent  of  100  per  cent,  or  more,  in  these  small  strains.  Fortunately  this  is  a  matter  of  little 
importance,  for  these  members  are  so  small  that  a  libeial  allowance  for  errors  involves  but  a  trifling 
waste  of  material.  In  the  larger  strains  errors  of  .1  or  .2  of  a  toft  are  of  no  consequence.  Never  con- 
aider  the  work  of  a  diagram  complete,  however,  until  after  testing  it  by  some  of  the  proofs  in  Art  19. 

Art.  19.  Example.  We  will  now  apply  the  foregoing;  re- 
marks to  the  bowstring:  truss.  Fi«f  2«.  Its  span  is  80  ft ;  its  rise  10 
ft.  The  bow  is  divided  into  8  equal  parts  ;  and  the  lower  apices  are  horizontally  half-way  between 
the  upper  ones.  The  trusses  are  assunied  to  be  7  ft  apart  from  center  to  center.  The  total  wt  of  the 
truss  and  its  load  is  supposed  to  be  equally  distributed  along  the  bow  only,  and  to  amount  to  10.4 
tons,  which  corresponds  to  40  tbs  per  square  ft  of  roof  covering.  This  gives  1:3  tons  for  a  full  panel 
load  ;  and  half  as  much,  or  .65  of  a  ton  resting  directly  (or  without  passing  along  the  web  members) 
on  each  abut ;  and  the  finding  of  these,  and  figuring  them  on  the  diagram  as  shown,  constitute  our 
"first  process,"  Art  11.*?  p  255. 

tfhis  done,  draw  at  the  abut  a  a  vert  line-ow  equal  by  scale  to  half  the  entire  wt  of  the  truss  and 
load,  minus  the  part  panel  load  which  rests  directly  upon  one  abut;  or  to  5.2  —  .65=4.55  tons.  This 
line  represents  the  vert  Upward  reaction  of  the  abut  against  that  portion  of  the  wt  of  the  half  truss 
and  load  that  causes  the  strains  which  we  are  about  to  seek.t  This  reacting  force,  which  is  a  known 
one,  balances  the  two  unknown  forces,  ae  along  the  bow,  and  ap  along  the  string.  To  find  these  we 
have  only  ( Rem  2)  to  consider  a  v  as  one  side  of  a  triangle,  and  from  its  two  ends  to  draw  two  meet- 
ing lines  on  and  vn  parallel  to,  or  in  the  same  directions  as,  the  unknown  forces.  Then  will  v  n 
give  by  the  same  scale  9.94  tons  strain  along  the  hor  string  ;  and  a  n  10.93  tons,  along  the  bow.  The 
side  an  coincides  with,  or  covers,  the  member  ae,  which  is  therefore  (Rem  6)  a  strut.  But  if  vn  is 
placed  with  its  arrow-head  at  a,  without  changing  its  direction,  il"will  come  to  the  opposite  side  of  a 
from  its  corresponding  member  ap,  which  is  therefore  a  tie. 

Now  let  us  go  to  the  apexp.  There  we  find  that  we  have  one  known  force,  fthe  9.94  tons  along 
ap)  balancing  three  unknown  ones,  namely  pq,pc,  and  p  e.  Hence  (Rem  2)  we 
cannot  now  find  these  last ;  therefore  we  leave  them  for  the  present,  and  try  at  the 
apex  e.  Here  we  have  two  known  forces,  namely  the  10:93  tons 

pushing  along  a  e,  and  the  1.3  tons  of  load  which  of  course  push  vert  downwards.  Both  these  being 
pushes,  neither  of  them  requires  to  be  reversed  ;  and  they  balance  two  unknown 
forces,  namely  ec  along  the  bow  ;  and  ep  along  the  oblique.  Hence  we  have  only  to  draw  fe  (see 
Fig  W)  to  represent  a  e  ;  and  x  e  to  represent  the  1.3  ton  load,  and  completing  the  parallelogram  fe 
xd.  draw  its  diagonal  de,  which  is  the  resultant  of/c  and  xe  ;  or  it  alone  would  balance  the  two 
unknown  forces.  By  measuring  d  e  by  scale  it  becomes  a  known  force.  Therefore  taking  it  as  one 
side  of  a  triangle,  from  its  two  ends  draw  d  s  parallel  to  the  bow  e  c,  and  e  a  parallel  to  the  oblique 

*;Remember  that  in  a  truss  of  any  form  it  is  only  when  the  stretches  along  which  a  load  is  uni- 
forriilv  distributed. are  equal,  that  the  panel  loads  are  also  equal;  or  that  the  portion  which  rests  di- 
rectly on  an  abut  is  just  half  &  panel  load. 

t  Kach  abut  of  course  reacts  vert  upwards  against  the  entire  half  wt  of  the  truss  and  its  load  :  but 
Inasmuch  as  that  portion  of  said  wt  which  rests  directly  on  an  abut,  does  not  reach  the  abut  by  way 
of  the  web  members,  and  therefore  has  nothing  to  do  with  their  •trains,  it  is  omitted  from  the  prwi- 
ure  iet  up  at  the  abut  for  ascertaining  said  strains. 


TRUSSES.  273 

ep,  thus  completing  the  triangle  dea.  Then  da  gives  by  scale  the  strain  10.56  along  ec ;  and  eg 
gives  .14  of  a  ton  along  ep.* 

Now  going  again  to  the  apex  p,  we  find  that  we  have  two  known  forces  p  a,  p  e,  both  pulls,  balanc- 
ing two  uniinown  forces  p  q,  p  c.  Therefore  as  at  Fig  Y.  take  pf  and  p  a  to  represent  the  two  known 
forces  ;  complete  the  parallelogram  fpsd;  draw  its  diagonal  p  d ;  and  taking  it  as  one  side  of  a  tri- 
angle, draw  d  o  parallel  to^>  q,  audp  o  parallel  toj>c.  Then  is  do  by  scale  10.01  tons  strain  along 

pq\  &ndpo  is  .10  of  a  ton  along  pc.    Going  to  c,  we  have  three  known 

forces,  ce,  cp,  and  the  1.3  ton  panel  load,  Jill  of  them  pushes,  so  that  none  of 
them  need  be  reversed;  and  balancing  co,  cq,  both  unknown.  In  this  case,  as  shown  at  Fig  V,  we 
must  draw  two  parallelograms,  begin uiug  with  any  two  of  the  known  forces ;  see  Art  36,  p  466.  Say 
we  begin  with  ex  representing  the  1.3  ton  load,  and  ce,  representing  the  10.56  tons,  on  these  two 
draw  the  parallelogram  cetx,  and  its  diagonal  c  t.  Then  draw  ca  to  represent  the  third  known  force 
cp  of  .10  of  a  ton  ;  and  on  it  and  the  diagonal  ct  draw  the  second  parallelogram  ctxa.  and  its  di- 
agonal ca.  This  last  diagonal  is  the  resultant  or  single  force  which  would  balance  the  two  unknown 
forces  co,  cq;  therefore  take  it  as  one  side  of  a  triangle,  and  from  its  two  ends  draw  two  meeting 
lines  parallel  to  said  unknown  forces,  and  measure  them  by  scale  to  obtain  the  amounts  of  those 
forces.  On  account  of  the  smallness  of  our  scale  we  have  not  shown  these  two  lines.  In  practice,  in 

first  diagonal.    In  this  manner  proceed  until  the  final  strains  on  08,  ar,  and  r  t  complete  the  whole. 

If  the  entire  uniform  wt  of  truss  and  load  is  assumed  to 
be  on  the  string  or  lower  chord,  as  in  Fig  23  a,  then  all  the  web  mem- 
bers become  ties ;  but  the  process  remains  unchanged.  Therefore  first  distribute  the  entire  wt 
among  the  lower  apice-*  and  abuts  by  our  "  first  process."  Then,  as  in  Fig  23,  draw  the  vert  line  o  v 
( —  half  the  enure  wt,  minus  what  rests  directly  on  one  abut)  and  from  it  find  the  strains  on  ae  and 
ap.  Then  going  to  e  we  have  one  known  force  ae  balancing  the  two  unknown  ones  ec  and  ep. 
After  finding  these  go  top.  Here  we  have  three  known  forces,  p  e,  p  a,  and  the  panel  load  ;  the  first 
two  of  which  are  pulls,  while  the  third  (resting  on  top  of  the  string)  is  a  push  vert  downwards. 
Therefore  we  will  reverse  this  last,  and  consider  it  as  being  a  vert  pull  pi;  and  draw  the  first  par- 
allelogram on  pi  and  p  a.  After  finding  the  resultant  of  the  three  pulls,  and  by  it  the  two  forces  p  q, 
p  c,  we  go  to  c.  Here  of  the  two  known  forces,  ec,  being  a  push ;  and  p  c  a  pull,  we  must  reverse 
one  of  them,  say  p  c ;  representing  it  by  an  arrow  t  c ;  and  complete  the  parallelogram  on  t  c  and  c  e. 

We  have  now  had  an  instance  of  all  the  cases  that  occur,  and  have  shown  how  to  manage  them. 

Proofs  of  accuracy  of  the  work.  The  resultant  of  the  strains  on 
those  members  (o  <  and  r  «),  Fig  23,  of  the  half  truss  that  meet  at  the  center  *,  of  the  bow,  and  of  half 
the  panel  load,  8,  at  the  center,  should  come  out  to  be  a  hor  line,  and  equal  to  the  hor  .strain  along 
the  center  stretch,  r  t,  of  the  string.  With  all  the  care,  however,  that  can  be  taken,  it  will  be  very 
difficult  to  make  the  coincidence  exact.  In  some  trusses  the  half  truss  will  have  three  members 
meeting  at  the  center  of  the  bow  ;  one  of  them  (a  center  vert  one)  belonging  partly  to  each  half  of 
the  truss.  Then  only  half  the  strain  on  this  one,  aa  well  as  half  the  center  panel  load,  is  to  be  used 
in  the  proof.  All  this  applies  to  any  form  of  truss  however  loaded. 

Again,  if  all  the  uniform  wt  of  the  truss  and  its  load  is  as- 
sumed to  be  on  the  hor  string,  and  if  the  string  •*  divided  into  equal, 

or  nearly  equal,  parts  by  the  web  members,  the  hor  strain  at  the  center 
should  be  equal,  or  about  equal,  to 

Wt  of  half  truss  and  load  X  .25  of  the  span 

Depth  of  truss. 

But  with  very  unequal  divisions  of  the  string,  such  as  will  rarely  occur,  this  formula  is  not  even 
roughly  approx. 

With  all  the  wt  uniformly  on  the  bow,  unequal  divisions  of  the 

string  have  no  effect  on  the  center  hor  strain ;  and  if  the  rise  does  not  exceed  about  one  tenth  to  one 
eighth  of  the  span,  and  the  bow  is  about  evenly  divided,  the  above  formula  will  be  nearly  as  approx 
as  when  the  load  is  on  the  string.  For  greater  rises,  however,  multiply  the  span  by  the  following 
multipliers  ^instead  of  by  the  .25  of  the  above  formula),  when  the  load  is  on  the  bow  ;  and  the  half  bow 
about  equally  divided  into  at  least  two  parts. 

Hise,  in  Farts  of  the  Span.  (Original.; 

.1  or  less      |      .15      |       .2       |      .25      |       .3       |       .333       |      .35      |       .4       |      .45      |       .5 

Multipliers. 
.246         |      .243     |      .239     |      .233     |      .226     {        .222       j      .219     |      .211     |      .202     |      .192 

RKM.  The  multipliers  for  intermediate  rises  may  be  taken  in  simple  proportion.  When  multiplied 
by  the  span  they  give  the  hor  dist  from  the  abut  to  the  center  of  gravity  of  one  half  the  loaded 
bow,  (as  the  .25  of  the  formula  gives  that  of  one  half  the  loaded  string,)  assuming  the  load  to  be 
concentrated  at  the  points  of  support  on  the  abut  and  at  the  apices.  It  is  this 
center  of  gravity  of  half  load  so  concentrated  that  must  be 
used  for  finding  the  strains  in  the  truss,  and  not  that  of  half  load  aa 

actually  evenly  distributed;  for  these  two  centers  of  gravity  under  these  two  aspects  may  differ 
greatly  from  each  other,  not  only  in  the  evenly  loaded  bow,  but  in  the  string,  if  divided  into  very  un- 
equal parts  by  the  web  members.  Even  the  number  of  divisions  of  the  loaded  bow  makes  some  dif- 
ference in  this  respect,  but  not  so  great  but  that  the  multipliers  in  the  above  table  will  be  correct  to 
within  about  three  per  ct  at  most  in  any  case  in  which  the  entire  bow  has  at  least  four  nearly  equal 
divisions. 

*  It  is  usual,  and  far  more  convenient,  to  draw  all  such  lines  at  the  apices  of  the  diagram  itself;  but 
on  account  of  the  smallness  of  the  scale  of  Figs  23,  and  23  a,  we  have  drawn  them  at  W,  T,  and  V,  to 
prevent  confusion.  Also  our  lines  are  not  drawn  to  scale  because  some  of  the  forces  are  too  small 
to  be  appreciable  with  so  small  a  scale. 


274 


TRUSSES. 


If  both  the  bow  and  the  string  are  uniformly  loaded,  it  is 

plain  that  the  multiplier  for  any  given  rise  must  be  somewhere  between  the  .25  of  the  formula,  and 
the  decimal  for  that  rise  in  our  table;  and  this  furnishes  an  easy  method  of  finding,  approx  enough 
for  practice,  the  hor  dist  from  the  abut  to  the  cen  of  grav  of  a  half  truss  thus  loaded.  Thus  afier 
allotting  to  the  bow  and  string  their  respective  proportions  of  the  entire  wt  of  the  truss  and  load, 
find  the  hor  dist  for  each  of  the  two  separately  ;  and  then  combine  them  ;  see  p  442. 

Table  of  approximate  strains  in  tons  on  Bowstring  trusses 

of  80  ft  span.  Trusses  7  ft  apart  from  center  to  center.  Load  (including  wt  of  trusses)  40  Ibs  per 
square  ft  of  roof  covering;  all  assumed  tj  be  uniformly  distributed  on  the  bow.  Bow  divided  into  8 
equal  parts;  like  Fig  23;  and  straight  fiom  apex  to  apex.  Lower  apices  half-way  hor  between  the 
upper  ones.  Each  column  of  the  table  commences  near  an  abut,  or  end  of  truss.  The  first  or  end 
web  member  in  the  columns  is  a  tie,  the  next  one  a  strut,  and  so  on  alternately  towards  the  center, 
as  in  Fig  23.  Below  the  table  is  given  the  wt  of  each  entire  truss  and  its  load. 


Rise  20  ft,  or  y±  Span. 

Rise  13J*  ft,  or  K  Span. 

Rise  10  ft,  or  ft  Span. 

Rise  8  ft,  or  y1^-  Span. 

Bow. 

TIB. 

WEB. 

Bow. 

TIE. 

WEB. 

Bow. 

TIB. 

WEB. 

Bow. 

TIE. 

WKB. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

7.01 

4.85 

0.38 

8.80 

7.50 

0.22 

10.9 

y.«4 

0.14 

]3.3 

.  12.5 

0.07 

6.02 

5.18 

0.35 

8.30 

7  72 

0.18 

10.6 

10.01 

0.10 

13.0 

12.8 

0.05 

5.56 

5.34 

0.25 

8.03 

7.78 

0.15 

10.4 

10.16 

0.07 

12.8 

12.7 

0.07 

5.34 

5.38 

0.25 

7.90 

7.88 

0.15 

10.3 

10.22 

0.07 

12.7 

12.7 

0.07 

0.10 

0.10 

0.05 

0.03 

0.10 

0.10 

0.04 

0.03 

(See  Note,  p.  263.) 

Total  wt  =  11.6  tons. 

Total  wt  —  10.75  tons. 

Total  wt  —  10.375  tons. 

Total  wt  —  10.25  tons. 

Art.  20.    The  braced  arch,  uniformly  loaded  each  way  from  the 

center,  Fig  23  W.  In  all  the  preceding  trusses,  as  also  in  the  left  hand  half  Fig  23  Y,  there  is  com- 
pression in  the  upper  member,  whether  straight  or  curved,  and  tension  in  the  lower  one.  But,  as  in 
either  half  of  Fig  23  W,  (supposing  m  n  to  be  one  continuous  straight  line  ;  and  a  gp  c  one  continu- 
ous unbroken  curve;)  or  in  the  right  band  half  of  Fig  23  Y.  (supposing  the  curve  e,  s  c  to  be  com- 
plete,)  the  truss  may  be  so  constructed  that  both  members  shall  be  in  compression. 


When  this  is  the  case,  in  order  to  be  able  to  begin  at  an  abut  and  work  up  the  strains  towards  the 
center,  we  must  have  not  only  as  before  the  vert  line  c  v  Cwhich  represents  the  half  wt  of  the  entire 
truss  and  its  load,  minus  the  part  panel  load  that  rests  directly  on  the  abut,)  but  also  the  horizontal 
line  h  c,  which  represents  the  hor  pres  at  the  center  of  the  truss;  for  when  the  lower  member  is  not 
a  tie,  the  abuts  take  the  place  of  a  tie,  and  react  against  this  hor  pres,  as  well  as  against  the  weight 
of  the  truss.  When  only  the  upper  member  is  compressed,  we  have  seen  that  it  is  easy  to  find  this 
hor  pres  by  either  the  formula  or  the  table  for  that  purpose  in  the  preceding  Art,  under  the  head 
"  Proofs  of  Accuracy."  When  both  members  are  compressed,  we  must,  in  order  to  find  the  hor  line 

h  c,  assume  that  all  the  compression  at  the  centre  is  borne  by 
only  one  of  them  ;  and  fiud  its  amount  by  the  formula  or  the  table  just  al- 
luded to. 


TRUSSES.  275 

Thus,  as  on  the  left  hand  half  of  Fig  23  W,  we  may  assume  it  all  to  be  sustained  at  d  ;  or  as  on  the 
right  hand  half,  at  o,  between  d  and  the  supposed  curve  of  the  arch.  On  the  right  hand  half  of  Fig 
23  Y,  it  is  all  supposed  to  be  borne  at  t.  In  Fig  23  W,  the  upper  or  bor  member  is  then  omitted  from 
•d  to  1;  as  well  us  the  portion  from  g  top  of  the  arch,  unless  I  o  be  inclined  downward  until  o  coin- 
cides with  the  actual  arch.  In  Fig  23  Y  also  the  portion  e  s  of  the  arch  is,  or  may  be,  omitted.  In 
either  flg  sham  arches  may  be  used  to  fill  up  the  intervals,  if  appearance  requires  it.  In  Fig  23  Y, 
all  to  the  right  of  y  c,  and  to  the  left  of  n  a,  is  considered  as  forming  no  part  of  the  truss  proper. 

Having  found  the  hor  strain  (h  c,  Figs  23  W,  23  Y)  at  the  center,  and  knowing  the  wt  (v  c)  of  the 
half  truss  and  load  minus  the  part  panel  load  that  rests  directly  on  the  abut,  draw  them  as  shown, 
and  find  their  resultant  r  c.  Then  use  this  resultant  precisely  as  the  vert  line  a  v,  Figs  23  and  23a, 
is  used:  namely,  by  drawing  from  its  ends  two  lines  for  finding  the  first  two  unknown  forces;  then 
proceed  precisely  as  in  those  figs.  The  strains  are  given  in  Figs  23  W  and  23  Y,  to  enable  the  stu- 
dent to  work  them  out  for  himself. 

Proof  of  accuracy  of  the  'work.    If  all  has  been  done  correctly,  the 

last  resultant  near  the  center  of  the  truss  will  be  in  a  straight  line  with  the  last  member;  the  strain 
alons;  which  it  represents.  Also,  the  resultant  of  those  members  of  the  half  truss  which  meet  at  the 
center  of  the  truss,  and  of  half  the  center  panel  load,  should  come  out  a  hor  line  equal  to  the  calcu- 
lated hor  strain.  But  as  in  the  bowstring,  &c.  it  is  almost  impossible  to  secure  a  perfect  agreement. 
It  would  be  well  to  test  the  strains  near  the  couter,  especially  the  very  small  ones,  by  the  principle 
indicated  by  the  following  remark. 

RKM.  It  is  not  necessary  to  begin  at  an  abut  in  order  to 
work  out  the  strains:  for  after  having  disposed  the  load  properly  among 
the  apices,  and  calcuhited  the  hor  strain  at  the  center  of  a  truss,  we  may  employ  Haid  strain,  and  one 
half  of  the  panel  load  at  the  center,  as  two  known  forces  acting  against  the  half  truss  at  the  center; 
find  their  resultant;  and  resolve  it  along  the  two  members  of  said  half  truss  that  meet  there;  and 
thus  work  on  down  to  an  flbut.  The  line  representing  the  hor  strain  must  evidently  be  drawn  as  if 
pnslii/iy  nyaitixt  the  half  truss  whose  strains  are  sought.  Tllis  remark  ap- 
plies also  to  bowstring  trusses,  and  others,  as  Figs  10  and  11,  (fee., 

in  which  we  have  two  or  more  known  forces  acting  against  our  half  truss  ai  the  center;  and  not 
more  than  two  unknown  forces  of  our  half  truss  meeting  there  also. 

Cantilevers.  Suppose  the  half  o,  w,  c.  of  the  braced  arch  Fig  23  W,  to  be  a 
trussed  cur.tilever,  with  n  and  c  firmly  built  into,  or  attached  to  a  wall ;  and  loaded  either  along  the 
top  o  »,  or  otherwise.  Then  to  calculate  the  strains,  begin  with  only  the  load  concentrated  at  the 
outer  end  or  apex  o,  as  a  given  known  force,  and  resolve  said  force  along  o  I  and  op;  and  so  on  to  n 
and  c  taking  in  on  the  way  the  loads  at  the  other  apices  as  before.  The  same  with  cantilever  d  m  a, 
Fig  23  W;  or  with  one  of  the  Warren.  &c.,  truss  with  parallel  chords;  or  one  with  both  chords 

curved.    The  upper  chord  in  all  will  be  in  tension;  the  lower  in 

compression.    A  revolving  truss  drawbridge,  when  open,  assimilates 

to  two  cantilever*  joined  back  to  back. 

Art.  2OJ4.    This  fig  represents  an  „ 

opened  swing-bridge  sup- 
ported on  rollers  on  the  pier  jj,  and  by 
tie-rods  a  o,  at,  &c.  All  the  wt  of  oe 
is  upheld  by  a  n ;  that  of  e  s  by  a  i,  «fec. ; 

and  that  of  st  by  the  rollers.    Draw  o        ^  /      T     \          »X'i          ^^i 

6  and  ik  vertical  to  represent  the  wts  "^xsjy^/x/^/^ 
of  of.,  and  e  s ;  and  draw  6  w,  k  n  hori-  liil 

zontaJ.    Then  will  ow,andtn  give  the  '"\ 

strains  along  a  o,  and  a i.  Also  bin  will  give  a  hor  compressive  strain  reaching 
from  o  to  c;  and  kn  one  reaching  from  i  to  c.  If  this  truss  should  be  suspended 
from  the  top  of  the  post  ac  instead  of  resting  on  rollers,  then  the  tie-rod  at  would 
uphold  all  the  wt  from  e.  to  c  ;  and  the  post  would  sustain  the  wt  of  the  entire  truss. 
If  the  truss  be  inverted  the  strains  will  remain  the  same  in  amount,  but  reversed  in 
kind.  If  one-half  of  the  Fig  represent  a  projecting  platform,  with  c  fixed  in  a  wall, 
the  calculation  will  be  as  before. 

Art.  20%.  Moving  loads,  and  counterbraciiig.  We  shall  merely 
glance  at  this  subject  in  a  superficial  manner;  without  pretending  to  discuss  the 
comparative  merits  of  diff  forms  of  bridge  truss.  As  already  stated  in  Remark,  Art 
10,  counters  are  not  needed  n\  a  truss  uniformly  loaded  from  end  to  end.  The  mode 
which  we  propose  for  finding  the  strains  produced  in  beam  trusses  by  moving  loads, 
differs  from  that  commonly  used.  It  renders  the  principle  more  apparent,  and  is 
equally  reliable  and  safe. 

To  assist  in  illustrating  the  calculations  for  strains  produced  by  moving  loads,  we  shall  suppose 
the  following  Fig  23 /,  to  represent  the  two  trusses,  combined  into  one;  of  a  span  m  z  of  120  feet; 
divided  into  6  panels,  each  20  ft  long,  and  30  ft  high.  Or,  in  other  words,  we  (for  convenience  of 
calculating)  suppose  all  to  be  borne  by  one  truss  only.  The  weight  of  this  double  truss  is  48  tons; 
or  8  tons  per  panel ;  or  A  ton  per  ft  run.  The  floor.'and  its  several  timbers,  such  a*  cross  girders, 
hor  bracing,  &c,  which,  although  not  portions  of  the  truss  proper,  are  essential  to  the  bridge;  and 
to  be  considered  as  so  much  permanent  load,  equally  distributed  along  the  truss,  are  assumed  to 
weigh  an  additional  '<J4  tons  ;  or  4  tons  per  panel ;  or  .2  ton  per  ft  run  of  the  double  truss.  There- 
fore, the  truss  and  its  accompaniments  together,  or  in  other  words,  the  bridge  superstructure,  weighs 
72  tons ;  or  12  tons  per  panel ;  or  .6  of  a  ton  per  ft  run.  Since  each  panel  is  20  ft  long,  and  30  ft  high, 
each  oblique  is  1/202  -}-  30S  —  3fr  ft  long;  and  the  nat  sec  of  the  angle  which  it  forms  with  a  vert,  (or 
of  the  brace  angle,)  t-^Jr£^S  =M  =  Uj  Md  **  nai'toBg  °f  lh*  "°"5  »Dgle  to  %w£* 


276  TRUSSES. 


of  loaded  cars  . 
even,  exceeded 


i  ion;  is  an  ordinary  allowance,  anu  a  saie  one ;  iiiaismucii  as  intiue 
never,  perhaps,  weigh  1  ton  per  foot.  Until  lately  but  few  engines, 
1  that  limit.  Now  many  heavy  freight  locos  weigh  1.4  tons  per  ft. 


thus  bringing  a  great  strain  upon  each  individual  web-member,  as  it  crosses  the 
bridge.  This  point  is  one  of  the  greatest  importance  in  arranging  the  preliminary 
data  on  which  to  form  the  basis  for  calculating  the  strains  on  a  bridge  truss ;  and  we 
should  allow  liberally  for  the  greatest  load  that  can  possibly  be  brought  upon  one 
panel  at  a  time.  When  the  panels  are  quite  short,  each  one  will  have  of  course 


will  by  degrees  ue  mcreatseu.     mis  siiouiu  ue  coiiBiuereu  in  uesigmng  orit 
do  not  think  they  should  be  proportioned  for  less  than  6  tons  on  a  driver 

The  fact  that  the  engine  produces  upon  each  panel  in  succession,  a  greater  strain 
than  an  equal  length  of  loaded  cars  can  do,  causes  a  modification  of  the  calcula- 
tions; as  will  be  seen  as  we  proceed. 

Such  data  must  be  prepared  before  beginning  the  calculations.  Our  assumptions 
in  the  case  before  us  have  been  made  entirely  with  reference  to  ease  of  calculating 
our  example  with  but  few  figures.  Our  truss,  together  with  a  load  of  cars  extending 
from  end  to  end,  will  weigh  198  tons;  or  l.t>5  tons  per  ft  run :  or  33  tons  per  panel. 

Having  prepared  a  diagram,  begin  by  finding  the  strains  caused  in  only  the  verts 
and  obliques,  uy  the  bridge  itself,  half  the  wt  of  the  truss  aione  being  considered 
to  be  on  the  top  chord,  or  8  tons  at  each  lower  apex,  and  4  tons  at  each  upper  one. 
Then 

Oa         tp  8  tons.  On  co,  eg  each    7.2  tons. 

"  Ao,  cq  each  14    "  "/in,  cr      "    21.6    " 

"in,  ar     "    26    "  "  im,ax      "    36.0    " 

All  of  which  are  set  down  at  their  feet.    We  shall  write  the  moving  load  strains 

near  their  tops ;  and  the  total  strains  at  their  centers. 

We  now  have  the  strains  on  the  web  members,  resulting  from  the  truss  itself,  and  from 
its  uniformly  distributed  permanent  load  of  floor,  &c ;  and  are  prepared  to  begin  with 
the  additional  strains  resulting  from  the  moving  load.  We  find  those  on  the  coun- 


placed  uniformly  along  y  z,  its  whole  wt  may  be  assumed  to  be  concentrated  at  the 
middle  point  r ;  and  that  ^  of  it  will  be  borne  by  the  nearest  abut  x\  and  only  \  by 
the  farthest  abut  m.  So  also  with  the  panel  length  y  w  of  cars ;  its  wt  being  supposed 
concentrated  at  q,  f  of  it  must  be  borne  by  x ;  and  |  of  it  by  m.  Of  the  cars  w  v,  £ 
are  borne  by  x ;  and  -|  by  m.  Of  those  along  u  w,  f  are  borne  by  x ;  and  %  by  m ;  of  those 


ana  is  strained  by  it  all ;  out  wnen  mis  enure  wi  readies  m«  top  ui  me  vc.ii,  umj 
the  portion  f ,  as  above  stated,  go  down  to  ax  to  the  abut  x.    The  remaining  %  gooa 


*  Not  strictly  correct;  but  a  safe  and  advisable  assumption  in  practice,  when  the  entire  wt  of  tbt 
engine  oan  stand  on  one  panel. 


TRUSSES. 


•277 


to  the  other  abut  m.  By  this  process,  then,  we  find  that^  of  the  engine,  or  30  tons ;  £  of  the 
cars  y  u>,  or  14  tons  ;  -|  of  the  cars  wr,  or  10.5  tons;  f-  of  WM,  or  7  tons;  %  ofws,or3.5 
tons;  and  ^  of  s  »i,  or  .44  of  a  ton;  making  65.44  tons  in  all,  are  borne  by  the  abut 
x,  when  the  moving  load  is  in  the  position  shown  in  our  fig.  This  load  of  651  tons 
passes  from  r  to  a;  and  from  a  to  x.  The  calculation  is  made,  as  s.hown-under  our 
fig;  below  the  heading  "Engine  at  r." 

We  next  suppose  the  engine  to  back  into  the  position  y  w  ;  with  the  train  reaching 
to  m\  and  with  no  load  between  y  and  x.  In  this  position  the  strains  on  q c,  and  cr, 
are  greater  than  in  any  other ;  and  by  a  process  precisely  as  before,  and  as  shown 
under  our  fig,  below  the  heading  "  Engine  at  7,"  we  find  that  45.44  tons  of  the  moving 
load  go  to  the  abut  x ;  passing  from  </  to  c ;  and  from  c  to  r,  on  their  way  to  it.  Then 
placing  the  engine  at  w  r,  the  cars  reaching  to  m,  and  no  load  between  w  and  x,  the 
strains  on  p  e  and  e  7,  are  the  maximum  that  they  can  be  subjected  to  by  an  engine 
and  train  ;  and  are  found  like  the  others,  as  shown  under  "  Engine  at  p."  We  haVe 
BOW  reached  the  center  of  the  truss,  and  have  obtained  data  for  the  greatest  strains 
that  can  occur  on  the  verts  and  main  obliques  on  one-half  of  it.  So  far  as  regards 
the  corresponding  members  on  the  other  half,  we  stop  here  ;  for  it  is  manifest  that 
an  engine  and  train  crossing  the  bridge  in  the  opposite  direction,  must  produce  the 
same  maximum  strains  upon  them  as  we  have  already  found  for  the  others.  That  is, 
i  m  will  be  strained  the  same  as  a  x ;  n  i  the  same  as  r  a ;  e  o  the  same  as  e  q,  and  so 
on,  by  a  returning  train. 

But  as  yet,  we  have  not  sufficient  data  for  determining  where  counters  will  be 
required.  To  obtain  it,  we  draw  the  two  lines  h  p,  and  i  o,  parallel  to  the  obliques 
on  the  opposite  side  of  the  truss ;  and  taking  it  for  granted  for  the  present,  that  these 
two  lines  are  counters,  we  back  the  engine  to  ?>  M,  and  then  to  u  s ;  performing  each 
time,  calculations  precisely  similar  to  the  former  ones :  and  as  shown  under  the  head- 
ings "  Engine  at  o,"  &c.  We  have  stated  in  a  former  Art,  that  a  counter  is  most 
strained  when  the  moving  load  extends  from  it  to  the  nearest  abut.  Therefore,  a 
counter  hp  would  be  most  strained  with  the  engine  on  v  u,  and  the  train  reaching 
to  m ;  and  i  o,  by  the  engine  on  u  s,  with  a  train  to  m. 


s     n     u     o 

Span  120  ft. 


With  Engine  at  p, 

there  go  to  x. 
Tom. 
18      =  3-6  engine. 
7      =  2-6  v  u. 
3.5    =1-6  us. 
.44  =  1-24  B  m. 

28  94  ao  to  x 

MAIN    BRACES. 
With  Engine  at  q, 

there  go  to  x. 
Tons. 
24       =  4-6  engine. 
10.5    =  3-6  w  v. 
7       =  2-6  v  u. 
3.5    =  1-6  u  s. 
.44  =  1-24  s  m. 

;, 

With  Engine  at  r, 

there  go  to  x. 
Tons. 
30      =  5-6  engine. 
14       =  4-6  y  w. 
10.5    =3-6wv. 
7       =  2-6  v  u. 
3.5    =1-6  us. 

1  2  nat  sec 

45  44  go  to  x 

84  728  on  e  q 

65.44  go  to  x. 

froTYi  load  only. 

54  528  on  c  r 

from  load  only. 

78.528  on  a  x 
from  load  only. 

278 


TRUSSES. 


Vertical  e  p. 
28.94  go  to  X. 
18  =  remaining 
3-6  engine. 

VERTICALS. 

Vertical  c  q. 
45.44  go  to  x. 
12  =  remaining 
2-6  engine. 

See  Rem,  p  375. 
Vertical  a  r. 
65.44  go  to  x. 

6      =  remaining 
1-6  engine. 

46.94  on  e  p 
from  load  only. 

57.44  on  c  q 
from  load  only. 

71.44  on  a  r 
/row  Zoad  o»Zy. 

COUNTERS. 


Counter  i  o. 
Engine  at  »i. 

6      =1-6  engine. 
.44  =  1-24  s  m. 

6.44  gotnx. 
1.2  noi  stc. 

7J28  on  i  o 


Counter  h  p. 
Engine  at  o. 

12       =  2-6  engine. 
3.5    =  1-6  u  s. 
.44  =  1-24  s  m. 


15.94  go  to  x. 
1.2  not  sec. 


19.128  on  h  p 
from  load  only.* 


Supposing  the  calculations  to  have  been  made  as  above,  we  have  thereby  found  the 
several  loads,  say  65.4;  45.4;  28.9;  15.9;  and  (5.4;  which  go  to  the  abut  a?,  from  the 
several  points  r,  q,  p,  o,  n,  when  the  engine  is  at  each  of  those  points  in  succession; 
with  a  train  reaching  in  each  case  to  m.  Each  of  these  loads,  in  travelling,  as  it  were, 
from  its  particular  point,  to  the  abut  x,  first  ascends  to  the  top  of  its  own  vert ;  and 
then  descends  along  the  adjacent  oblique ;  producing  in  said  oblique  a  strain  as  much 
greater  than  the  load  itself,  as  the  length  of  the  oblique  w  greater  than  its  vert 
spread.  Now,  each  of  our  obliques,  as  before  stated,  is  36  ft  long,  or  1.2  times  as  long 

36 

as  its  vert  spread  30  ft,  (or  the  height  of  truss;)  that  is,  —  =  1.2;  which  1.2  is  there- 
fore the  nat  sec  of  the  angle  which  any  of  our  obliques  forms  with  a  vert  line.  There- 
fore, to  find  the  strain  which  each  of  our  moving  loads  produces  on  the  oblique  next 
to  it  on  its  way  to  the  abut  x,  mult  each  said  load  by  1.2,  as  in  the  above  calculations. 
We  thus  obtain  for  these  strains,  78.5  tons  on  a  x  ;  54.5  on  c  r ;  and  34.7  on  e  q :  all 
which  write  near  the  tops  of  the  obliques,  as  in  our  fig,  in  which  we  omit  some  of 
the  decimals,  to  avoid  crowding  it. 

Next,  as  regards  the  strains  from  the  moving  load  alone,  upon  the  verts,  we  must 
bear  in  mind  that  these  last  have  to  sustain  not  only  the  loads  just  considered,  which 
pass  up  them  on  their  way  to  the  abut  x;  but  also  the  remaining  portion  of  the  wt 
of  the  engine;  which  also  passes  up  them,  on  its  way  to  th*  farthest  abut  m.  Tims, 
the  end  vert  a  r  sustains  the  entire  wt  of  the  engine ;  whereas  our  load  of  65.4  tons 
before  found,  at  r,  includes  only  -|  of  it.  The  remaining  %,  or  6  tons,  must  there- 
fore be  added  to  it,  as  is  done  under  the  head,  "  Vertical  a  r,"  making  71.4  tons  strain 
on  a  r,  from  the  moving  load  only.  Write  this  near  the  top  of  a  r.  For  the  same 
reason,  at  the  next  vert  c  q,  we  add  to  the  load  45.4  before  found,  that  goes  from  q  to 
a;,  tha  remaining  |-,  or  12  tons,  of  the  engine,  that  go  to  the  abut  m,  after  ascending 
to  c.  This  we  have  done  under  the  head  "  Vertical  c  g,"  thus  getting  f,7.4  tons  for 
the  total  moving  loud-strain  on  c  q.  And  so  at  the  center  vert  ep,  by  adding  18  tons, 
or  the  remaining  f  of  the  engine,  that  go  to  aw,  we  get  46.9  tons  total  load-strain 
on  that  member;  to  be  written  near  its  top.  The  strains  on  the  verts  of  the  other 
half,  p  m,  of  the  truss,  are  not  to  be  calculated.  They  will  be  the  same  as  those 
already  found.  Having  now  the -loud-strains  written  near  the  tops  of  the  verts ;  and 

*  Mr  De  Volson  Wood,  the  accomplished  professor  of  Mathematics  and  Mechanics  in  the  Stevens 
Institute  of  Technology,  author  of  Treatise  on  Bridges  and  Roofs,  has  shown  that  our  method  for 
moving  loads  is  not  strictly  correct,  inasmuch  as  it  makes  the  strains  on  the  web  members  somewhat 
too  great.  It  is  therefore  at  least  safe. 


TRUSSES.  279 

the  truss  strains  near  their  feet  ;  we  have  only  to  add  these  together  to  find  the  total 
strain  on  each  ;  namely,  97.4  tons  on  a  r  ;  71.4  on  c  q  ;  and  54.9  on  ep.  We  have  now 
done  with  the  main  obliques,  and  the  verts. 

For  the  counters,  we  go  to  the  other  side,  p  m,  of  the  truss.  Beginning 
with  the  panel  chop,  we  examine  its  two  diags  /<p,eo,  and  see  that  with  the  engine 
at  v  u,  and  the  cars  reaching  to  w,  there  is  produced  in  the  counter  h  p  its  maximum 
strain  of  19.1  tons  ;  which  tends  to  cause  in  the  truss  the  kind  of  derangement  shown 
in  Figs  9^,  9%,  and  9  6.  Now,  to  resist  this  derangement,  there  is  nothing  but  the 
7.2  tons  produced  by  the  truss,  floor,  &c,  upon  the  opposite  diag  eo  of  the  same  panel. 
Since,  therefore,  the  deranging  effect  of  the  load  is  greater  than  the  preventive  effect 
of  the  wt  of  the  truss,  there  must  be  a  counter  at  ftp,  able  to  bear  a  strain  equal  at 
least  to  the  diff  between  the  two,  or  to  19.1  —  7.2  =  11.9  tons. 

We  now  go  to  the  next  panel,  hino.  But  here  we  find  that  the  deranging  effect 
of  the  load  on  the  counter  i  o,  with  the  engine  at  u  s,  is  but  7.7  tons;  while  the  pre- 
ventive effect  of  the  wt  of  the  truss,  exerted  through  the  opposite  diag  h  n,  is  21.6 
tons.  Hence,  the  moving  load  can  produce  no  derangement  of  the  truss  ;  and  con- 
sequently the  counter  i  o  may  be  omitted.  On  this  same  principle  each  panel  on  one 
side  of  the  truss  must  be  examined,  when  there  are  many  of  them  ;  and  the  insertion 
or  omission  of  counterbraces  be  determined  upon.  When  we  thus  arrive  at  a  panel 
at  which  no  counter  is  reqd,  none  will  be  needed  between  it  and  the  nearest  end  of 
the  bridge.  Similar  counters  will,  of  course,  be  needed  on  the  other  side  of  the  truss. 
In  practice  it  is  better  to  retain  the  first  apparently  unnecessary  counterbrace  ; 
counting  from  the  center  of  the  truss.  Thus,  although  calculation  shows  i  o  to  be 
unnecessary,  it  is  well  to  retain  it.  The  lighter  a  bridge  is,  in  proportion  to  its 
moving  load,  the  greater  will  be  the  number  of  panels  requiring  counters. 

The  strains  on  the  chords  are  greatest  when  the  truss  is  loaded 
from  end  to  end  ;  and  for  Fig  23  /,  as  well  as  for  Fig  23  ^,  may  readily  be  calculated 
by  Art  12,  p  258  ;  or  found  by  a  diagram  with  a  max  load.  Or  sufliciently  close  for 
most  purposes,  the  hor  strain  along  either  chord  at  the  center  of  the  truss  where 
the  strain  is  greatest,  will  be  equal  to 

Total  weight  of  truss  and  load  X  span. 
8  times  the  depth  or  height  of  truss. 


Finally,  each  of  all  the  strains  in  our  double  truss  must  bo  div  by  2  ;  for  propor- 
tioning them  among  the  two  actual  trusses,  which  we  have  all  along  supposed  (for 
convenience)  to  be  combined  into  one. 

When  the  load  at  the  center  of  the  truss  (as  the  load  a  a,  Figs  23  c,  and  23  d,)  is 
sustained  directly  by  the  center  obliques  dn,  df,  first  find,  and  write  upon  the  dia- 
gram, precisely  as  in  our  foregoing  case,  all  the  strains  produced  by  the  bridge  itself; 
all  the  portions  of  load  that  go  to  the  abut  s,  Figs  23  c,  23d;  the  total  strains  on  the 
main  obliques  of  the  side  cs,  of  the  truss  ;  and  those  on  the  counters  of  the  other 
side  ;  in  short,  all  but  the  strains  on  the  verts.  Finally,  for  the  total  strain  on  the 
end  vert  k  o,  (*  £,  «  f,  are  not  truss-verts,  but  uprights  for  upholding  the  truss,)  add 
together  the  strain  produced  on  it  by  the  truss,  and  the  load  that  goes  to  the  abut  s,  (not 
the  strain  on  the  oblique,)  from  the  next  point  of  support  h  toward  the  center  of  the 
truss.  For  the  total  strain  on  the  next  vert  hg,  add  together  its  strain  from  the  truss  ; 
and  that  of  the  load  that  goes  to  s  from  the  next  point  e  toward  the  center.  And  so 
with  the  total  strain  on  «/;  and  last,  for  that  on  the  center  vert  c  d,  add  together  its 
truss-stiain,  and  the  load  that  goes  to  s  from  the  point  I  on  the  opposite  side  of  the 
truss. 
,  We  will  now  proceed  to  the  Warren  truss. 

Art  2O%.  The  Warren  or  triangular  truss.  In  this  truss  the  pro- 
cess for  finding  the  strains  from  moving  loads,  is  not  precisely  the  same  as  in  the 
preceding  one.  The  following  Fig  23#,  represents  such  a  truss  ;  in  which  the  span, 
height,  weight,  and  number  of  panels  will  be  taken  to  be  the  same  as  in  the  foregoing 
Fig  '/3/.  Here  the  dotted  web-members  which  supplant  the  ties  in  Fig  23/,  are  not 
vert  ;  but  inclined  to  the  same  extent  as  the  struts  or  braces.  Hence  the  hor  stretch 
of  each  oblique  will  be  but  //a//  the  length  of  a  panel,  or  10  ft  ;  or  only  half  as  great  as  in 
Fig  23/.  Consequently,  the  length  of  each  oblique  will  be  V  10*  -f  30a  =  31.6  ft  ; 

and  the  nat  sec  of  the  brace  angle  will  be  -r^-=  1.05;  and  the  nat  tang  of  the  same 

10  ^" 

angle  will  be  —  =  .333.    All  the  other  data  being  the  same  as  in  the  foregoing  ex- 


280 


TRUSSES. 


ample,  prepare  a  diagram,  and  from  it  find  the  strains  on  the  obliques  from  the 
wt  of  the  bridge  alone ;  one-half  of  the  wt  of  the 


truss  only  being  supposed  to  be  on  the  top  chord,  making  4  tons  at  each  upper 
apex,  while  the  other  half  wt  of  truss,  with  the  whole  wt  of  floor,  make  8  tons  at 
each  lower  apex.  The  strains  then  will  be  as  follows : 

On  p  d,    4.2  tons  On  e  n,  21     tons 

"   d  o,    8.4    "  "  n  i.  29.4    " 

"  oe,  16.8    "  "  i  in,  33.6    " 

as  set  down  at  their  feet  in  the  Figure. 

Having  now  the  correct  strains  arising  from  the  weight  of  the  truss  and  floor ; 
next  find,  precisely  as  in  the  preceding  example,  how  much  of  the  moving  load  will 
go  to  x  when  the  engine  is  at  r,  as  in  the  fig,  with  the  train  reaching  to  m;  and 
afterward  with  the  engine  at  q,p,  o,  and  n,  in  succession.  These  loads  will  of  course 
be  the  same  as  in  the  former  example,  namely  : 

Engine  at  n.        Engine  at  o.         Engine  at  p.         Engine,  at  q.         Engine  at  r. 

6.44  go  to  x.        15.94  go  to  x.        28.94  go  to  x.        45.44  go  to  x.        65.44  go  to  x. 
Multiplying  each  of  these  by  the  nat  sec  1.05,  we    get   the   compressing   strain* 
which  they  produce  on  the  end  oblique  strut  a  x\  and  on  the  other  obliques  that 
are  parallel  to  it ;  namely  : 

One  o.          On  d  p.  On  c  q.  On  b  r.  On  a  x. 

6.76  16.73  30.39  47.71  68.71 

which  write  near  the  tops  of  said  obliques,  as  done  in  our  fig.  But  they  also 
produce  precisely  the  same  amount  of  strains,  in  the  shape  of  tensions  or  pulls,  on 
the  respective  dotted  ties  which  carry  them  to  the  struts  between  x  and  p ;  and  on  the 
struts  which  carry  them  to  the  dotted  ties  between  m  and  p.  Write  them  all  near 
the  heads  of  said  obliques  also,  as  is  done  in  our  fig ;  except  that  we  have  there  used 
but  one  decimal,  to  avoid  crowding. 

Now,  if  on  the  half  truss  x  p,  we  add  together  the  strains  written  at  the  head  and 
foot  of  each  oblique  separately,  the  sums  will  be  the  total  or  max  strain  (compres- 
sive  on  the  struts,  arid  tensile  on  the  ties,)  which  said  obliques  will  be  subjected  to  by 
the  passage  of  the  tra»n;  approximate  enough  for  practice ;  but  a  little  in  excess  on 
some  of  them,  and  therefore  safe.*  These  max  strains  are  written  at  the  middle  Df 
each  oblique  between  x  and  p.  They  will  of  course  be  the  same  on  the  other  half 
of  the  truss,  when  the  train  crosses  in  the  opposite  direction. 

Finally,  as  to  con  liter  bracing,  (which,  in  a  Warren  truss,  consists  simply  in 
finding  which  obliques  may  have  to  act  as  both  ties  and  struts,)  we  go  to  the  other 
half,  m  p,  of  the  truss.  Beginning  with  the  oblique  d  p,  we  see  that  with  the  engine 
at  o,  and  the  train  reaching  to  m,  the  deranging  compressive  strain,  1C.73  tons,  of  the 

*  This  arises  from  our  here  neglecting  what  in  the  foregoing  example,  of  Art  20)4,  we  called  "  the 
remaining  %,  |,  &c,  of  the  Engine,"  which  go  to  the  abut  m.  Thus,  whe»  the  engine  is  at  r,  and 
the  train  reaching  to  m,  %  of  the  ena;  alone  goes  to  m.  It  first  passes  up  from  r  to  b  ;  and  since  the 
oblique  r  b  is  a  strut,  ami  since  the  %  eng  acts  on  it  as  a  pull,  it  follows  thr>,t  r  b  is  thereby  relieved 
from  pres  to  the  amount  of  %  eng.  From  6  this  %  eng  passes  down  6  q  as  a  pre» ;  but  since  b  q  is  a 
tic,  it  thus  becomes  relieved  of  that  much  pull.  From  q.  this  ^  passes  up  the  strut  q  c,  as  pull ; 
thus  relieving  it  from  that  much  pres;  an4  so  on  alternately,  until  the  %  eng  reaches  m.  In  the  same 
way,  when  the  eng  is  at  q  &  of  it  alone  pass  up  q  c,  down  c  p,  and  so  on  to  m,  relieving  each  oblique 
as  before;  but  to  twic-i  the  extent.  Arriving  at  q,  (or  in  any  ease,  at  the  panel  next  before  the  center 
ene  p.)  and  making  the  allowance  for  the  remainder  of  the  engine,  we  need  go  no  farther;  for  the 
same  result  will  take  place  on  the  other  half  of  the  truss  when  the  train  crosses  it  in  the  opposite 
direction.  In  practice  we  should  recommend  the  omission  of  small  considerations  like  this,  when,  M 
here,  it  conduces  to  safety  to  do  §o.  This  note  is  added  merely  to  show  the  principle. 


TRUSSES.  281 

moving  load,  is  greater  than  the  preservative  tensile  strain,  4.2  tons,  of  the  truss 
and  floor,  acting  on  it  at  the  same  time.  Therefore,  dp,  although  a  tie,  the  same  as 
c  jo,  is  liable  at  times  to  be  compressed  rather  than  pulled.  Therefore,  it  must  be  so 
arranged  as  to  act  also  as  a  strut ;  at  least  so  far  as  to  bear  a  pres  equal  to  the  diff 
between  the  16.73  tons  of  pres  from  the  load,  and  the  4.2  tons  of  tension  from  the 
wt  of  the  truss  and  floor ;  or  to  12.53  tons. 

On  the  next  oblique  o  d,  which  is  a  strut,  the  same  as  c  q,  the  moving  load  on  o  wt 
produces  a  pull  of  16.73 ;  while  the  truss  and  floor  produce  on  it  a  pres  of  only  8.4 
at  the  same  time.  Therefore,  although  it  is  a  strut,  it  is  liable  at  times  to  be  pulled 
rather  than  compressed ;  and  consequently  it  must  be  made  able  to  bear  a  pull  also, 
equal  at  least  to  16.73  —  8.4  =  8.33  tons.  On  the  tie  e  o,  the  deranging  compressive 
load  strain  6.76  is  less  than  the  preservative  tensile  strain  16.8  of  the  truss  and 
floor  acting  upon  it  at  the  same  time.  Therefore,  it  may  remain  as  a  tie  only  ;  or  in 
other  words,  it  requires  no  counterbracing.  When  this  is  the  case,  no  other  oblique 
between  it  and  the  nearest  abutment  m  needs  counterbracing.  It  is  almost  needless 
to  remark,  that  the  half  xp  of  the  truss  requires  the  same  as  the  half  mp,  when  the 
engine  crosses  in  the  opposite  direction. 

The  strains  oil  the  chords  are  found  as  directed  on  p  279. 
In  the  Fink  truss,  Figs  26,  27,  the  effects  of  a  moving  load, 
may  be  calculated  as  for  a  full  uniform  max  load  from  end  to  end.  Thus  assuming 
at  first,  as  in  the  preceding  cases,  that  everything  is  borne  by  one  truss  only;  then, 
when  the  load  is  upon  the  top  chord  uf  the  truss,  each  vert  post  may  in  practice  be 
regarded  as  upholding  one-half  of  that  portion  of  the  entire  wt  of  bridge  and  dis- 
tributed load  which  is  between  the  two  extreme  ends  of  the  two  obliques  which 
uphold  said  post.  Thus,  in  Fig  26,  the  half-way  post  dc,  bears  half  of  all  between 
a  and  b.  The  post  m  g,  half  of  all  between  a  and  d ;  the  post  h  o,  half  of  all  between 
m  and  d.*  This  is  equivalent  to  saying  that  the  half-way  post  bears  half  the  entire 
wt  of  the  bridge  and  load :  each  quarter-way  post,  one-quarter ;  each  eighth-way 
post,  one-eighth,  &c,  &c,  of  this  same  entire  wt  of  bridge  and  load;  and  these  consti- 
tute theoretically  the  strains  on  the  several  posts.  But  after  having  got  thus  far,  it 
is  necessary  to  examine  whether  some  of  the  smaller  ones  may  not  have  to  be  in- 
creased, for  the  following  plain  reason  :  Suppose  we  have  assumed  our  max  load  to  be 
a  string  of  heavy  engines,  weighing  1  ton  per  foot  run ;  or,  including  the  wt  of  the 
bridge  itself,  say  1.4  ton  per  ft ;  and  suppose  our  posts  to  be  as  close  together  as  5 
ft ;  then  the  least  loaded  posts  would  each  bear  5  X  1.4  =  7  tons.  But  we  know  that 

from  16  to  20  tons  may 

n  ,      be  concentrated  within  a 

length  of  5  ft,  on  four 
drivers  of  an  engine ;  and 
half  of  it  will  have  to  be 
supported  by  each  post  in 
succession  as  a  train  passes. 
When  we  thus  find  by  trial, 
which  posts  will  be  more 
strained  by  an  engine  than 
by  our  assumed  max  per  ft 
run  of  the  whole  truss,  wo 

must  increase  the  load  first  found,  correspendingly.  In  the  Fink,  and  Bollmar: 
trusses,  the  verts  are  always  struts  or  posts.  Having  fixed  upon  the  load  for  each 
post,  as  p  o,  Fig  23  h,  then  for  the  strain  which  said  load  will  produce  upon  each  of 
the  obliques,  or  ties,  p  c,  p  A,  upholding  said  post,  take  any  distp  d  on  the  post,  to 
represent  the  load  by  scale;  and  draw  dw,  d  r»,  parallel  to  the  ties;  then  p  w, p  n, 
measured  by  the  same  scale,  will  respectively  give  the  strains  on  each ;  whether  they 
be  equally  inclined  as  usual,  or  not.  The  two  hor  lines  n  a,  w  a,  by  the  same  scale, 
give  the  two  hor  forces  which  the  load  at  the  top  o  of  the  post,  acting  through  the 
ties,  produces  upon  the  chord  at  c  and  h\  which  two  equal  and  op  posing  forces  pro- 
duce along  the  intermediate  stretch  c  h  of  chord,  a  strain  equal  to  one  of  them.  In 
other  words,  either  n  a,  or  w  a,  gives  the  hor  strain  produced  along  c  h,  by  the  load 
at  o  only.  See  "  Strain,"  Art  2,  of  Force  in  Rigid  Bodies,  p  444. 

Strain  on  the  chord.  This,  from  a  uniformly  distributed  load,  is  the  same 
throughout  the  entire  length  of  a  Fink  chord.  To  find  it,  observe  which  obliques, 
(as  m  e,  g  e,  ot.  Fig  23 1',)  of  one-half  of  the  truss,  terminate,  at  t,nr.  end,  e,  of  the  chord. 
Then,  having  previously  found  the  loads  on  the  posts,  c  o,  u  g,  t  w,  which  pertain  to 

*  In  Fig  26,  the  load  is  really  on  the  bottom  chord.  But  in  this  case  also,  the  same  loads  art 
carried  to  the  tops  of  the  posts  by  the  obliques :  except  at  the  four  posts  h  e,  h  o.  &c,  which  have  no 
obliques  at  their  heads.  With  a  bottom  load,  these  may  in  practice  be  considered  to  sustain  each 
only  one  half-panel  wt  of  trusi  alone;  but  their  obiiguet  sustain  the  same  load  as  if  the  train  wer» 
OQ  the  top  chord. 


282 


TRUSSES. 


Ro-23.i 


those  obliques,  ascertain  by  the  pro- 
cess in  Fig  23  h,  the  hor  forca  n  a, 
(in  both  figs,)  which  each  of  those 
loads  produces  on  one  oblique.  Add 
together  these  forces  n  a,  (there 
will  be  but  three  of  them  in  Fig  23 1, 
as  marked  by  the  dark  lines ;)  their 
sum  will  be  the  strain  along  the  en- 
tire chord.  The  obliques  m  u,  u  r, 
r  c,  g  c,  do  not  terminate  at  e;  and 
are,  therefore,  omitted  in  finding 

the  chord  strain.     The  process  is  the  same  whether  the  verts  are  all  of  the  same 
length  or  not. 


Or  the  hor  chord-strain  produced  by  each  of  the  loads  on  the  posts  c  o,  u  & 
may  be  calculated  thus,  and  added  together. 

entire  load  vy  hor  dist  from  post 


t  in, 


Horizontal 
strain 


on  post 


X 


to  end  e  of  chord 


twice  the  length  of  the  post. 

In  the  Ho  II  m  a  ii.  Figs  22,  24,  25,  for  a  moving-  load,  having  first  pre- 
pared the  working  diagram,  determine  the  max  weight  that  can  come  upon  a  post. 
This  will  be  the  same  for  each  post.  If  the  moving  load  is  on  top  of  the  truss,  this 
load  on  each  post  will  consist  of  the  greatest  wt  of  engine  that  can  stand  upon  one 
panel-length  of  truss;  together  with  (approximately  enough  for  practice)  the  wt  of 
one  panel-length  of  truss,  floor,  &c,  if  the  posts  uphold  the  oblique  ties ;  (see  Boll- 
man  truss,  in  Art  17;)  or  one  panel-length  of  floor;  and  the  half  of  a  panel-length 
of  truss,  if  the  posts  do  not  uphold  the  ties.  If  the  load  is  at  the  bottom  of  the  truss, 
the  posts  bear  no  part  of  either  the  moving  load,  or  of  the  floor;  but  each  of  them 
will  be  strained  to  the  amount  of  the  wt  of  say  one  panel  of  truss,  if  the  posts  up- 
hold the  rods ;  or  of  half  a  panel  of  truss,  if  they  do  not. 

The  loads  on  the  posts  may  then  be  written  upon  the  diagram. 

The  obliques  or  ties,  however,  when  the  load  is  at  the  bottom,  bear  (as  in  the 
Fink)  the  same  amount  of  strain  from  the  moving  load  and  floor,  as  when  it  is  on  top. 
Therefore,  when  it  is  at  the  bottom,  each  pair  of  ties  sustains  not  only  the  load  rest- 
ing on  the  post  which  they  uphold ;  but  the  wt  of  one  panel-length  of  floor,  and  the 
max  panel-wreight  of  engine.  In  other  words,  lohether  the  load  be  on  t'>p,  or  at  bottom, 
the  two  ties  at  the  foot  of  each  post,  sustain  a  wt  equal  to  a  full  panel-length  of  truss 
and  floor;  together  with  the  max  panel-wt  of  engine.  Having  added  these  wts  to- 
gether, lay  off  their  sum  by  scale  at  each  post,  as  shown  at  1 i\  k  v,j  v,  i  v.  Fig  22 ;  com- 
plete I  g  v  u,  k  g  v  u,  &c ;  and  measure  the  strains  I  u,  I  g ,  k  u,  k  g,  &c,  along  the  ties. 

The  strains  on  any  pair  of  ties,  may  also  be  calculated  thus;  having 
the  load  they  sustain. 


Strain  on 
short  tie 


i    /?  N/  hor  dist  from  post  to       length  of 
""  X  farthest  end  of  chord   v  short  tie 


total  length  of  truss 


deptfTof 
truss. 


7     ,         hor  dist  from  post  to        length  of 
'   x    nearest  end  of  chord        long  tie 


Strain  on  ______ 

long  tie.    =  tutai  iength  Of  trms     '  '  *  depth  of 

truss. 

The  hor  strain  on  the  chord  will  be  uniform  throughout,  as  in  the  Fink 
truss  ;  and  will  depend  upon  the  max  uniform  load  that  can  cover  the  whole  bridge; 
and  not,  as  in  the  case  of  the  ties,  upon  the  greatest  load  which  each  pair  of  ties  may 
have  to  sustain  in  succession;  unless  we  assume  our  max  uniform  load  to  be  a  string 
of  engines  which  may  bring  a  max  panel-wt  of  engine  upon  every  pair  of  ties  at 
once.  In  that  case  wre  have  only  to  measure  upon  one-half  of  our  working  diagram, 
the  several  hor  lines  corresponding  to  MO,  &c,  in  Fig  22 •  and  their  sum  will  be  the 
reqd  hor  strain  on  the  chord.  But  if  we  take  our  max  uniform  load  on  the  whole 
truss,  to  be  a  string  of  cars,  we  must  diminish  the  chord-strain  thus  found,  in  this 
manner :  Add  together  a  full  panel-weight  of  truss,  floor,  and  cars ;  then,  as  the  full 
panel-wt  of  truss,  floor,  and  engine,  (which  we  before  assumed  as  the  straining  load 
of  each  pair  of  ties,)  is  to  the  panel-wt  of  truss,  floor,  arid  cars,  just  found,  so  is  the 
hor  chord-strain  before  found,  to  the  one  reqd. 

We  will  repeat,  that  chords  must  be  strong  enough  to  bear  not  only  the  hor  pull 
or  push  to  which  they  are  exposed ;  but  also  to  sustain  safely,  as  beams,  the  trans- 
Terse  strains  from  the  floor,  and  from  the  moving  load,  when  these  rest  upon  them. 


TRUSSES. 


282i 


Art.  21.  If  a  long*  beam  a  ft,  Figs  25  and  27,  requires  to  be 
strengthened,  this  may  be  done  by  the  addition  of  a  vert  post  d  c;  and  two  in- 
clined tie-rods  c  a,  c  6.  And  if  after  this,  the  two  halves,  d  a,  d  b  of  the  beam  still 
are  found  to  be  too  weak,  additional  intermediate  posts  o,  o,  may  be  introduced ;  with 
other  ties  i,j,  to  sustain  them. 

In  Figs  25  and  27,  the  roadway 
is  at  the  chord  a  6;  no  parallel 
lower  chord  being  necessary,  it 
may  be  omitted.  The  inclined 
ties  act  as  substitutes  for  it.  But 
if  the  bridge  is  so  near  the  water 
as  not  to  allow  the  posts  and  ties 
to  be  placed  beneath  the  roadway 
a  6,  we  may  raise  the  entire  truss 
upon  two  posts  or  piers  s  s,  Figs 
24,  26 ;  and  place  the  roadway 
n  n,  at  the  lower  ends  of  the  posts 

and  ties ;  instead  of  letting  it  rest  on  top  of  the  chord,  as  in  Figs  25  and  27.    In  Figs 
24  and  26,  the  truss  and  its  load  do  not  then  rest  directly  upon  the  abuts  y  y,  but 

upon  the  tops  of  the  posts 
-i  i  », « ;  and  the  only  part  that 

s *t does  rest  directly  on  the 

abuts,  is  one-half  of  that 
small  portion  of  the  road- 
way comprised  at  each  end, 
between  e  and  n ;  in  other 
words,  only  one  half  the  wt 
of  the  roadway  of  the  end 
panels;  the  other  half  being 
sustained  by  the  inclined 
ties  which  meet  at  e. 
When  the  tie-rods  all  pass 
from  the  feet  of  the  posts  to  the 
ends  of  the  chords,  as  in  Figs 
24  and  25,  we  have  the  Boll- 
man  truss.   And  when,  as 
in  Figs  26  and  27,  only  those 
which  sustain  the  center  post 
d  c,  both  pass  to  the  ends  of 
the  chord,  while   the  others 
are  disposed  as  in  said  figs,  the 
Fink  truss  is  the  result. 


BOLLMAN 

Kg  Z5 


Kg  27. 


TRUSSES. 


283 


Art.  22.  Fig  28,  shows  the  general  arrangement  of  a  small 
wooden  Howe  bridge-truss;  Fig  29,  some  of  its  details ;  and  Fig  30.  those 
of  an  iron  truss.  High  trusses  are  sometimes  made  as  in  Fig  1.  The  top  and  bottom 
chords  of  the  wooden  one  are 

each  made  up  of  three  or  more  j-       9^         ^_        ^        Ji      jvv      ^b 

parallel  timbers  c  c  c,  placed  a 
email  dist  apart,  to  let  the  vert 
tie-rods  r  r  pass  between  them. 
The  main  braces,  o  o,  are  in  pairs 
or  in  threes.  The  pieces  com- 
posing them,  abut  at  top  and  bot- 
tom, against  triangular  angle 
blocks,  s ;  which  if  of  hard 
wood,  are  solid ;  and  if  of  cast- 
iron,  hollow;  as  shown  at  T, 
Figs  30;  strengthened  by  inner  ribs. 
These  extend  entirely  across  the  three 
or  more  chord-pieces.  Against  their 
centers,  abut  also  the  counterbraces  e. 
These  are  single  pieces  in  small  bridges ; 
or  in  pairs,  in  large  ones ;  and  pass  be- 
tween the  pieces  which  compose  a  main 
brace.  Where  the  wooden  braces  and 
counters  cross  each  other,  they  are 
bolted  together.  For  wooden  chords, 
the  angle- blocks  are  cast,*  as  at  T.  The 
dotted  lines  show  the  strengthening 
ribs;  and  x  serves  to  keep  the  block  in  place.  The  vert  tie-rods  r  r,  of  iron,  are  in 
pairs,  threes,  or  fours,  &c,  according  to  size  of  bridge ;  with  a  screw  and  nut  at  each 
end.  The  heads  and  feet  of  the  braces  and  counters,  butt  square  against  the  angle- 
blocks  :  and  are  kept  in  place  only  by  the  tightening  of  the  screws  of  the  vert  ties. 
When  the  floor  is  below,  as  in  Fig  28,  the  end  posts  p  d ;  and  the  ends  g  i  and  w  6,  of 
the  upper  chord,  may  be  omitted ;  also  i  c  and  b  y ;  but  it  is  seldom  done. 


E 


M 


~W 


In  Figs  30,  of  an  iron  Howe  truss,  the  top  chord  P,  M,  and  W,  is  cast  in  one  piece 
transversely,  as  at  P.  Its  separate  lengths  are  connected  together  by  flanges  and 
bolts,  somewhat  as  shown  at  W ;  where,  a  a,  are  cast  longitudinal  flanges  for  strength- 
ening the  transverse  bolting-flanges  g.  Instead  of  separate  angle-blocks  at  the  upper 
chord,  solid  ones  may  be  cast  in  the  same  piece  with  the  chord  itself,  as  shown  at  M. 
The  lower  chord  usually  consists,  as  in  other  iron  bridges,  of  four  or  more  flat  bars  of 
rolled  iron,  c,  placed  on  edge :  and  some  dist  apart,  as  at  R.  On  top  of  them  rest  the 
lower  angle-blocks  «,  which  have  shallow  channels  below,  for  receiving  the  chord 
pieces;  and  thus  securing  them  from  lateral  motion.  A  cast  washer,  a,  below  the 
chords,  is  provided  with  similar  channels  on  top,  for  the  same  purpose.  The  braces 


-£  In  large  spans,  to  prevent  the  pressure  of  the  heads  and  feet  of  the  obliques  from  crushing 
the  chords,  the  angle-blocks  are  cast  with  deep  projecting  Ganges  under  their  bases ;  and  which,  pass- 
ing between  the  pieces  which  compose  a  chord,  extend  to  the  opposite  face  of  the  chord.  There  the 
flanges  bear  upon  broad  washers  at  the  ends  of  the  vert  rods.  By  this  means  the  strains  along  the 
obliques  are  transferred  directly  to  the  verts,  without  at  all  affecting  the  chords.  Angle  block*  of 
•curse  have  openings  for  the  passage  of  the  vert  rods. 


284 


TRUSSES. 


and  counters,  o,  e,  in  moderate  spans  are  usually  cast  in  a  star-shape,  as  at.;.  The 
following  table  gives  dimensions  sufficient  for  a  strong  Howe  bridge ;  although  in 
wooden  bridges  it  is  customary  to  add  arches  when  the  span  exceeds  about  150  feet. 
IHmensioiis  for  each  of  two  trusses  of  a  Howe  bridg-e  for  a 
single-track  railway.  Timber  not  to  be  strained  more  than  800  ft>s  per  sq 
inch  ;  nor  iron  more  than  5  tons  per  sq  inch.  Iron  supposed  to  be  of  rather  superior 
quality,  requiring  from  25  to  27  tons  (60450  Ibs)  per  sq  inch  to  break  it.  The  rods 
to  be  upset  at  their  screw-ends.  To  each  of  the  two  sides  of  each  lower  chord  is  sup- 
posed to  be  added,  and  firmly  connected,  a  piece  at  least  half  as  thick  as  one  of  the 
chord-pieces ;  and  as  long  as  three  panels ;  at  the  center  of  the  span. 


9 

£* 

1 

es 
P-i 

An  upper 
Chord. 

A  lower 
Chord. 

An  End 
Brace. 

A  Center 
Brace. 

A  Counter. 

End  Rod. 

Center 
Rod. 

fe* 
I 

s! 

°§ 

.§ 

o£ 

jj 

o| 

V 

5| 

8 

SI 

I 

A 

i 

ij 

3 

i 

525 

*£ 

OS 

SK 

BO 

Si 

02 

** 

CQ 

£s 

02 

** 

5 

£« 

ft 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

25 

6 

8 

I 

4X  5 

i 

4X10 

2 

4X  0 

2 

4X  5 

i 

4X  5 

•2 

15-16 

2 

H 

50 

9 

9 

3 

6X  7 

i 

6X10 

2 

6X  7       2 

5X  6 

i 

5X  6 

2 

1^6 

2 

1  1-16 

75 

12 

10 

3 

6X  9 

3 

6X11 

2 

6X  8|     2 

6X  6 

i 

6X  6 

2 

1  Ji 

2 

1  3  16 

100 

13 

11 

I 

6X10 

3 

6X12 

2 

8X  9       2 

6X  8 

i 

6X  8      2 

2  3-16 

2 

1  ^16 

125 

1H 

1? 

4 

6X10 

4 

6X13 

2 

9X101     2 

6X  9 

i 

6X  9!     2 

2^ 

2 

\Y 

150 

21 

13 

4 

8X10 

4 

8X14 

8 

8X10!     3 

6X  8 

2 

6X  8 

3 

2  $£ 

3 

13-16 

175 

?4 

H 

4 

10X12 

4 

10X15 

3 

8X11 

3 

8X  8 

2 

8X  8 

3 

2^g 

3 

1J4 

200 

27 

15 

4 

12X12 

4 

12X16 

3 

9X12 

3 

8X10 

2 

8X10 

3 

w 

3 

1% 

The  rods  in  onr  table  are  somewhat  larger  than  customary. 
The  same  dimensions  will  serve  for  a  double  road  for  common  travel. 
For  bridges  of  iron,  assuming  the  safe  strain  for  iron  to  be  5  tons  per  sq  inch,  or 
14  times  as  great  as  the  800  fbs  assumed  for  wood  ;    the  areas  of  the  cross-sections 
of  the  individual  members  will  as  a  general  rude  approximation,  be  about  one- 
fourteenth  part  as  great  as  those  of  wooden  ones.  Eq  ually  strong:  wooden, 

differ  very  materially  in  weight. 
Art.   23.     Fig  31, 


anil  iron  bridsres  of  the  same  span,  will  not  dif 
T. 


/SIIXXXIIXIIX 


PRATT 


shows  in    like   manner, 
a  wooden   Pratt 

truss  :  and  Fig  32,  some 
details  of  a  small  iron 
one.  After  the  foregoing, 
they  do  not  need  much 
explanation.  Since  the 
angle-blocks  have  to  re- 
sist tension,  instead  of 
thrust,  they  ai-e  placed 
above  the  top  chord,  and 
below  the  bottom  one. 
The  main  obliques  are 
in  pairs ;  and  the  smaller 
single  counters  pass  be- 
tween them,  as  in  the 
Howe.  In  large  bridges 
they  are  in  threes,  fours, 
&c.  The  vertical  posts, 
which,  when  of  iron,  are 
hollow,  are  retained  in 
their  positions  both  by 
the  strains  on  the 
obliques,  which  termi- 
nate above  and  below 
their  ends ;  and  by  being 
let  into  the  chords.  In 
large  spans,  tho  details 
generally  vary  more  or 
loss  from  those  in  the  figs.  In  Fig  31.  c  c  c  are  the  main  braces ;  and  o  n  o  the  counters. 


TRUSSES. 


285 


When  the  roadway  is  below,  as  in  Fig  31,  the  ends,  r  6,  y  x,  of  the  upper  chord ;  the 
end  verticals/)  and  u ;  and  the  two  tension  obliques  in  each  end  panel,  may  be  omit- 
ted ;  and  two  diagonal  struts  from  6  and  y  must  then  be  substituted,  extending  to 
the  abutments,  for  upholding  the  upper  chord,  &c.  Ill  tile  Pratt  the  chords 
may  be  of  the  same  dimensions  as  in  the  foregoing  table  for  Howe's.  The  posts  may 
have  about  ^th  less  area  than  the  main  braces  of  the  Howe.  The  main  brace  rods, 
and  others,  (of  the  same  number  as  the  main  brace  pieces  of  the  Howe,)  may  hava 
the  following  diams  in  ins ;  allowing  the  safe  strain  to  be  five  tons  per  sq  inch. 

For  each  Truss  of  a  Pratt  Bridge. 


•25  Ft. 

50  Ft. 

75  Ft. 

Spi 

100  Ft. 

ins. 
125  Ft. 

150  Ft. 

175  Ft. 

200  Ft. 

End  Main-brace  Hods. 

Ins. 
3  of  2% 

Ins. 
3  of  3Ji 

2  of  1% 

2  of  IX 

2  of  2^ 

2  of  2J^ 

2  of  1% 

3  of  IX 

Center  Main-brace  Hods. 

2  of  1 

2<>flfV 

•-1& 

2  of  IX 

3  of  lT5ff 

3or,« 

3  of  1% 

Ins. 
1  of  ly^" 

Ins. 
loflH 

Cc 

Ins. 

>unter  Ro< 
Ins. 

Iof2 

IB  at  Cent 
las. 

3r. 
2  of  IX 

Ins. 
2  of  1-j-g- 

2  of  lyf 

In  Pratt's  truss  the  directions  of  the  main  braces  and  counters  are  respectively 
the  reverse  of  the  Howe.  Many  of  the  remarks  in  the  preceding  Art,  apply  equally 
in  this.  Neither  the  Howe  nor  the  Pratt  possesses  any  special  advantage  over  the 
other  as  regards  ease  of  adjustment,  &c.  In  both  trusses,  arches  are  frequently  added 
in  wooden  railroad  bridges  when  the  span  exceeds  about  150  ft.* 

Art.  24.  Town's  lattice  truss,  Fig  33,  as  originally  introduced,  and  very 
extensively  employed,  was  of  extremely  simple  construction ;  being  composed  en- 
tirely of  planks  from  2  to  3  ins 

thick;  and  from  9  to  12  wide;  de-  t    t     ll  V    i 

pending  on  the  span.  Two  sets  -  „.  ' 
of  these  were  placed  crossing 
each  other  at  angles  of  about 
90°;  and  were  connected  to- 
gether at  their  intersections  by 
either  2  or  4  treenails  of  locust, 
or  other  hard  wood,  about  2 
ins  diam.  At  the  top  and  bot- 
tom, similar  planks,  a  a,  c  c, 
were  treenailed  hor,  to  form  the 
chords ;  or  in  large  spans,  (many 
exceeded  150  ft,)  there  were  two 
upper  and  two  lower  chords,  (sometimes  of  timber  6  ins  thick,)  as  shown  in  the  fig, 
by  n  w,  o  o.  The  transverse  section  A  shows  the  two  upper  chords  on  a  larger  scale ; 
each  chord  consisting  of  two  planks ;  one  on  each  side  of  the  lattices.  Two  trusses 
of  this  kind,  with  a  depth  equal  to  %  or  ^  of  any  clear  span  not  exceeding  about  175 
ft ;  planks  of  3  X  12  white  pine ;  the  open  squares  2^  ft  on  a  side,  in  the  clear,  were  con- 
sidered sufficient  for  a  common-road  bridge  20  ft  wide.  Many  of  these  bridges  warped 
sidewaya  very  badly ;  and  when  applied  to  railroad  purposes,  failed  entirely.  In 
some  cases  the  better  mode  of  three  lattices  was  employed;  two  of  them  running  in 
one  direction ;  and  the  third  in  the  other  direction,  passing  between  them.  A  funda- 
mental defect  was  that  the  parts  were  of  equal  size  throughout  the  span  ;  whereas  the 
chords  should  be  stoutest  at  the  center ;  and  the  lattices,  near  the  ends.  These  de- 
fects caused  the  lattice  to  fall  into  unmerited  neglect,  in  the  United  States ;  whereas 

*  The  Baltimore  Bridge  Co  (Smith,  Latrobe,  Ac)  have  built  a  wooden  Howe  of  two  trusses,  of  300 
ft  span.  30  high,  26  wide,  from  center  to  center  of  chords,  without  any  arch.  It  has  a  wrought-iron 
lower  chord,  and  is  prooortioried  for  a  moving  load  of  but  1000  Ibs  per  ft  run. 

19 


e     x  sy 

LATTICE 


Pig  33 


286 


TRUSSES. 


in  Europe  it  is,  when  properly  proportioned,  highly  esteemed ;  especially  for  iron 
bridges ;  some  of  which,  on  this  principle,  have  been 
constructed  of  more  than  300  ft  span.  Another  detect 
was  lateral  weakness ;  or  a  tendency  to  warp  sideways, 
owing  to  the  thinness  of  the  trusses.  This  is  obviated 
in  the  large  bridges  referred  to,  by  placing  a  double 
truss,  Fig  34,  (instead  of  a  single  one,)  at  each  side  of 
the  bridge.  The  trusses  T,  D,  composing  a  double  one, 
are  placed  a  foot  or  more  apart ;  and  are  connected  to- 
gether at  proper  intervals,  by  short  pieces  riveted  to 
each  one,  for  stiffening  them.  At  T  and  D  are  seen  the 
three  lattices  or  lattice-bars,  of  each  truss ;  two  of 
which,  on  the  outside,  constitute  the  main  braces;  while  the  center  one  is  thecoun- 
terbrace.  Such  a  double  truss  bears  some  resemblance  to  the  Fairbairn  box-girder; 
the  diif  being  chiefly  that  in  the  former  the  sides  are  composed  of  lattice-bars ;  and 
in  the  latter  of  solid  plates. 

The  Bowstring1  truss,  Fig  35,  is  an  excellent  one  as  regards  strength,  and 
economy  of  construction.  It  has,  however,  the  disadvantage,  in  large  spans,  of  a  diffi- 
culty in  connecting  together  overhead  the  two  trusses  of  a  span,  so  as  to  be  as  free 

from  lateral  vibration  as  a  bridge 
with  parallel  upper  and  lower 
chords.  In  the  latter,  this  con- 
nection can  be  made  from  end  to 
end  of  the  span  ;  but  in  the  Bow- 
string it  can  be  done  only  for 
some  distance  each  way  from  the 
center;  from  want  of  headway 
near  the  ends.  In  short  spans 
with  low  trusses,  this  defect  is 
not  felt.  The  verts  and  obliques 
shown  in  the  fig,  may  all  be 
struts,  or  all  ties;  or  either  a 
Howe,  a  Pratt,  or  a  Warren  ar- 
rangement. 

The    pnll    along:    the 
chord    y  y   in    a  Bowstring 

truss,  is  uniform  throughout  its  length ;  and  is  the  same  in  amount  as  the  hor  pres 
at  the  center  or  crown,  4,  of  the  arch.  Either  may  be  found  thus ;  remembering 
that  in  our  calculations  of  bridge  trusses,  we  assume  at  first,  as  in  previous  exam- 
ples, that  all  the  weight  of  the  bridge  and  its  load  is  sustained  by  one  truss  only ;  so 
that  the  resulting  strains  must  finally  be  divided  by  2,  to  distribute  them  among  the 
two  actual  trusses :  * 


BOWSTRING 


Hor  pres  at  crown; 
or  pull  on  chord     '• 


half  the  united  wt  o/v  ^ 
bridge  and  max  load  x  ^  span 


rise  measured  between  the  half-depths  of 

the  arch  itself,  at  its  foot  and  crown. 
Which  is  the  same  as  for  the  Howe,  and  other  beam-trusses ;  but  not  for  Fink's,  and 
Bollman's ;  in  both  of  which  it  is  greater.    If  the  chord  has  also  to  act  as  a  beam  for 
supporting  cross-floor  girders,  it  must  be  made  stronger. 
This  may  be  avoided  by  placing  such  girders  only  close  to  the  verts. 

For  the  pres  along:  the  direction  of  the  arch,  at  its  foot,  o9 
square  half  the  united  wt  of  the  bridge  and  its  max  load.  Also  square  the  hor  press 
at  crown  just  found.  Add  these  two  squares  together.  Take  the  sq  rt  of  their  sum. 
This  is  the  same  process  as  for  a  stone  arch.  The  pres  does  not  increase  regularly 
from  the  crown  to  the  foot ;  but  in  bridge  practice  it  may  safely  be  assumed  to  do  so. 

Strains  on  the  obliqnes.  Whether  these  (Fig  35)  are  all  struts,  or  all 
ties,  the  strains  upon  them,  according  to  Prof.  Kankine,  remain  unaltered;  and  that 
along  any  one  of  them  is  found  as  follows,  using  the  numbers  of  the  two  verts  betAvcen 
which  it  is  situated,  after  having  first  numbered  the  verts  &\ong  one-half  of  the  span, 
as  at  1,  2,  3,  4,  in  Fig  35.  The  moving  load  is  supposed  to  be  uniform  ;  and  hence 
the  Professor's  formulas  do  not  allow  for  the  extra  load  concentrated  upon  the  driv- 
ing-wheels of  an  engine. 


Strain  along 
any  oblique 


weight  of  one 


No  of  one 


Jfnnth  of  ^         ^-tvo  oj  (me  », 

?f^X  the  oblique\^f  __ ^ 

load  J   \       twice  the  9 

'etch  of  the  oblique    ^       ^     panels  in  i 


JVo.  of  other- 
vert 


vert  stretch  of  the  obliq 


number  of 
one  truss. 


*  This  and  what  follows  on  Bowstring  trusses  is  deduced  from  Rankine's  "  Ciy  Eng,"  as  the  writer 
understand*  it,  (edition  of  1862,  pages  483,  563,  to:)  not  having  himself  studied  this  form  of  truss. 


TRUSSES.  287 

Ex.  What  is  the  strain,  along  the  oblique  e  c,  Fig  35,  situated  between  verts  No.  2 
and  No.  3  ;  the  weight  of  a  panel-length  of  the  moving  load  alone  being  10  tons  :  the 
length  e  c,  12%  ft  ;  and  its  vert  stretch  c  n,  7.5  ft?  There  are  8  panels  in  one  truss. 


Here,  (ll^JH)  X  (^-?)  -  16.7  X  .375  =  6.26  tons,  Ans;  or  ~  =  3.13 
tons,  for  the  oblique  ec  of  each  of  the  two  actual  trusses  of  a  span. 

Strains  oil  the  verticals.  When  the  arrangement  of  verts  and  obliques 
is  as  in  Fig  i>5,  then  the  strains  on  the  verts  will  depend  considerably  upon  whether 
the  obliques  are  all  struts,  or  all  ties.  In  either  case,  however,  they  will  be  greatest 
near  the  center  of  the  truss,  and  least  near  its  ends  ;  resembling  in  this  respect  the 
roofs  Figs  14,  15,  and  16;  but  the  reverse  of  the  beam  truss,  such  as  a  Howe,  Pratt, 
or  Warren.  In  the  Bowstring,  neither  the  verts  nor  the  obliques  sustain  any  portion 
of  the  bow  or  arch  itself;  but  on  the  contrary,  they  are  all  upheld  by  it. 

"When  the  obliques  are  all  struts,  first  find  the  wt  of  one  panel-length 
of  the  bridge  and  its  m;tx  load  ;  and  from  this  take  the  weight  of  one  panel-length 
of  the  arch  or  bow  itself.  We  will  call  the  remainder,  the  "reduced  total  panel- 
weight;"  then  the 

No.  of  vert  X  next  Jess  No.\ 
—  —  — 


Strain  on        (  reduced  total  \    v      ..   ,  —  — 

any  vert    =  I  panel-weight  )    X  11+  twice  the  number  of  panels 
*  in  one  truss. 


.\ 
\ 
) 

/ 


Ex.   What  is  the  greatest  strain  that  can  come  upon  vert  No.  3,  Fig  35,  the  weight 
of  a  reduced  total  panel-weight  being  13  tons  ?     Here 


13  X     l  +       -     =  13  X     l  +         =  13  X  1.375  =  17.875  tons. 

Or,   —  '-—•  =  8.9375  tons  on  the  vert  No.  3  of  each  of  the  two  actual  trusses. 

But  if  the  obliques  in  Figr  35  are  all  ties,  the  greatest  strain  upon 
any  one  of  the  verts,  is  simply  equal  to  one  reduced  total  panel-weight. 

Proi.  Rankine  also  says  that  in  some  cases,  under  a  moving  load,  if  the  obliques  in 
Fig  35  are  tie;*,  some  of  the  verts  may  have  to  act  at  times  as  struts,  as  well  as  ties. 
To  ascertain  if  this  is  the  case  with  any  given  vert,  use  the  following  : 


/No.  of  vert  X   next  greater  No. ^  ,  wt  of  one  panel  of     \ 

per  panel      *     (       twice  the  number  of  pandg       )     —    (          bridge  only,  ) 

\  in  one  truss  /  \without  the  arch  itself./ 

If  the  result  is  minus,  the  vert  will  act  as  a  tie  only  ;  but  if  the  result  is  either  0 
or  plus,  it  will  show  to  what  extent  the  vert  may  havu  to  act  as  a  strut  also. 

When  a  Bowstring*  truss  has  no  obliques,  but  only  verts,  the  arch 
being  supposed  to  be  sufficiently  stiff  in  itself  to  resist  change  of  form  by  a  moving 
load,  the  greatest  strain  on  each  vert  from  a  uniform  moving  load,  is  equal  to  a  re- 
duced total  panel-weight;  but  in  practice  we  must  allow  for  the  greatest  load  that 
can  come  upon  a  panel.  The  proper  shape  for  the  arch,  under  a  uniform  quiescent 
load,  is  a  parabola;  but  when  the  rise  does  not  exceed  about  %  of  the  span,  (which 
it  should  not,)  a  circular  segment  is  sufficiently  approximate  for  common  practica 
Under  a  uniform  quiescent  load,  obliques  are  not  needed. 

For  an  arch  trussed  as  in  Fig-  35%,  with  vertical  struts,  and  with  ob* 
liques  all  struts,  or  all  ties ;  the  strains 

on  the  arch  and  obliques  may,  according          0      1      2      3      A-  "vr 

to  Prof.  Rankine,  be  found  by  the  rules 
just  given  for  the  Bowstring.  The  upper 
piece,  owt  need  only  be  strong  enough 
to  bear  the  load  upon  it. 

When  the  obliques  are  all 
ties,  the  verts  will  always  be  struts  or 
posts;  and  the  strains  on  any  one  of 
them  will  be  given  by  the  foregoing  formula  for  any  vert  of  a  Bowstring. 

When  the  obliques  are  all  struts,  the  strain  on  each  vert  is  equal  to 
the  redueed  total  panel-weight ;  and  in  this  case  it  is  possible  that  some  of  the  verts 
may  at  times  during  the  passage  of  a  moving  load,  have  to  act  as  ties,  as  well  as 
posts.  To  ascertain  if  this  is  the  case  with  any  given  vert,  use  the  last  preceding 
formula.  If  th«  result  is  plus,  it  gives  the  pull  or  tension  which  the  vert  may  have 
to  resist  as  a  tie;  but  if  it  is  either  0  or  minus,  the  vert  acts  as  a  post  only.  Prof. 
Rankine  says  that  the  dotted  obliques  in  the  end  panels  are  not  absolutely  necessary, 


288  TRUSSES. 

as  the  formula  will  show ;  but  that  it  is  still  well  to  insert  them,  for  the  sake  of 
greater  stiffness. 

The  Lock  Ken  viaduct,  England,  of  130  ft  clear  span,  and  18  ft  high, 
has  two  trusses,  13  ft  8  ins  apart  clear,  for  single-track  railroad,  on  the  Bowstring 
principle,  Fig  35,  omitting  only  the  verts;  which,  however,  affects  the  strains  on  the 
obliques.  For  convenience  of  construction,  it  was  built  chiefly  of  rolled  channel- 
iron,  (see  tt,)  of  8  ins  by  4,  by  4,  by  %  inch.  At  Fig  35,  d  shows  a  transverse  section 
of  the  arch  or  bow;  which  is  uniform  throughout.  That  of  the  uniform  chord  or 
string  is  of  the  same  fig  and  size  as  the  arch.  Each  has  an  area  of  33  sq  ins  in  each 
truss.  The  top  strip,  a,  a,  of  rolled  iron,  is  24  ins  by  %;  and  is  riveted  to  the  upper 
flanges  of  the  two  channel-irons  1 1 ;  which  are  8  ins  apart,  so  as  j ust  to  allow  the  passage 
between  them  of  the  obliques  d ;  which  also  are  of  the  same  channel-iron.  The  panels 
are  about  12  ft  long  near  the  center  of  a  truss ;  and  8  ft  at  its  ends.  The  wt  of  the 
two  trusses  alone,  without  the  roadway,  is  very  nearly  50  tons.  The  wt  was  increased 
beyond  the  theoretical  requirements,  to  save  the  trouble  and  expense  of  preparing 
and  fitting  together  many  pieces  of  diff  dimensions;  yet,  although  the  bridge  is  a 
strong  one,  the  trusses  alone,  weigh  together  but  .37  of  a  ton  per  foot  run.  Where 
two  obliques  cross,  one  is  in  two  pieces,  riveted  to  the  other  by  straps. 

In  the  State  of  New  York  are  many  much -used  bridges  for  common 
travel,  by  Mr.  Whipple,  of  100  feet  span,  and  12%  ft  rise ;  with  two  trusses  19 
ft  apart  from  center  to  center  ;  for  two  roadways :  and  having  two  outside  footways, 
each  6  ft  wide.  Each  truss  has  9  panels,  braced  altogether  by  vert  and  oblique  tie- 
rods  (no  struts)  arranged  as  in  Fig  35.  The  verts  next  the  center  of  each  truss,  consist 
each  of  two  rods  of  1%  ins  diam,  welded  together  at  top ;  and  straddling  2  ft  at  bottom. 
The  other  verts  are  single,  and  2  ins  diam.  The  obliques  are  all  single,  and  l^ins  diam. 
The  arches  are  of  cast-iron.  The  transverse  area  of  metal  of  each  arch  is!8sq  ins  at  the 
crown ;  and  21  sq  ins  at  the  spring.  The  shape  of  arch  transversely  resembles  a  channel- 
iron  with  its  back  upward;  the  total  depth  of  flange  7  ins ;  the  width  of  arch  on  top,  11 
ins  at  center  of  span;  but  increasing  uniformly  by  means  of  wide  open-work  on  top, 
to  3  ft  at  springs.  Each  consists  of  9  straight  segments,  held  together  at  their  but- 
ting flanges,  by  the  verts  themselves ;  which  pass  through  them,  and  have  screws  and 
nuts  at  their  ends.  The  screw-ends  are  not  upset.  The  thickness  of  metal  in  the 
arches  nowhere  varies  much  from  %  inch.  Under  the  floor,  and  between  the  trusses, 
are  horizontal  diagonal  braces  of  rods  •%  inch  diam  ;  two  of  them  to  each  panel ;  each 
of  them  with  a  tightening  swivel.  The  chord  of  each  arch  consists  of  4  rods  of  2  ins 
diam.  In  the  same  State,  are  also  many  similar  common  road  bridges  of  72  ft  span. 
Rise  9  ft ;  two  trusses,  19  ft  apart  from  center  to  center  ;  *  and  two  outside  footways 
of  6  ft  each  in  addition.  Each  truss  has  7  panels,  with  vert  and  oblique  ties,  as  in  Fig  35. 
Each  cast-iron  arch  is  in  7  straight  segments,  of  the  same  shape  as  the  foregoing; 
with  a  cross-area  of  metal  of  about  12  and  15  sq  ins.  Its  width  at  center  of  truss  10 
ins:  at  springs,  30  ins.  The  two  verts  next  the  center  of  each  truss,  consist  each  of  2 
rods  of  1%  diam ;  the  other  verts  are  single,  each  1%  diam.  The  obliques  are  all 
single,  1  inch  diam.  The  chord  or  string  of  each  arch,  is  4  rods  of  1%  inch  diam. 
Horizontal  diag  bracing  of  9£  inch  rods  under  the  floor,  as  in  the  foregoing. 

Some  cast-iron  bridges  of  the  Severn  Valley  Railroad, 
England,  of  200  ft  clear  span,  consist  of  arches  rising  20  ft,  and  supporting  the 
railroad  on  a  level  with  the  tops  of  the  arches,  instead  of  above,  them  as  in  Fig  35%. 
There  are  no  diags  between  the  arches  and  the  roadway,  as  in  that  fig;  but  cast- 
iron  verts  only,  placed  4  ft  apart.  The  railroad  is  double  track  ;  and  there  are  four 
arches,  one  under  each  line  of  rails.  The  transverse  section  of  an  arch  is  I ;  each 
flange  is  lr>%  ins  wide,  by  2  ins  deep ;  the  web  is  2  ins  thick.  Total  depth  at  center 
of  span,  4  ft;  and  at  the  skewbacks,  4  ft  9  ins  Transverse  area  of  each  rib  at  crown, 
150  sq  iqs.  Each  arch  is  cast  in  9  segments  of  equal  length. 

The  cast-iron  bridge  across  the  Schiiylkill  at  Chestnut  St, 
Phila.  Strickland  Kneass,  Esq,  Engineer,  roadway  on  top,  has  two  arches  of  18ift 
clear  *pan  each,  and  20  ft  rise.  Clear  width,  42  ft.  Each  arch  has  fi  ribs,  about  8  ft 
apart  in  the  clear;  and  of  the  uniform  depth  of  4  feet,  including  a  hor  top  rib  8  ins 
wide ;  and  a  similar  one  at  the  bottom.  Thickness  everywhere  2%  ins ;  thus  giving  to 
each  rib  a  transverse  area  of  147%  sq  ins.  The  standards  are  vert,  with  ornamenta- 
tion. It  is  a  city  street  bridge.  The  roadway  consists  of  cast-iron  plates,  which  sup- 
port a  pavement  of  cubi'-al  blocks  of  granite,  laid  in  gravel.  The  arches  are  cast  in 
segments  12  ft  10  ins  long;  each  with  end  flanges  12  ins  wide,  for  bolting  them  to- 
gether with  four  1%  inch  diam  screw-bolts  at  each  end.  For  a  change  of  tempera- 
ture from  12°  to  99°  Fah.  the  crowns  of  the  arches  rise  2j^  ins.  Under  a  uniform 
extraneous  load  of  100  fos  per  sq  foot,  the  greatest  pres  on  the  arches  is  but  36UO  R>s 

*  With  only  two  trusses,  the  width  between  them,  in  the  clear,  should  not  be  less  than  16  ft.  to 
allow  two  ordinary  vehicles  to  pass  each  other  readily ;  but  18  or  20  ft  is  still  better;  more  would  be 
unnecessary  when  there  are  outside  footways.  The  headway  should  not  be  lest  than  13  ft. 


TRUSSES.  289 


per  sq  Inch  of  their  cross-section ;  or  not  more  than  ^y  of  the  ultimate  crushing 
strength  of  average  cast  iron,  in  short  blocks. 

The  Moseley  Bridge.  Figs  35%,  by  Thos.  W.  H.  Moseley,  of  Kentucky,  is 
essentially  a  wrought-iron  Bowstring,  with  a  hollow  plate-iron  arch  of  triangular 
cross-section,  apex  up;  and  formed  of  three  plates  riveted  together;  the  two  side- 
plates,  a 6,  ac,  having  their  top  and  bottom 
edges  bent  to  form  flanges  for  this  purpose. 
The  chords  o  e ;  the  verts  ;  and  the  two  counter- 
arches  tt\  are  also  of  iron.  These  counter- 
arches  are  intended  as  a  substitute  for  the  ob- 
liques of  Fig  35.  Each  of  them  consists  of  two 
angle-irons,  back  to  back,  riveted  together, 
and  to  the  verticals,  which  pass  between  them. 
Each  of  them  has  a  sectional  area  equal  to 
half  that  of  the  main  arch.  The  verticals  are 
placed  about  2  ft  apart.  They  are  flat  (not 
square)  bars,  for  convenience  of  riveting.  They 
pass  through  holes  in  the  bottom-plate  of  the 
main  arch,  (see  dotted  line  of  top  Fig,)  and  are 
fastened  at  a  by  the  same  rivets  which  connect 

the  upper  flanges  of  the  two  side-plates,  ab,  ac.  The  chords,  oe,  are  also  flat  bars ; 
and  have  a  transverse  area  half  as  great  as  that  of  the  main  arches.  At  their  ends 
they  are  attached  to  strong  wrought-iron  shoes  upon  which  the  feet  of  the  main 
arches  rest  and  abut.  The  rise  of  the  main  arches,  measured  to  the  bottom  of  the 
arch,  is  ^  or  y1^  of  the  clear  span. 

The  following  are  the  principal  dimensions  of  a  single  track  bridge  of  93  ft  clear 
span,  (97  ft  from  out  to  out  of  arch,)  carrying  the  "  Iron  Railroad  "  at  Ironton,  Ohio. 
It  was  built  in  I860,  and  is  traversed  by  heavy  engines  with  trains  of  pig  iron,  coal, 
&c.  Rise  to  bottom  of  arch  10%  ft;  to  middle  of  arch  11  ft.  Bottom-plate  ss  of  arch, 
16%  ins,  by  .44  inch.  Side-plates  a  6,  a  c,  in  clear  of  flanges,  14%  ins,  by  .29  inch. 
Top  flanges  at  a.  each  3  ins.  Bottom  flanges  b  and  c,  each  1.88  ins.  Vert  rods,  3  ins, 
by  %  inch ;  and  2  ft  apart  from  center  to  center.  The  chord  of  each  arch  consists  of 
two  flat  bars,  each  of  4%  sq  ins  of  cross-section.  The  bridge  was  tested  for  three 
weeks  by  a  dead  load  of  y^  a  ton ;  and  a  rolling  load  of  1  ton  at  the  same  time,  per 
foot  run ;  and  deflected  only  %  inch.  With  a  load  of  1  ton  per  foot,  the  pressure  at 
the  center  of  the  arches  would  be  about  full  4  tons  per  sq  inch  of  metal ;  and  a  trifle 
more  at  the  feet.  The  sectional  area  of  metal  in  each  main  arch  is  18%  S(l  ins-  These 
bridges  are  of  easy  construction,  and  consequently  cheap.  For  long  spans,  vert 
diagonal  bracing  (see  Fig  35)  would  probably  be  essential  for  preserving  the  form, 
of  the  arches  under  heavy  moving  loads.* 

An  iron  arch  roof  in  Philadelphia,  clear  span  80  ft,  rise  16ft,  con- 
sists of  a  uniform  arch  of  single  7-inch  Phosnix  beam  of  6  sq  ins  sectional  area ;  weigh- 
ing '20  Ibs  per  foot.  This  rests  on  cast-iron  shoes  on  the  walls.  The  hor  chord  or  tie 
at  the  feet,  is  of  two  rods  of  1%  diam.  At  the  center  of  this  tie  is  an  arrangement 
similar  to  No.  15,  of  Figs  21%,  from  which  diverge  upward,  to  the  arch,  a  central  vert 
rod  1  inch  diam ;  two  struts  of  6-inch  Phoenix  beam,  20  ft  apart  at  the  arch  ;  and  two 
ties  each  1%  diam,  reaching  the  arch  at  half-way  between  the  struts  and  the  feet  of 
the  arch.  There  are  10  such  trusses,  each  of  which  by  itself  weighs  about  3900  fibs; 
they  are  placed  16  ft  apart;  and  by  means  of  purlins  resting  upon  them,  support  the 
entire  weight  of  the  roof,  which  is  of  inch  boards,  covered  by  thin  sheet-iron. 

The  iron  roof  of  a  rolling-mill  near  Boston,  Mass,  of  about  80 
ft  tpan.and  16  ft  rise,  has  arches  of  the  Moseley  section  a,  />,  c.  Fig  35%;  but  without 
counter  arches.  The  trusses  are  12  ft  apart.  Sides  of  the  arches,  clear  of  the  flanges, 
7  ins ;  upper  flanges,  2  ins ;  lower  ones.  1  inch ;  total  of  each  side,  10  ins,  by  .19  inch  thick. 
Bottom  plate  8%  inch  by^  inch  thick.  Total  sectional  area  of  an  arch,  5.925  sq  ins. 
There  are  besides,  a  chord;  and  24  vert  suspending  rods ;  but  no  obliques.  The  roof 
is  covered  with  corrugated  iron,  on  purlins.  When  required,  the  heavy  iron  rolls  of 
the  mill  are  lifted  by  tackle  supported  by  a  roof-truss. 

Figs  36,  represent  the  Burr  truss;  which  was  formerly  more  used 

*  Although  this  bridge  seems  to  hare  stood  very  well  for  several  years,  the  writer  would  prefer  not 
to  exceed  two  tons  of  compressive  strain  per  sq  inch  on  plate-iron  in  such  structures.  In  several 
bridges,  General  Moseley  has  used  a  continuous  web  of  %  inch  iron,  instead  of  the  vert  suspenders. 
But  in  such  cases  the  triangular  tube  is  not  applicable  for  the  arch  ;  and  he  substitutes  two  Z  bars, 
riveted  together,  and  to  the  web,  w,  which  passes  between  them,  as  in  the  foregoing  fig.  The  counter 
arches  being  here  unnecessary,  are  omitted.  This  web  would  be  objectionable  in  large  spans,  espe- 
cially of  draw-bridges,  on  account  of  the  wind.  More  recently  still,  he  has  also  introduced 
lattices,  instead  of  vert  bars,  in  some  of  his  bridges  ;  together  with  many  innovations  on  the  arrange- 
ment first  described.  At  1  ton  per  ft  run,  the  pull  on  the  chord  above,  =  8  tons  per  sq  in. 


290 


TRUSSES. 


than  any  other  in  the  United  States.  It  is  at  present  regarded  with  disfavor  by  soma, 
because  many  early  ones  tailed  under  railroad  traffic,  in  consequence  of  bad  propor- 
tions, and  the  absence  (as  in  our  Fig  3G)  of  couriterbracing.  When  properly  con- 
structed it  makes  an  excellent  bridge.  The  common  objection  to  it,  and  not  without 
reason,  is  that  a  truss  and  an  arch  cannot  be  so  combined  as  to  act  entirely  in  con- 
cert; yet,  as  soon  as  any  ordinary  truss  begins  to  fail,  the  almost  invariable  remedy 
is  to  add  an  arch.  When,  however,  the  two  are  to  be  united,  it  is  better  to  so  pro- 
portion the  arch  as  to  be  capable  by  itself  of  safely  sustaining  the  max  load  at  rest; 


LJ 


and  to  confine  the  duty  of  the  truss  to  preventing  the  arch  from  changing  its  form 
under  a  moving  load.  Counterbracing  may  be  effected  by  strapping  the  heads  and 
feet  of  the  braces  to  the  chords;  or  by  iron  rods  parallel  to  the  braces;  two  to  a 
brace ;  with  screws  and  nuts,  as  at  v  f.  Or  by  similar  rods  across  the  other  diags  of 
the  panels.  The  following1  dimensions  answer  lor  a  single-track  rail- 
road bridge  of  about  150  feet  span.  Rise  from  out  to  out  of  chords  *%  of  the 
span ;  about  fourteen  or  sixteen  panels.  Width  in  clear  of  arches,  13  ft.  Six  arch- 
pieces  t,  of  8"  X  1^"  each,  to  each  truss.  Upper  chord  c,  12"  X  14".  Lower  chord  a  a, 
two  pieces  each  8"  X  !•*"•  Posts  ;>,  12"  transversely  of  the  bridge,  as  in  the  right- 
hand  fig;  by  8";  except  at  the  heads  and  feet,  where  enlarged  to  receive  the  ends  of 
the  braces.  Braces  8"  deep,  by  12"  wide.  Floor  girders  o,  8"  X  W  \  and  2%  to  3  ft 
apart  from  center  to  center.  Suspending  rods  (shown  at  s;  and  dotted  in  x)  1% 
diam.  Counterbrace  rods  in  pairs  parallel  to  braces,  about  1V£"  diam.  Bolts  for 
arches,  lower  chords,  &c,  \%"  diam.  Theoretically,  the  posts,  braces,  and  arches 
should  gradually  diminish  from  the  ends,  toward  the  center  of  the  truss ;  while  the 
chords  should  increase ;  but  in  practice,  the  additional  labor  of  getting  out  and  fitting 
pieces  of  diff  sizes,  frequently  makes  it  more  economical  to  use  uniform  sizes.*  The 
same  amount  of  arch  would  answer  also,  if  trussed,  as  in  Kig  35 ;  and  the  arch  by 
itself  with  a  full  max  load,  would  be  strained  less  than  800  Ibs  per  sq  inch,  at  its  feet. 
For  a  span  of  200  ft,  with  the  same  proportion  of  rise,  the  transverse  areas  of  the 
several  arch  and  truss  pieces  should  be  increased  33  per  cent ;  and  for  one  of  100  ft, 
they  may  be  diminished  the  same.  None  of  these  dimensions  are  the  result  of  close 
calculation.  The  dimensions  just  given  will  answer  for  common  travel,  for  a  span  of 
200  ft ;  with  a  depth  from  out  to  out  of  chords,  of  %  the  span ;  panels  10  to  12  ft  long. 
Many  such  spans  have  been  built  with  timbers  of  about  ^  less  transverse  section; 
and  without  counterbracing.  The  heads  of  the  posts  are  notched  about  2"  to  3"  into 
the  bottom  of  the  upper  chords;  and  are  moreover  tenoned  into  it  some  ins  further,' 
with  two  wooden  pins  through  the  tenon;  see  w,  Figs  36.  Their  feet  are  notched 
both  into  and  upon  the  lower  chords,  so  as  to  leave  the  two  chord-pieces  a  a  only 
about  2"  apart.  Through  these  and  the  post  pass  two  bolts  of  about  1"  to  1V£  diam. 

Since  the  upper  chord  resists  compression  only,  its  pieces  may  come  together  with 
a  plain  butt  joint,  d.  To  this  may  be  added  fishes  e  d,  of  stout  plank,  on  the  sides 
of  the  chord,  bolted  through  by  4  or  8  bolts. 

The  lower  chords  resist  pull;  and  the  pieces  composing  each  lower  chord  must 
therefore  be  joined  together  somewhat  as  in  Fig  37,  &c.  These  pieces  should  be  as 

*  It  will  be  borne  tn  mind  that  our  examples  are  not  intended  to  illnstrate  perfectly  proportioned 
structures.  None  of  them  would  endure  strict  criticism.  There  is  more  waste  of  timber  in  an  arch 
built  with  a  uniform  transverse  section  throughout,  than  in  the  straight  upper  chord  of  a  Howe  or 
Pratt  similarly  built;  for  both  must  be  proportioned  to  the  greatest  strain.  This  is  at  the  center  of 
the  two  last;  but  while  that  at  the  center  of  the  arch  is  as  great  as  ia  these,  that  at  its  feet  ia  muoh 
greater.  See  Example  2,  Art  33,  of  Force  ia  Rigid  Bodies,  p  468. 


TRUSSES. 


291 


long  as  possible,  and  should  never  be  jointed  opposite  to  each  other,  but  one  opposite 
the  middle  of  the  other ;  see  Fig  39. 

The  braces  are  merely  cut  to  fit  to  the  heads  and  feet  of  the  posts,  after  these  last 
have  been  fixed  in  their  places ;  and  usually  have  no  other  connection  to  them  than 
one  or  two  spikes  at  each  end,  for  small  bridges ;  or  screw-bolts  for  large  ones. 

The  ends  of  the  timbers  composing  the  arches,  butt  full  square  against  each  other; 
and  may  also  have  a  wooden  dowel.  The  joints  should  occur  at  the  posts,  as  shown 
at  W.  The  arches  are  screw-bolted  to  the  posts,  as  shown  in  the  figs,  by  bolts  of  1  to 
1*4  ins  diam.  Where  the  arches  pass  the  lower  chord,  both  are  notched,  and  well 
bolted  together.  The  feet  of  the  arches  abut  against  cast-iron  plates. 

When  suspension  rods  (dotted  in  Fig  36)  are  used  for  assisting  to  support  the  road- 
way, they  are  placed  as  shown  at  s  s ;  m  being  a  strong  block  of  wood  slightly  notched 
on  top  of  the  upper  arch-pieces.  The  rods  are  suspended  by  a  washer  and  nut  on  top 
of  the  block ;  and  after  passing  down  between  the  arches  and  chord,  have  a  similar 
arrangement,  but  inverted,  below  the  last;  as  shown  at  g. 

Floor-girders  not  exceeding  14  ft  clear  span,  may  be  8  ins,  by  15  ins  deep; 
and  placed  not  more  than  about  *2%  ft  apart  from  center  to  center.  Upon  them 
should  be  notched  and  spiked  stout  string-pieces,  say  12"  wide,  by  9"  deep,  to  carry 
the  rails;  and  to  distribute  the  pressure  of  the  load. 

Ilor  diagr  bracing:  for  diminishing  lateral  motion,  can  be  used  only  under 
the  floors  of  low  bridges ;  but  in  high  ones  it  is  introduced  also  at  the  top  of  the 
trusses.  When  of  timber,  these  braces  are  about  4  to  b  inches  thick  ;  by  6  to  9  deep ; 
and  form  a  hor  cross  between  each  two  opposite  panels  of  the  two  trusses.  If  the 
bridge  is  roofed,  and  has  girders  r,  Figs  36,  06^,  upon  and  well  secured  to  the  upper 
chords  c  c,  the  upper  lateral  bracing  may  consist  simply  of  4  iron  rods  n  w,  passing 
through  the  chords  about  midway  of  their  depth  ;  and  having  heads  and  washers  on 
their  outer  sides.  At  the  center  of  the  cross  the  rods  terminate  in  an  adjusting-ring ; 
see  No  14,  of  Figs  '21%.  In  a  bridge  of  l.;>0  ft  span,  these  rods  need  not  exceed  % 
inch  diam  at  the  center  panel,  and  1%  at  the  end  ones.  If  the  bridge  is  high,  and 
not  roofed,  but  open  at  top,  then  cross-struts  r  r,  Figs  36%,  must  be  inserted  pur- 
posely, when  this  rod-bracing  is  used.  If  it  is  also  used  at  the  lower  chords,  the  floor- 
girders  perform  the  duty  of  these  strutsT  Iron-bracing  is  not  liable  to  catch  tire 
from  the  locomotives.  Hor  diag  bracing  is  also  called  sway  bracing. 

A  favorite  mode  of  lateral  bracing-.  W,  Figs  3C%.  resembles  a  Howe 
truss  laid  flat  on  its  side.  In  it  the  diags  of  the  cross  are  struts  of  timber;  and  the 
pieces  r  r  are  round  rods.  One  of  the  struts  is  whole,  with  the  exception  of  a  slight 
mortice  on  each  vert  side,  at  its  center,  for  receiving  tenons  cut  on  the  inner  ends  of 
the  two  pieces  which  compose  the  other  diag.  At  the  sides  of  the  chords,  the  ends 
of  the  diags  rest  upon  a  ledge,  (shown  by 
the  dotted  line  i  i,)  about  1%  ins  wide,  cast 
at  the  bottom  of  the  cast-iron  angle-block. 
The  tie-rod  r  r,  passing  through  the  chords 
of  both  trusses,  being  tightened  by  means 
of  the  nut  s,  holds  the  diags  firmly  in  place ; 
and  in  case  of  their  shrinking  a  little  in 
time,  can  be  again  tightened  up  by  the  same 
means. 

Various  modifications  of  these  methods 
are  in  use ;  but  we  cannot  afford  them  space 
here.    The  cast  angle-block  is  as  deep  as  a 
brace ;  its  thickness  need  not  exceed  y2  inch,  in  a  large  bridge.  The  dark  triangle  if 
a  top  view  of  it.     It  has  holes  for  the  passage  of  the  rod  r  r. 

We  have  entered  somewhat  into  the  details  of  the  Burr  truss,  because  many  of 
them  are  more  or  less  applicable  to  others.  A  good  light  water-and-fire-proof  ma- 
terial for  bridge-roofs ;  and  one  not  corroded  by  the  smoke  from  the  engines,  is  yet  a 
desideratum. 

Art.  25.  Lengthenings-scarfs,  splices,  or  Joints.  The  lower  chords 
of  bridges,  being  exposed  to  great  pulling  strains,  require  much  care  in  connecting 
together  the  ends  of  the  several  pieces  of  which  they  are  composed.  There  is  much 
uncertainty  regarding  the  strength  of  the  joint-fastenings  in  common  use  for  this 
purpose.  Experiments  on  the  subject  are  much  needed.  When  only  two  pieces,  as 
t  and  y,  Fig  37,  or  38,  are  joined  by  any  of  the  ordinary  methods,  it  is  probably  not 
safe  to  depend  on  their  possessing  more  than  %  of  the  tensile  strength  of  a  s'ingle 
solid  beam  of  equal  cross-section.  When  the  chord,  as  in  Fig  39,  is  composed  of  two 
parallel  parts  a  a,  n  n,  made  up  of  long  pieces,  breaking  joint  with  each  other,  as  at 
jjj.eB.ch  of  the  two  parts  may  be  made  somewhat  stronger  than  either  one  of 
them  would  be  by  itielf.  This  is  owing  to  the  opportunity  afforded  of  connecting 


if 


292 


TRUSSES. 


them  also  by  bolts  b  b,  and  packing-blocks,  cc,  of  wood  or  iron,  intermediate  of  tha 
joints  jjt  &c.  By  this  means  the  strength  of  the  entire  chord  may  probably  be  prac- 
tically rendered  equal  to  one-half  of  what  it  would  be  if  solid.  If  the  chord  con- 
sists of  3  or  4  parallel  parts,  of  long  pieces,  breaking  joint,  and  connected  in  thtf 
same  way,  it  will  probably  have  about  %  of  the  strength  of  the  solid.  Care  must- 
of  course  be  taken  that  the  serviceable  area  of  the  pieces  shall  not  be  reduced  at  any 
intermediate  point,  to  less  than  it  is  at  the  joints. 


TOP 


Pig  38  is  a  simple  and  efficient  form  of  scarf.  Its  length  i  i  may  be  about  3  to  4 
times  the  greatest  transverse  dimension  of  the  beam.  At  the  center  is  a  block  t  of 
hard  wood,  with  a  thickness  equal  to  %  that  of  the  beam  ;  a  width  of  2  or  3  times  its 
thickness;  and  a  length  just  sufficient  to  reach  entirely  through  the  beam.  The 
beams  are  connected  by  4  screw-bolts  nn  ;  or  by  8  of  them,  if  the  length  requires  it. 
Plates  of  stout  rolled  iron,  a  a,  cc,  with  their  ends  bent  down  into  the  beams,' are 
occasionally  added.  These  require  bolts  o  o,  beyond  the  ends  i  i  of  the  scarf.  These 
bolts  are  not  shown  in  the  side  view. 

Fig  37  is  another  excellent  joint  with  splicing -bJ-ocks  e  e,  instead  of  the  block  t  of 
Fig  38.  The  indentations  v  v,  may  each  be  about  Y&  as  deep  as  the  beam  is  thick.  The 
length  of  each  splice-block,  about  6  times  s  s.  From  4  to  8  screw-bolts,  as  the  case 
may  require.  Length  of  each  indent  about  ^  that  of  the  block  itself. 

Fig  40  is  a  joint  formed  by  two  flat  iron  links  or  rings  1 1,  let  flush  into  the  tim- 
bers, and  retained  in  place  by  spikes.  The  iron  may  vary  from  \±  to  1  inch  in  thick- 
ness ;  from  1  to  4  or  5  ins  in  width ;  and  2  to  6  ft  in  length ;  as  occasion  may  require. 

Fig  40  6,  is  a  joint  formed  by  two  blocks  c  c,  of  hard  wood,  passing  through  the 
timbers ;  and  connected  by  bolts,  a  a,  n  n. 

In  Fig  40  a,  s  s  are  cast-iron  packing-blocks,  sometimes  used  instead  of  plain  wooden 
ones,  at  points  6  &,  Fig  39,  intermediate  of  the  joints  jj.  The  openings  in  the  centers 
of  the  blocks  are  needed  only  when  vertical  truss-rods  have  to  pass  through  those 
points.  At  e  e,  of  the  same  fig,  is  shown  another  form,  much  used  in  chords  composed 
of  two  or  more  parallel  strings.  Both  these  are  as  deep  as  the  chord;  and  their 
cross-sections,  or  end  views  shown  in  the  fig,  may  be  from  4  to  10  ins  long:  2  to  4  ins 
wide;  and  from  %  to  1%  ins  thick;  according  to  size  of  bridge,  Ac. 

REM.  In  selecting  hard  wood  for  splicing-blocks,  treenails,  or  for  any  part  of  a 
bridge,  it  is  well  to  remember  that  the  oaks  when  in  contact  with  the  pines,  expedite 
the  decay  of  the  latter;  therefore,  it  is  generally  better  to  employ  the  best  southern 
yellow  pine  heart  wood  for  such  blocks,  &c,  or  interpose  sheet  iron.* 

*  The  tendency  of  some  kinds  of  timber  to  produce  rapid  decay  when  brought  into  close  contact 


TRUSSES. 


293 


Eye-Bars  and  Pins.  The  lower  chords  of  iron  bridges  usually 

•ousist  of  flat  links  or  bars  c  and  o,  W  and  H,  Figs  41,  on  edge  and  connected  by  tight-fitting  wrought- 
iron  pins  b  and  P.  After  deciding  on  the  size  of  the  body  W  or  H  of  the  bars  to  bear  safely  the  pull 
upon  them,  the  proper  proportioning  of  their  heads  or  eyes  and  pins  is  an  abstruse  and  difficult  point 
upou  which  much  has  been  written.  It  was  formerly  supposed  that  the  diam  of  the  pin  should  be 
governed  by  its  resistance  to  shearing,  but  experience  has  shown  that  this  was  entirely  insufficient. 


n 


Pigs  41. 
Js 


We  give  a  table  of  practical  conclusions  arrived  at  by  that  accomplished  expert,  Chs.  Shaler  Smith, 


,         .  , 

from  111  experiments  by  himself  on  a  working  scale.  The  table  shews  some  irregularities,  for  as  Mr. 
Smith  remarks  "  the  bars  declined  to  break  by  formula."  The  pin  is  more  strained  at  the  outer  links 
o  o  than  at  the  inner  ones  c  c  c,  so  that  the  latter  would  not  require  so  large  a  diam,  but  that  this 
must  be  uniform  throughout  in  order  to  secure  tight  fitting  for  all  of  them.  When  web  members  as 
well  as  chords  are  held  by  the  same  pin  the  diara  and  head  must  be  proportioned  for  that  bar  of  them 
all  which  is  most  strained.  When  the  heads  are  made  by  pressure  in  one  piece  with  the  body,  the 
metal  v  a  and  u  x  at  the  sides  of  the  pin  b  must  be  wider  than  when  the  heads  are  first  made  in  sepa- 
rate pieces  by  hammering  and  then  welded  to  the  body.  But  the  welded  one  W  requires  more  iron 
back  of  the  pin  as  shown  at  1  1.  This  width  1  1  must  be  equal  to  the  diam  of  the  pin. 

The  links  are  supposed  to  be  of  uniform  thickness. 

Having  drawn  a  circle  b  for  the  pin,  lay  off  on  each  side  of  it  as  at  v  s,  u  x,  half 
the  width  of  metal  in  the  table  for  the  head  of  W  or  H  as  the  case  may  be. 
Then  for  forming  the  head  of  H  use  only  the  rad  b  s  as  shown.  For  the  head  of 
W  lay  off  also  1  1  =  diam  of  pin.  Find  by  trial  the  rad  g  n  or  g  t  and  use  it,  except 
for  uniting  the  head  to  the  body,  where  use  a  rad  =  1.5  g  n  as  shown. 


Metal  in  head 

Metal  in  bead 

Width 

Thicks. 

Diam. 

across  pin. 

Width 

Thicks. 

Diam. 

across  pin. 

of  bar. 

of  bar. 

of  pin. 

W. 

H. 

of  bar. 

of  bar. 

of  pin. 

W. 

H. 

I. 

.2 

.67 

1.33 

1.50 

1. 

.55 

1.28 

1.50 

1.60 

1. 

.25 

.77 

1.33 

1.50 

1. 

.60 

1.36 

1.55 

1.72 

1. 

.30 

.86 

1.40 

1.50 

1. 

.65 

1.43 

1.60 

1.76 

1. 

.35 

.95 

1.50 

1.50 

1. 

.70 

1.50 

1.67 

1.85 

I. 

.40 

1.04 

1.50 

1.50 

1. 

.80 

1.64 

1.67 

1.95 

1. 

.45 

1.12 

1.50 

1.53 

1. 

.90 

177 

1.70 

2.05 

1. 

.50 

1.20 

1.50 

1.56 

1. 

1.00 

1.90 

1.76 

2.21 

Art.  26.    Figs  42,  exhibit  joints  adapted  to  most  of  the  cases  that 
occur  in  practice  with  wooden  beams,  &c.    They  need  but  little  explanation.    Fig  a 
is  a  good  mode  of  splicing  a  post ;  in  doing  which  the  line  o  o  should  never  be  in- 
clined or  sloped,  but  be  made  vert;  otherwise,  in  case  of  shrinkage,  or  of  great 
pressure,  the  parts  on  each  side  of  it  tend  to  slide  along  each  other,  and  thus  bring 
a  great  strain  upon  the  bolts.    When  greater  strength  is  reqd,  iron  hoops  may  be 
i     used,  as  at  6,  h.  and.;,  instead  of  bolts.     Fig  ft,  a  post  spliced  by  4  fishing  pieces: 
!     which  may  be  fastened  either  by  bolts,  as  in  the  upper  part ;  or  by  hoops,  as  in  the 
i     lower.    The  hoops  may  be  tightened  by  flanges  and  screws,  as  at  s ;  or  thin  iron 
wedges  may  be  driven  between  them  and  the  timbers,  if  necessary.     Fig  C  shows  a 
,    good  strong  arrangement  for  uniting  a  straining-beam  Ar,  a  rafter  £,  and  a  queen-post 
u ;  by  letting  k  and  I  abut  against  each  other,  and  confining  them  between  a  double 
queen-post  1 1 ;  n  n  are  two  blocks  through  which  the  bolts  pass.    A  similar  arrange- 
ment is  equally  good  for  uniting  the  tie-beam  w,  with  the  foot  u,  of  the  queens ;  with 
•    the  addition  of  a  strap,  as  in  the  fig.     Fig  ?  is  a  method  of  framing  one  beam  into 
'    another,  at  right  angles  to  it.    An  iron  stirrup,  as  at/,  may  be  used  for  the 
i    same  purpose  ;  and  is  stronger.    Figs  g  h,  ij  are  built  beams.     When  a  beam 
I    or  girder  of  great  depth  is  required,  if  we  obtain  it  by  merely  laying  one  beam  flat 

with  other  kinds,  is  a  subject  of  great  practical  importance;  but  one  which  hitherto  has  received  but 
little  attention.  Black  walnut  and  cypress  are  said  to  cause  mutual  rot  within  a  year  or  two.  Ob- 
served cases  of  this  kind  should  be  reported  to  the  leading  professional  journals. 


294 


TRUSSES. 


u 


TRUSSES. 


295 


^NS 


then  bolt  or  strap  them  firmly  together  to  create  friction ;  we  obtain  nearly  the  strength 
of  a  solid  beam  of  the  tolal  depth  ;  which  strength  it,  as  the  square  ot  the  depth. 

Tne  strength  of  a  built  beam  is  increased  by  increasing  its  depth  at  its  center,  where  it  is  most 
strained  ;  as  in  the  upper  chords  of  a  bridge.  This  may  be  done  by  adding  the  triangular  strip  y  y 
between  the  two  beams. 

Tredgold  directs  that  the  combined  thicknesses  of  all  the  keys  be  not  less 

than  1.4  times  the  entire  depth  of  the  girder;  or  when  indents  are  used,  as  in  ij.  that  their  combined 
depths  be  at  least  %  that  of  the  girder.  If  the  girder  ij  be  inverted,  it  will  lose  mnch  of  its  strength. 

A  piece  of  plate-iron  may  be  placed  at  the  joints  of  timbers  wneu  there  is  a  great  pressure  ;  which 
is  thus  more  equalized  over  the  entire  area  of  the  joint;  or  cast  iron  may  be  used. 

Frequently  a  simple  strap  will  not  suffice,  when  it  is  necessary  to  draw  the  two  timbers  very 
tightly  together.  In  such  cases,  one  end  of  each  strap  may,  as  at  x,  terminate  as  a  screw  ;  and  after 
passing  through  a  cross-bar  Z,  all  may  be  tightened  up  by  a  nut  at  x.  Or  the  principle  of  the  DOU- 
BLE KEY,  shown  at  K,  may  be  applied.  Sometimes,  as  at  A,  the  hole  for  the  bolt  is  first  bored  ;  then 
a  hole  is  cut  in  one  side  of  the  timber,  and  reaching  to  the  bolt-hole,  large  enough  to  allow  the  screw 
nut  to  be  inserted.  This  being  done,  the  hole  is  refilled  by  a  wooden  plug,  which  holds  the  nut  in 
place.  Then  the  screw-bolt  is  inserted,  passing  through  the  nut.  By  turning  the  screw  the  timbers 
may  then  be  tightened  together.  Figs  21^,  page  268,  may  be  consulted  also. 

When  the  ends  of  beams,  joists,  &c,  are  inserted  into  walls  in  the  usual  square  manner,  there  is 
danger  that  in  case  of  being  burnt  in  two,  they  may,  in  falling,  overturn  the  wall.  This  may  be 
avoided  by  cutting  the  ends  into  the  shape  shown  at  TO. 

When  a  strap  o.  Fig  R,  has  to  bear  a  strain  so  great  as  to  endanger  its  crushing  the  timber  p,  on 
which  it  rests,  a  casting  like  v  may  be  used  under  it.  The  strap  will  pass  around  the  back  r  of  the 
casting.  The  small  projections  in  the  bottom  being  notched  into  the  timber,  will  prevent  the  casting 
from  sliding  under  the  oblique  strain  of  the  strap.  The  same  may  be  used  for  oblique  bolts,  and 
below  a  timber  as  well  as  above  it.  When  below,  it  may  become  necessary  to  bolt  or  spike  the  casting 
to  the  under  side  of  the  timber.  When  the  pull  on  a  strap  is  at  right  angles  to  the  timber,  if  then 
is  much  strain,  a  piece  of  plate  iron,  instead  of  a  casting,  may  be  inserted  between  the  strap  and  tht 
timber,  to  prevent  the  latter  from  being  crushed  or  crippled ;  see  I  and  I. 

Art.  27.  Expansion  rollers,  Fig  43,  or  planed  iron  Slides;  or 
rockers,  Fig  44 ;  or  suspension-links.  Figs  45, 46;  must  be  provided  when 
an  iron  span  exceeds  about  t  Oft;  in  order  to  allow  the  trusses  to  contract  and  expand 
freely  under  changes  of  temperature,  without  undue  strain  upon  some  of  its  mem- 
bers. Fig  43  shows  the  general  arrangement  of  roll- 
ers; which  are  cylinders  of  cast  iron  or  steel,  from  3 
to  6  ins  diam ;  and  1  to  4  ft  long ;  planed  smooth.  From 
4  to  8  or  more  of  these  are  connected  together  by  a 
kind  of  framing,  n  n;  and  one  such  frame  is  placed 
under  at  least  one  end,  of  the  truss.  The  rollers  rest 
upon  a  strong  planed  cast  bed-plate  00;  bolted  to  the 
masonry  below.  Under  the  end  of  the  truss  is  a  sim- 
ilar plate  s  s,  by  which  it  rests  on  the  rollers.  Since 
a  truss  of  even  200  ft  span  will  scarcely  change  its 
length  as  much  as  3  ins  by  extremes  of  temperature, 
the  play  of  the  rollers  is  but  small.  They  are  kept  in 
line  by  flanges  cast  along  the  side  of  the  bed-plate.  Flanges  should  also  project 
downward  from  s  s,  so  as  completely  to  protect 
the  rollers  from  dust,  rain,  &c. 

In  Fig  44,  r  r  gives  a  general  idea  of  a  KOCKER  ;  and  Fig  45, 
«  «,  of  a  SUSPENSION-LINK.  U  U  in  each  ng  is  aside  view  of 
a  cast-iron  Fink  upper  chord,  through  each  end  of  which 
passes  around  pin  o,  which  sustains  the  entire  weight  of  the 
truss  and  its  load;  and  which  is  sustained  by  from  4  to  6 
rockers  or  links  as  the  case  may  be.  In  a  railroad  bridge  of 
205  ft  span,  across  the  Mouongahela,  the  links  are  3^  ft  long ; 
and  the  pins  5  ins  diain  ;  and  in  others  of  the  same  size,  over 
Barren  and  other  rivers,  the  rockers  are  a  foot  wide  from  r 
tor;  and  about  5  ins  wide  transversely  on  the  curved  tread 
or  rim.  For  the  accommodation  of  these  several  links  and 
rockers ;  as  well  as  of  the  various  bars  b  b,  which  constitute 
the  oblique  ties  of  a  Fink  truss;  the  ends  of  the  octagonal 
cast-iron  upper  chords  are  widened  out,  as  shown  by  Fig  46; 
which  is  atop  view  of  a  longitudinal  section  of  such  an  end. 
The  rockers,  or  links,  and  bars,  6,  occupy  the  spaces  n  n,  &c, 
between  the  several  partitions  of  the  chord  ;  and  the  pin  oo 
passes  through  them  all.  except  when  it  is  expedient  to 
attach  some  of  the  bars  to  the  sides,  or  to  the  top  of  the 
chord,  as  at  t.  These  figs  are  intended  merely  to  illustrate 
the  general  principle,  without  regard  to  detail  of  construction. 

In  some  English  bridges  of  considerable  size,  such  as  the 
Crumlin  Viaduct,  of  150  ft  spans ;  and  the  Newark  Dyke 
bridge,  of  240  span;  (both  of  them  Warren  girders,)  and 
sustained  like  the  foregoing,  by  the  ends  of  the  upper 
chords  ;  no  further  precaution  is  taken  with  regard  to  expan- 
sion and  contraction,  than  merely  to  rest  the  ends  of  said 
chords  upon  smoothly  planed  iron  plates,  upon  which  they 
may  slide.  So  also  several  American  bridges. 

Art.  28.    The  weight  of  bridges  of 

the  game  span,  designed  by  different  persons,  va- 
ries considerably,  from  several  causes  ;  such  as  the 


U 


296 


TRUSSES. 


form  of  truss;  quality  of  iron;  proportions  of  wrought  and  cast  iron;  coefficient 
adopted  for  safety,  and  for  strength  of  materials  ;  whether  the  roadway  is  on  the  top 
chord,  or  the  bottom  one,  <fcc,  &c. 

For    a    mere    approx   Wt  to  be  assumed  as  a  preliminary  in  calculating  the  strength 

idea  of  the  quantity  of  iron  reqd, 
r  foot  run  of  bridge,  of  only  the  tw» 


and  proportioning  the  parts  of  a  bridge,  or  for  forming  some  rude  idea  of  the  quantity  of  iron  reqd, 
gest  the  following  purely  empirical  rule.    It  is  for  the  wt  per  foot  run  of  bridge,  of  only  the  tw» 
trusses  together,  and  their  lateral  bracing,  for  a  single-track  railway  bridge.    The  wt  of  cross-beams, 


•we  suggest  the  following  purely  empirical  rule.    It  is  for  the  wt 
trusses  together,  and  their  lateral 
flooring,  rails,  &c.,  is  not  included. 
Jf  unde 


Wt  of  only  the  two  trusses 
together  ("and  their  lateral 


Span        Sq  rt  of  span         But  if  over        Sq  of  span 

—  Tt       =  • J-  .1 


10 


112000 


As  a  rude  average,  we  may  set  down  for  spans  not  exceeding  about  200  ft,  cross  floor  girders  of 
about,  7'  X  15",  and  about  2J^  to  3  ft  apart  from  center  to  center;  clear  bearing  14  ft;  together  with 
substantial  string-pieces  of  about  10"  X  12",  for  supporting  the  rails;  the  rails  themselves;  and  a 
plank  pathway  between  the  rails,  all  complete,  at  about  .14  ton,  or  314  fts  per  ft  of  span  ;  or  with  a 
full  floor  of  3"  plank,  14  ft  wide,  about  .2  ton,  or  448  fts.  For  greater  spans,  with  the  trusses  farther 
apart,  increase  this  to  .25  ton,  up  to  300  ft  ;  .3  ton,  to  400  ft  ;  .35  ton,  to  500  ft  and  .4  ton,  to  600  ft  ; 
as  is  done  in  the  following  table. 

If  trussed  Phoenix  beams,  p  210;  or  Fairbairn  beams,  page  214;  of  Strength  of  Materials,  be  used 
instead  of  wooden  ones,  the  weight  of  roadway  will  not  be  seriously  increased  thereby. 

When  a  bridge  is  to  be  roofed  and  weather-boarded,  an  addition  must,  of  course,  be  made  to  our 
weights. 

Wooden  bridges  weigh  about  the  same  as  iron  ones  of  equal  strength. 

Table  of  approximate  average  weights  per  foot  run  of 
single  track  railroad  bridges  of  iron,  by  above  rule.  (Original.) 


c 

Wt  of  only  the 
two  trusses 

Wt  of  trasses, 

•°T3 

0 

Wt  of  only  the 
two  trusses 

Wt  of  trusses, 

Is, 

§, 

together;  with 
their  lateral 

roadway,  Ac, 
complete;  per 

J3j 

1 

together  ;  with 
their  lateral 

roadway,  Ac, 
complete  ;  per 

*! 

I 

bracing:  per 

foot  run. 

£  3 

s 

bracing;  per 

foot  run. 

^    SJ 

£ 

foot  run. 

111 

5 

foot  run. 

js| 

Feet. 
5 

Tons. 
.032 

Lbs. 
72 

Tons. 
.232 

Lbs. 
520 

Tons. 
4.23 

Feet. 
120 

Tons. 
.350 

Lbs. 
784 

Tons. 
.550 

Lbs. 
1232 

Tons. 
1.72 

7Jl» 

.042 

94 

.242 

542 

3.50 

130 

.374 

838 

.574 

1286 

1.73 

10 

.052 

116 

.252 

564 

3.50 

140 

.398 

891 

.598 

1339 

1.74 

12}* 

.060 

134 

.260 

582 

3.50 

150 

.422 

945 

.622 

1393 

.75 

15 

.069 

155 

.269 

603 

2.94 

160 

.446 

999 

.646 

1447 

.76 

.077 

172 

.277 

620 

2.80 

170 

.470 

1053 

.670 

1501 

.78 

20 

.085 

190 

.285 

638 

2.79 

180 

.494 

1107 

.694 

1555 

.80 

25 

.100 

224 

.300 

672 

2.30 

190 

.518 

1160 

.718 

1608 

.82 

30 

.115 

258 

.315 

706 

2.00 

200 

.541 

1212 

.741 

1660 

.84 

35 

.129 

289 

.329 

737 

1.95 

225 

.600 

1344 

.850 

1904 

.94 

40 

•143 

320 

.343 

768 

1.90 

250 

.658 

1474 

.908 

'J034 

.99 

45 

.157 

352 

.357 

800 

1.85 

275 

.775 

1738 

1.025 

2296 

.10 

50 

.171 

383 

.371 

831 

1.80 

300 

.904 

2025 

1.154 

2585 

.22 

60 

.197 

441 

.397 

889 

1.80 

350 

1.194 

2674 

1.494 

3346 

.55 

70 

.224 

502 

.424 

950 

1.75 

400 

1.529        3425 

1.829 

4097 

.88 

80 

.249 

558 

.449 

1006 

1.70 

450 

1.908 

4274 

2.258 

5058 

.30 

90 

.275 

616 

.475 

1064 

1.70 

500 

2.332 

5224 

2.682 

6008 

3.72 

100 

.300 

672 

.500 

1120 

1.70     I 

550 

2.801 

6274 

3.201 

7170 

4.24 

110 

.325 

728 

.525 

1170 

1.70     i 

600 

3.314 

7423 

3.714 

8319 

4.75 

The  Fink  trass,  when  the  roadway  is  oil  top,  requires  but  one 
chord,  and  makes  a  lighter  bridge  than  any  of  the  beam-systems.  The  following  are 
the  weights  per  ft  run,  of  some  single-track  Fink  railroad  bridges  of  that  kind,  by  those 
first-class  builders,  Smith  &  Latrobe,  of  Baltimore.  These  gentlemen  adopt  high 
coefficients  of  safety;  and  use  only  iron  of  superior  quality.  The  wts  include  the 
roadway ;  which,  however,  is  open,  and  consequently  considerably  lighter  than  our 
assumed  .18  ton  per  ft  run.  See  Art  31. 

Clear  span  50  ft.    Wt  per  ft  .28  ton.    I    Clear  span  150  ft.    Wt  per  ft  .45  ton. 
"        «   100        .    "    "     "  .38    "       I        "        *«     210          "       "    "  .57    " 

Of  late  years,  however,  several  casualties  resulting  from  open  floors  are 
inducing  our  engineers  to  reject  them  in  favor  of  close  ones,  sufficiently  strong 
to  sustain  a  derailed  train ;  beside  affording  passengers  in  such  cases  a  safe 
egress  from  the  bridge  on  foot. 

Two  trusses  of  the  Newark  Dyke  bridge,  England,  Warren  girder,  240%  ft  span, 
weigh  1.02  tons.  The  single  track  tubes  of  the  Victoria  bridge  at  Montreal,  244  ft 
span,  1.14  tons :  the  330  ft  span,  2  tons.  The  single-track  Britannia  tube,  Eng.,  460 
ft  span,  3.43  tons.  Two  trusses  alone,  of  the  Penn'a  Cen.  R  R,  at  Phila.,  180  ft  span, 


TRUSSES. 


297 


(Pratt's  system,)  by  Mr  Linville,  .52  ton.     The  fine  320  ft  span  across  the  Ohio  at  Steu- 
benville,  (  Pratt, )  also  by  Mr  Linville,  1.6  tons  per  ft.  All  these  are  of  iron;  single  track. 
1'or  some  dimensions  for  short  bridge  trusses,  see  Art  31,  p  304. 

Tbe  greatest  load  that  can  come  upon  a  bridge, 

If  for  a  single-track  rail- 
road, can  scarcely  exceed 
that  of  a  string  of  heavy 
locomotives  coupled  to- 
gether,  without  their  ten- 
ders. Such  engines,  with 
very  few  exceptions,  do 
not  weigh  more  than  a 
ton  per  foot  of  their  ex- 


me  length  ;  this  being   *• 
the  proportion  of  the  27-    * 
ton   coal-haulers   on   the  r* 
Reading  R  R.     A  45-ton  ^ 
auxiliary   engine    on    12 
drivers,  on  that  road,  is  36 
ft  long  from  out  to  out;  * 
and  conseqaently  weighs 
1  %  tons  per  foot ;  but  a  , 
long  string  of  such,  with-  ~ 
out  their  tenders,  is  hard-   ** 
ly    probable.     The    diffs   ^ 
between  our  4th  and  6th    • 
cols  will  give  our  assumed  7j! 
max  loads.  But  inasmuch 
as  8  tons   is  a  common 
weight  on  a  pair  of  driv- 
ers ;  and  (though  rarely) 
eveu  12  tons;  it  is  plain 
that  a  span  of  only  I  ft 
may  have  to  sustain  8  or 
12  tons  per  ft  run.    Our 
table  makes  sufficent  al- 
lowance for  the  greater  © 
loads  per  ft  run  that  may 
probabl  v  come  upon  small   I 
spans. 

On  very  small  spans  i 
the  loads  cannot  be  as- 
sumed to  be  equally  dis- 
tributed. On  bridges  for 
turnpikes  and  common 
roads,  no  probable  contin- 
gency could  crowd  people 
upon  them  to  such  an  ex- 
tent as  to  weigh  more  than 
80  Ibs  per  sq  ft  of  floor. 
The  French  standard  in- 
deed is  but  half  of  this, 
or  42  Ibs  per  sq  ft ;  and 
is  sufficient  for  probabil- 
ity, but  not  for  possibility. 
The  latter  may  increase  it 
to  80  Ibs ;  and  this  may 
safely  be  taken  as  the 
maximum  load  on  spans 
of  20  or  more  feet.  To 
compensate,  however,  ( 
for  momentum,  we  re-  i 
commend  to  adopt  100  Ibs,  K 
or  .045  of  a  ton,  as  the  *** 
limit  for  crowds,  t  A 
bridge  for  a  single-track 
carriage-way;  with  room 
between  the"  trusses  for  a 
footway  also,  should  not 
be  less  than  12  ft  wide  in 
the  clear ;  iu  which  case 
its  greatest  load  at  100 
fts  per  pq  ft,  would  be  J^ 
a  ton  per  ft  run  ;  or  if  24 
ft  wide,  with  but  two  trus- 
ses, the  load  would  be  full 
Iton  per  ft  run:  or  as  great 
as  that  on  a  single-track 
railway  bridge  more  than 


*  All  the  wt  of  such  an  engine  is,  however,  concentrated  within  about  20ft  length  of  road:  and 
would  therefore  produce  in  a  20  ft  span  a  load  of  '2^  tons  per  ft,  independently  of  the  wt  of  the  bridge 
itself,  or  of  any  allowance  for  momentum. 

t  The  engineers  of  the  Chelsea  bridge,  London,  packed  picked  men  upon  the  platform  of  a  weigh 
bridge;  with  a  result  of  84  fl>s  per  sq  ft.  Mr  Nash,  architect  of  Buckingham  Palace,  experimenting 
with  reference  to  fire-proof  floors  for  that  building,  wedged  men  together  as  closely  as  they  could  pos- 


298 


TRUSSES. 


60  ft  span.  But  in  a  common  bridge  also,  the  greatest  load  per  ft  run,  on  a  very  short  span,  will  be 
greater  than  in  a  long  one  ;  as  in  the  case  of  two  wheels  of  a  truck  hauling  a  large  block  of  stone,  &o  ; 
and  this  must  be  taken  iuto  consideration  in  building  such. 

It  must  also  be  remembered  that  each  transverse  floor-girder  must  bear  at  least  all 
the  weight  resting  upon  two  wheels;  ho  matter  how  close  together  the  girders  may 
be  placed.  If  they  are  farther  apart  than  the  dist  between  two  axles  of  a  vehicle, 
they  will  have  to  bear  more  than  the  load  on  one  pair  of  wheels. 

The  allowance  for  safety  in  a  trim*  bridge. 

As  the  result  of  a  long-continued  series  of  deflections  applied  to  an  experimental  plate-iron  girder 
of  20  ft  span,  Mr  Fairbairn  concludes  that  a  bridge  subject  to  100  deflections  per  day,  each  equal  to 
that  produced  by  %  of  its  extraneous  breaking  load,  would  probably  break  down  in  about  8  years  ; 
while,  with  100  daily  deflections  equal  to  that  arising  from  but  y^  of  its  breaking  load,  it  would  last 
fully  300  years.  We  are  of  the  opinion  that  a  bridge  should  not  have  a  safety  of  less  than  4  for  its 
max  extraneous  loud,  and  Its  own  weight,  combined;  nor  do  we  see  any  use  in  exceeding  6.  From 
4  to  5  may  be  used  in  temporary  structures,  or  in  those  rarely  exposed  to  maximum  strains ;  and  6  in 
more  important  ones  frequently  so  exposed.  The  last  will  (roughly  speaking)  generally  give  a  safety 
of  about  2  against  reaching  the  elastic  strength,  which  is  the  true  guide  in  such  matters.  But  4,  d, 
&c,  usually  refer  to  the  ultimate  or  breaking-down  strength;  so  that  a  truss  with  such  a  safety 
of  2  would  in  fact  be  very  unsate. 

Art.  29.  Remarks  on  king:  and  queen ;  and  on  Fink  trusses, 
for  roofs.  The  following  comparison  is  founded  upon  total  spans,  or  lengths 
of  truss,  of  151  ft.  Rise  30.8  ft ;  or  i  of  the  total  span.  Trusses  7  ft  apart  from  cen- 
ter to  center.  Each  rafter  83  ft  long.  Total  load,  including  the  truss  itself,  40 
fts  per  sq  ft  of  roof;  or  20.8  tons  to  each  truss.  There  are  seventeen  points  of  sup- 

20.8 
port  in  each  truss;  consequently  a  full  panel-load  (Art  11)  is  —  ==  1.3 tons.  Trusses 

as  shown,  one-half  of  each,  in  Figs  47  and  48.  The  strain  in  tons  (calculated 
as  if  all  the  weight  of  truss  and  load  were  on  the  rafters)  is  marked  on  each 
member.  The  assumed  coefficient  of  safety  for  ties  is  3.  Iron  is  supposed  to  be  used 
that  will  not  break  with  a  less  pull  than  20  tons  per  sq  inch ;  the  assumed  safe  allow- 
able pull  being  therefore  here  taken  at  -^  =  6%  tons  per  sq  inch.  The  safe  pressure 
along  the  rafters  is  taken  at  3^  tons  per  sq  inch.  The  struts  are  assumed  to  be  wrought 
cylindrical  tubes,  with  an  outer  diam  equal  to  ^  of  their  length;  and  of  such 
thickness  as  will  give  them  a  metal  area  of  1  sq  inch  for  each  2  tons  of  strain.  The 
rafters  are  in  the  present  case  supposed  to  be  9-inch  rolled  Phoenix  beams ;  7%  sq  ins 
transverse  area;  weighing  25  ft»s  per  foot  run.  The  ends  of  all  ties  are  supposed 
to  be  enlarged,  or  upset;  so  that  the  cutting  of  the  screw-threads  shall  not  diminish 
their  effective  area.  The  purlins  are  supposed  to  be  at  or  near  the  "points  of  sup- 
port," so  as  to  produce  no  cross-strains  on  the  rafters. 
Table  1.  Weight  of  the  Fink  truss,  of  which  Fig  47  shows 

one-half.     See  also  Figs,  p  264. 

Length  154  ft.    Rise  £  length.     Trusses  7  ft  apart.    Load  40  fts  per  sq  ft  of  roof;  including  trusi. 
See  Note,  p  263.  (Original.) 


Name  of  part. 

Number 
of 
parts. 

Area  of 
each  part, 
sq  ins. 

Lbs.  per 
foot  run 

each  part. 

Total 
weight  of 
all  the 
parts. 
Lbs. 

Lbs. 

Rafter  

2 

7  50 

(n  

2 

1  95 

6  5 

4150 

2 

jrfl 

2 

3  41 

'  '    '  ' 

979  f 

1446 

If  '.'.'.'.:*.:::::  

2 

3  66 

1  '     99 

(U    

4 

' 

1  

4 

n-4 

2 

2 

0  98 

3  27 

•2 

1  47 

4  89 

us' 

676 

\k  :::::. 

2 

1  71 

5  70 

TOO 

U  :.::. 

g 

0  25 

•JO     } 

Cj   

2 

' 

Struts  <  w  
(i    

4 

• 

1.20 

4.0 

136  [ 

476 

Center  vertical  y  say 
Joint  and  splicing-pieces,  nuts,  &c,  &c...  say 

1 

40 
400 

40 
400 

Shoes  at  ends  of  rafters,  say  
Wt  of  purlins  not  included.  Total  wt  of  truss 

400 
=7588 

400 
7588 

erea  down  from  """"• amo1*  ' 


TRUSSES. 


299 


With  the  same  total  load  per  sq  ft,  including  the  trusses,  (with  trusses  7  ft  apart ;  rise  £ 
span)  the  area,  and  wt  per  foot  run  of  each  part,  as  well  as  the  strain  upon  it, 
will  vary  directly  as  the  spans;  but  the  total  wts,  as  the  squares  of  the  spans. 
Hence,  it  is  easy  to  deduce  from  the  table  the  areas  reqd  for  smaller  spaus.  The 
rafters  for  small  spans  are  frequently  made  of  round  iron  rods  from  \VA  to  1%  ins 
diam  ;  or  of  ordinary  flat  bars.  Tubes  with  the  name  area  of  metal,  would  be  better. 
For  trusses  also  of  different  spans,  and  rise  of  \  the  span,  7  ft  apart,  in  which  the 
rafters  and  struts  are  of  wood,  with  ties  of  iron,  the  strains  may  be  deduced  quite 
closely  from  those  in  Figs  47  and  4*.  They  will,  however,  be  somewhat  greater, 
because  wooden  struts,  not  being  hollow  like  our  assumed  iron  ones,  must  be  heavier 
than  the  latter  to  prevent  bending.  The  weight  of  the  load,  however,  is  generally  so 
much  greater  than  that  of  the  truss,  that  this  consideration  of  the  strut  is  not  very 
material ;  so  that  a  roof  partly  of  wood  may  be  assumed  in  practice  to  weigh,  together 
with  its  load,  but  little  more  than  an  iron  one ;  and  the  strains  on  the  several  parts 
will  be  nearly  the  same  in  both  cases. 

Table  2.    Woijrht  of  the  king  and  qneen  truss,  of  which 

Fig:  48  shows  one-half.    See  also  Fig  14,  p  260. 

Length  154  ft.    Rise  i  length.    Trusses  7  ft  apart.    Load  40  B>s  per  sq  ft  of  roof;  including  truss. 
See  Note,  p  263.  (Original.) 


Name  of  part. 

Number 
of 
parts. 

Area  of 
each  part, 
sq.  ins. 

Lbs.  per 
foot  run 
of 
each  part. 

Total 
weight  of 
all  the 
parts. 
Lbs. 

• 

Kafter  

2 

7.5 

25 

4150 

4150  00 

IH      . 

2 

2.2 

7  33 

146  70 

G 

2 

2.44 

8  14 

162  80 

P 

2 

2.68 

8  94 

178  80 

i, 
*•  

2 

2.92 

9.74 

194.80 

1606.30 

0  ... 

2 

3.41 

10.54 
11  34 

210.80 
226  80 

B 

2 

3.65 

12  14 

242  80 

A  :....  y  :::::::::::: 

2 

3.65 

12  14 

242  80 

a 

2 

.0 

.0 

o  ^ 

\'J    • 

2 

33 

5  34 

\i  .  :     :::::::::: 

2 

.2 

67 

1600 

Verticals                •(  I    .  .         

2 

.3 

1  00 

32  00  /• 

298  66 

1  m 

2 

.4 

1  33 

53  3S  1 

„         .   . 

2 

.5 

1  67 

80  00 

U  .............. 

2 

.6 

2  00 

112  00  j 

Center  vertical        p         

1 

1  4 

4  67 

150  00 

150  00 

f  n 

2 

88 

2  92 

64  20^ 

\r 

2 

1  05 

8  50 

91  00 

\\  .............. 

2 

1  28 

4  25 

136  00 

Struts.      .            <  t                

2 

1  55 

5  17 

196  37  ^ 

1587  50 

1  V 

2 

1  82 

6  08 

267  63  f 

1"  ;      .:.:;:  

2 

212 

7  08 

368  30 

(w  

2 

2  40 

8  00 

464  00  J 

Joint  and  splicing-pieces,  nuts,  &c,  &c  ..  say 

400  00 

400  00 

400  00 

400  00 

Total  weight  of  truss  — 

8592  46 

8592  46 

Wt  of  purlins  not  included. 

Hence,  the  wt  of  the  king  and  queen  truss  in  this  instance  is  equal  to   -~2— 6 
=  1.132  times  (say  U  times)  that  of  the  equally  strong  Fink;  or  the  Fink  is  about 


r  upper  enns.  instead  ot  being  proportioned  throughout  with  reference  to 
the  max  strain  at  their  feet.  If  the  theoretical  diminution  toward  the  tops  of  the 
rafters,  were  made  in  both  cases,  the  wts  of  the  two  forms  of  truss  would  be  nearly 
equal.  But  in  practice,  on  the  score  of  inconvenience,  this  is  rarely  done  in  roofs 
of  moderate  span  ;  say  less  than  about  100  ft.  No  such  diminution,  or  but  very  slight, 
would  be  admissible  even  theoretically,  when  the  purlins  are  not  placed  at  the  points 
of  support  only.  With  same  total  load  per  sq  ft,  ineiudiii* 
trusses  themselves  at  same  dist  apart,  the  total  wts  of  trusses 
are  as  the  squares  of  their  spans;  but  their  wts  per  ft  of  span, 
as  well  as  the  cross  areas,  wts  per  ft  run,  and  strains  along 


300 


TRUSSES. 


individual  members,  are  directly  as  the  spans.    But  see  Note. 

p  263.  When  the  dist  apart  of  the  trusses  is  7  ft  from  center  to  center ;  the 
rise  -J-  of  the  span ;  assumed  load,  including  the  wt  of  the  trusses  themselves,  40 
Ibs  per  sq  ft  of  roof  covering ;  and  the  various  parts  proportioned  for  the  several 
strains  per  sq  inch  assumed  in  Tables  1  and  2;  the  weight  of  a  properly  con- 
structed Fink  truss  will  be  approximately  as  follows : 

Total  wt  in  Ibs  of  _  square  of  span  in  ft 
a  Fink  roof-truss  —  3.1 ; 

and  the  wt  in  Ibs  per  ft  of  span  =  sPaninfeet 

O.I 

A  total  K  and  Q  truss,  will  be  about  \  part  more ;  or  ^^  ^-L^^L^L^ 

span  in  feet 
Or  per  foot  of  span,  =  — — : 

23716  _  „'  ' 


These  rules  give 


—  — 


2*?n  fi 
650  ft>8,  for  the  foregoing  Finland  ~-^  =  8784  Ibs, 


for  the  K  and  Q  truss.     From  these  rules  we  have  drawn  up  the  following 

Table  3.  Approximate  weights  of  roof-trusses  of  the  Fink 
system.    (Original.) 

•  Rise  4-  span.     Trusses  7  ft  apart.     Load  40  tt»s  per  sq  ft  of  roof,  including  truss. 


Total 
Span. 

Total  wt  of 
a  Truss. 

Wt  per  ft. 
of  Span. 

Wt  per  sq  ft 
of  ground 
covered. 

Total 

Span. 

Total  wt  of 
a  Truss. 

Wt  per  ft. 

of  Span. 

Wt  per  sq  ft 
of  ground 
covered. 

Feet. 

Lbs. 

Lbs. 

Lbs. 

Feet. 

Lbs. 

Lbs. 

Lbs. 

20 

129 

6.46 

.92 

100 

3228 

32.3 

4.60 

25 

202 

8.08 

1.15 

105 

3557 

33.9 

4.83 

30 

290 

9.67 

1.38 

110 

3904 

35.5 

5.06 

35 

396 

11.3 

1.61 

115 

4267 

37.1 

5.29 

40 

516 

12.9 

1.84 

120 

4640 

38.7 

552 

45 

654 

14.5 

•2.07 

125 

5041 

40.4 

5.75 

50 

807 

16.1 

2.30 

130 

5452 

42.0 

5.98 

55 

976 

17.8 

2.33 

135 

5880 

43.6 

6.21 

60 

1160 

19.4 

2.76 

140 

6336 

45.2 

6.44 

65 

1363 

21.0 

2.99 

145 

6782 

46.8 

6.67 

70 

1584 

22.6 

3.22 

150 

7260 

48.4 

6.90 

75 

1815 

24.2 

3.45 

155 

7750 

50.0 

7.13 

80 

2064 

25.§ 

3.68 

160 

8256 

51.6 

7.36 

85 

2331 

27.^ 

3.91 

165 

8782 

53.3 

7.59 

90 

2616 

29.1 

4.14 

170 

9324 

54.9 

7.82 

95 

2912 

80.7 

4.37 

175 

9879 

56.5 

8.05 

For  king  and  queen  trusses  add  ^  part  to  the  tabular  wts ;  when  the  rafters  are 
as  usual  of  the  same  size  throughout.  See  Note,  p  263. 

The  wts  in  the  4th  column  will  remain  nearly  the  same,  whatever  may 
be  the  dist  apart.  For  if  this  be  increased  say  to  14  ft,  each  truss  will  sustain  twice 
as  many  sq  ft  of  roof;  and  must  itself  be  at  least  twice  as  strong  and  heavy,  in  order 
to  do  so.  We  say  "  at  least,"  because  if  the  dist  apart  is  increased,  the  wt  of  thejpwr- 
lins  will  generally  increase  more  rupidly  than  said  dist.  Thus,  if  the  dist  be  doubled, 
the  purlins  will  not  only  be  doubled  in  length,  which  alone  would  double  their  wt ; 
but  they  must  also  be  deeper.  In  practice,  however,  long  purlins  are  usually  pre- 
vented from  becoming  very  heavy,  by  trussing  them,  as  at  7,  Figs  21^,  page  268. 

The  cost  of  trusses  alone  for  iron  roofs,  generally  varies  between  10  and 
12  cts  per  ft)  put  up ;  depending  on  the  price  of  iron  and  labor.  The  putting  up  alone 
from  %  to  1^  cts  per  ft).  With  trusses  7  ft  apart,  iron  purlins  will  weigh  about  2  ft>s 
per  sq  ft  of  ground  covered  by  the  roof.  Therefore  to  any  wt  in  the  4th  col  add  2  Ibs. 
Mult  the  sum  by  from  10  to  12  cts,  for  the  cost  of  trusses  and  purlins  alone  per  sq  ft  of 
ground.  Add  for  covering  with  tin  or  slate  on  boards,  say  15  to  20  cts  per  sq  ft ;  or  for 
corrugated  iron  on  the  bare  purlins  say  35  cts ;  or  if  on  boards,  38  to  40  cts  per  sq  ft. 

Bridge  trusses,  at  shop,  wrought  iron  parts  8  to  11  cts  per  ft> ;  cast  iron,  5  to  7  cts. 

HEM.  1.     As  to  the  proper  total  weight,  or  load,  per  sq   ft 

of  roof,  (including  snow  and  wind,)  that  should  be  assumed  to  be  sustained  by  the  trusses,  engineers 
differ  considerably.  The  French  appear  to  consider  30  TJS  as  suflciont;  while  the  English  use  40. 
Since  roofs  are  not  subject  to  violent  vibrations  like  bridges,  they  do  not  require  so  high  a  coefficient 
of  safety;  this  should  not.  however,  in  our  opinion,  be  taken  at  leas  than  3;  and  this  we  consider 
sufficient.  The  load  is  evidently  influenced  by  the  character  of  the  roof-covering.  Within  ordinary 
limits,  for  spans  not  exceeding  a'bout  75  ft,  and  with  trusses  7  ft  apart,  the  total  load  per  sq  ft,  includ- 
ing the  truss  itself,  purlins,  &c,  complete,  may  be  safely  taken  as  follows; 


TRUSSES.  301 

Table  4. 
Span  75  ft  or  less.  and  snow.»    Total. 

Roof  covered  with  corrugated  iron,  unbearded,  t  8Ibs.  20  Ibs.  28  fts. 

If  plastered  below  the  rafters,  18  "  20  "  38    ' 

"          ••  "      corrugated  iron,  on  boards,  11  "  20  "  31    ' 

If  plastered  below  the  ratters,  21"  20"  41    ' 

'•          "  "      slate,  unbearded,  or  on  laths,  13  "  20  "  33    ' 

11          "          "         "      on  boards,  1J4  ins  thick.  16  "  20  "  35    ' 

"          "          "         "     if  plastered  below  the  rafters,  26"  20"  46    ' 

"      shingles  on  laths.  10  "  20  "  30  " 

If  plastered  below  rafters  or  below  tie  beam.  20  "  20  "  40  " 

For  spans  from  75  to  150  ft,  it  will  suffice  to  add  4  fts  to  each  of  these  totals. 

Example  of  use  of  foregoing  tables.  A  Fink  roof  60  feet  span ;  rise  £ ; 
trusses  1-4  ft  apart ;  and  to  be  covered  with  slate,  on  boards  \%  incn  thick.  Here  we 
see  at  once  from  Table  3  that  at  7  ft  apart,  its  wt  would  be  about  1160  Ibs ;  therefore, 
at  14  ft  apart,  it  would  be  2320  Ibs.  But  our  table  is  for  40  Ibs  per  sq  ft  of  roof:  while, 
for  slate  on  boards,  35  Ibs,  or  ^  part  less,  is  sufficient.  Therefore,  we  may  reduce  the 
weight  of  the  truss  y8  part,  making  it  only  2030  fts.  See  Note,  p  263. 

Ex.  2.  Roof  as  before,  60  span ;  trusses  only  1  ft  apart.    Turn  to  Table  1,  where  the 

areas  are  given  for  a  total  length  or  span  of  154  ft.    But  60  ft  is  the  —  =  say  the 

.4  part  of  154  ft ;  therefore,  the  areas,  and  the  wts  per  foot  run  of  each  member  of 
the  60  ft  span,  will  be  .4  of  those  of  the  154  ft  one.  Thus,  the  area  of  a  rafter  will 
be  7.5  X  -4  =  3.  sq  ins;  which  corresponds  with  a  rolled  T  iron  of  3  X  3%  ins,  and 
}4  inch  average  thickness.  Its  wt  per  foot  run  will  be  25  X  -4  =  10  Ibs.  The  area 
of  the  part  n  of  the  main  tie  will  be  1.95  X  .4  =  .78  sq  inch,  which  we  see  at  once 
from  a  table  of  circular  areas,  is  equal  to  a  round  rod  very  nearly  1  inch  diam.  Its 
wt  per  ft  run  —  6.5  X  -4  =  2.6  Ibs  ;  and  so  with  all  the  other  members.  But  the  total 
wts  will  be  as  the  squares  of  the  span.  The  square  of  154  is  23716 ;  and  that  of  60  is 

3600.   And  jj¥£-  =  .152;  therefore,  the  total  wts  will  be  .152  of  those  in  Table  1. 

Thus,  the  two  rafters  will  weigh  4150  X  -152  =  631  Ibs.  The  main  tie,  1446  X  -152  = 
220  Ibs,  &c.  Lastly,  if  for  35  fibs  per  sq  ft,  reduce  each  area  and  wt  }/&  part. 

Since  the  rafters  are  generally  made  of  T  or  I  iron,  a  pattern  precisely  adapted  to  the  calculated 
strains,  will  not  always  be  procurable  ;  and  in  that. case  we  may  either  take  the  nearest  one  in  excess  : 
or  change  the  dist  apart  of  the  truss  to  suit  the  pattern  on  ha'nrt.  Owing  to  the  variety  of  modes  of 
arranging  the  details  of  the  junctions.  Ac.  an  exact  coincidence  between  the.calculated  and  the  actual 
wts,  is  not  to  be  expected ;  but  we  suspect  that  in  properly  proportioned  roofs,  the  discrepancy  will 
rarely  be  found  to  vary  more  than  about  5  per  ct  from  the  results  of  our  rules. 

It  might  be  supposed  that  with  iron  of  a  tensile  quality  considerably  higher  than 
our  assumption  of  20  tons  per  sq  inch ;  as  say  of  25  to  30  tons,  the  truss  might  be 
made  much  lighter.  But  this  is  not  the  case ;  because  the  superiority  would  affect 
the  ties  only ;  inasmuch  as  the  compressive  strength  of  iron  does  not  increase  with 
its  tensile  strength  ;  but  to  a  certain  extent  rather  the  reverse.  Now,  by  Table  1,  it 
appears  that  the  ties  in  a  Fink  roof-truss,  constitute  less  than  T%  of  its  entire  wt. 
Therefore,  iron  of  even  30  tons,  would  reduce  the  weight  of  the  truss  less  than  % 
part  of  j3^  part;  or  y1^ ;  and  25  ton  iron,  about  ^V  part. 

Short  spans  need  not  have  as  many  subdivisions,  or  "points  of  support,  '  as  a  large  one;  and  this 
will  lessen  the  number  of  parts  of  the  truss ;  but  inasmuch  as  the  remaining  parts  will  require  to  be 
proportionally  stronger,  this  consideration  will  not  materially  affect  the  wts.  While  on  this  subject, 
we  will  remark  that  too  few  points  of  support  are  probably  used  at  times ;  owing  to  either  an  under- 
valuation, or  an- ignorance  of  the  effect  of  the  transverse  strains  produced  by  the  load  on  the  parts 
of  the  rafter  between  said  points.  These  parts  must  be  regarded  as  so  many  separate  beams  sup- 
ported at  both  ends;  or  rather,  as  firmly  fixed  at  both  ends,  when  the  pieces  composing  a  rafter  are, 
as  usual,  strongly  connected  together ;  in  which  case  the  beam  is  about  twice  as  strong  as  when 
merely  supported.  If  the  separate  parts  be  trussed,  like  the  purlin  at  7,  Figs  21%,  to  neutralize  this 
transverse  action,  it  must  be  remembered  that  additional  compression  will  be  thereby  produced 
lengthwise  along  the  rafter.  The  best  practice  is,  as  far  as  practicable,  to  increase  the  number  of 
points  of  support,  so  that  the  purlins  may  rest  upon  them  alone,  or  near  them ;  and  thus  relieve  Che 
rafters  entirely,  or  in  part,  from  transverse  strain. 

HEM.  2.  As  to  the  effect  produced  on  the  weight  of  a  truss,  by 
changing*  its  rise,  no  short  correct  rule  can  be  laid  down.  Although  as  a 
roof  becomes  flatter,  its  area  becomes  less,  so  that  each  truss  has  less  total  wt  of  roof- 
covering,  snow,  and  wind,  to  sustain,  still  the  strains  on  most  of  its  members  become 
greater;  requiring  greater  wt  of  truss.  To  find  this  increase  with  accuracy,  it  is 


*  See  Snow  and  Wind,  p  519,  520. 

t  The  corrugated  iron  itself  will  weigh  from  1 J^  to  2  Tbs  per  sq  ft  on  the  roof.  If  not  plastered  under- 
neath, the  condensed  moisture  of  the  air,  especially  from  crowded  rooms,  will  fall  from  the  iron  into 
the  rooms  below.  Mere  boarding  will  not  prevent  this,  even  if  tongued  and  grooved,  unless  the  circu- 
lation of  air  against  the  under  side  of  the  iron  is  effectually  cut  off. 

20 


302 


TRUSSES. 


necessary  to  make  a  diagram,  and  perform  all  the  calculations.  The  strains  on  a 
Fink  rafter  become  more  nearly  uniform  throughout  its  length,  as  the  pitch  of  the 
roof  becomes  less ;  while,  with  a  rise  of  %  span,  the  strain  at  its  foot  is  about  1-fy 
times  that  at  its  head.  On  the  contrary,  the  strains  on  its  struts  remain  nearly  the 
game  in  amount  for  all  ordinary  rises. 

In  the  king  and  queen  truss  the  strains  at  the  heads  and  feet  of  the  rafters  retain  the  same  pro- 
portions to  each  other,  at  all  rises ;  the  strains  on  the  verticals  become  less  as  the  roof  becomes  flatter ; 
while  those  on  the  obliques  vary  according  to  their  several  obliquities.  Under  these  irregularities, 
which  affect  the  K  and  Q,  much  more  than  the  Fink,  we  can  do  nothing  more  than  say  that  when  it 
is  merely  wished  at  the  moment  to  form  a  rough  idea  of  the  effect  of  changing  the  rise,  we  may 
assume  the  weight  of  a  Fink  truss  to  increase  about  in  the  same  proportion  as  we  diminish  the  rise; 
or  to  diminish  as  we  increase  the  rise.  Thus,  if  we  increase  the  rise  of  the  roof  in  Table  I,  one-fourth 
part,  so  as  to  make  it  equal  to  .25  or  }£  of  the  span,  instead  of  .2  or-^-  of  the  span,  we  may  diminish  its 
wt  y±  part ;  making  it  about  6000  fts,  instead  of  8000.  Or  if  we  reduce  the  rise  from  ^  to  y1^,  making 
it  only  half  as  great,  we  shall  double  its  weight,  making  it  16000  0>s;  as  rude  approximations. 

Art.  3O.  Oil  the  camber  of  truss  bridges.  In  practice,  the  upper  and 
lower  chords  of  bridges  are  not  made  perfectly  straight,  but  are  curved  slightly  up- 
ward ;  and  this  curve  is  called  the  camber  of  the  truss  or  bridge.  Its  object  is  to 
prevent  the  truss  from  bending  down  below  a  hor  line  when  heavily  loaded.  A  cam- 
bered chord  is  of  course  longer  than  a  straight  line  uniting  its  ends  ;  but  in  practice 
the  camber  is  so  small  that  this  diflf  is  inappreciable,  and  may  be  entirely  neglected. 
But  when  the  chords  are  cambered,  (see  y  s  and  c  d,  Fig  51,)  they  become  concentric 
arcs  of  two  large  circles,  of  which  the  center  is  at  t\  and  the  upper  one  plainly  be- 
comes longer  than  the  lower,  to  an  extent  which,  although  much  exaggerated  in  our 
fig,  cannot  be  overlooked  in  practice.  The  verticals,  instead  of  remaining  truly  vert, 
become  portions  of  radii  of  the  aforesaid  large  circles;  and  although  their  lengths 
remain  the  same,  yet  their  tops  become  a  little  farther  apart  than  their  feet ;  and 
this  renders  it  necessary  to  lengthen  the  obliques  or  diags  a  trifle.  Therefore,  we 
must  find  how  great  is  this  increase  of  length  of  the  upper  chord  beyond  the  lower 
one;  and  divide  it  equally  among  all  the  panels,  along  said  chord;  otherwise  the 
several  parts  of  the  truss  will  not  fit  accurately  together. 

1st.  To  find  the  amount  of  camber  of  the  lower  chord.   Di- 
vide the  span  in  feet,  (measured  from  center  to  center  of  the  outer  panel-points,) 
by  50.    The  quot  will  be  a  sufficient  camber,  in  inches ;  as  shown  in  the  following 

Table  of  cambers  for  bridge  trusses. 


Span. 

Camber. 

Span. 

Camber. 

Span. 

Camber. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

25 

0.5 

100 

2.0 

250 

5.0 

50 

1.0 

150 

3.0 

300 

6.0 

75 

1.5 

200 

4.0 

350 

7.0 

Rem.  1.  It  is  by  no  means  necessary  to  adhere  strictly  to  this  rule ;  and  the 
camber  by  experienced  builders  of  iron  bridges  is  often  but  one-half  the 
above,  or  1  inch  per  100  ft  of  span. 

Rem.  2.  A  well  built  bridge  of  good  design  should  not,  under  its  greatest 
load,  deflect  more  than  about  1  inch  for  each  100  feet  of  its  span.  The  deflec- 
tion is  frequently  much  less  than  this. 

2d.   To  find   the  increase  of  length  in  the  upper  chord. 

beyond  the  lower  one,  having  the  span ;  the  depth  of  truss ;  and  the  camber ;  (all 
in  feet,  or  all  in  ins.) 

With  any  camber  not  exceeding  J^  of  the  span ;  (which,  however,  is  about  7  times  as  great  as  is 
usually  given  to  trusses  ;)  mult  together  the  depth  of  truss,  the  camber,  and  the  number  8 :  div  the 
prod  t>y  the  span.  The  quot  will  be  the  increase,  in  ft,  or  in  ins,  as  the  case  may  be.  Or  as  a  formula, 

Increase  in  ft,  —   depth  X  camber  *  8    all  in  feet;  or 
or  in  ins,  span,  all  in  inches. 

This  rule  may  be  considered  practically  perfect  with  any  camber  not  exceeding  T^J- 
of  the  span.  Based  upon  this  principle,  we  have  prepared  the  following  table,  which 
may  be  used  instead  of  making  the  above  calculations. 


TRUSSES. 


303 


Table  for  finding:  increase  of  length  of  upper  chord  beyond 
lower  one. 


Depth 
of 

Mult 
Camber 

Depth 
of 

Mult 
Camber 

Depth 

Mult 
Camber 

Depth 
of 

Mult 
Camber 

Truss. 

by 

Truss. 

by 

Truss. 

by 

Truss. 

by 

y±  span. 

2.00 
1.60 

H  span. 
1-9    " 

1.00 

.888 

1-12  span. 
1-13    » 

.666 
.614 

1-16  span. 
1  17     " 

.500 
.470 

&  - 

1.33 
1.15 

1-10  " 
1-11  " 

.800 
.727 

1-14     •• 
M5    " 

.571 

•OT    1 

1-18     «• 
1-20     " 

.444 
.400 

O    d 


Ex.  How  much  longer  is  the  upper  chord  than  the  lower  one,  when  the  depth  of 
the  truss  is  \  of  the  span  ;  and  the  camber  5  ins?  Here  in  the  table,  and  opposite  \ 
span,  we  find  the  multiplier  1.15.  Therefore,  5  ins  X  1.15  =  5.75  ins,  Ans.  If  the 

truss  has  say  8  panels,  then  -^—  =  .72  inch  of  this  increase  must  be  given  to  each 

o 
panel,  along  the  upper  chord. 

The  length  of  a  diagr,  or  oblique,  &  c,  Fig  50,  may  readily  be  found 
thus  :  Let  a  s  n  c  in  this  fig  represent  a  panel  when  there 
is  no  camber;  then  o  b  n  c  will  represent  a  panel  when 
there  is  a  camber  ;  and  o  a  and  s  It  together  are  the 
portion  of  the  increased  length  of  upper  chord  given  to 
each  panel  ;  but  to  an  exaggerated  scale.  Now,  to  find 
b  c,  we  have  the  right-angled  triangle  a  b  c,  in  which  we 
know  the  side  a  c,  (the  depth  of  truss  ;)  and  the  side  a  b, 
(equal  to  the  panel  width  en  on  the  lower  chord;  added 
to  s  6,  or  half  the  portion  of  the  increased  length  of 
upper  chord  given  to  one  panel.)  Hence,  we  have  only 
to  square  each  of  those  two  sides;  add  the  two  squares 
together;  and  take  the  sq  rt  of  the  sum.  This  sq  rt  is  b  c. 

Example.  Span  200  ft.  Height  (a  c)  of  truss  ys  of  the 
span,  or  25  ft,  or  300  ins.  Camber  5  ins  ;  10  panels  each 
20  ft,  or  240  ins,  (c  n,)  measured  on  the  lower  chord  or 
span.  Now,  the  height  being  ^  of  the  span,  the  increase 
of  length  of  upper  chord  will  be  equal  to  the  camber,  5  ins  ;  and  this  divided  among 

10  panels,  will  be  —  =  .5  inch  to  each  panel  ;  or  o  b  will  be  .5  inch  longer  than  en; 

ind  o  a  and  s  b  will  each  be  .25  inch.    Hence,  in  the  right-angled  triangle  a  b  c,  we 
Have  a  6  =  240.25  ins  ;  and  a  c  =  300  ins.    Hence, 


Pit/ 50 


6  c 


57720.0625  +  90000  =  ^/  147720^0625  =  384.344  ins  ; 


or  32  ft,  .344  ins.    Without  any  camber,  b  c  would  be  in  the  position  s  c,  which  is  32 
ft,  .187  ins  long  ;  or  .157  of  an  inch  (about  %  inch)  shorter  than  b  c. 

An  error  to  this  extent  would  prove  seriously  inconvenient  if  the  oblique  were  a  cast-iron  strut 
with  carefully  planed  ends,  intended  to  fit  closely  between  planed  bearings  at  the  chords  ;  or  a  bar 
with  a  drilled  hole  at  each  end  for  fitting  over  pins  whose  position  was  fixed  and  unalterable.  In 
many  cases,  as  when  the  obliques  are  merely  rods  with  screw-ends,  it  is  only  necessary  to  be  sure 
that  they  are  long  enough;  because  their  exact  length  can  then  be  adjusted  when  put  into  place,  by 
means  of  the  nuts  on  their  ends.  So  also  when  the  obliques  or  other  pieces  are  flat  bars  intended  to  be 
bolted  or  riveted  to  the  sides  of  the  chords;  for  the  final  rivet-holes  may  be  made  when  the  pieces 
come  to  be  finally  fitted  in  place.  In  the  case  of  wooden  obliques,  &c,  if  too  long,  they  can  readily  be 
reduced  by  the  saw  or  chisel. 

When  the  panels  are  all  of  one  size,  as  is  generally  the  case,  it  is  usual  for  builders 
to  draw  one  of  them  full  size  on  a  board  platform  or  floor,  to  guide  in  fitting  the 
parts  together. 

In  raising  a  trass,  or  in  other  words,  when  putting  its  parts  together  hi 
their  proper  position  on  the  abutments  and  piers,  a  scaffold  or  false-works, 
must  first  be  erected  for  sustaining  the  parts  until  they  are  joined  together  so  as  to 
form  the  complete  self-sustaining  truss.  Upon  the  false  works  'the  bottom  chords 
are  first  laid  as  nearly  level  as  may  be  ;  and  the  top  chords  are  then  raised  upon  tem- 
porary supports  which  foot  upon  the  one  that  carries  the  lower  chord.  The  upper 
chords  are  at  first  placed  a  few  inches  higher  than  their  final  position,  or  than  the 
true  height  of  the  truss,  in  order  that  the  obliques  and  verts  may  be  readily  slipped 
into  place.  After  this  is  done,  the  top  chords  are  gradually  let  down  until  all  rests 
upon  the  lower  chords.  The  screws  are  then  gradually  tightened  to  bring  all  the 
surfaces  of  the  joints  into  their  proper  contact  ;  and  by  this  operation  (the  upper 
chord  being  supposed  to  have  the  increased  length  given  by  the  foregoing  rule)  the 


304 


TRUSSES. 


camber,  as  it  were,  forms  itself;  and  lifts  the  lower  chords  clear  off  from  their  false- 
works ;  leaving  the  truss  resting  only  upon  the  abuts  or  piers,  as  the  case  may  be. 

As  a  support  for  the  falseworks  themselves  on  soft  bottoms  piles 
may  be  driven,  to  which  the  uprights  of  the  falseworks  may  be  notched  and  bolted 
or  banded.  In  some  cases,  as  of  rock  bottom  in  a  strong  current,  it  may  become 
expedient  to  sink  cribs  filled  with  stone,  as  a  support  for  the  falseworks. 

The  falseworks  should  be  well  protected  by  fender-piles  or  otherwise  from  passing 
boats,  ice  and  other  floating  bodies,  especially  in  positions  liable  to  sudden  floods : 
and  numerous  accidents  have  shown  the  expediency  of  guarding  the  unfinished 
truss  itself  against  hig*li  winds.  This  last  remark  applies  as  well  to  roofs  as  to 
bridges  ;  and  is  too  frequently  neglected. 

Art.  31.  For  very  short  spans,  the  rolled  iron  I  beam,  page 
210,  answers  every  purpose.  A  single  20  ft>,  7-inch  beam  under  each  rail,  will  suffice 
for  3  or  4  ft  span  :  and  the  52ft),  15-inch  one,  for  8  to  10  ft.  Two  52  ft),  15-inch  ones, 
under  each  rail,  for  15  ft.  By  employing  a  greater  number  of  beams,  or  by  intro- 
ducing a  truss-rod,  ?*,  Fig  52,  the  spans  may  be  increased;  or  lighter  beams  be  used. 
The  beams  should  be  fitted  at  their  ends  into  wide  cast-iron  shoes,  well  bolted  to  the 
abuts.  Care  must  be  taken  to  insure  lateral  stability,  by  means  of  hor  cross-ties  and 
diag  bracing.  This  may  generally  be  secured  by  notching  and  bolting  the  cross-ties 
to  the  beams;  and  by  diag  rods  passing  through  the  chords;  and  having  their  inner 
ends  confined  by  screws  to  such  a  ring  as  is  shown  at  14,  Figs  '21%,  page  268;  or  by 
the  diag  bracing  shown  at  W,  Fig  36%,  p  291. 

To  prevent  ail  overturning  tendency  in  a  whole  truss  when  it  is 
not  high  enough  to  admit  of  being  horizontally  braced  overhead,  we  may  introduce 
wooden  knees,  or  short  straight  struts  or  ties,  of  either  wood  or  iron ;  which  may 
foot  upon  the  cross-girders  of  the  floor;  and  head  against  either  some  of  the  web 
members,  or  the  upper  chord.  These  braces  or  ties  may  be  placed  either  between 
the  two  trusses  of  a  span  ;  or  outside  of  them :  or  both.  When  outside,  some  of  the 
floor-girders  may  be  lengthened  out  a  few  feet  beyond  the  lower  chord,  for  receiving 
the  feet  of  the  braces  or  ties. 

See  Art  41,  of  Strength  of  Materials,  for  other  iron  beam  bridges  of  still  greater  spans. 

Or,  a  single  beam  of  wood  under  each  rail,*  and  firmly  braced 
against  lateral  motion,  will  suffice.  Assuming  the  weight  of  entire  bridge  and  load, 
at  two  tons  per  foot,  the  following  dimensions  may  be  used : 


Span  in  Ft. 

Size  of  Beam. 

Span  in  Ft. 

Size  of  Beam. 

5 
10 

8X  10  ins. 
9  X  12    » 
10  X  H    " 
11  X  16    " 

15 

20 

22}* 

12X18 
13  X20 
14  X  22 
16X24 

The  greatest  dimension  to  be  the  depth.    The  ends  should  be  well  bolted  down  to 

'(jfeiyS^ 
iV%o 


*  If  single  beams  of  sufficient  depth  cannot  be  procured,  built-beam*  may  be  used ;  see  g,  and  ij, 
Figs  42,  p  294. 


TRUSSES. 


305 


bolsters.  These  are  long  stout  sticks  of  timber,  from  10  to  15  ins  square,  (accord- 
ing to  the  span.)  laid  across  the  abuts  at  the  bridge-seat,  for  the  chords  to  rest  on. 
See  h,  Fig  31.  Frequently  two  are  used  at  each  abut,  even  in  small  spans;  and  \ve 
have  seen  but  one,  under  railroad  spans  of  150  feet.  Large  spans  may  require  three 
or  more.  They  are  not  necessarily  placed  in  contact  with  each  other;  but  may  be 
some  feet  apart,  if  required. 

Or  for  spans  of  about  15  to  30  ft,  we  may  use  somewhat  lighter  beams ;  and  truss 
each  of  them  as  in  Fig  52,  by  an  iron  bar  e  s  e ;  and  a  center  post  p.  In  this  case  the 
following  dimensions  will  answer;  the  total  deflection  of  the  rod  being  1^  of  the 
clear  span.  The  screw  ends  of  the  bars  are  supposed  to  be  upset;  but  the  areas  are 
given  for  the  body  of  the  rods. 

For  each  beam. 


Span. 
Ft. 

Beam. 
Ins. 

Section  of  Rod. 
Sq  Ins. 

Section  of  Post. 
Sq  Ins. 

15 
20 

25 
30 

12  X  15 
13  X  IT 
14  X  18 
15  X  20 

3^ 
4% 
6 

7 

25 
33 
42 
50 

It  is  better  to  have  two  rods  instead  of  one  under  each  beam  ;  each  rod  being  of  half 
the  section  here  given  ;  and  the  two  placed  several  ins  apart.     This  affords  a  better 
footing  for  the  post.    The  ends  of  the  beam  should  be  at  right  angles  to  the  direction 

of  the  rod ;  and  be  provided  with  ample  washers  e  e,  of  wood  or  iron,  for  distributing 
the  pressure  from  the  rod,  over  the  whole  area  of  the  ends.  The  ends  of  these  wash- 
ers may  extend  a  few  ins  each  way  beyond  the  ddes  of  the  beam,  as  shown  on  a  larger 
scale  at  g.  This  allows  the  rods  to  be  outside  of  the  beam  ;  instead  of  requiring  holes 
to  be  bored  in  the  latter,  for  passing  the  rods  through  them.  They  may  be  nearer 
together  at  the  foot  of  the  post. 

The  head  of  the  post  may  be  tenoned  into  the  bottom  of  the  beam  ;  and  be  further  united  to  it  by 
ron  straps.  To  prevent  the  foot  from  being  worn  by  the  rods  it  should  be  shod  with  iron.  A  cast- 
ron  shoe,  as  at  s,  may  be  bolted  to  it;  having  ribs  for  keeping  the  bars  in  place.  Or  a  stout  wrought- 
ron  shoe  may  be  well  secured  to  it.  In  either  case  the  rod  at  a  should  be  so  united  to  the  shoe  as  to 
heck  any  tendency  in  the'foot  to  slide  toward  r  or  r,  under  the  vibration  of  passing  loads.  Perhaps 
his  can  "be  most  conveniently  done  by  making  each  rod  e  s  e,  iu  two  separate  lengths,  r,  r;  and  by 
uniting  their  lower  ends  to  the  shoe  at  «  by  hooks  and  eyes :  or  by  eyes  and  bolts,  &c.  Various 
methods  are  in  use  for  the  heads  and  feet  of  the  posts  of  large  spans ;  but  we  cannot  here  treat  upon 
details  which  pertain  more  to  the  professional  bridge-builder. 

This  mode  of  trussing  is  also  well  adapted  to  long  floor  beams;  and  has  been  used  in  long  oblique 
web  members ;  as  well  as  in  long  stretches  of  chords  from  one  point  of  support  to  another.  Fig. 2  is  a 
vertical,  thus  trussed  in  more  than  one  direction. 

The  following  dimensions  for  single-track  Fink  bridges, 

with  chords  and  posts  of  wood;  and  iron  suspension  bars;  are  on  the  assumption 
that  all  the  bars  deflect  YB  of  the  span  ;  that  the  road  is  on  top ;  that  the  bars  shall 
not  be  strained  more  than  10000  ft>s,  or  4%  tons  per  sq  inch,  under  a  weight  of  bridge 
and  load,  amounting  in  all  to  two  tons  per  running  foot.  Assumed  wt  on  each 
driving-wheel  of  engine,  5  tons.  Screw  ends  upset. 

Dimensions  for  one  truss  only,  of  a  single-track  Fink  bridge. 
The  spans  are  in  feet ;  the  other  dimensions  are  square  inches  of  cross-section  of 
each  member. 


Areas  in  sq  in 


Areas  in  sq  ins. 


21 

23  « 


225 
280 
335 
390 


We  have  given  the  area  of  the  1st  post  only.     For  that  of  the  2.1,  we  may  take  %  of  the  first :  for 
the  3d,  %  of  the  second  ;  and  for  the  4th.  %  of  the  third  ;  without  pretending  to  any  great  accuracy. 


The 

be  best  to  h 


iay  be  flat,  square,  or  round,  so  that  the  proper  area  be  maintained.    It  will  usually 
e  them  flat. 


Our  assumed  total  weight  of  2  tons  per  foot  of  span,  for  bridge 
and  load  together,  up  to  spans  of  200  ft,  is  greater  than  is  usually  adopted.    Still  we 


306 


TRUSSES. 


would  recommend  not  to  diminish  the  areas  in  the  last  three  short  tables,  or  in  thb 
following  one,  more  than  %  part.  They  will  then  be  about  up  to  ordinary  practice 
for  railroads;  and  will  also  be  sufficient  for  bridges  for  common  travel,  with  a  clear 
width  of  1$  ft  between  the  two  trusses ;  and  two  outside  footpaths,  each  5  ft  wide,  in 
addition.  A  width  of  18  ft  is  necessary  for  allowing  two  ordinary  vehicles  to  pass 
each  other  readily*  It  should  never  be  less  than  16  ft ;  and  nothing  is  gained  by 
exceeding  20  ft. 

It  will  of  course  be  understood  that  each  member,  especially  in  large  spans,  will  consist  of  two  or 
more  pieces,  side  by  side.  Thus,  in  a  span  of  200  ft,  the  800  sq  inches  ot  each  chord,  will  probably 
consist  of  four  beams  of  about  10"  X  20",  or  12"  X  16",  placed  side  by  side  ;  but  with  sufficient  inter- 
vals between  them  to  allow  the  several  oblique  bars  to  pass.  Or  it  may  consist  of  six  beams  of  smaller 
size.  So  also,  the  1st,  or  main  rod,  of  46%  sq  ins,  will  probably  be  made  up  of  from  4  to  8  bars,  of  11.7, 
or  5.85  sq  ins  each,  placed  side  by  side;  occasionally  some  inches  apart.  And  so  with  the  others. 
The  feet  of  the  opposite  posts  of  the  two  trusses  of  a  Fink  spau,  are  connected  together  by  ties  of  wood 
or  iron,  to  prevent  lateral  motion;  and  for  the  same  purpose,  diagonals  are  carried  from  each  upper 
chord  to  the  foot  of  the  opposite  post ;  as  is  usually  done  in  top-road  bridges  of  any  kind.  These  had 
better  be  tie-struts. 

A  simple  truss  like  Fig  6,  of  30  ft  span,  and  10  ft  high,  may  have  a  chord 
of  In"  by  18";  rafters  10"  by  10";  one  rod  of  2%"  diam;  or  two  rods  of  l%"  diam, 
and  several  inches  apart  transversely  of  the  bridge ;  which  is  far  better  than  one. 

The  Pratt  truss,  when  put  together  as  in  Fig  31;  each  member 
(except  the  m-iin  obliques  x  w.yn,  &c,)  being  of  one  piece,  is  perhaps  as  easy  of  con- 
struction, and  as  suitable  for  small  spans,  as  any  other.  With  the  same  assumptions 
as  before  with  regard  to  total  load,  upset  rods,  <fec,  the  following  dimensions  may  be 
used.  They  are,  however,  by  no  means  the  result  of  close  calculation  ;  but  are  pur- 
posely full. 

Table  of  dimensions  for  small,  single-track  Pratt  trusses. 

Height  from  top  of  top  chord,  to  bottom  of  bottom  chord,  y  of  the  clear  span. 
Twelve  panels  to  each  truss.  Each  main  oblique  consists  of  two  parallel  rods  or 
bars.  Each  counter  oblique  is  a  single  rod  passing  between  the  mains. 

For  one  truss  only. 


Diam  in  ins  of  each  of 

Clear 

Chords. 

Center 

End 

the  two  bars  of  a 

Center 

End 

Span. 

Each. 

Post. 

Posts. 

main  oblique. 

Counter. 

Counter. 

Diam. 

Diam. 

Ft. 

Ins. 

Ins. 

Ins. 

At  center 

At  ends 

Ins. 

Ins. 

of  Truss. 

of  Truss. 

30 

9X  11 

4X    9 

7  X    9 

1 

1^ 

IX 

1 

40 

10  X  12 

4X  10 

8X  10 

1H 

1  K 

1 

50 

10  X  14 

5  X  10 

9  X  10 

\y± 

2H 

1% 

1 

60 

12  X  15 

5  X  12 

9  X  12 

1% 

2% 

2 

1 

70 

12  X  17 

6X  12 

11  X  12 

^ 

2* 

1 

The  diams  of  the  main  rods  may  increase ;  and  those  of  the  counters  diminish,  regularly  from  the 
center  each  way  to  the  abuts. t  The  chords  and  posts  are  supposed  to  have  the  same  dimension  trans- 
versely of  the  truss.  When  more  convenient,  all  the  posts  may  be  of  the  same  size  as  the  t:nd  ones; 
their  ends  tenoned  an  inch  or  two  into  each  chord.  The  angle-blocks  above  and  below  the  posts, 
sliould  be  of  oak,  or  other  hard  wood ;  or  of  cast  iron.  In  such  small  spans  they  need  not  be  notched 
more  than  an  inch  into  the  chords ;  or  at  most,  1%  ins,  near  the  ends  of  the  larger  ones  in  the  table. 

Fig  9  b,  or  Fig  1O,  without  any  special  couiiterb races,  will 
answer  very  well  for  spans  of  25  to  45  ft.  The  height  of  truss,  number  of  panels,  and 
size  of  chords,  to  be  the  same  as  in  the  preceding  table  for  Pratt.  W,  at  Fig  52,  shows 
the  arrangement  at  the  head  and  foot  of  each  brace  6;  except  that  the  heads  of  the 
two  center  braces  abut  against  each  other. 

The  vert  rods,  v,  are  in  pairs;  like  the  main -brace  rods  of  the  Pratt  above  ;  but 
their  areas  of  section  may  be  ^  less  than  those  given  for  the  Pratt  braces.  The  areas 
of  the  wooden  braces  or  obliques,  6,  should  be  %  greater  than  those  of  the  vert  posts 
of  the  Pratt.  The  notches  of  the  heads  and  feet  into  the  chords,  may  be  2]4  to  3  ins 
deep.  Two  long  spikes,  n,  may  be  used  at  each  head  and  foot.  At  i  is  a  wooden  block 
washer,  across  the  chord,  Fig  52,  W. 

REM.  1.  It  may  appear  strange  that  we  refer  to  Fig  9  6,  as  being  appropriate ;  after  having  used  it 
for  illustrating  the  necessity  for  counterbraces  under  moving  loads.  But  experience  has  shown  that 
this  necessity  is  not  always  so  urgent  as  some  suppose ;  and  that  in  short  spans  at  least  with  stout 
chords,  well  connected  with  each  other  at  short  panel  lengths,  it  may  be  dispensed  with.  In  that 
case  it  is  true  no  special  provision  is  made  for  the  counter  strains,  and  the  stiffness  of  the  joints  is 


*  These  remarks  of  course  apply  to  railroad  bridges  of  any  form  of  truss. 

t  Not  theoretically  correct  under  moving  loads  ;  but  in  all  cases  safe.    The  same  remark  applies  to 
verticals. 


TRESTLES. 


307 


relied  ou  to  supply  the  omission.  See  footnote,  p  252.  Although  this  Is  not  to  be  commended,  yet 
the  writer  has  known  such  bridges  of  45  ft  span  carry  safely  for  many  years  an  immense  traffic  drawn 
by  heavy  engines;  and  that  without  any  spikes  or  bolts  at  the  heads  and  feet  of  the  braces,  and  by 
•which  considerable  counterbracing  power  is  imparted  to  them. 

The  Newark  Dyke  Warren,  England,  span  240  ft,  has  no  counterbracing;  and  so  with  some  other 
shorter  bridge  trusses  of  various  kinds.  The  writer  is  inclined  to  believe  that  many  failures  which 
have  been  ascribed  to  want  of  counterbracing,  have  in  fact  been  owing  to  the  common  mistake  alluded 
to  near  the  bottom  of  p.  174;  and  to  not  giving  the  main  braces  a  sufficient  depth  of  insertion  into  the 
Chords ;  thus  causing  an  undue  yielding  of  the  timber  at  these  points ;  as  he  has  frequently  seen. 

For  wooden  arches,  with  only  a  lower  chord  ;  or  in  other  words, 
for  a  bowstring  bridge,  Fig  35,  the  following  rule  will  give  quite  sufficient  dimensions 
for  the  arches  alone,  for  a  single-track  railway.  The  rise  of  the  lower  arc  of  the  arch 
is  supposed  to  be  ^  of  the  clear  span. 

Area  of  arch  of  one       clear  span  in  ft  +  25 
truss  only,  in  sq  ft.  =  37.5. 

This  gives  as  follows,  for  each  truss  of  the  two  which  constitute  a  span ;  each  truss 
having  two  arches,  side  by  side ;  or  within  say  about  a  foot  of  each  other. 


Clear  Span. 

Inches. 

Clear  Span. 

Inches. 

50ft 
75ft 
100ft 
110ft 

2  of  10  X  14 
2  of  10  X  19 
2  of  10  X  24 
2  of  11  X24 

125ft 
150ft 
175  ft 
200ft 

2  of  11  X  26 
2  of  12  X  28 
2  of  12  X  32 
2  of  12  X  36 

The  span  to  be  divided  into  about  16  panels:  diminishing  in  horizontal  length 
from  the  center,  toward  the  abuts.  Either  Howe  or  Pratt,  or  the  Warren  trussing 
for  the  arches  will  be  suitable.  The  arches  may  be  made  of  planks  3  or  4  ins  thick; 
bent  to  form  the  curve ;  and  well  bolted  together. 

Railroad  trasses  for  4  ft  SK  inch  gaage,  should  be  at  least 
13  ft  apart  in  the  clear;  14  ft  is  better.  For  lateral  stability  in  long  spans  the 
width  from  out  to  out  should  not  be  less  than  about  -fa  of  the  span. 

A  headway  of  at  least  16  ft,  should  be  allowed  for  clearing  smoke-stacks. 


TRESTLES. 
.2.  ,  3 


FIGS  1,  2,  3,  5,  6,  7,  are  elevations  of  trestles:  taken  across  the  track  or 
roadway.    We  may  consider  Fig  1  as  adapted  to  a  height  of  about  10  to  20  ft;  Figs 2 


308  TRESTLES. 

and  3,  from  20  to  30 ;  Pig  5,  from  30  to  40 ;  Fig  6,  from  40  to  60 :  as  mere  approxima- 
tions. These  figs  of  course  admit  of  many  modifications.  They  are  called  bents. 
These  bents  are  usually  supported  by  bases  of  masonry,  as  in  the  figs;  to  preserve 
the  lower  timbers  from  contact  with  the  earth,  which  would  hasten  decay.  It  is  also 
well  to  make  these  bases  high  enough  to  prevent  injury  from  cattle,  or  passing 
vehicles,  &c.  Up  to  heights  of  about  40  or  50  ft,  a  single  row  of  posts  or  uprights, 
a,  a,  a,  Figs  1  to  9,  as  shown  at  e  e  under  Figs^I  and  6,  will  answer.  But  as  the  height 
becomes  greater,  more  posts  should  be  introduced,  as  shown  at  x  x  under  Fig  5;  or 
two  entire  rows  of  them  ;  or  three  rows,  as  under  Fig  7  :  and  as  also  in  Fig  8,  which 
is  an  end  view  of  Fig  7.  Figs  7  and  8  bear  much  resemblance  to  the  trestles  190  ft 
high,  with  masonry  bases  30  ft  high.  (S.  Seymour,  C.  E.,)  which  carry  the  N  York 
and  Erie  R  R  over  Geiiesee  R  at  Portage.  There  each  bent  has  21  posts,  14 
ins  square  at  its  base ;  and  15  posts  of  12  X  12,  at  its  top.  The  other  timbers  are 
6  X  1-;  many  of  them  are  in  pairs,  embracing  the  posts.  Frequently  the  posts  of 
trestles  are  in  pairs ;  and  the  other  timbers  pass  between ;  all  bolted  together. 

Tn  Fig  4,  the  posts  a,  a,  a,  are  end  views  of  three  trestles  or  bents,  such  as  Fig  3; 
and  1 1  are  diag  braces  extending  from  trestle  to  trestle  ;  the  two  outer  ones  inclining 
in  one  direction ;  and  the  central  one  crossing  them.  These  may  be  placed  either 
intermediate  of  the  posts,  as  in  Fig  3 ;  with  the  heads  of  the  two  outer  ones  confined 
to  the  cap  c  c  of  one  trestle;  and  their  feet  to  the  sill  y  y  of  the  next  one;  or  they 
may  all  be  spiked  or  bolted  to  the  posts  themselves,  as  in  Fig  4.  The  last  is  the  best, 
as  it  serves  also  directly  to  stiffen  the  posts :  as  do  also  the  braces  o  o,  n  n,  Fig  2. 
Such  bracing  is  too  frequently  omitted.  During  the  passage  of  trains  the  reaction 
(so  to  speak)  of  the  rails  against  the  adhesion  of  the  driving-wheels  of  the  engines, 
produces  a  serious  strain  lengthwise  of  the  road,  and  tending  to  upset  the  trestles; 
and  this  becomes  more  dangerous  as  the  height  increases.  Hence  the  necessity 
for  such  braces.  Usually  the  outer  posts  may  lean  1.5  to  2.5  ins  to  a  ft. 

The  posts  should  not  be  less  than  about  12  ins  square,  except  in  quite  low  trestles ; 
and  even  then  not  less  than  about  10  X  10.  The  diag  bracing  may  generally  be  about 
as  wide  as  the  posts ;  and  half  as  thick.  The  dist  apart  of  the  bents,  when  the  road- 
way is  supported  by  simple  longitudinal  beams,  should  not  exceed  10  or  12  ft,  for 
railroads.  But  if  these  beams  receive  support  from  braces  beneath,  like  s  ,<?,  Fig  8 ;  or 
from  iron  truss  rods,  as  at  Fig  52,  page  304 ;  the  dist  may  be  extended  to  15  or 
20  or  more  ft.  But  when  the  trestles  become  very  high,  and  contain  a  great  deal  of 
timber,  it  becomes  cheaper  to  place  them  farther  apart,  say  30  to  60  feet ;  and  to  carry 
the  railway  upon  regular  framed  trusses,  as  at  u  u,  Figs  7  and  8;  as  in  a  bridge  with 
stone  piers.  In  the  foreraentioned  Genesee  river  viaduct,  the  trestles  are  50  ft  apart 
from  center  to  center. 

When  such  a  trestle  as  Fig  8  becomes  very  narrow  in  proportion  to  its  height,  we 
may  add  to  its  stability  by  introducing  beams  w,  extending  from  trestle  to  trestle; 
and  still  further  by  inserting  diag  Uraces  v  u,  as  at  the  Geiiesee  river. 

Fig's  42,  p  29  1,  "  Trusses,"  will  show  how  the  timbers  may  be  joined.  In  designing  trestles, 
(as  in  wooden  bridges,)  it  is  advisable,  as  far  as  practicable,  to  arrange  the  pieces  so  that  any  one 
may  be  removed  if  it  becomes  decayed :  and  another  put  in  its  place.  ON  CURVES,  additional  strength 
sho'uld  be  given  on  the  convex  side;  as  suggested  by  the  dotted  lines  in  Fig  5.  On  very  high  trestles 
especially,  (.as  well  as  on  bridges,}  wheel-guards,  g  y,  Fig  10,  either  inside  or  outside  of  the  rails, 
should  never  be  omitted,  as  is  commonly  done. 

There  are  as  yet  but  few  iron  trestles  in  the  United  States ;  but  they  are  common 
for  piers  in  Europe.  At  the  Crumlin  iron  .viaduct,  (spans  150  ft,)  they  are  about  180 
ft  high;  60  by  27  ft  at  base;  30  by  18  at  top;  each  composed  of  14  cast-iron  posts, 
arranged  as  a  long  hexagon ;  each  post  being  formed  of  17  ft  lengths  of  iron  pipes,  1 
ft  outer  diam,  by  1  inch  thick.  At  each  17  ft  length,  the  pipes  are  firmly  connected 
by  hor  iron  pieces;  and  between  these  diff  stages  is  diag  bracing  of  4  X  ^  X  %  J»ch 
rolled  T  iron,  arranged  as  in  Fig  7. 

The  rolled  Phoenix  column,  Fig  B,  of  page  233,  has  lately  been  use  I 
for  trestles,  and  is  well  adapted  to  the  purpose  *  In  marshy  ground,  piles  may  be 
driven  to  support  the  trestles ;  or  may  be  left  so  far  above  ground,  as  themselves  to 
constitute  the  posts.  Such  trestles  may  often  be  used  advantageously,  even  when 
to  be  afterward  filled  in  by  embkt.  They  then  sustain  the  rails  at  their  proper  level 
until  the  embkt  has  reached  its  final  settlement. 

They  are  generally  used  to  avoid  the  expense  of  embkt ;  especially  when  earth  can  only  be  ob- 
tained" from  a  great  dist.  Even  when  earth  and  timber  are  equally  convenient,  they  will  rarely  much 


*  The  Baltimore  Bridge  Co  have  recently  used  it  in  a  single-track  trestle-work  about  800  ft  long, 
on  a  R  R  in  New  York.  The  bents  are  30~ft  apart ;  the  tallest  ones  being  120  ft  high.  Each  bent 
consists  of  but  two  4-segment  columns,  about  8  ft  apart  at  top ;  and  each  spreading  outward  about 
1%  ins  per  foot  of  height,  toward  the  base.  Each  of  these  long  columns  is  in  four  lengths  of  30  ft 
each.  The  top  length  has  a  transverse  section  of  10  square  ins  of  metal ;  the  2d.  11  ins;  the  3d,  12 
ins  ;  and  the  4th,  or  bottom  one,  13  ins.  See  Table,  p.  234.  There  are  no  flats  between  the  segments. 

Beams  trussed  like  Fig  52,  of  p  304,  extend  from  bent  to  bent,  to  sustain  the  rails.  All  is  very 
/irmly  braced  ii  every  direction,  especially  at  all  the  joints  of  the  columns.  The  same  Co  have 
since  built  piera  or  Phoenix  columns  to  a  height  of  252  It. 


THERMOMETERS. 


309 


exceed  about  half  the  cost  of  embkt ;  even  when  but  about  30  ft  high ;  but  owing  to  their  liability  to 
decay,  they  should  be  resorted  to  ouly  in  case  of  necessity  ;  or  as  a  temporary  expedient. 


THERMOMETEBS. 


TABLE  of  corresponding  temperatures  by  the  Fahrenheit, 
and  by  the  Centigrade  thermometers. 

To  change  degrees  of  Fall,  to  deg  of  Cent;  if  above  the  Fah  zero, 
from  the  deg  of  Fah,  subtract  32 ;  mult  the  remainder  by  5;  divide  the  product  by 
9.  If  below  the  Fah  zero,  first  add  32;  then  mult  by  5 ;  and  divide  by  9. 

To  change  deg  of  Cent,  to  deg  of  Fah ;  if  above  the  Cent  zero  mult 
by  9 ;  divide  by  5 ;  add  32.  If  below  the  Cent  zero,  mult  by  9 ;  divide  by  5 ;  sub- 
tract 32. 


Deg  P. 

DegC. 

DegF. 

DegC. 

Deg  F. 

DegC. 

Deg  F. 

DegC. 

DegF. 

DegC. 

212 

100 

158 

70.0 

104 

40.0 

50 

10.0 

—  4 

—  20.0 

211 

99.4 

157 

69.4 

103 

89.4 

49 

9.4 

—  5 

—  205 

210 

98.9 

156 

68.9 

102 

38.9 

48 

8.9 

—  6 

—  21.1 

209 

98.3 

155 

68.3 

101 

38.3 

47 

8.3 

—  7 

—  21.7 

208 

97.8 

154 

67.8 

100 

37.8 

46 

7.7 

—  8 

—  22.2 

207 

97.2 

153 

67.2 

99 

37-2 

45 

7.2 

—  9 

—  22.7 

206 

96.7 

152 

66.7 

98 

36.7 

44 

6.7 

—  10 

—  23.3 

205 

96.1 

151 

66.1 

97 

36.1 

43 

6.1 

—  11 

—  23.9 

204 

95.6 

150 

65.6 

96 

35.6 

42 

5.5 

—  12 

—  24.4 

203 

95.0 

149 

65.0 

95 

35.0 

41 

5.0 

—  13 

—  25.0 

202 

94.4 

148 

64.4 

94 

34.4 

40 

4.4 

—  14 

—  25.5 

201 

93.9 

147 

63.9 

93 

33.9 

39 

3.9 

—  15 

-  26.1 

200 

93.3 

146 

63.3 

92 

33.3 

38 

8.3 

—  16 

—  26.7 

199 

92.8 

145 

62.8 

91 

32.7 

37 

2.7 

—  17 

—  27.2 

198 

92.2 

144 

62.2 

90 

32.2 

36 

2.2 

—  18 

—  27.7 

197 

91.7 

143 

61.7 

89 

31.7 

35 

1.7 

—  19 

—  28.3 

1% 

91.1 

142 

61.1 

88 

31.1 

34 

1.1 

—  20 

—  28.9 

195 

90.6 

141 

60.6 

87 

30.5 

33 

0.6 

—  21 

—  29.4 

194 

90.0 

140 

60.0 

86 

30.0 

32 

0.0 

—  22 

—  30.0 

193 

89.4 

139 

59.4 

85 

29.4   . 

31 

—  0.6 

—  23 

—  30.5 

192 

88.9 

138 

58.9 

84 

28.9 

30 

—  1.1 

—  24 

—  31.1 

191 

88.3 

137 

58.3 

83 

28.3 

29 

-  1.7 

—  25 

—  31.7 

190 

87.8 

136 

57.8 

82 

27.7 

28 

—  2.2 

—  26 

—  32.2 

189 

87.2 

135 

57.2 

81 

27.2 

27 

—  2.7 

—  27 

—  32.7 

188 

86.7 

134 

56.7 

80 

26.7 

26 

—  3.3 

—  28 

—  33.3 

187 

86.1 

133 

56.1 

79 

26.1 

25 

—  3.9 

—  29 

—  33.9 

186 

85.6 

132 

55.6 

78 

25.5 

24 

—  4.4 

—  30 

—  34.4 

185 

85.0 

131 

55.0 

77 

25.0 

23 

—  5.0 

—  31 

—  35.0 

184 

84.4 

130 

54.4 

76 

24.4 

22 

—  5.5 

—  32 

—  35.6 

183 

83.9 

129 

53.9 

75 

23.9 

21 

—  6.1 

—  33 

—  •86.1 

182 

83.3 

128 

53.3 

74 

23.3 

20 

—  6.7 

—  34 

—  36.7 

181 

82.8 

127 

52.8 

73 

22.7 

19 

—  7.2 

—  35 

—  37.2 

180 

82.2 

126 

52.2 

72 

22.2 

18 

—  7.7 

—  36 

—  37.8 

179 

81.7 

125 

51  7 

71 

21.7 

17 

—  8.3 

—  37 

—  38.3 

178 

81.1 

124 

51.1 

70 

21.1 

16 

—  8.9 

—  38 

—  38.9 

177 

80.6 

123 

50.6 

69 

20.5 

15 

—  9.4 

—  39 

—  39.4 

176 

80.0 

122 

50.0 

68 

20.0' 

14 

—  10.0 

—  40 

—  40.0 

175 

79.4 

121 

49.4 

67 

19.4 

13 

—  10.5 

—  41 

—  40.6 

174 

78.9 

120 

48.9 

66 

18.9 

12 

—  11.1 

—  42 

—  41.1 

173 

78.3 

119 

48.3 

65 

18.3 

11 

—  11  7 

—  43 

—  41.7 

172 

77-8 

118 

47.8 

64 

177 

10 

—  12.2 

—  44 

—  42.2 

171 

77.2 

117 

47.2 

63 

17.2 

9 

—  12.7 

—  45 

—  42.8 

170 

76.7 

116 

46.7 

62 

16.7 

8 

—  133 

—  46 

—  43.3 

169 

76.1 

115 

46.1 

61 

16.1 

7 

—  3.9 

—  47 

-43.9 

188 

75.6 

U* 

45.6 

60 

15.5 

6 

—  4.4 

—  48 

—  44.4 

167 

75.0 

113 

45.0 

59 

15.0 

5 

—  5.0 

—  49 

—  45.0 

166 

74.4 

112 

44.4 

58 

14.4 

4 

—  5.5 

—  50 

—  45.6 

165 

73.9 

111 

43.9 

57 

13.9 

3 

—  6.1 

—  51 

—  46.1 

164 

73.3 

110 

43.3 

56 

133 

2 

—  6.7 

—  52 

—  46.7 

163 

72.8 

109 

42.8 

55 

12.7 

1 

—  7.2 

—  53 

—  47.2 

162 

72.2 

108 

42.2 

54 

12.2 

0 

7_f 

—  54 

—  47.8 

161 

71.7 

107 

41.7 

53 

11.7 

—  1 

—  8.3 

—  55 

—  48.3 

160 

71.1 

106 

41.1 

52 

11.1 

—  2 

—  8.9 

—  56 

—  48.9 

159 

70.6 

105 

40.6 

51 

10.5 

—  3 

—  19.4 

—  57 

—  49.4 

Below  about  35°  below  zero  of  Fah,  the  mercurial  thermometer  and  barometer  become  too  irregular 
to  be  depended  on.  Mercury  begins  to  freeze  at  about  -40°  Fah.  Below  -40°  pure  alcohol  is 
used. 


310 


STONEWOIiK. 


Approximate  expansion  of  solid*  by  heat:  and  their  melt» 
ing:  points  by  Fahrenheit's  thermometer.  J 


For  1  degree. 

For  180  degrees.* 

Melting 
point 
in  Deg.t 

4593 

955 
1920  to 
2800 
2370  to 
2550 

3000  to 
3500 

506 
2016 
2000 

1873 

1861 
444 
612 

680  to  772 

Fire-brick  

1  part  in 
365220 
187560 
228060 
221400 
214200 
211500 
209700 
208800 
173000 
128000 
405000 
166500 
162000* 
173000 
151200 
159840 
167400 
131400 
146880 
147600 
149940* 
147420 
146340 
129600 
123120 
104400 
103320 
97740 
94140 
95040 
87840 
63180 
78840 
61920 
440530 

Ji  inch  in 
3804ft. 
1U54 
2375 
2306 
2231 
2203 
21K4 
2175 
1802 
1333 
4219 
1722 
1688 
1802 
1575 
1665 
1744 
1369 
1530 
1537 
1562 
1536 
1524 
1350 
1282 
1088 
1076 
1018 
981 
990 
915 
658 
821 
645 
4588 

1  part  in 
2029 
1042 
1267 
1230 
1190 
1175 
1165 
1160 
961 
711 
2250 
925 
900 
961 
840 
888 
930 
730 
816 
820 
833 
819 
813 
720 
684 
580 
574 
513 
523 
528 
4«8 
351 
438 
344 
2447 

H  inch  in 
21.14  ft. 
10.85 
13.20 
12.81 
12.40 
12.24 
12.13 
12.08 
10.00 
7.41 
23.44 
9.63 
9.38 
10.00 
8.75 
9.25 
9.69 
7.60 
8.50 
8.54 
8.68 
8.53 
855 
7.50 
7.12 
6.04 
5.98 
5.66 
5.45 
5.50 
5.08 
3.66 
4.56 
3.58 
25.49 

Granite                         .      J  from 

I   to 
Glass  rod                            

Glass  tube 

•'      plate 

Platina  ..  

Marble,  granular,  white,  dry.. 
"            "     moist. 
"  black,  compact 

Cast  iron 

Slate                                                 * 

Steel  

"     blistered  

"     unterupered  

"     hardened 

I  •        r  lied 

'  '     soft,  forged  

"     wire  

Bismuth     

Gold   annealed  

S-     d  -t   ne'-r' 

.,                                            average 

"     wire  

miver         

Pewter         

Zinc  (most  of  all  metals)  

i  epi   e 

Heat  of  a  common  wood-fire,  variously  estimated  from  800  to  1140  deg.    That  of  a  charcoal 

one  about  2200J  ;  coal  about  2400°. 

*  Hence  wrought  expands  by  heat  about  one  eleventh  part  more  than  cast j  whereas  under 
tension  within  elastic  limits  cast  stretches  twice  as  much  as  wrt. 

Each  12°  to  16°  of  expansion  of  wrt  iron  is  equivalent  to  about  1  ton  tension  per  sq  inch 
of  section ;  varying  with  quality  of  iron. 


STONEWOKK. 


POWDEH.     Explosive  force  about  40000  fts,  or  18  tons,  per  sq  inch.     Weight  aver 
ah -HU  i  hf  s. .me  as  ttiat  of  water,  or  62}^  fts  per  cub  ft;  hence,  1  fi>  ~  about  28  cub  ins.    In  ordin 


tverasrea 
linary 

quarrying,  a  cub  yard  of  solid  rock  in  place,  (or  about  1.9  cub  yds  piled  up  after  being  quarried.) 
requires  from  J^  to  %  ft.  In  very  refractory  rock,  lying  badly  for  quarryiug,  a  solid  yard  may  require 
from  1  to  2  fts.  In  some  of  the  most  successful  great  blasts  for  stone  for  the  Holyhead  Breakwater, 
England,  (where  several  thousands  of  fts  of  powder  were  usually  exploded  by  galvanism  at  a  single 
Mast,)  from  2  to  4  cub  yds  solid  were  loosened  per  ft;  but  in  many  instances  not  more  than  1  to  1J4 
yds.  Tunnels  and  shafts  require  2  to  6  fts  per  solid  yard:  usually  3  to  5  fts.  Soft,  partially  decom- 
posed rock  frequently  requires  more  than  harder  ones.  Usually  sold  in  kegs  of  25  fts.g 

The  fluid  called  nitro-glycerine  is  coming  into  use  instead  of  powder.  Its  strength  is  from  10 
to  13  times  that  of  an  equal  bulk;  or  from  8  to  10  times  that  of  an  equal  weight,  of  powder. 

Gun-cotton  has  from  3  to  6  times  the  strength  of  an  equal  weight  of  powder. 

*  By  adding  y1^  part  to  the  lengths  in  the  two  cols  under  180°,  we  get  the  lengths  corresponding  to 
a  number  of  degrees  y1^  less  than  180°;  or  to  163°.63  deg ;  which  may  be  taken  as  about  the  extremes 
of  temp  in  the  colder  portions  of  the  United  States.  In  the  Middle  States  the  extremes  rarely  reach 
135°,  or  Yi  part  less  than  180°. 

No  dependence  whatever  is  to  be  placed  on  results  obtained  by  Wedgewood's  pyrometer. 

f  The  table  shows  that  the  contraction  and  expansion  of  stone  will  cause 

open  joints  in  winter;  and  crushing  of  the  mortar  in  summer,  at  the  ends  of  long  coping-stones. 

i  The  melting  points  are  quite  uncertain.    We  give  the  mean  of 

the  best  authorities.  Assuming  that  with  a  change  of  temp  of  about  163°,  wrought  iron  will  alter  its 
length  1  part  in  916;  this  in  a  mile  amounts  to  5.764  ft,  or  about  5  ft  9^  ins ;  and  in  100  ft  to  .109  of  a 
foot;  or  1%  ins;  so  that  a  diff  of  5  ft,  or  more,  can  readily  result  from  measuring  a  mile  in  winter 
and  in  summer  with  the  same  chain ;  and  a  25  ft  rail  will  change  its  length  full  %  of  an  inch. 

\  Price,  1880,  in  Atlantic  cities,  about  $3  to  $3.25  per  25ft  keg ;  according  to 

quantity. 


STONEWORK.  311 

Weight  of  powder  in  one  foot  depth  of  hole. 

Diameter  of  hole  in  inches. 

1      I    IX    1    1H    I      2      |    2^    |      3      |    3^    |      4      |    4^    |      5      |    5J4    |      6 

Weight  of  powder  in  pounds  and  ounces  avoir. 
0"5    |  0"8    I  0"11  |   1"4   |      2      |2"13   1  3"14  |  5"0    |  6"6    |  7"14  |  9"8    |  11"5 


holes  for  blasting*.  The  holes  are  generally  from  2^  to  4  ft 
deep;  and  from  1%  to  2  ins  diam.  Churn-drilling;  is  much  more  expeditious 
and  economical  than  that  by  jumping,  mentioned  below.  The  churn-drill  is  merely 
a  round  iron  bar,  usually  about  1%  ins  diam,  and  6  to  8  ft  long;  with  a  steel  cutting 
edge,  or  bit,  (weighing  about  a  tb,  and  a  little  wider  than  the  diam  of  the  bar,)  welded 
to  its  lower  end.  A  man  lifts  it  a  few  inches;  or  rather  catches  it  as  it  rebounds, 
turns  it  partially  around;  and  lets  it  fall  again.  By  this  means  he  drills  from  5  to 
15  feet  of  hole,  nearly  2  ins  diam,  in  a  day  of  10  working  hours,  depending  on  the 
character  of  the  rock.  From  7  to  8  ft  of  holes  1%  ins  diam,  is  about  a  fair  day's 
work  in  hard  gneiss,  granite,  or  compact  siliceous  limestone  ;  5  to  7  ft  in  tough  com- 
pact hornblende  ;  3  to  5  in  solid  quartz  ;  8  to  9  in  ordinary  marble  or  limestone  ;  9  to 
10  in  sandstone;  which,  however,  may  vary  within  all  these  limits.  When  the  hole 
is  more  than  about  4  ft  deep,  two  men  are  put  to  the  drill.  Artesian,  and  oil  wells, 
in  rock,  are  bored  on  the  principle  of  the  churn-drill. 

The  jumper,  as  now  used,  is  much  shorter  than  the  churn-drill.  One  mnn  (the  holder)  sitting 
down,  lifts  it  slightly,  and  turns  it  partly  around,  during  the  intervals  between  the  blows  trom  about 
8  to  12  ft  hammers,  wielded  by  two  other  laborers,  the  strikers.  It  can  be  used  for  holes  of  smaller 
diameters  than  can  be  made  by  the  churn-drill  ;  because  the  holder  can  more  readily  keep  the  cutting 
end  at  the  exact  spot  required  to  be  drilled.  It  is  also  better  in  conglomerate  rock  ;  the  hard  siliceous 
pebbles  of  which  deflect  the  churn-drill  from  its  vertical  direction,  so  that  the  hole  becomes  crooked, 
and  the  tool  becomes  bound  in  it.  The  coal  conglomerates  are  by  no  means  hard  to  drill  with  a 
jumper.  The  jumper  was  formerly  used  for  large  deep  holes  also,  before  the  superiority  of  the  churn- 
drill  became  established. 

Either  tool  requires  resharpening  at  about  each  6  to  18  inches  depth  of  hole;  and  the  wear  of  the 
steel  edge  requires  a  new  one  to  be  put  on  every  2  to  4  days.  With  iron  jumpers,  the  top  also  be- 
comes battered  away  rapidly.  As  the  hole  becomes  deeper,  longer  drills  are  frequently  used  than  at 
the  beginning.  .The  smaller  the  diameter  of  the  hole,  the  greater  depth  can  be  drilled  in  a  given 
time;  and  the  depth  will  be  greater  in  proportion  than  the  decrease  of  diam.  Under  similar  circum- 
stances, three  laborers  with  a  jumper  will  about  average  as  much  depth  as  one  with  a  churn-drill. 

The  hand-drill,  in  which  the  same  man  uses  both  the  hammer  and  the  short  drill,  is  chiefly  used 
for  shallow  holes  of  small  diam.  With  it  a  fair  workman  will  drill  about  as  many  feet  of  hole  from 
6  to  12  ins  deep,  and  about  %  inch  diam,  as  one  with  a  churn-drill  can  do  in  holes  about  3  ft  deep,  and 
2  ins  diam,  in  the  same  time.  Only  the  jumper  or  the  hand-drill  can  be  used  for  boring  holes  which 
are  horizontal,  or  much  inclined. 

The  Bnrleig-h  Rock  Drill,  (Co  at  Fitchburg,  Mass  ,)  is  much  more  rapid  and 
economical  than  the  foregoing  if  the  work  ia  so  great  as  to  justify  the  preliminary  outfit.  It  drills  at 
any  angle  ;  and  is  worked  either  by  steam  directly  ;  or  by  air  compressed  by  steam  into  a  tank,  and 
thence  led  to  the  drills  through  iron  pipes.  The  air  is  best  for  tunnels  and  shafts,  because  after 
leaving  the  drills  it  aids  ventilation.  The  cost  of  each  drill  and  its  support  is  about  $450  to  $1200  at 
the  shop,  depending  on  the  size  of  the  holes,  (%  to  6  ins  diam.)  Each  compressor,  with  attached 
engine,  $1100  to  $2800.  Add  boilers,  air-tank,  air-pipes,  house,  extra  parts,  &c.  A  $2000  compressor 
will  work  two  drills  ;  and  require  3  inch  air-pipes  ;  and  a  tank  15  ins  diam.  by  4  ft  long.  The  boiler 
and  compressor  together  require  1  man;  each  drill  2  men,  and  will  drill  from  one  to  two  holes  of 
about  3  ins  diam,  and  5  ft  deep  per  hour,  depending  on  the  kind  of  rock  ;  and  about  -i  more  of  2  inch 
hole.  In  limestone  from  25  to  45  ft  of  hole  with  one  sharpening.  One  blacksmith  and  helper  can 

sharpen  for  5  or  6  drills.    prof.  Wood's  drill  also  is  a  first-class  machine. 

Cost  of  quarrying-  stone.  After  the  preliminary  expenses  of  purchasing 
the  site  of  a  good  quarry;  cleaning  off  the  surface  earth  and  disintegrated  ti.p  rock  ; 
and  providing  the  necessary  tools,  trucks,  cranes,  &c  ;  the  total  neat  expenses  for 
getting  out  the  rough  stone  for  masonry,  per  cub  yard,  ready  for  delivery,  may  be 
roughly  approximated  thus:  Stones  of  such  sizes  as  two  men  can  readily  lift,  meas- 
ured in  piles,  will  cost  about  as  much  as  from  %  to  %  the  daily  wages  of  a  quarry 
laborer.  Large  stones,  ranging  from  %  to  1  cub  yd  each,  got  out  by  blasting,  from 
1  to  2  daily  wages  per  cub  yd.  Large  stones,  ranging  from  1  to  1%  cub  yds  each,  in 
which  most  of  the  work  must  be  done  by  wedges,  in  order  that  the  individual  stones 
shall  come  out  in  tolerably  regular  shape,  and  conform  to  stipulated  dimensions  ; 
from  2  to  4  daily  wages  per  cub  yard.  The  smaller  prices  are  low  for  sandstone, 
while  the  higher  ones  are  high  for  granite.  Under  ordinary  circumstances,  about 
\}/A  cub  yds  of  good  sandstone  can  be  quarried  at  the  same  cost  as  1  of  granite  ;  or, 
in  other'words,  calling  the  cost  of  granite  1,  that  of  sandstone  will  he  %:  so  that 
the  means  of  the  foregoing  limits  may  be  regarded  as  rather  full  prices  for  sandstone; 
rather  scant  ones  for  granite  :  and  about  fair  for  limestone  or  marble. 

€ost  of  dressing*  stone.  In  the  first  place,  a  liberal  allowance  should  be 
made  for  waste.  Kven  when  the  stone  wedges  out  handsomely  on  all  sides  from 
the  quarry,  in  large  blocks  of  nearly  the  required  shape  and  size,  from  %  to  %  of 
tne  rough  block  will  generally  not  more  than  cover  waste  when  well  dressed.  In 
moderate-sized  blocks,  (say  averaging  about  %  a  cub  yard  each.,)  and  got  out  by 


312 


STONEWORK. 


blasting,  from  ^  to  %  will  not  be  too  much  for  stone  of  medium  character  as  to 
straight  splitting.  About  the  last  allowance  should  also  be  made  for  well-scabbled 
rubble.  The  smaller  the  stones,  the  greater  must  be  the  allowance  for  waste  in 
dressing.  In  large  operations,  it  becomes  expedient  to  have  the  stones  dressed,  as 
far  as  possible,  at  the  quarry ;  in  order  to  diminish  the  cost  of  transportation,  which, 
when  the  distance  is  great,  constitutes  an  important  item  —  especially  when  by  land, 
and  on  common  roads. 

A  Stonecutter  will  first  take  out  of' wind;  and  then  fairly  patent-hammer  dress,  about  8 
to  10  sq  ft  of  plain  face  in  hard  granite,  in  a  day  of  8  working  hours;  or  twice  as  much  of  such  infe- 
rior dressing  as  is  usually  bestowed  on  the  beds  and  joints ;  and  generally  on  the  faces  also  of  bridge 
masonry.  <fec,  when  a  very  fine  finish  is  not  required.  In  good  sandstone,  or  marble,  he  can  do  about 
J4  more"  than  in  granite.  Of  finest  hammer  finish,  granite,  4  to  5  sq  ft. 

Cost  of  masonry.  Every  item  composing  the  total  cost  is  liable  to  much 
variation;  therefore,  we  can  merely  give  an  example  to  show  the  general  principle 
upon  which  an  approximate  estimate  may  be  made;  assuming  the  wa^es  of'  a 
laborer  to  be  $*2.00  per  day  of  8  working  hours;  and  $3.50  for  a  mason.  The 
monopoly  of  quarries  affects  prices  very  much.* 

Cost  of  ashlar  facing  masonry.  Average  size  of  the  stones,  say  5. ft 
long,  2  ft  wide,  and  1.4  thick ;  or  two  such  stones  to  a  cub  yd.  Then,  supposing  the 
stone  to  be  gj-anite  or  gneiss,  the  cost  per  cub  yd  of  masonry  at  such  wages 

will  be,      Getting  out  the  stone  from  the  quarry  by  blasting,  allowing  y±  for  waste  in 

dressing  ;  1  %  cub  yds,  at  $3.00  per  yard $4.00 

Dressing  14  sq  ft  of  face  at  35  cts 4.90 

"        52      "        beds  and  joints,  at  18  cts 9.36 

Neat  cost  of  the  dressed  stone  at  the  quarry 18.26 

Hauling,  say  1  mile;  loading  and  unloading 1.20 

Mortar,  say 40 

Laying,  including  scaffold,  hoisting  machinery,  superintendence,  &c 2.00 

Neat  cost 21.86 

Profit  to  contractor,  say  15  per  ct 3.28 

Total  cost 25. H 

Dressing  will  cost  more  if  the  faces  are  to  be  rounded,  or  moulded.  If  the  stones  are  smaller  than 
•we  have  assumed,  there  will  be  more  sq  ft  per  cub  yd  to  be  dressed,  Ac. 

If  in  the  foregoing  case,  the  stones  be  perfectly  well  dressed  on  all  sides,  including  the  back,  the 
cost  per  cub  yd  would  be  increased  about  $10;  and  if  some  of  the  sides  be  curved,  as  in  arch  stones, 
say  $12  or  $14;  and  if  the  blocks  be  carefully  wedged  out  to  given  dimensions,  $16  or  $18  j  thus 
making  the  neat  cost  of  the  dressed  stone  at  the  quarry  say  $28,  $31,  or  $35  per  cub  yd. 

The  item  of  laying  will  be  much  increased  if  the  stone  has  to  be  raised  to  great  heights ;  or  if  it  has 
to  be  much  handled ;  as  when  carried  in  scows,  to  be  deposited  in  water-piers,  &c.  Almost  every 
large  work  presents  certain  modifying  peculiarities,  which  must  be  left  to  the  judgment  of  the  engi- 
neer and  contractor.  The  percentage  of  contractors'  profit  will  usually  be  less  on  large  works  than 
on  small  ones.  See  App  p  630. 

Cost  of  large  scabbled  granite  rubble,  such  as  is  generally  used  as 
backing  for  the  foregoing  ashlar  ;  stones  averaging  about  %  cub  yd  each  : 

Cost  per 
At  above  rates  of  wages.  cub  yd  of 

masonry. 
Getting  out  the  stone  from  the  quarry  by  blasting,  allowing  X  for  waste  in 

scabbling;  J^-  cub  yds  at  $3.00 $3.43 

Hauling  1  mile,  loading  and  unloading  1.20 

Mortar ;  (2  cub  ft,  or  1.6  struck  bushels  quicklime,  either  in  lump  or  ground  ; 

and  10  cub  ft,  or  8  struck  bushels  of  sand,  or  gravel ;  and  mixing) 1.50 

Scabbling ;  laying,  including  scaffold,  hoisting  machinery,  &c 2.50 

Neat  cost 8.63 

Profit  to  contractor,  say  15  per  ct 1-30 

Total  cost 9.93 

Common  rubble  of  small  stones,  the  average  size  being  such  as  two 
men  can  handle,  costs,  to  get  it  out  of  the  quarry,  about  80  cts  per  yard  of  pile ; 
or  to  allow  for  waste,  say  $1.00.  Hauling  1  mile,  $1.00.  It  can  be  roughly  scabbled, 
and  laid,  for  $1.20  more ;  mortar  as  foregoing,  $1.50.  Total  neat  cost,  $4.70  ;  or,  with 
15  per  ct  profit,  $5.40,  at  the  above  wages  for  labor.  See  p  630. 

*  The  blocks  of  granite  for  Bunker  Hill  monument  averaging  2  cub  yds  each,  were 
quarried  by  wedging,  and  delivered  at  the  site  of  the  monument,  at  a  neat  actual  cost  of  $5.40 
per  cub  yd  ;  by  the  Monument  Association  ;  from  a  quarry  opened  by  themselves  for  the  purpose.  The 
Association  received  no  profit ;  their  services  being  voluntary.  The  average  contract  offers  for  the 
same  were  $24.30 !  The  actual  cost  of  getting  out  the  rough  blocks  at  the  quarry  was  $2.70.  Load- 
ing upon  trucks  at  quarry,  about  15  cts.  Transportation  8  miles  by  railway  and  common  road,  $2.56. 
Total,  $5.40.  In  1825  to  1845  ;  common  unskilled  labor  averaging  $1  per  day. 

In  1873  large  blOCkS  Of  grailite  of  about  a  cub  yd  each,  with  dressed  beds  and 
join"  but  with  only  a  2  inch  draft  around  the  showing-face,  (which  is  left  rough  )  are  deld  on  the 
inarf'at  Philada  from  Maine  &c,  at  $27  per  cub  yd.  And  from  Port  Deposit  at  $23. 


STONEWORK.  313 

With  smaller  Stones,  such  as  one  man  can  handle,  we  may  sav,  stone  70  cts;  hauling  $1 ; 
laying  and  scaffold,  tools  &c,  SI;  mortar  $1.50.  Making  the  neat  cost  $4/20;  or  with  15  per  ct  profit,  $4.83. 
Neat  scabbled  irregular  range- work  costs  from  $2  to  $3  more  per  yd  than  rubble:  according  to  the  charac- 
ter of  the  stone  &c.  The  layingof  thin  walls  costs  more  than  that  of  thick  ones,  such  as  abutments  &c.* 

The  cost  of  plain  8  inch  thick  ashlar  facings  lor  dwellings  &c  in 
Philada,  in  1873,  is  about  as  follows  per  square  foot  showing,  put  up,  including  everything.  Sand- 
stone, $1.50  to  $2.25.  Pennsylvania  warble,  $2.50.  New  England  marble,  $2.75  to  $3.25.  Granite, 
$2.25  to  $2.75.  If  6  ins  thick,  deduct  one-eighth  part.  First  Class  artificial  StOIie 
could  be  made  and  put  up  at  one-third  the  price.  See  p  508.  North  River  blue  Stone 
flags,  3  ins  thick,  for  footwalks,  put  down,  including  gravel  Ac,  70  cts  per  sq  foot.  Belgian 

street  pavement,  with  gravel,  complete,  $3.50  per  sq  yard  in  Eastern  cities. 

When  dressed  ashlar  facing  is  backed  by  rubble,  the  expense  per  cub  yard  of  the 
entire  mass  will  of  course  vary  according  to  the  proportions  of  the  two.  Thus,  if 
ashlar  at  $12  per  yd,  is  backed  by  an  equal  thickness  of  rubble  at  $5,  the  mean  cost 
will  be  ($12  +  $5)  -s-  2  =  $8.50;  or  if  the  rubble  is  twice  as  thick  as  the  ashlar  then 
($12  +  $5  -f-  $5)  -5-  3  =  $7.33,  &c.  Such  compound  walls  are  weak  and 
apt  to  separate  in  time,  as  also  walls  of  cut  stone  backed  by  concrete,  or  by  brick  ; 
from  unequal  settlement  of  the  two  parts. 

At  times  the  contractor  must  be  allowed  extra  in  opening  new  quarries;  in  forming 
short  roads  to  his  work  ;  in  digging  foundations  ;  or  for  pumping  or  otherwise  draining  them,  wh*-n 
springs  are  unexpectedly  met  with  ;  for  the  centers  for  arches,  &c;  unless  these  items  are  expressly 
included  in  the  contract  per  cub  yd. 

FOUNDATIONS, 

A  VOLUME  might  be  occupied  by  this  important  subject  alone.  We  have  space  for 
only  a  few  general  hints  ;  leaving  it  to  the  student  to  determine  how  far  they  may 
be  applicable  in  any  given  case.  In  ordinary  cases,  as  in  culverts,  retaining  walls, 
Ac,  if  excavations,  or  wells,  <tc,  in  the  vicinity,  have  not  already  proved  that  the  soii 
is  reliable  to  a  considerable  depth,  it  will  usually  be  a  sufficient  precaution,  after 
having  dug  and  levelled  off  tlie  foundation  pits  or  trenches  to  a  depth  of  3  to  6  ft.  to 
test  it  by  an  iron  rod,  or  a  pump-augur ;  or  to  sink  holes,  in  a  few  spots,  to  the  depth 
of  4  to  8  ft  farther;  (depending  upon  the  weight  of  the  intended  structure:)  to  ascer- 
tain if  the  soil  continues  firm  to  that  distance.  If  it  does,  there  will  rarely  be  any 
risk  in  proceeding  at  once  with  the  masonry;  because  a  stratum  of  firm  soil,  from  4 
to  8  ft  thick,  will  be  safe  for  almost  any  ordinary  structure;  even  though  it  should 
be  underlaid  by  a  much  softer  stratum.  If,  however,  the  firm  upper  stratum  is  ex- 
posed to  running  water,  as  in  the  case  of  a  bridge-pier  in  a  river,  care  must  be  taken 
to  preserve  it  from  gradually  washing  away;  or  from  becoming  loosened  and  broken 
up  by  violent  freshets ;  especially  if  they  bring  down  heavy  masses  of  ice,  trees,  and 
other  floating  matter.  These  are  sometimes  arrested  by  piers,  and  accumulate  so  as 
to  form  dams  extending  to  the  bottom  of  the  stream  ;  thus  creating  an  increase  of 
velocity,  and  of  scouring  action,  that  is  very  dangerous  to  the  stability  both  of  the 
bottom  and  of  the  structure.  When  the  testing  has  to  be  made  to  a  considerable 
depth,  it  may  be' necessary  to  drive  down  a  tube  of  either  wrought  or  cast  iron,  to 
prevent  the  soil  from  falling  into  the  unfinished  hole.  If  necessary,  this  tube  may 
be  in  short  lengths,  connected  by  screw  joints,  for  convenience  of  driving;  and  the 
earth  inside  of  it  may  be  removed  by  a  small  scoop  with  a  long  handle.f 

BoringS  in  Common  SOilS  or  clay  may  be  made  100  ft  deep  in  a  day  or  two  by  a 
common  wood  auger  l^.ins  diam,  turned  by  two  to  four  men  with  3  ft  levers.  This  will  bring  up 
samples.  For  this  and  other  earth-boring  tools  see  p  636. 

In  starting  tiie  masonry,  the  largest  stones  should  of  course  be  placed  at  the  bot- 
tom of  the  pit,  so  as  to  equalize  the  pressure  as  much  as  possible;  and  care  should 
be  taken  to  bed  them  solidly  in  the  soil,  so  as  to  have  no  rocking  tendency.  The 
next  few  courses  at  least  should  be  of  large  stones,  so  laid  as  to  break  joint  thoroughly 
with  those  below  The  trenches  should  be  refilled  with  earth  as  soon  as  the  masonry 
will  permit;  so  as  to  exclude  rain,  which  would  injure  the  mortar,  and  soften  the 
foundation.  It  is  well  to  ram  or  tread  the  earth  to  some  extent  as  it  is  being  deposited. 

If  the  tests  show  that  the  soil  (not  exposed  to  running  water)  is  too  soft  to  support 
the  masonry,  then  the  pits  should  be  made  considerably  wider  and  deeper:  and  after- 
ward be  filled  to  their  entire  width,  and  to  a  depth  of  from  3  to  6  or  more  ft,  (de- 
pending on  the  weight  to  be  sustained.^  with  rammed  or  rolled  layers  of  sand,  gravel, 
or  stone  broken  to  turnpike  size ;  or  with  concrete  in  which  there  is  a  good  propor- 
tion of  cement.  On  this  deposit  the  masonry  may  be  started.  The  common  practice 
in  such  cases,  of  laying  planks  or  wooden  platforms  in  the  foundations,  for  building 

*  In  Philada  in  1873.  cellar  and  other  walls  of  rough  rubble,  $5  to  $6  per  perch  of  22  cnb  ft  of  wall. 
Outside  walls  with  a  facing  of  broken  range  rock-work  of  sandstone,  (as  common  in  Gothic  churches,) 
$8  to  $10  per  22  cub  ft.  including  everything. 

t  Subterranean  cavern*  in  limestone  regions  are  a  frequent  source  of  trouble,  against  which  it  is 
difficult  to  adopt  precautions. 


314  FOUNDATIONS. 

upon,  is  a  very  bad  one.  For  if  the  planks  are  not  constantly  kept  thoroughly  wet, 
they  will  decay  in  a  few  years;  causing  cracks  and  settlements  in  the  masonry. 

Some  portions  of  the  circular  brick  aqueduct  for  supplying  Boston  with  water, 
gave  a  great  deal  of  trouble  where  its  trenches  passed  through  running  quicksands, 
and  other  treacherous  soils.  Concrete  was  tried,  but  the  wet  quicksand  mixed  itself 
with  it,  and  killed  it.  Wooden  cradles,  &c,  also  failed  ;  and  the  difficulty  was  finally 
overcome  by  simply  depositing  in  the  trenches  about  two  feet  in  depth  of  strong 
gravel.*  Sand  or  gravel,  when  prevented  from  spreading  sideways,  forms  one  of  the 
best  of  foundations.  To  prevent  this  spreading,  the  area  to  be  built  on  may  be  sur- 
rounded by  a  wall;  or  by  squared  piles  driven  so  close  as  to  touch  each  other;  or  in 
less  important  cases,  by  short  sheet  piles  only.  But  generally  it  is  sufficient  simply 
to  give  the  trenches  a  good  width;  and  to  ram  the  sand  or  gravel  (which  are  all  the 
better  if  wet)  in  layers;  taking  care  to  compact  it  well  against  the  sides  of  the  trench 
also.  Under  heavy  loads,  some  settlement  will  of  course  take  place,  as  is  the  case 
in  all  foundations  except  rock.  If  very  heavy,  adopt  piling,  Ac.  See  GRILLAGE,  p.  634. 

When  an  unreliable  soil  overlies  a  firm  one,  but  at  such  a  depth 

that  the  excavation  of  the  trenches  (which  then  must  evidently  be  made  wider,  as  well  as  deeper,) 
becomes  too  troublesome,  and  expensive ;  especially  when  (as  generally  happens  in  that  case)  water 
percolates  rapidly  into  the  trenches  from  the  adjacent  strata,  we  may  resort  to  piles.  See  "  Piling." 

When  making  deep  foundation  pits  in  damp  clay,  we  must  remember  that 

this  material,  being  soft,  has,  to  a  certain  degree,  a  tendency  to  press  in  every  direction,  like  water. 
This  causes  it  to  bulge  inward  at  the  sides;  and  upward  at  the  bottom  The  excavations  for  tunnels, 
or  for  vertical  shafts,  often  close  in  all  around,  and  become  much  contracted  thereby  before  they  can 
be  lined;  therefore  they  should  be  dug  larger  than  would  otherwise  be  necessary.  The  bottoms  of 
canal  and  railroad  excavations  in  moist  clay  are  frequently  pressed  upward  by  the  weight  of  the  sides. 

Dry  clay  rapidly  absorbs  moisture  from  the  air,  and  swells,  producing  effects 

similar  to  the  foregoing.  Its  expansion  is  attended  by  great  pressure  :  so  that  retaining- walls  backed 
with  dry  rammed  clay  will  be  in  danger  of  bulging  if  the  clay  should  become  wet.  It  is  a  treacherous 

material  to  work'in.    For  concrete  foundations,  see  p  507. 

As  to  the  greatest  load  that  may  safely  be  trusted  on  an  earth  founda- 
tion, Rankine  advises  not  to  exceed  1  to  1.5  tons  per  sq  ft.  But  experience  proves  that  on  good  com- 
pact gravel,  sand,  or  loam,  at  a  depth  beyond  atmospheric  influences,  2  to  3  tons  are  safe,  or  even  4 
to  6  tons  if  a  few  ins  of  settlement  may  be  allowed,  as  is  often  the  case  in  isolated  structures  without 
tremors.  Years  may  elapse  before  this  settlement  ceases  entirely.  Pure  clay,  especially  if  damp,  is 
more  compressible,  and  should  not  be  trusted  with  more  than  1  to  2.5  tons,  according  to  the  case.  All 

earth  foundations  must  yield  somewhat.  Equality  of  pressure  is  a  main 
point  to  aim  at.  Tremor  increases  settlements,  and  causes  them  to  continue 

for  a  longer  period,  especially  in  weak  soils,  great  care  must  be  taken  not  to  overload  in  such  cases, 
even  if  piled.  Foundations  In  sllty  soils  will  probably  settle,  in  years,  at  the  rate  of  from  3  to 
12  ins  per  ton  (up  to  2  tons)  per  sq  ft  of  quiet  load,  if  not  on  piles. 

Fig  2  shows  an  easy  mode  of  obtaining  a  foundation  in  certain  cases.  It  is  the 
" pierre  perdue"  (lost  stone)  of  the  French;  in  English,  " random 
stone,"  or  rip- rap. 

It  is  merely  a  deposit  of  rough  angular  quarry  stone  thrown  into  the  water:  the  largest  ones  being 
at  the  outside,  to  resist  disturbance  from  freshets,  ice,  floating  trees,  &c.  A  part  of  the  interior  may 
be  of  small  quarry  chips,  with  some  gravel,  sand,  clay,  &c.  When  the  bottom  is  irregular  rock,  this 
process  saves  the  expense  of  levelling  it  off  to  receive  the  masonry.  For  2  or  3  feet  below  the  surface 
of  the  water,  the  stones  may  generally  be  disposed  by  hand  so  as  to  lie  close  and  firmly.  Small  spawls 
packed  between  the  larger  ones  will  make  the  work  smoother,  and  less  liable  to  be  displaced  by  violence. 
Cramps  or  chains  may  at  times  be  useful  for  connecting  several  of  the  large  stones  together  for  greater 

stability.    Rip-rap,  however,  is  apt  to  settle. 

If  the  bottom  is  so  yielding*  as  to  be  liable  to  wash  away  in 

freshets,  it  may,  in  addition,  be  protected,  as  in  Fig  2,  by  a  covering  of  the  same  kind 

of  stones,  as  at  c :  extend- 
ing all  around  the  struc- 
ture. Or  the  main  pile 
of  stones  may  be  extend- 
ed as  per  dotted  line  at  d ; 
so  that  if  the  bottom 
should  wash  away,  as  per 
dotted  line  at  o,  the 
stones  d  will  fall  into 

•O-       «  V  ^  the  cavity,  and  thus  pre- 

'  j  vent     further     damage. 

Sheet-piles,  s  s,  may  be 
driven  as  an  additional  precaution.  For  greater  security,  the  bed  of  the  river  may 
be  dredged  or  scooped  under  the  entire  space  to  be  covered  by  the  main  deposit,  as 
per  dotted  lines  in  Fig  3,  to  as  great  a  depth  as  any  scouring  would  be  apt  to  reach; 

*  Smeaton  mentions  a  stone  bridge  built  upon  a  natural  bed  of  gravel  only  about  2  ft  thick,  over- 
lying deep  mud  so  soft  that  an  iron  bar  40  ft  long  sank  to  the  head  by  its  own  weight.  One  of  the 
piers,  however,  sank  while  the  arches  were  being  turned;  and  was  restored  by  Smeaton.  Although 
a  wretched  precedent  for  bridge  building,  this  example  illustrates  the  bearing 'power  of  a  thick  layer 
of  well -compacted  gravel. 


FOUNDATIONS. 


315 


Fig  3 


thig  excavation  also  to  be  filled  with  stone.  Such  foundations  are  evidently  beet 
adapted  to  quiet  water.  The  masonry  should  rest  on  a  strong  platform. 

Large  deposits  of  stone,  as  in  these 
two  figs,  greatly  increase  the  velocity, 
and  the  scouring  action  of  the  stream 
around  them,  especially  in  freshets  ;  un- 
less the  bottom  on  each  side  from  the  de- 
posit he  dredged  out  to  such  an  extent 
that  the  original  area  of  water  shall  not 
be  reduced.  If  the  bottom  is  treacherous, 
this  should  be  done  before  depositing  the 
covering  stones  c,  Fig  2.  Judgment  and 
experience  are  necessary  in  such  matters, 
as  in  all  others  connected  with  engineer- 
ing, IVfiere  study  will  not  guard  apainst 
constant  failures.  Theory  and  practice 
must  guide  each  other. 

Fig  3  is  another  simple  method;  and 
when  it  does  not  create  too  great  an  ob- 
struction to  the  navigation  of  the  stream,  or  to  the  escape  of  its  waters  In  time  of  high  freshets,  is  a 
very  effective  one.  Here  th«  piles  are  first  driven  into  the  river  bottom,  for  the  support  of  the  pier; 
then  the  deposit  of  stone  is  thrown  in,  for  the  support  and  protection  of  the  piles ;  preventing  them 
from  bending  under  their  loads ;  and  shielding  them  from  blows  from  floating  bodies.  The  top.*  of 
the  piles  being  cut  off  to  a  level,  a  strong  platform  of  timber  is  laid  on  top  of  them,  as  a  base  for  the 
masonrv.  The  top  of  the  platform  should  not  be  less  than  about  12  or  18  ins  below  ordinary  low 
water,  to  prevent  decay.  Mitchell's  iron  screw  pile ;  or  hollow  piles  of  cast  iron,  may  be  used  instead 
•f  wooden  ones.  See  PIMNO,  p.  320. 

Figs  4  represent  a  convenient  method  of  establishing  a  foundation  in  water,  by 
means  of  a  timber  crib,  A  A,  without  a  bottom.  It  should  be  built  of 
squared  timbers,  notch- 
ed together  at  their 
crossings,  as  shown  at 
Fig  5 ;  each  notch  being 
Y±  of  the  depth  of  the 
stick.  By  this  means 
each  timber  is  support- 
ed throughout  its  entire 
length  by  the  one  below 
it;  and  resists  pulling 
in  both  directions.  Bolts 
also  are  driven  at  the 
intersections;  at  least 
in  the  sides  of  the  crib, 
to  prevent  one  portion 
from  being  floated  off 
from  the  other.  The 
crib  is  thus  divided  into 
square  or  rectangular 
cells,  from  2  to  4  or  5  ft 
on  a  side,  according  to 
the  requirements  of  the 
case.  The  partitions 
between  the  cells  are 
put  together  in  the 
same  manner  as  those 
at  the  sides  of  the  cribs ; 
and  consequently,  like 
the  latter,  form  solid 

wooden  walls. 

\& 

The  crib  may  be  framed  afloat,  at  any  convenient  spot;  and  when  finished,  maybe  to^ed  to  its 
final  place,  where  it  is  carefully  moored  in  position,  and  then  suuk  by  throwing  stone  into  a  few 
cells  provided  with  platforms,  as  at  c  c,  for  that  purpose.  Thesu  platforms  should  be  placed  a  little 
above  the  lower  edge  of  the  cells,  so  as  not  to  prevent  the  crib  from  settling  slightly  into  the  soil,  and 
thus  coming  to  a  full  bearing  upon  the  bottom.  After  it  has  been  sunk,  all  the  cells  are  filled  with 
rough  stone.  A  stout  top  platform  may  be  added  or  not,  as  the  case  may  be  ;  also,  a  protection,  1 1,  of 
random  stone,  to  prevent  undermining  by  the  current.  If  the  sides  are  exposed  to  abrasion  from 
ice,  &c,  they  may  be  covered  in  whole  or  in  part  with  plank,  or  plate  iron  ;  and  the  angles  strength- 
ened by  iron  straps,  &c.  In  deep  water,  a  foundation  may  be  made  partly  of  random  stone,  as  in 
Figs  2  and  3 ;  and  on  top  of  this  may  be  sunk  a  crib,  with  its  top  about  2  ft  under  low  water,  as  a  base 
for  the  masonry.  This  is  much  safer  than  random  stone  alone. 

On  uneven  rock  bottom  it  may  be  necessary  to  scribe  the  bottom  of  the 

crib  to  fit  the  rock;  or  the  crib  may  first  be  sunk  by  means  of  a  loaded  platform  on  its  top,  or  by 
filling  some  of  its  cells,  until  its  lowest  timbers  are  within  a  short  distance  above  the  bottom.  Being 
there  kept  in  a  horizontal  position,  small  stones  may  be  thrown  into  the  cells,  and  allowed  to  find 
their  way  under  the  timbers  of  the  crib,  thus  forming  a  level  support  for  it.  The  cells  may  then  be 
filled  ;  and  rip-rap  deposited  outside  around  the  crib  to  prevent  the  small  stones  from  being  displaced. 


FidsJJ. 


316 


FOUNDATIONS. 


A  crib  with  only  an  outside  row  of  cells  for  sinking  it  may  be 

built  i  and  the  interior  chamber  may  be  filled  with  concrete  underwater.  The  masonry  may  then 
rest  on  the  concrete  alone.  If  the  crib  rests  upon  a  foundation  of  broken  stone,  the  upper  interstices 
of  this  stone  should  first  be  levelled  off  by  small  stone  or  coarse  gravel  to  receive  the  concrete  of  the 
inner  chamber. 

Or  a  crib  like  Fig**  4  may  be  sunk,  and  piles  be  driven  in  tbe  cells,  which 
may  afterward  be  filled  with  broken  stone  or  concrete.  The  masonry  may  then  rest  on  the  piles  only. 
which  in  turn  will  be  defended  by  the  crib.  If  the  bottom  is  liable  to  scour,  place  sheet-piles  or 
rip-rap  around  the  base  of  the  crib. 

By  all  means  avoid  a  crib  like  e,  Fig  5^,  much  higher  at  one  part 
than  at  another,  if  the  superstructure,  s  is  to  rest  on  tlv.  timber  of  the  crib  instead  of  on 
piles,  or  on  concrete  independent  of  tbe  timber  ;  for  the  high  part  of  the  crib  will  compress  more  under 
its  load  than  the  low  part  ,  and  will  thus  cause  the  superstructure  to  lean  or  to  crack. 

A  crib  either  straight  sided  or  circular,  with  only  an  outer  row  of  cells  for  pud- 
dling may  be  used  as  a  cofferdam  (see  cofferdams,  p  317).  The  joints 

between  the  outer  timbers  should  be  well  caulked  ;  and  care  be  taken,  by  means  of  outside  pile-planks, 
gravel,  &c,  to  prevent  water  from  entering  beneath  it. 

The  cast-iron  Bridge  across  the  Schuylkill  at  Chestnut  St, 

Phi  la,  Mr.  Strickland  Kneass,  Engineer,  affords  a  striking  example  of  crib 
foundation.  The  center  pier  stands  on  a  crib,  an  oblong  octagon  in  plan  ;  31  by  87  feet  at  base  ;  24 
by  80  ft  at  top;  and  (with  its  platform)  29  ft  high.  Its  timbers  are  of  yellow  pine,  hewn  12  ins 
square  :  and  framed  as  at  Fig  5.  The  lower  timbers  were  carefully  cut  or  scribed  to  conform  to  the 
irregularities  of  the  tolerably  level  rock  upon  which  it  rests.  These  were  ascertained  (after  the  8  ft 
depth  of  gravel  had  been  dredged  off)  in  the  usual  manner  of  mooring  above  the  site  a  large  floating 
wooden  platform,  composed  of  timbers  corresponding  in  position  with  all  those  of  the  lower  course 
of  the  intended  crib,  both  longitudinal  and  transverse.  Soundings  were  then  taken  close  together 
along  all  these  lines  of  timber.  Most  of  the  cells  are  about  3  by  4  ft  on  a  side,  in  the  clear.  A  few 
of  them  had  platforms  at  the  level  of  the  second  course  from  the  bottom,  for  receiving  stone  for  sink- 
ing the  crib;  the  others  are  open  to  the  bottom. 

The  crib  was  built  in  the  water  ;  and  was  kept  floating,  during  its  construction,  with  its  unfinished 
top  continually  just  above  water,  by  gradually  loading  it  with  more  stone  as  new  timbers  were  added. 
The  stone  required  for  this  purpose  alone  was  300  tons.  When  the  crib  was  towed  into  position,  and 
moored,  150  tons  more  were  added  for  sinking  it.  All  the  cells  were  afterward  filled  with  rough  dry 
stone,  and  coarse  gravel  screenings  ;  making  a  total  of  1666  tons.  A  platform  of  12  by  12  inch  squared 
timber  covered  the  whole  ;  its  top  being  2Ji  ft  below  low  water.  The  pier  alone,  which  stands  on  this 
crib,  weighs  3255  tons;  and  during  its  construction  it  compressed  the  crib  6^  ins.  The  weight  of 
superstructure  resting  on  the  pier,  may  be  roughly  taken  at  1000  tons  more. 

An  ordinary  caisson  is  merely  a  strong  scow,  or  a  box  with- 

out a  lid;  and  with  sides  which  may  at  pleasure  be  readily  detached  from  its  bottom.  It  is  built  on 
land,  and  then  launched.  The  masonry  may  first  be  built  in  it,  either  in  whole  or  in  part,  while 
afloat;  and  the  whole  being  then  towed  into  place,  and  moored,  may  be  sunk  to  the  bottom  of  the 
river,  to  rest  upon  a  foundation  previously  prepared  for  it,  either  by  piling,  if  necessary;  or  by 
merely  levelling  off  the  natural  surface,  &c.  The  bottom  of  the  caisson  constitutes  a  strong  timber 

latform,  upon  which  the  masonry  rests  ;  and 


, 

)  so  arranged,  that  after  it  is  sunk,  the  sides 
may  be  detached  from  it,  and  removed  to  be 
rebottomed  for  use  at  another  pier,  if  needed. 
This  detaching  may  be  effected  by  some  such 
contrivance  as  that  shown  in  Fig  6,  where 
P  P  w  is  the  bottom  of  the  caisson,  to  which 
are  firmly  attached  at  intervals  strong  iron 
eyes  t  ;  which  are  taken  hold  of  by  books  d,  at 
the  lower  end  of  long  bolts  E  n,"  reaching  to 
the  top  timbers  S  of  the  crib,  where  they  are 
confined  by  screw  nuts  n.  By  loosening  the 
nuts  n,  the  hooks  d  can  be  detached  from  the 
eyes  t;  and  the  sides  can  then  be  removed 
from  the  bottom;  there  being  no  other  connec- 
tion between  the  two.  These  hooks  and  eyes 
are  usually  placed  outside  of  the  caisson  ;  the 
screw  nuts  n  being  sustained  by  the  projecting 
ends  of  cross  pieces,  as  tt,  Fig  9.  The  im- 
proper position  given  them  in  our  Fig  was 
merely  for  convenience  of  illustrating  the  prin- 
ciple. It  will  sometimes  be  necessary  to  have 

one  side  detachable  from  the  others,  in  order  to  float  the  caisson  away  clear  from  the  finished  pier; 
unless  it  be  floated  away  before  the  masonry  has  been  built  so  high  as  to  render  the  precaution  use- 
less. Fig  6  shows  one  of  many  ways  of  constructing  a  caisson  ;  with  sides  consisting  of  upright 
corner-posts,  I;  cap  pieces  S,  on  top  ;  and  sills  g  at  bottom,  resting  on  the  bottom  platform  P  P  w  ; 
intermediate  uprights  T,  framed  into  the  caps  and  siMs  :  the  whole  being  covered  outside  by  one  or 
two  thicknesses  of  planking  B,  which,  as  well  as  the  platform,  should  be  well  calked,  to  prevent 
leaking.  Tarpaulin  also  may  be  nailed  outside  to  assist  in  this.  The  greatest  trouble  from  leaking 
is  where  the  sides  join  the  platform.  On  top  of  the  platform  is  firmly  spiked  a  timber  o  o,  extending 
all  around  it  just  inside  of  the  inner  lower  edge  of  the  sides  of  the  caisson.  Its  use  is  to  prevent 
the  sides  from  being  forced  inward  by  the  pressure  of  the  water  outside.  The  details  of  construction 
•will  of  course  vary  with  the  requirements  of  the  case.  In  deep  caissons,  inside  cross-braces  or  struts 
from  side  to  side,  as  at  c  c,  Fig  7,  will  be  required  to  prevent  the  sides  from  being  forced  inward  by 
the  pressure  of  the  water,  as  the  vessel  gradually  sinks  while  the  masonry  is  being  built  within  it. 
As  the  masonry  is  carried  up.  the  struts  are  removed;  and  short  ones,  extending  from  the  sides  of 
the  caisson  to  the  masonry,  are  inserted  in  their  place.  When  the  caisson  is  shallow,  only  the  upper 
course  of  braces  will  be  required;  they  also  support  a  platform  for  the  workmen  and  their  materials. 
In  deep  caissons,  in  order  not  to  be  in  the  way  of  the  masons,  the  outer  planking  of  the  sides  may, 
in  part,  be  gradually  built  up  as  the  masonry  progresses.  It  may  sometimes  be  expedient  to  build 
the  masonry  hollow  at  first,  with  thin  transverse  walls  inside  to  stiffen  it  if  necessary  ;  and  to  com- 


FOUNDATIONS. 


317 


pletc  the  interior  after  sinking  the  caisson.  Indeed,  masonry  or  brickwork,  in  cement,  may  thus  be 
built  hollow  at  first,  resting  ou  the  platform;  the  masonry  itself  forming  the  sides  of  the  caisson. 
Or  the  sides  may  consist  of  a  water-tight  casing  of  iron,  or  wood,  of  the  shape  of  the  intended  pier, 
&c.  This  casing  being  confined  to  the  platform,  becomes,  in  fact,  a  mould,  in  which  the  pier  may  be 
formed,  and  sunk  at  the  same  time  by  filling  it  with  hydraulic  concrete.  For 
concrete  foundations,  see  p  507. 

On  rock  bottom  the  under  timbers  of  the  platform  may  be  cut  to  suit  the  irregularities 
as  already  stated  under  '•  Cribs."  Or  the  bottom  may  be  levelled  up  by  first  depositing  large  stones 
around  the  area  upon  which  the  caisson  is  to  rest ;  and  then  filling  between  these  with  smaller  stones 
and  gravel;  testing  the  depth  by  sounding.  Or  a  level  bed  of  cement  concrete  may,  with  care,  be 
deposited  in  the  water.  If  there  are  deep  narrow  crevices  in  the  rock,  through  which  the  concrete 
may  escape,  they  may  be  first  covered  with  tarpaulin.  Diving  bells  may  often  be  used  to  advantage 
in  all  such  operations.  But  in  the  case  of  very  irregular  rock,  it  will  often  be  better  to  resort  to  cof- 
fer-dams. The  draft  of  a  caisson  (the  depth  of  water  which  it  draws)  whether  empty  or  loaded  can 
be  found  by  Art  19  of  page  534.  Valves  for  the  admission  of  water  for  sinking  the  caisson  are 

usually  introduced.  If,  after  sinking,  it  should  be  necessary  to  again  raise  the  whole  it  is  onlv 
necessary  to  close  the  valves,  and  pump  ont  the  water.  Guide  piles  may  be  driven  and  braced  along- 
side of  the  caisson,  to  insure  its  sinking  vertically,  and  at  the  proper  spot.  Or  it  may  be  lowered  by 
screws  supported  by  strong  temporary  framework. 

Assuming  the  uprights  I,  T,  Ac.  Fig  6,  to.be  sufficiently  braced,  as  at  cc,  Pig  7,  the  following  table 
will  show  the  thickness  of  planking  necessary  for  different  distances  apart  of  the  uprights  (in  the 
clear,)  to  Insure  a  safety  of  six  against  the  pressure  of  the  water  at  different  depths;  and  at  the 
same  time  not  to  bend  inward  under  said  pressure,  more  than  ?  1^  part  of  the  distance  to  which 
they  stretch  from  upright  to  upright;  or  at  the  rate  of  J^  inch  in  10  ft  stretch  ;  \i  inch  in  5  ft,  &c. 
Such  a  table  may  be  of  use  m  other  matters. 

Table  of  thickness  of  white  pine  plank  reqnired  not  to  bend 
more  than  -£$-$  part  of  its  clear  horizontal  stretch,  under 
different  heads  of  water.  (Original.) 


Stretch 
in  Ft. 

3 
4 
6 
8 
10 
12 
15 
20 

40 

HEADS  IN  FEET. 

30      !        20      j        10 

5 

* 

Thicki 

j 

10 

15  * 

20 

less  in  I 

2% 
3}* 

7 
8% 
10% 
13 

aches. 
1% 

7 

14 

Jg 

6% 
11 

Coffer-dams  are  enclosures  from  which  the  water  may  he  pumped  out.  so  a* 
to  allow  the  work  to  be  done  in  the  open  air.  Their  construction  of  course  varies 
greatly.  In  still  shallow  water,  a  mere  well-built  bank  of  clay  and  gravel ;  or  of 
bags  partly  filled  with  those  materials  when  there  is  much  current,  will  answer 
every  purpose ;  or  (depending  on  the  depth)  a  single  or  double  row  of  sheet-piles ;  or  of 
squared  piles  of  larger  dimensions,  driven  touching  each  other;  their  lower  ends  a 
few  feet  in  the  soil ;  and  their  upper  ones  a  little  above  high  water,  and  protected 
outside  by  heaps  of  gravelly  soil  or  puddle,  (as  at  P  in  Fig  7,)  to  prevent  leaking. 
The  sheet-piles  may  be  of  wood;  or  of  cast  iron,  of  a  strong  form. 

The  sufficiency  of  a  mere  bank  of  well-packed  earth  in  still 
water,  is  shown  by  the  embankments  or  levees,  thrown  up  in  all  countries,  to  pre- 
vent rivers  from  overflowing  adjacent  low  lands.  The  general  average  of  the  levees 
along  700  miles  of  the  Mississippi,  is  about  6  ft  high ;  only  3  ft  wide  on  top ;  side- 
slopes  1%  to  1.  In  floods  the  river  rises  to  within  a  foot  or  less  of  their  tops  ;  and 
frequently  bursts  through  them,  doing  immense  damage.  They  are  entirely  too  slight. 

The  method  of  a  singlo  row  of  12  by  12  inch  squared  piles,  driven  in  contact  with 
each  other,  (close  pilfs,)  and  simply  backed  by  an  outer  deposit  of  impervious  soil, 
is  very  effective;  and  with  the  addition  of  interior  cross-braces  or  struts,  like  cc,  Fig 
7,  to  prevent  crushing  inward  by  the  outside  pressure  of  the  water  and  puddle  when 
pumped  out,  has  been  successfully  employed  in  from  20  to  25  ft  depth  of  water,  in 
which  there  was  not  sufficient  current  to  wash  away  the  puddle.  The  cross-braces 
are  inserted  successively,  as  the  water  is  being  pumped  out;  beginning,  of  course, 
with  the  upper  ones.  The  ends  of  these  braces  may  abut  on  longitudinal  timbers, 
bolted  to  the  piles  for  the  purpose.  Another  method  is  a  strong'  crib,  com- 
posed of  uprights  framed  into  caps  and  sills ;  and  covered  outside  with  squared 
timbers  or  plank,  laid  touching  each  other,  and  well  calked  ;  as  in  the  caisson,  Fig 
6 ;  but  without  a  bottom.  Between  the  opposite  pairs  of  uprights  are  strong  interior 
struts,  as  c  c,  Fig  7,  reaching  from  side  to  side,  to  prevent  crushing  inward.  The 

21 


318 


FOUNDATIONS. 


upper  series  of  these  usually  supports  a  platform  for  the  workmen,  windlasses,  &c. 
The  crib  having  been  built  on  land,  is  launched,  taken  to  its  final  place,  and  sunk  by 
piling  stones  on  a  temporary  platform  resting  on  the  cross-struts  ;  the  bottom  of  the 
stream  having  been  previously  levelled  off,  if  necessary,  for  its  reception. 

To  prevent  leaking  under  the  bottom  of  the  crib,  sheet-piles  may  be  driven  around  it,  their  heads 

as  shown  at  the  stone  deposits  t  t,  Fig  4.  Or  a  broad  flap  of  tarpauiiu  may  be  closely  nailed  around 
and  a  little  above  the  losver  edge  of  the  crib:  so  arranged  that  it  may  be  spread  out  loosely  ou  the 
river  bottom,  to  a  width  of  a  few  feet  all  around  the  outside  of  the  crib ;  and  the  puddle  may  be  placed 
upon  it.  Such  a  tarpaulin  is  also  very  useful  in  case  the  river  bottom  is  somewhat  irregular,  and 
cannot  be  levelled  off  without  too  great  expense;  in  which  case  the  crib  cannot  come  to  a  full  bearing 
upon  it;  and  consequently  the  water  would  leak  or  flow  beneath  freely.  It  is  especially  adapted  to 
uneven  rock  ;  where  sheet-piles  cannot  be  driven.  An  artificial  stratum  of  impervious  soil  may,  how- 
ever, be  deposited  on  bare  rock  :  in  which  case  the  sinking  of  the  crib,  and  the  subsequent  operations 
will  be  the  same  as  on  a  natural  stratum.  These  expedients  are  evidently  more  or  less  applicable  in. 
other  cases,  where,  to  avoid  repetition,  they  are  not  specially  mentioned. 


E.o-«7 


Ran  at  one  end. 

Fig:  7  Is  another  crib  coffer-dam;  in  which  the  sides,  instead  of  being 
planked  longitudinally,  as  in  the  last  instance,  are  sheathed  with  vertical  sheet-piles 
s,  driven  after  the  crib  is  sunk.  It  is  much  inferior  to  the  last,  owing  to  its  greater 
liability  to  leak.  In  one  of  this  description,  Fig  ?,  successfully  used  in  16  ft  water, 
the  dimensions  of  the  crib  were  34  ft  by  80  ft.  Along  each  long  side  were  7  uprights  1. 1, 
19  ft  long,  12  ins  square,  12%  ft  apart.  Into  each  opposite  pair  of  these  were  notched, 
and  held  by  dog-irons,  6  cross-braces  c  c,  of  12  ins  square.  The  distance  between  the 
two  upper  ones  was  3  ft  in  the  clear;  gradually  diminishing  to  18  ins  between  the 
two  lower  ones,  on  account  of  the/increased  pressure  of  the  water  in  descending  On 
the  outside  of  the  uprights,  and  opposite  the  ends  of  the  braces,  were  bolted  longi- 


SECTION. 


tudinal  timbers  to  support  the  outside  pressure  against  the  3-inch  sheet-piling  **. 
Other  longitudinal  pieces  o  o,  confine  the  heads  of  the  sheet-piles  to  the  top  of  the 
crib  after  they  are  driven.  The  feet  of  the  sheet-piles  were  cut  to  an  angle,  as  at  m ; 
to  make  them  draw  close  to  each  other  at  bottom  in  driving. 

The  sheet-piles  will  drive  in  a  far  more  regular  and  satisfactory  manner,  with  the 
arrangement  shown  In  Figs  8.    Here  o  o  are  the  uprights ;  c  c  arc  pairs  of  longitudinal 


FOUNDATIONS. 


319 


pieces,  notched  and  bolted  to  the  uprights,  near  both  their  tops  and  their  feet ;  and 
at  as  many  intermediate  points  as  may  be  desired.  The  sheet-piles  I,  are  inserted 
between  these ;  and  of  course  are  guided  during  their  descent  much  more  perfectly 
than  in  Fig  7.  The  crib  at  top  of  p  316  may  be  used  as  a  cofferdam. 

When  the  current  is  too  strong  to  permit  the  use  of  outside  puddle,  P,  Fig  7,  th« 
principle  of  coffer-dam  shown  in  Fig  9,  is  generally  used  ;  in  which  both  sides  of  the 
puddle  are  protected  from  washing  away.  '1  he  space  to  be  enclosed  by  the  dam  is  sur 
rounded  by  two  rows  of  firmly-driven  main  piles  p  p,  on  which  the  strength  chiefly 
depends.  They  may  be  round.  In  deciding  upon  their  number,  it  must  be  remem- 
bered that  they  may  have  to  resist  floating  ice,  or  accidental  blows  from  vessels,  &c. 
"With  reference  to  this,  extra  /e/'t^r-pihs  may  be  driven.  A  little  below  the  tops  of 
the  main  piles  are  bolted  two  outside  longitudinal  pieces  iv  w,  called  wales ;  and  oppo- 
site to  them  two  inner  ones,  as  in  the  fig.  The  outer  ones  serve  to  support  cross- 
timbers  1 1,  which  unite  each  pair  of  opposite  piles,  and  steady  them  ;  and  prevent 
their  spreading  apart  by  the  pressure  of  the  puddle  P.  The  inner  ones  act  as  guides 
for  the  sheet  piles  s  s,  while  being  driven ;  after  which  the  heads  of  the  theet-piles 
Are  spiked  to  them.  In  deep  water  these  sheet-piles  must  be  very  stout,  say  12  ins 
square  ;  to  resist  the  pressure  of  the  compacted  puddle. 

A  gangway  m,  is  often  laid  on  top  of  the  cross-pieces  1 1,  for  the  use  of  the 
workmen  in  wheeling  materials,  &c.  The  puddle  P  is  deposited  in  the  water  in  the 
space,  or  boxing,  between  the  sheet-piles.  It  should  be  put  in  in  layers,  and  com- 
pacted as  well  as  can  be  done  without  causing  the  sheet-piles  to  bulge,  and  thus  open 
their  joints.  The  bottom  of  the  puddle-ditch  should  be  deepened,  as  in  the  fig,  in 
case  it  consists,  as  it  often  does,  of  loose  porous  material  which  would  allow  water  to 
leak  in  beneath  it  and  the  sheet-piles.  This  leaking  under  the  dam  is  frequently  a 
source  of  much  trouble  and  expense.  Water  will  find  its  way  readily  through  almost 
any  depth  and  distance  of  clean  coarse  gravelly  and  pebbly  bottom,  unmixed  with 
earth.  Sand  is  also  troublesome;  and  if  a  stratum  of  either  should  present  itself  ex- 
tending to  a  great  depth,  it  will  generally  be  expedient  to  resort  to  either  simple 
cribs,  Fig  4;  or  to  caissons;  with  or  without  piles  in  either  case,  according  to  cir- 
cumstances. But  if  such  open  gravel,  or  any  other  permeable  or  shifting  material, 
as  soft  mud,  quicksand,  &c,  is  present  in  a  stratum  but  a  few  feet  in  thickness,  and 
underlaid  by  stiff  clay,  or  other  safe  material,  leaking  may  be  prevented,  or  at  least 
much  reduced,  by  driving  the  sheeting-piles  "2  or  3  ft  into  this  last ;  and  by  deepening 
the  puddle-trench  to  the  .same  extent.  It  may  sometimes  be  better,  and  more  con- 
venient, to  dredge  away  the  bad  material  entirely  from  all  the  space  to  be  enclosed 
by  the  dam,  and  for  a  short  distance  beyond,  before  commencing  the  construction  of 
the  latter.  If  the  dam,  Fig  9,  is  (as  it  should  be)  well  provided  with  cross-braces, 
like  c  c.  Fig  7,  extending  across  the  enclosed  area,  the  thickness  or  width  o  r>  of  the 
puddle,  need  not  be  more  than  4  or  5  feet  for  shallow  depths  ;  or  than  5  to  1 0  ft  for  great 
ones:  because  its  use  is  then  merely  to  prevent  leaking.  But  if  there  are  no  1  races, 
it  must  be  made  wider,  so  as  to  resist  upsetting  bodily;  and  then, with  good  puddle, 
o  o  may,  as  a  rule  of  thumb,  be  %  of  the  vertical  depth  o  /  below  high  water;  except 
when  "this  gives  less  than  4  ft;  in  which  case  make  it  4  ft;  unless  more  should  be 
required  for  the  use  of  the  workmen,  fordepositing  materials,  &c.  Or  if  the  excavation 
for  the  masonry  is  sunk  deeper  than  the  puddle,  the  dam  must  be  wider;  else  it  may 
be  upset  into  the  excavated  pit. 

Tlie   excavated    soil  may  be 

raised  in  buckets  by  windlasses,  or  by  hand,  in 
successive  stages.  The  pumps  may  be  worked 
by  hniid,  or  by  steam,  as  the  case  may  require; 
as  also  the  windlasses  generally  Deeded  for 
lowering  mortar,  stone.  &c.  More  or  less  leak- 
ing may  always  be  anticipated,  notwithstanding 
every  precaution. 

Where  a  coffer-dam  is  exposed  to  a  violent 
current,  and  great  danger  from  icp,  &c,  the  ex- 
pensive mode  shown  in  Figs  10  may  become 
necessarv.  The  two  black  rectangles  c  c.  repre- 
sent two  lines  of  rough  cribs  filled  with  stone, 
and  sunk  in  position;  one  row  being  enclosed 
by  the  other:  with  a  space  several  feet  wide  be- 
tween them.  Sheet-piles  p  p  are  then  driven 
around  the  opposite  faces  of  the  two  rows  of 
crihs :  and  the  puddle  is  deposited  within  the  boxing  thus  provided  for  it,  as  shown  in  the  fig. 

Where  the  current  is  not  strong  enouch  to  wash  away  gravel  bnckine,  we  may.  on  rock  especially, 
enclose  the  space  to  he  built  on.  by  a  siriirle  qnadrnngle  of  cribs  sunk  by  stone;  nnd  after  adopting 
precautions  to  prevent  the  gravel  from  being  pressed  in  beneath  the  cribsi  apply  the  backing.* 

Fi.a;s  10^  show  the  plan,  outside  view,  and  transverse  section,  to  a  scale  of  20  ft  to 
an  inch,  of  a  coffer  dam  on  rock,  in  8  to  9  ft  water,  used  successfully  on  the  Schuylkill 
Navigation. 


PLAN 


*  A  pure  clean  coarse  gravel  is  entirely  unfit  for  such  purposes.     A  considerable  proportion  of 
«arth  is  essftntia)  for  preventing  leaks.    For  another  crib-oonferdam.  s*e  p  316. 


320 


FOUNDATIONS. 


Its  construction  Is  very  simple.  Uprights  5,  about  1  ft  square,  and  10  ft  apart  from  center  to  center 
along  the  sides  of  the  dam ;  and  10  ft  in  the  clear,  transversely  of  the  dam,  support  two  lines  of  hori- 
zontal stringers,  i  i;  inside  of  which  are  the  two  lines  of  sheeting- piles,  x  s,  enclosing  between  them 
a  width  of  7  ft  of  gravel  puddle.  Two  flat  iron  bars  (t  t,  of  the  transverse  section)  tie  together  each 
pair  of  uprights  6  6.  These  bars  are  %  inch  thick,  by  '2^  ins  deep,  and  9  It  long.  Their  hooked  ends 
fit  into  eye-bolts  c,  which  pass  turougu  the  uprights  b;  outside  of  which  they  are  fastened  by  keys,  A:, 
'.see  detail  sketch.)  Between  the  keys  aud  6,  were  washers.  At  the  corners  of  the  dam  (see  plan) 
•were  additional  tie-bars,  as  shown.  A  small  band  of  straw,  as  seen  at  y,  wrapped  around  the  tie- 
bars  just  inside  of  the  sheet-piles  ;  and  kept  in  place  by  the  puddle  ;  effectually  prevented  the  leaking 
which  generally  proves  so  troublesome  ia  such  cases  The  stout  oblique  braces,  o  o,  were  merely 
spiked  to  the  outside  faces  of  the  uprights  6.  They  are  not  shown  in  the  transverse  section.  This  dam 
was  built  on  shore;  in  sections  30  to  40  ft  long.  These  were  floated  into  place,  and  weitrhteddown, 
sheet-piled,  and  puddled  with  gravel.  The  dam  had  sluices  by  which  water  was  admitted  when 
necessary  for  preventing  the  outside  head  from  exceeding  9  ft.  The  lengths  of  the  uprights  6  b  were 
first  found  by  careful  soundings. 


,    TR.SEC 
b  b 


OUTSIDE. 

b  b 


PLAN 


Valuable  hints  for  coffer-dams  may  be  taken  from  what  is  said  under  the  head  of 
"  Dams,"  where  Fig  I  affords  useful  suggestions  for  coffer-dams  also,  on  rock  in  shallow 
•water;  p  583. 

The  mooring  of  larjje  caissons  or  cribs,  preparatory  to  sinking 
them,  is  sometimes  troublesome,  especially  in  strong  currents.  It  may  be  neces- 
sary to  drive  clumps  of  piles;  or  to  temporarily  sink  rough  cribs  filled  with  stone, 
to  which  to  attach  the  long  guide-ropes  by  which  the  manoeuvring  into  position,  &c, 
is  done.  Frequently  dams  are  left  standing  after  the  work  is  done  ;  if  not  in  the  way 
of  navigation,  or  otherwise  objectionable ;  inasmuch  as  the  materials  are  rarely  worth 
the  expense  of  removal.  But  if  removed,  the  piles  should  not  be  drawn  out  of  the 
ground ;  but  be  cut  off  close  to  river  bottom ;  for  if  drawn,  the  water  entering  their 
holes  may  soften  the  soil  under  the  masonry.  It  is  often  expedient  to  drive  two 
rows  of  piles  from  the  dam  to  the  shore,  for  supporting  a  gangway  for  the  workmen  ; 
or  even  for  horses  and  carts ;  or  for  a  railway  for  the  easy  delivery  of  large  stones,  &c. 

Coffer-dams  may  be  sunk  through  a  soft  to  a  firm  soil,  in 
shape  of  a  box  of  cribwork,  either  rectangular  or  circular,  and  without  a  bottom. 
This  being  strongly  put  together,  and  provided  with  proper  temporary  internal 
bra -ing,  (to  be  gradually  removed  as  the  masonry  is  built  up,)  is  floated  into  place; 
and  after  being  loaded  so  as  to  rest  on  the  soft  bottom,  is  sunk  by  dredging  out  the 
soft  material  from  inside.  Additional  loading  will  sometimes  be  required  for  over- 
>iiiin^  the  friction  of  the  soil  against  the  outside;  or  it  may  even  become  necessary 
hi  drudge  away  some  of  the  outer  material  also.  On  rock  it  may  at  times  be 
expedient  to  drill  holes  in  deep  water,  for  receiving  the  ends  of  piles,  or  of  iron  rods, 
&c.  This  may  be  done  by  means  of  long  drill-rods,  working  in  an  iron  tube  or  pipe 
sunk  as  a  guide  to  the  rod ;  with  its  lower  end  over  the  spot  to  be  bored  Or  a  diving- 
bell  may  be  used.  Or  a  cylinder  of  staves  4  to  12  inches  thick,  long  enough  to 
reach  above  the  surface,  and  having  abroad  tarpaulin  flap  or  apron  around  its  lower 
edge,  to  be  covered  with  gravel  to  prevent  leaking:  maybe  sunk,  and  the  water 
pumped  out,  to  allow  a  workman  to  descend,  and  work  in  the  open  air. 

Piles.  When  driven  in  close  contact. as  on  Fig  11,  for  preventing  leakage;  for 
confining  puddle  in  a  coffer-dam:  or  for  enclosing  a  piece  of  soft  or  sandy  ground,  to 
prevent  its  spreading  when  loaded ;  or  if  the  outside  soil  should  wash  away  from 

COSt  Of  piles  delivered  at  wharf.  Philada.  1873.  Hemlock,  6  to  8  ct«  per  foot  lineal,  Bay 
yellow  pine,  10  to  15  cts,  Southern  yellow  pin,.-,  18  to  25  cts. 


FOUNDATIONS. 


321 


around  them,  &c,  they  are  called  sheet-piles.  Generally  these  are  thinner  than 
they  are  wide ;  but  fre- 
quently they  are  square; 
and  as  large  as  bearing 
piles;  and  are  then  called 
close  piles.  To  make 
them  drive  tight  to- 
gether at  foot,  they 
are  cut  obliquely  as  at 
/.  Occasionally,  when 
driven  down  to  rock 
through  soft  soil,  their 
feet  are  in  addition  cut 
to  an  edge,  as  at  r,  so  as 
to  become  somewhat 
bruised  when  they  reach 
the  rock,  and  thus  fit  closer  to  its  surface.  Their  heads  are  kept  in  line  while  driv- 
ing, by  means  of  either  one  or  two  longitudinal  pieces  a  and  o,  called  wales  or 
stringers.  These  wales  are  supported  by  gauge-piles,  or  guide-piles,  previously  driven 
1:1  the  required  line  of  the  work,  and  several  ft  apart,  lor  this  purpose.  See  Figs  8. 

A  dog-iron  d,  of  round  iron,  may  also  be  used  for  keeping 
the  edges  of  the  piles  close  at  top  to  those  previously  driven  both 
during  and  after  the  driving.  Its  sharp  ends,  cc,  being  driven  into 
the  tops  of  the  wales  ww,  (shown  in  plan,)  it  holds  the  descending 
pile  o  firmly  m  place.  A  t  n,  d,  p,  Fig  11,  are  other  modes  occasionally 
used  for  keeping  the  piles  in  proper  line.  At^>,  the  letters  s  s  denote 
small  pieces  of  iron  well  screwed  to  the  piles,  a  little  above  their  feet 
to  act  as  guides;  very  rarely  used.  At  m  are  shown  wooden  tongues 
tt,  sometimes  driven  down  between  the  piles  after  they  themselves 
have  been  driven ;  to  assist  in  preventing  leaks.  In  some  cases 
sheet-piles  are  employed  without  being  driven.  A  trench  is  first 
dug  to  their  full  depth  for  receiving  them  ;  and  the  piles  are  simply 

placed  m  these,  which  are  then  refilled.  Closer  joints  can  be  secured  in  this  manner  than  by  driving. 
^  When  piles  are  intended  to  sustain  loads  on  their  tops,  whether  driven  all  their 
length  into  the  ground,  or  only  partly  so,  as  in  Fig  3,  they  are  called  bearing 
P  •*?:  1lheyiai'e  generally "round;  from  9  to  ]8  ins  diam  at  top;  and  should  be, 
straight,  but  the  bark  need  not  be  jemoved.  White  pine,  spruce,  or  even  hem- 
lock, answer  very  well  in  soft  soils  ;  good  yellow  pine  for  firmer  ones;  and  hard 
oaks,  elm,  beach,  &c,  for  the  more  compact  ones.  They  are  usually  driven  from  about 
2%  to  4  ft  apart  each  way,  from  center  to  center,  depending  on  the  character  of  the 
soil,  and  the  weight  to  be  sustained.  A  tread-wheel  is  more  economical  than 
the  winch  for  raising  the  hammer,  when  this  is  done  by  men.  Morin  found  that 
the  work  performed  by  men  working  8  hours  per  day,  was  3900  foot-pounds  per  man 
per  minute  by  the  tread-wheel ;  and  only  2600  by  a  winch. 


The  gunpowder  pile  driver  invented  by 
mechanical  engineer  of  Philada,  is  a  very  meritorious  machi 
' 


the  well-known 


,    ..  >rked  bv  small 

cartridges  of  powder,  placed  one  by  one  in  a  receptacle  on  top  of  the  pile;  and  exploded  by  the  ham- 
mer itself.  It  can  readily  make  30  to  40  blows  of  5  to  10  ft,  per  minute  ;  and,  since  the  hammer  does 
not  come  into  actual  contact  with  the  piles,  it  does  not  injure  their  heads  at  all ;  thus  dispensing 
with  iron  hoops  <fec,  for  preserving  them.  When  only  a  slight  blow  is  required,  a  smaller  cartridge 
is  used.  This  machine,  however,  does  not  assist  in  raising  the  pile,  and  placing  it  in  position,  as  is 
done  by  ordinary  steam  pile  drivers  ;  the  latter,  however,  average  but  from  3  to  6  blows  per  minute. 

The  forcfe  ill  Ibs  with  which  a  pile-hammer  strikes  the  head  of  a  pile,  can- 
not be  calculated.  All  rules  for  that  purpose  are  founded  in  error.  We  must 
here  depend  upon  experience  alone.  But,  unfortunately  the  recorded  facts  are  so 
few,  and  so  incompletely  described,  as  not  to  furnish  grounds  for  reliable  rules.  We 
however  have  rules  by  Maj  Sanders,  and  others:  and  to  these  we  venture  to  add  one 
of  our  own,  which  is  also  purely  empirical.  Mol<-s\vorth\s  rule  is  Sanders. 

They  differ  very  much.  No  rule  can  apply  correctly  to  all  conditions.  The  ground  itself  between 
the  piles,  in  most  cases,  supports  a  part  of  the  load ;  although  the  whole  of  it  is  usually  assigned  to 
the  piles.  Again,  in  very  clayey  soils,  there  is  greater  liability  to  sink  somewhat  with  the  lapse  of 
time,  in  consequence  of  the  admission  of  water  between  the  pile  and  the  clay;  thus  diminishing  the 
friction  between  them.  The  les«  firm  the  soil,  the  more  will  the  piles  be  affected  by  tremors  ;  which 
also  tend  in  time  to  cause  sinking.  In  some  cnses  this  sinking  will  not  be  that  of"  the  piles  settling 
deeper  into  the  earth  around  them  ;  but  of  that  of  the  entire  compacted  mass  of  piles  and  earth  into 
which  they  were  driven,  settling  down  into  the  less  dense  mass  below  them.  Squared  piles,  and  ta- 
pering round  ones,  will  not  bear  equal  loads.  Piles  are  sometimes  blamed  for  settlements  which  are 
reallv  due  to  the  crushing  (flatways)  of  the  timbers  which  rest  immediately  upon  their  heads. 

In  the  fine  London  bridge  across  the  Thames,  each  pile  under  some  of  the 
piers  sustains  the  very  heavy  load  of  80  tons.  They  are  driven  but  20  feet  into  the 
stiff  blue  London  clay  ;  and  are  placed  nearly  4  ft  apart  from  center  to  center ;  which 

»  For  cost  and  performance,  see  foot  of  page  323. 


322  FOUNDATIONS. 

is  too  much  for  such  piers  and  arches.  At  3  ft  apart  scant,  they  would  have  had  hut 
45  tons  to  sustain.  They  are  1  ft  in  diam  at  the  middle  of  their  length.  Ugly  set- 
tlements some  of  them  to  the  extent  of  about  a  ft,  have  occurred  under  these  piers. 
Blacltfriarm  bridge,  in  the  same  vicinity,  exhibits  the  same  defect.  By  some 
this  is  ascribed  in  both  cases  to  the  gradual  admission  of  water  between  the  clay  and 
the  piles,  perhaps  by  capillary  action  of  the  piles  themselves;  or  perhaps  by  direct 
leaking.  It  may,  however,  be  owing  in  part  to  the  crushing  of  the  platforms  on 
top  of  the  piles;  or  to  a  bodily  settlement  of  the  entire  mass  of  piled  clay,  into 
the  unpiled  clay  beneath,  under  the  immense  load  that  rests  upon  it.  This  here 
amounts  to  5  %  tons  per  sq  foot  of  area  covered  by  a  pier  ;  and  is  probably  too  much 
to  trust  upon  damp  clay,  when  even  the  slightest  sinking  is  prejudicial. 

Maj  J.  Sanders,  U.  S.  Engs,  experimented  largely  at  Fort  Delaware  in  river 
mud  ;  and  gave  the  following  in  the  Jour.  Franklin  Inst,  Nov  1851.  For  the  sake 
load  for  a  common  wooden  pile,  driven  until  it  sinks  through  only  small  and 
nearly  equal  distances,  under  successive  blows,  divide  the  height  of  the  fall  in  ins, 
by  the  small  sinking  at  each  blow  in  ins.  Mult  the  quot  by  the  weight  of  the 
hammer,  ram,  or  monkey,  in  tons  or  pounds,  as  the  case  may  be.  Divide  the 
prod  by  8.  He  does  not  state  any  specific  coefficient  of  safety. 

Example.  At  the  Chestnut  8t  Bridge,  Philada,  the  greatest  weight  on  any  pile  is  18  tons. 
Mr  Kiieass  had  the  piles  driven  uut.il  they  sank  %,  or  .75  of  an  inch  under  each  blow  from  a  1200  B 
hammer,  falling  20  ft.  Was  he  safe  in  doing  so  ?  Here  we  have  the  fall  in  ins  =  20  X  12  -  240.  And 

240  384000 

-jij-  =  3-20  ;  and  320  X  1200  =  384000  B>s  ;  and  —  g  -  =  48000  fts,  =  21.4  tons  safe  load  by  Maj  San- 

ders'  rule.     The  soil  was  river  mud. 

Our  own  rule  is  as  follows.  Mult  together  the  cube  rt  of  the  fall  in  ft  ;  the  wt  of  hammer  in  fts  ; 
and  the  decimal  .023.  Divide  the  prod  by  the  last  sinking  in  ins.  -f-  1.  The  quotient  will  be  the 
extreme  load  that  will  be  just,  at  the  point  of  causing  more  sinking.  For  the  safe  load  take  from 
one  twelfth  to  one  half  of  this,  according  to  circumstances.  Or,  as  a  formula, 

Cube  rt  of  v  Wt  of  hammer  v   02, 
Extreme  load  __  fall  in  feet  *      in  pounds 

i»  tous  Last  sinking  in  inches  +  1 

Example.  The  same  as  the  foregoing  at  Chestnut  St  Bridge.  Here  the  cube  rt  of  20  ft  fall  it 
2.714  ft.  Hence  we  have 


Extreme  M  =  MMXIWXJW  =  „  , 

A  in  tons  .75  -|-  1  1.7o 

Or  say  half  of  tliis,  or  21.4  tons,  the  load  for  a  safety^>f  2.  Major  Sanders'  rule  makes  the  safe 
load  21.4.  The  actual  one  is  18  tons. 

A  safety  of  '2  is  not  enough  for  river  mud.  See  "  Proper  load  for  safety,"  below. 
But  although  Major  Sanders'  rule  and  our  own  agree  very  well  in  this  instance  if  a  safety  of  2  Z>« 
taken  for  each,  they  differ  widely  in  some  others.  Thus  at  Neullly  Bridge,  France,  Perronet's 
heaviest  hammer  weighed  2000  fts,  fall  5  ft,  sinkage  .25  of  an  iucli  in  the  lust  16  blows;  or  say  .016 
inch  per  blow.  The  piles  sustain  47  tons  each.  Our  rule  gives  38.8  tons  for  a  safety  of  2  ;  while  San- 
ders' rule  gives  515  tons  safe  load  !  If,  as  we  think  probable,  there  was  no  actual  sinking  at  the  last 
blow,  then  onr  rule  gives  39.3  tons  for  a  safetv  of  2  ;  while  Sanders'  gives  0. 

At  the  Hull  Docks,  England,  piles  10  ins'square.  driven  16  ft  into  alluvial  mud,  by  a  1500  ft  ham- 
mer.  falling  24  ft,  sank  2  ins  per  blow  at  the  end  of  the  driving.  They  sustain  at  least  20  tons  each, 
or  according  to  some  statements  25  tons.  Our  rule  gives  33.2  tons  for  the  extreme  load  ;  or  16.6  for  a 
safety  of  only  2.  Sanders  gives  for  safety  12.06  tons.  As  before  remarked,  2  is  not  safetv  enough  for 
mud.  In  mud,  it  is  not  primarily  the  piles,  but  the  piled  soil  that  settles,  bodily,  for  vears. 

At  the  Royal  Border  Bridge,  England,  piles  were  very  firmly  driven  from  30  to  40  ft  in  sand 
and  gravel,  iu  some  cases  wet.  Pine  was  first  tried,  but  it  split  and  broomed  so  badly  under  the  hard 
driving,  that  American  elm  was  substituted,  with  success.  They  were  driven  until  they  sank  but  .05 
inch  per  blow,  under  a  1700  Ib  monkey,  falling  16  ft.  They  support  70  tons  each.  Our  rule  gives  47 
tons  for  a  safety  of  2  ;  while  Sanders"  gives  364  tons  safe  load  1 

It  is  the  writer's  opinion,  however,  that  the  piles  did  not  nciually  sink,  as  was  (and  always  is,  in 
such  cases)  taken  for  granted  by  the  observers  ;  but  that  they  were  merely  compressed  or  partially 
crushed  by  overdriving.  Most  of  the  piles  were  driven  until  they  sank  (?)  only  an  inch  under  150 
blows  ;  but  we  doubt  whether  they  were  any  safer,  or  farther  in  the  ground,  thnn  when  they  had  re- 
ceived only  one  of  them  ;  and  consider  such  extreme  precaution  worse  than  useless. 

In  some  experiments  (1873)  at  Philada,  a  trial  pile  was  driven  15  ft  into  soft  river  mud,  by  a 
1600  tt>  hammer  ;  its  last  sinking  being  18  ins  under  a  fall  of  36  ft.  Only  5  hours  after  it  was  driven 
it  was  loaded  with  65  tons  ;  which  caused  a  sinking  of  but  a  very  small  fraction  of  an  inch.  Our  rule 
gives  6.4  tons  as  the  extreme  load.  Under  9  tons  it  sank  .75  of  an  inch  ;  and  under  15  tons,  5  ft.  By 
Miij  Sanders'  rule  its  safe  load  would  be  2.14  tons. 

A  U.  8.  Govt  trial  pile,  about  12  ins  sq,  driven  29  ft  through  layers  of  silt,  sand,  and  clay,  ham- 
mer y  10  fbs,  fall  5  ft,  last  sinking  .375  of  an  inch,  bore  26.6  tons  ;  but  sank  slowly  under  27.9  tons. 
Our  rule  gives  26  tons  extreme  load. 

French  engineers  consider  a  pile  safe  for  a  load  of  25  tons,  when  it  is  driven  to  the  refusal  of 
1344  fts,  falling  4  ft  ;  our  rule  gives  24.2  tons  for  safety  2.  They  estimate  the  refusal  by  its  not  sink- 
ing  more  than  .4  of  an  inch  under  30  blows.  In  many  important  bridges  &c  they  drive  until  there  is 
no  sinking  under  an  800  R>  hammer,  falling  5  ft.  Our  rule  here  gives  31.5  tons  extreme  load  ;  or  15.7 
for  safety  2. 

As  to  the  proper  load  for  safety,  we  think  that  not  more  than  one-half  the  extreme  load  given 
by  our  rule  should  be  taken  for  piles  thoroughly  driven  infirm  soils;  nor  more  than  one-sixth  when 
in  liver  mud  or  marsh  ;  assuming,  as  we  have  hitherto  done,  that  their  feet  do  not  rest  upon  rock. 
If  liable  to  tremors,  take  only  half  these  load*. 


FOUNDATIONS. 


Piles  may  be  made  of  any  required  size  as  regards  either  length  or  cross  section,  by  bolt- 
ing and  fishing  together  side  wise  and  length  wise,  a  number  of  squared  timbers. 
Piles  with  blunt  ends.     At  South  Street  Bridge,  Phila,  1200  stout  piles  of  Nova  Scotia  spruce 

total  cost  (piles  and  driving)  of  $7  to  $8  each.  At  Wilmington  Harbor,  Cal,  Mr.  C.  B.  Sears, 
U.  S.  Army.  (Jour.  Am.  Soc.  C.  E.,  Dec  1876)  found  that  m  firm  compact  ^et  smud,  after  the  first  few 
blows  the  piles  would  not  penetrate  moie  than  .5  to  1.5  ins  at  a  blow,  no  natter  how  far  the  2400  ft 
hammer  fell.  The  unpointed  ones  of  which  there  were  many  thousands,  drove  quite  as  readily  to  aver- 
age depths  of  15  ft  in  this  sand  as  the  pointed  ones,  and  with  much  less  tendency  to  cant.  As  a  high 
fall  had  no  farther  effect  than  to  batter  the  heads  he  reduced  it  to  10  ft.  which  drove  an  average  of 
nbnnt  .72  inch  to  a  blow.  To  insure  straight  driving,  the  ends  must  be  at  right  angles  to  the  length. 
Instead  of  driving  piles  to  moderate  depths  it,  mav  at  times  be  better  to  mereh  plant  them  hutt 
down  in  holes  bored  by  an  auger  like  Pierce's  Well  Borer.  See  p  636.  This'will  avoid  shaking 
adjacent  buildings.  See  "  In  Mobile  Bay,"  p  325. 

The  ultimate  friction  of  piles  even  with  the  bark  on,  and  driven  about  3  ft  apart  from  cen 
to  cen  probably  never  much  exceeds  about  1  ton  per  sq  ft  even  when  well  driven  into  dense  moist 
sand  or  loamy  gravel ;  nor  more  than  .5  to  .75  of  a  ton  in  common  soils  and  clays;  or  than  .1  to  .'2 
of  a  ton  in  silt  or  wet  river  mud  depending  on  the  depth  and  density. 

The  friction  of  cast  iron  cylinders  seems  to  be  about  .3  that  of  piles. 

There  is  a  great  difference  in  the  penetrability  of  different 

•ands.  Thus,  in  the  Lary  bridge,  no  special  uitticulty  was  found  in  driving  piles  ;io  ft  into  deep  wet 
sand  :  while,  in  other  wet  localities,  piles  of  very  toi.gh  wood,  well  shod  with  iron,  cannot  be  driven 
6  ft  into  sand,  without  being  battered  to  pieces.  The  same  difference  has  been  found  in  the  case  of 
screw-piles.  At  the  Brandywine  light-house  these  could  not  be  forced  more  than  10  ft  into  the  clean 
wet  sand.  Stiff  wet  clay  (and  clean  gravels)  also  differ  very  much  in  this  respect.  Generally  they 
are  penetrable  to  any  required  depth  with  comparative  ease  :  but.  we  have  seen  stout  hemlock  piles 
battered  to  pieces  iti  driving  6  ft  through  wet  gravel;  and  Mr.  Rendel  found  that  at  Plymouth  he 
"could  not  by  any  force  drive  screw-piles  more  than  about  5  ft  into  the  clay,  which  is  not  ILK  stiff  as 
the  London  clay,"  on  which  the  foremen tioned  new  London  and  Blackfriars  bridges  were  lounded; 
and  into  which'even  ordinary  wooden  piles  were  driven  20  ft  without  special  difficulty. 
A  mixture  of  mud  with  the  sand  or  gravel  facilitates  driving  very  much  ;  but  before  beginning  aii 

trouble  and  expense  that  may  be  anticipated.    Mere  boring  will  often  be  but  a  poor  substitute  for  this. 

As  a  general  rule,  a  heavy  hammer  with  a  low  fall,  drives  more  pleasantly  than  a  light  one  with  a 
high  fall.  Where  a  hammer  of  %  ton  (1500  fi>s)  falling  25  ft,  in  a  very  strong  ground,  shattered  the 
piles;  one  of  2  tons,  (4500  Ibs,)  with  7  ft  fall,  drove  them  satisfactorily.  More  blows  can  be  made  in 
the  same  time  with  a  low  fall;  and  this  gives  less  time  for  the  soil  to  compact  itself  around  the  piles 
between  the  bkrtvs.  At  times  a  pile  may  resist  the  hammer  after  sinking  some  distance;  but  start 
again  after  a  short  rest;  or  it  may  refuse  a  heavy  hammer,  and  start  under  a  lighter  one.  It  may 
drive  slowly  at  first,  and  more  rapidly  afterward,  from  causes  that  may  be  difficult  to  discover.  The 
driving  of  one  sometimes  causes  adjacent  ones  previously  driven,  to  spring  upward  several  feet.  A 
pile  is  in  the  most  favorable  position  when  its  foot  rests  upon  rock,  after  its  entire  length  has  been 
driven  through  a  firm  soil,  which  affords  perfect  protection  against  its  bending  like  an  overloaded 
column;  and  at  the  same  time  creates  great  friction  against  its  sides;  thus  as^sting  much  in  sus- 
taining the  load,  and  thereby  relieving  the  pressure  upon  the  foot.  A  pile  may  rest  upon  rock,  and 
yet  be  very  weak  ;  for  if  driven  through  very  soft  soil,  all  the  pressure  is  borne  by  the  sharp  point ; 
and  the  pile  becomes  merely  a  column  in  a  worse  condition  than  a  pillar  with  one  rounded  end.  See 
Fig  I,  page  233,  Strength  of  Iron  Pillars.  In  such  soils  the  piles  need  very  little  sharpening ;  indeed, 
had  better  be  driven  without  any ;  or  even  butt  end  down. 

The  driving  of  a  pile  in  soft  ground  or  mud  will  generally  cause  an  adjacent  one  previously  driven, 
to  lean  outwards  unless  means  be  taken  to  prevent  it. 

In  piling  an  area  of  firm  soil  it  is  best  to  begin  at  its  center  and  work  outwards;  otherwise  the  soil 
may  become  so  consolidated  that  the  central  ones  can  scarcely  be  driven  at  all. 

Elastic  reaction  of  the  soil  has  been  known  to  cause  entire  piled  areas 
to  rise,  together  with  the  piles,  before  they  were  built  upon. 

In  very  firm  soil,  especially  if  stony;  or 
even  in  soft  soil,  if  the  piles  are  pointed,  and 
are  to  be  driven  to  rock  ;  tlieir  feet  should 
be  protected  by  shoes  of  either  wrought 
iron,  as  at  a,  a,  and  /;,  Figs  13 ;  spiked  to  the 
pile  by  means  of  the  iron  straps  r?,  forged 
to  them;  or  of  cast  iron,  as  at  c,  where  the 
shoe  is  a  solid  inverted  cone,  the  wide  flat 
base  of  which  affords  a  good  bearing  for  the 
flat  bottom  of  the  pile-point.  The  dotted 
line  is  a  stout  wrought-iron  spike,  well  se- 
cured in  the  cone,  which  is  cast  around  it; 
this  holdd  the  shoe  to  the  pile.  Regular 


Fit/13 


The  cost  of  a  floating  steam  pile  driver,  in  Phiiada,  scow  2*  ft  by  50  ft. 

draft  18  ins,  with  one  engine  for  driving,  and  one  (to  save  time)  for  getting  another  pile  ready  ;  with 
one  ton  hammer,  is  about  $6000;  and  $500  more  will  add  a  circular  saw.  Ac.  for  sawing  off  piles  at  any 
reqd  depth.  Requires  engineman,  cook,  and  4  or  5  others.  Will  burn  about  half  a  ton  of  coal  per  day. 
Driving  20  ft  into  gravel,  and  sawing  off.  will  average  from  15  to  '20  piles  per  day  of  10  hours.  In 
mud  about  twice  as  many.  Ou  land  about  half  as  many  as  in  water.  A  gunpowder  driver,  scow,  Ac, 
costs  about  $1500  more;  but  will  do  about  twice  as  much  work  as  a  steam  driver.  To  drive  a  pile 
20  ft  into  mud  averages  about  one  third  ft>  of  powder:  into  gravel  4  times  as  much.  One  for  use  on 
land  $1000  to  $3000.  The  U.  S.  Gunpowder  Pile  Driving  Co.  No.  10  S.  Delaware  Avenue,  Pailada, 
contract  for  both  kinds  of  machines ;  also  for  dredging  and  piling. 


324 


FOUNDATIONS. 


wrought-iron  shoes  will  generally  weigh  18  to  30  l%s ;  but  sheet  iron  may  be  used  when  the  soil  Is  but 
moderately  compact;  plate  iron  when  more  so  ;  and  solid  iron  or  steel  points,  from  2  to  4  ins  square 
at  the  butt,  and  4  to  8  ins  long,  when  very  compact  and  stony.  Holes  may  be 
drilled  in  rock  for  receiving  the  points  of  piles,  and  thus  preventing  them 
from  slipping  ;  by  first  driving  down  a  tube,  as  a  guide  to  the  drill,  after  the  earth  is  cleaned  out  of 
the  tube.  To  preserve  the  heads  to  some  extent  from  splitting  under  the 
blows  of  the  hammer,  they  are  usually  surrounded  by  a  hoop  h,  Fig  d ;  from  %  to  1  inch  thick  ;  and 
l?i  to  3  ins  wide.  These  are,  however,  sometimes  but  imperfect  aids  ;  for  in  hard  driving  the  head 
will  crush,  split,  and  bulge  out  on  all  sides,  frequently  for  many  feet  below  the  hoop:  moreover,  the 
hoops  often  split  open.  The  heads,  therefore,  often  have  to  be  sawed,  or  pared  off  several  times 
before  the  pile  is  completely  driven;  and  allowance  must  be  made  for  this  loss  in  ordering  piles  for 
any  giveu  work;  especially  in  hard  soil.  Capt  Turnbull,  U  S  Top  Eng,  states  that  at  the  Potomac 
aqueduct,  his  pileheads  were  preserved  from  injury  by  the  simple  expedient  of  dishing  them  out  to  a 
depth  of  about  an  inch,  and  covering  them  by  a  loose  plate  of  sheet  iron ;  as  shown  in  section  at  c, 
Figs  13.  A  verv  slight  degree  of  brooming  or  crushing  of  the  head,  materially  diminishes  the  force 
of  the  ram.  Piles  may  be  driven  through  small  loose  rubble  without  much  labor.  Shaw's  driver 
does  not  injure  the  heads.  Piles  which  foot  on  sloping  rock  may  slide  when  loaded. 

To  drive  a  pile  head  below  water  a  wooden  punch,  or  follower,  as 

atp,  Figs  13,  may  be  used.  The  foot  of  this  punch  fits  into  the  upper  part  of  a  casting  //.  round  or 
square,  according  to  the  shape  of  the  pile;  and  having  a  transverse  partition  o  o.  The  lower  part 
«»f  the  casting  is  fitted  to  the  head  of  the  pile  t;  and  the  hammer  falls  on  top  of  the  punch.  AVhea 
driving  piles  vertically  in  very  soft  soil,  to  support  retainiug-walls,  or  other  structures  exposed  to 
horizontal  or  inclined  forces,  care  must  be  taken  that  these  forces  do  not  pu>h  over  the  piles  them- 
selves ;  for  in  such  soils  piles  are  adapted  to  resist  vertical  forces  only,  unless  they  be  driven  at  an 
inclination  corresponding  to  the  oblique  force. 

A  broken  pile  may  be  drawn  out,  or  at  least  be  started,  if  not  very 

firmly  driven,  by  attaching  scows  to  it  at  low  water,  depending  on  the  rising  tide  to  loosen  it.  Or  a 
long  timber  may  be  used  as  a  lever,  with  the  head  of  an  adjacent  pile  for  its  fulcrum.  Or  a  crab 
worked  bv  the  engine  of  the  pile  driver.  In  very  difficult  cases  the  method  devised  by  Mr  J.  Monroe, 
C  E,  may"  be  used.  A  4  inch  gas  pipe  15  ft  long,  shod  with  a  solid  steel  point,  and  having  an  outer 
shoulder  for  sustaining  a  circular  punch,  was  thereby  driven  close  to  and  2  or  3  ft  deeper  than  two 

Rilea  driven  12  ft,  in  37  ft  water,  and  broken  off  by  ice.  Four  pounds  of  powder  were  then  deposited 
i  the  lower  end  of  the  pipe,  and  exploded,  lifting  the  piles  completely  out  of  place.  It  will  often  be 
best  to  let  a  broken  pile  remain,  and  to  drive  another  close  to  it.  May  be  drawn  by  hydraulic  press. 

Ice  adheres  to  piles  with  a  force  of  about  30  to  40  Iba  per  sq  inch,  and  iii 

rising  water  may  lift  them  out  of  place  if  not  sufficiently  driven. 

Iron  piles  and  cylinders.    Cast  iron  in  various  shapes  has  been  much 

used  in  Europe  for  sheet  piles;  especially  when  intended  to  remain  as  a  facing  for  the  protection  of 
concrete  work,  filled  in  behind  and  against  them.*  Cast-iron  cylinders,  open  at  both  ends,  may  be 
used  as  bearing  piles ;  and  may  be  cleaned  out,  and  filled  with  concrete,  if  required.  The  friction  in 
driving  is  greater  than  in  solid  piles,  inasmuch  as  it  takes  place  along  both  the  inner  and  the  outer 
surfaces.  This  may  be  diminished  by  gradually  extracting  the  inside  soil  as  they  go  down.  Thev 
require  much  care,  and  a  lighter  hammer,  or  less  fall  than  wooden  ones,  to  prevent  breaking;  to 
which  end  a  piece  of  wood  should  be  interposed  between  the  hammer  and  the  pile ;  or  the  ram  may  be 
of  wood.  But  it  is  better  to  use  them  in  the  shape  of  screw  cylinders,  which, 
moreover,  gives  them  the  advantage  of  a  broad  base,  as  in  the  following.  See  foot  of  p  631. 

Brunei's  process.  He  experimented  with  an  open  cast-iron  cylinder,  3  ft 
outer  diarn  ;  1%  ins  thick  ;  in  lengths  of  10  ft,  connected  together  by  internal  socket  and  joggle  joints, 
secured  by  pins,  and  run  with  lead.  It  had  a  sharp-edged  hoop  or  cutter  at  bottom :  and  a  little 
above  this,  one  turn  of  a  screw,  with  a  pitch  of  7  ins,  and  projecting  one  foot  all  around  the  outside 
of  the  cylinder.  By  means  of  capstan  bars  and  wincbe.s,  he  screwed  this  down  through  stiff  clay  and 
sand,  58  feet  to  rock,  on  the  bank  of  a  river.  In  descending  this  distance  the  cylinder  made  142 
revolutions  ;  sinking  on  an  average  about  5  ins  at  each.  The  time  occupied  in  actually  screwing  was 
48J4  hours;  or  about  ly^j-  ft  per  hour.  There  were,  however,  many  long  intervals  of  rest  for  clean- 
ing away  the  soil  in  the  inside.  After  resting,  there  was  no  great  difficulty  in  restarting.  The  next 
fig  will  give  an  idea  of  the  arrangement  of  the  screw. 

The  screw-pile  of  Alex.  Mitchell,  Belfast,  consists  usually  of  a  rolled  iron 
shaft  A,  Figs  14.  from  3  to  8  ins  diam ;  and  having  at  its  foot  a  cast-iron  screw 
S  S  S.  with  a  blade  of  from  18  ins  to  5  ft  diam.  The  screws  used  for  light-houses, 
exposed  to  moderate  seas,  or  heavy  ice-fields,  are  ordinarily  about  3  ft  diam,  have 
\y%  turns  or  threads,  and  weigh  about  600  Ibs.  The  round  rolled  shafts  sire  from 
5  to  8  ins  diam.  They  are  screwed  down  from  10  to  20  ft  into  clay,  sand,  or  coral,  by 
about  30  to  40  men,  pushing  with  6  to  8  capstan  bars,  the  ends  of  which  describe  a 
circle  of  about  30  to  40  ft  diam.'  For  this  purpose  a  platform  on  piles  has  frequently 
to  be  prepared.  In  quiet  water,  this  may  be  supported  on  scows ;  or  a  raft  well 
moored  may  be  used  when  the  driving  is  easy ;  or  the  deck  of  a  large  scow  with  a 
well-hole  in  the  center  for  the  pile  to  pass  through.  Roughly  made  temporary 
cribs,  filled  with  stone  and  sunk,  might  support  a  platform  in  some  positions.  The 
platform  must  evidently  be  able  to  resist  revolving  horizontally  under  the  great 
pushing  force  of  the  men  at  the  capstan  bars ;  and  on  this  account  it  is  difficult 
to  drive  screws  to  a  sufficient  depth,  in  clean  compact  sand,  by  means  of  a  floating 
platform.  The  feet  of  the  piles  must  be  firmly  secured  to  the  screws,  to  prevent 

*  Cast  iron,  intended  to  resist  sea-water,  should  be  close-grained, 

hard,  white  metal. '  In  such,  the  small  quantity  of  contained  carbon  is  chemically  combined  with  the 
metal;  but  in  the  <iarker  or  mottled  irons  it  is  mechanically  combined,  and  such  iron  soon  becomes 
soft,  (somewhat  like  plumbago,)  when  exposed  to  sea- water.  Hard  white  iron  has  been  proved  to 
resist  for  at  least  40  years  without  any  deterioration :  whether  constantly  under  water,  or  alternately 
wet  and  dry.  Copper  and  bronze  are  but  slightly  and  superficially  affected  by  sea-water ;  but  destruc- 
tive galvanic  action  takes  place  if  diff  metals  are  in  contact. 


FOUNDATIONS. 


325 


II 


their  being  lifted  out  of  them  by  the  upward  force  of  waves  against  the  super- 
structure. At  y  p,  Figs  14,  is  shown  a  mode  of  splicing  or  uniting  the  different 
lengths  or  sections  of  a  pile.  The  point  of  junction  is  at  v ;  r  r  is  a  stout  iron  ring 
forged  on  to  the  lower  pile  p, 
about  a  foot  or  18  ins  below  its 
top  v.  A  strong  cylindrical  cast- 
ing n  n,  enclosing  the  ends  of 
the  sections,  rests  on  this  ring, 
and  is  pinned  through  the  piles, 
as  at  1 1.  On  this  casting  are 
also  cast  projections  ccc.  for  at- 
taching rodsgg,  and  beams  ?',  &c, 
necessary  lor  bracing  the  struc- 
ture from  pile  to  pile.  The  time 
actually  required  for  driving  a 
screw  is  from  2  to  10  hours,  in 
favorable  circumstances. 

At  the  Brandywine  lighthouse,  on 
a  sand- bank  of  very  pure  sand,  cov- 
ered 6  or  8  ft,  at  low  water,  and  from 
11  to  13  ft  at  high,  they  could  not.  be 
forced  down,  from  a  fixed  platform, 
for  m  TH  than  10  ft.  At  other  places  20  ft  in  sand  is  reached  without  much  trouble,  where  the  sand 
contains  a  good  deal  of  mud,  but  its  bearing  power  is  then  less.  This  (ultimate)  ranges  between 
about  1  and  6  tons  per  sq  ft  according  to  purity,  depth,  compactness,  &c,  of  the  saud.  In  important 
cases  the  bearing  power  should  be  tested. 

Mitchell's  piles  have  been  screwed  about  40  feet  into  a  mixture  of  clay  and  sand,  with  screws 
4  ft  diam.  They  pass  through  small  broken  stone  and  coral  rock  without  much  difficulty  ;  and  will 
push  aside  bowlders  of  moderate  size.  Ordinarily,  clay  or  sand  will  present  no  great  obstruction  ; 
but  occasionally  either  of  them  will  do  so.  Perfectly  pure  clean  sand,  as  a  general  rule,  gives  most 
difficulty.  At  the  Brandywine  shoal  the  driving  was  aided  by  a  spur  and  pinion  placed  as  low  as  the 
water  permitted  ;  and  the  levers  were  worked  by  30  men.  The  danger  of  twisting  off  the  shaft  is 
the  limit  for  screwing  them.  They  are  much  used  for  the  anchoring  of  chains  for  mooring  buoys,  &c. 
On  laud,  small  screws,  with  short  hollow  shafts,  make  good  durable  supports  for  depot  pillars,  cranes, 
•wooden  telegraph  poles,  station  signals  in  marine  surveying,  &c,  &c.  They  can  readily  be  unscrewed 
for  removal.  Horses  or  oxen  may  be  used  in  driving  large  screws.  The  Brandywine  light-house 
stands  on  9  screw-piles,  which  are  surrounded  by  30  others  of  5  ins  diam,  as  fenders.  Thev  have  to 
resist  not  only  moderate  seas,  but  immense  fields  of  floating  ice,  miles  in  extent.  An  unfinished 
structure  was  destroyed  by  ice,  which  at  times  injures  the  bracing  of  the  standing  one. 

Test  borings  should  be  made  to  ensure  that  the  screws  do  not  stop  just 
above  a  very  weak  stratum  which  may  endanger  their  bearing  power.  So  with  any  piles. 

By  means  of  a  jet  of  water  forcibly  impelled  through  a  tube  by  a  force 

Eump,  the  most  obstinate  sands  (but  not  stiff  clay  or  cemented  gravel)  will  be  loosened,  and  the  sink- 
ng  of  screw-piles,  or  wooden  ones,  or  even  the  largest  cylinders,  be  greatly  facilitated.  In  a  govern- 
ment pier  at  Cape  Heiilopeii  in  very  compact  sand,  in  which  6  out  of  7 
screws  previously  broke  before  reaching  10  ft,  the  use  of  the  jet  was  found  to  remove  more  than 
three-fourths  of  the  resistance.*  The  pile  p  to  be  sunk  having  first  been  placed  in  position  as  in  Pig 
15,  the  lower  open  ends  1 1  of  a  bent  iron  tube  t  «  t  of  one  and  a 
quarter  ins  bore  were  stood  upon  the  upper  face  of  the  screw  disk,  and  .  * 

there  held  firmly  by  3  or  4  men  while  the  pile  was  being  screwed  down  7*      , \&=\^ 

by  the  capstan  c,  which  was  worked  by  a  leading  rope  r.     From  the 

bend  s  of  the  pipe,  a  hose  h,  2  ins  diam.  led  to  the  force  pump,  the       Jp/Vy  7^     fl  /• 

cylinder  of  which  was  5  ins  bore,  and  9  ins  stroke,  and  worked  about     *  ^  ^*^     /V 

80  full  strokes  per  minute,  by  a  mule  walking  on  a  tn-ad  wheel  on  a 

floating  platform/.     There  was  now  no  trouble  in  screwing  the  piles  to 

any  required  depth.    Previous  trials  by  playing  the  jet  beneath  the  disk 

gave  unsatisfactory  results. 

In  Mobile  Bay  several  thousands  of  wooden  piles, 
from  18  to  48  ins  diam,  were  sunk  from  10  to  20  ft  into  obstinate  sand, 
at  the  average  sinking  rate  of  about  1  ft  per  second,  entirely  by  means 
of  jets.  The  jet  was  propelled  by  a  city  steam  fire  engine,  on  a  steam- 
boat, through  its  own  hose,  with  a  one  and  a  quarter  inch  nozzle. 
During  the  descent  the  nozzle  n  n  was  held  loosely  in  its  place  near 
the  foot  of  the  pile,  by  two  staples  «  «  and  by  a  stritig  t  reaching  to  the 
surface.  The  piles  were  suspended  by  their'  heads  from  shears,  by  the 
tackle  of  which  their  descent  was  regulated.  The  sand  settled  firmly 
around  the  piles  in  a  few  minutes  after  they  were  sunk.t 

At  Teiisas  River.  Alabama,  for  iron  cylinders  6  ft  diam 

(enclosing  piles,  see  p  329),  in  deep  light  shifting  sand,  the  jet  was  forced  by  a  small 
rotary  pump  of  200  to  300  revolutions  per  minute,  through  a  canvas  hose  8  ins  diam, 
into  a  central  conical  cast  iron  vessel  10  ins  diam.  from  which  radiated  12  gas  pipes  1 
inch  diam,  and  about  30  ins  long.  At  the  outer  end  of  each  of  these  radii  was  an 
elbow  to  which  was  attached  a  long  vertical  pipe  reaching  down  into  the  cylinder, 
and  made  in  10  ft  lengths  with  screw  ends  for  prolonging  them  as  the  cylinder  went 
down.  This  apparatus  was  raised  and  lowered  by  a  light  block  and  line;  and  by  it 
alone  each  cylinder  was  sunk  about  16  ft  into  the  light  saud  in  a  few  hours.; 


*  Report  Sec  of  War  1872.  t  John  W.  Glenn,  C  E,  Van  Nostrand,  Juno  1874. 

J  Gabriel  Jordan,  C  E ;  Trans  Am  Soc  C  E,  Peb  1874. 


326 


FOUNDATIONS. 


At  the  I<evan  Viaduct.  Mr  James  Brnnlee,  England,  hi  a  light 

nandy  marl  of  great  depth,  sunk  hollow  cast  iron  cylinders  of  10  ins  outer  diatu.  to  a  depth  of  '20  ft, 
by  mentis  of  a  jet  pipe  2  ins  diam  passing  down  inside  of  the  cylinder,  and  through  a  hole  in  its  base, 
whicn  was  a  cast  iron  disk  30  ins  diam,  and  1  inch  thick,  strengthened  by  outside  flanges.  The  con- 
necting flanges  of  the  cylinder  sections  are  outside,  thus  impeding  the  descent,  as  did  also  the  broad 
bottom  disk  ;  still  3  or  4  hours  usually  sufficed  for  the  sinking  of  each,  to  20  ft  depth.  Actual  trial 
showed  that  their  safe  sustaining  power  was  about  5  tons  per  sq  ft  of  bottom  disk. 

At  Lock  Ken  viaduct  each  pier  consists  of  two  cylinders,  open  at  both 
ends;  of  cast  iron,  8  ft  in  diam;  1^  ins  thick;  in  lengths  of  6  ft,  weighing  4  tons 
each ;  and  bolted  together  by  inside  flanges,  with  iron  cement  between  them.  The 
cylinders  stand  8  ft  apart  in  the  clear;  and  are  in  36  ft  water.  *k  A  strong  staging 
was  erected;  and  4  guide-piles  driven  for  each  cylinder.  The  several  lengths  being 
previously  bolted  together,  these  were  lowered  into  their  places.  Each  cylinder  sank 
by  its  own  weight  one  or  two  ft  through  the  top  mud,  and  then  settled  upon  the  sand 
an>l  gravel  which  form  the  substratum  for  a  great  depth.  Into  this  last  they  were 
sunk  about  8  or  V  ft  farther,  by  excavating  the  inside  earth  under  water,  by  mea:>8 
of  an  inverted  conical  screw-pan,  or  dredger,  of  %  inch  plate  iron.  This  was 
2  ft  greatest  diam,  and  1  ft  deep ;  and  to  its  bottom  was  attached  a  screw  about  1  ft 
long,  for  assisting  in  screwing  it  down  into  the  soil.  Its  sides  had  openings  for  the 
entrance  of  the  soil ;  and  leather  flaps,  opening  inward,  to  prevent  its  escape.  From 
opposite  sides  of  the  pan,  3  rods  of  %  inch  diam  projected  upward  4  feet,  and  were 
there  forged  together,  and  connected  by  an  eye-and-bolt  joint  to  a  long  rod  or  shaft, 
at  the  upper  end  of  which  was  a  four-armed  cross-handle,  by  which  the  pan  was 
screwed  down  by  4  men  on  the  staging. 

"  "When  a  pan  was  full,  a  slide  which  passed  over  the  joint  at  the  bottom  -was  lifted ;  and  the  pan 
was  raised  by  a  tackle.  This  pan  raised  about  1  cub  ft  at  a  time.  A  smaller  one  of  only  1  ft  diam, 
and  1  ft  deep,  raising  about  y±  cub  ft,  was  used  when  the  material  was  very  hard.  By  this  means 
the  cylinders  were  sunk  at  the  rate  of  from  2  to  18  ins  per  day.  The  slow  rate  of  2  ins  was  caused 
by  stones,  some  of  them  of  50  H>s.  These  were  first  loosened  by  a  screw-pick,  which  was  a  bar  of 
iron  3  ft  long,  with  circular  arms  12  ins  long  projecting  from  the  sides.  After  being  loosened  by  this, 
the  stones  were  raised  by  the  pan.  The  expense  of  all  this  apparatus  was  very  trifling;  and  the  ex- 
cavation was  done  easily  and  cheaply.  After  the  excavation  was  finished,  aud  the  cylinder  sunk, 
before  pumping  out  the  water,  concrete  (gravel  2,  hydraulic  cement  1  measure)  was  filled  in  to  the 
depth  of  12  feet,  by  means  of  a  large  pan  with  a  movable  bottom ;  and  about  12  days  were  left  it  to 
harden.  The  water  was  then  pumped  out,  and  the  masonry  built  in  open  air.  In  some  of  the  cylin- 
ders, however,  the  water  rose  so  fast,  notwithstanding  the  12  ft  of  concrete,  that  the  pumps  could  not 
keep  them  clear ;  and  6  ft  more  of  concrete  had  to  be  added  in  those.  Finally  random-stone,  or  rough 
dry  rubble,  was  thrown  in  around  the  outside*  of  the  cylinders,  to  preserve  them  from  blows  arad 
undermining."  *  The  masonry  extends  20  ft  above  the  cylinders,  and  above  water. 

The  vacuum;  and  the  plenum  processes.  We  can  barely  allude 
to  the  general  principles  of  these  two  modes  of  sinking  large  hollow  iron  cylinders. 
In  the  vacuum  invented  by  Dr.  Lawrence  Holkor  Polls,  of  London,  the  cylinder 
c,  Fig  16,  while  being  sunk,  is  closed  air-tight  at  top,  by  a 
trap  door,  opening  upward.  A  flexible  pipe  p,  of  India- 
rubber,  long  enough  to  adapt  itself  to  the  sinking  of  the 
cylinder,  and  provided  with  a  stopcock  .<?,  leads  from  the 
cylinder  to  a  vessel  v ;  which  may  be  placed  on  a  raft,  or  a 
scow,  or  on  land,  as  may  suit  circumstances.  The  cylinder 
being  first  stood  up  in  position,  as  in  the  fig,  the  water  is 
Fid  16  pumped  out,  and  the  interior  soil  removed  if  the  cylinder 

has   sunk  some  distance  by  its  own  weight.    The  cock 
*  is  then  closed,  and  the  air  is  drawn  out  from  the  vessel  v 

by  an  air-pump.  The  cock  is  then  opened,  and  most  of  the  air  in  the  cylinder  rushes 
into  the  void  vessel  v\  thus  leaving  the  cylinder  comparatively  empty,  and  therefore 
less  capable  of  resisting  the  downward  pressure  of  the  external  air  upon  its  top. 
This  pressure,  as  is  well  known,  amounts  to  nearly  15  ft>s  on  every  sq  inch  ;  or  nearly 
1  ton  per  sq  ft  of  area  of  the  top.  Consequently  the  cylinder  is  forced  downward  in 
the  bed  of  the  river,  by  this  amount  of  pressure,  in  addition  to  its  own  weight.  At 
the  same  time,  the  pressure  of  the  air  upon  the  surface  of  the  water  is  transmitted 
through  the  water  to  the  soil  around  the  open  foot  of  the  cylinder:  so  that  if  this 
soil  be  soft  or  semi-fluid,  it  will  be  pressed  up  into  the  nearly  void  cylinder,  in  which 
is  no  downward  pressure  to  resist  it.  The  descent  varies  from  a  few  inches,  to  4  or  5 
ft  each  time.  The  process  is  then  repeated,  by  admitting  air  again  into  the  cylin- 
der, opening  the  trap-door,  removing  the  water  and  soil,  as  before,  <fcc.  Additional 
lengths  of  cylinder  may  be  bolted  on,  by  means  of  interior  flanges. 

It  is  adapted  011  ly  to  soft  soi  Is,  and  to  wet  sandy  ones :  but  is  not  sufficient- 
ly, powerful  in  very  com  pact  ones;  nor  does  it  answer  where  obstructions  from  bowlders,  logs,  &c.  occur; 

Hollow  Iron  Piles  either  cast  or  wrought  with  solid  pointed  feet,  to  be  driven  by  the  hammer 
falling  inside  of  them  and  striking  against  the  top  of  the  solid  foot,  are  a  recent  device  of  great  nse  in 
many  cases.  They  are  made  in  sections  of  which  enough  can  be  gradually  united  to  reach  any 
required  depth.  They  avoid  the  danger  of  bending  which  attends  striking  the  top.  The  iron  feet  are 
swelled  outwardly  a  little  to  diminish  earth-Snetion  against  the  pile  above  them. 


FOUNDATIONS. 


327 


mgi7 


the  removal  of  which  requires  men  to  enter  the  cylinder  to  its  foot ;  which  they  cannot  do  in  the  rarefied 
uir.  The  pipe  jp  should  be  of  sufficient  diam  to  allow  the  air  to  leave  the  cylinder  rapidly,  so  that  the 
outer  pressure  mav  act  upon  the  top  as  suddenly  as  possible. 

At  the  Goodwii/Sauds  light-house.  England,  hollow  cylinders  '1%  ft  in  diam,  were  sunk  34  ft  into 
sand  bv  this  process  in  about  6  hours;  where  a  steel  bar  could  be  driven  ouly  8  ft  by  a  sledge-ham- 
mt-r  Others,  T2  ins  in  diiun,  have  been  sunk  16  ft  into  sand  within  less  than  an  hour.  In  this  last 
instance  the  air-pump  had  two  barrels,  4^  ius  diaui,  16  inch  stroke,  worked  by  4  men.  The  pipe p 
was  of  lead,  an-\  ouly  %  inch  diam. 

The  plenum  process,  invented  by  Mr  Triger, 
of  France,  consists  in  forcing  air  into  the  cylinder 
C  C,  Fig  17,  to  such  an  extent  as  to  force  out  the 
water,  compelling  it  to  escape  beneath  the  open  foot, 
into  the  surrounding  water.  The  interior  of  the  cylin- 
der being  thus  left  dry  to  the  bottom,  men  pass  down  it 
to  loosen  and  remove  the  soil  atand  below  its  base.  When 
this  is  done,  they  leave ;  the  compressed  air  is  allowed  to 
escape ;  and  the  cylinder,  being  no  lunger  sustained  by 
the  upward  pressure  of  the  compressed  air  beneath  its 
top,  sinks  into  the  cavity,  or  the  loosened  material  at  its 
foot.  Fig  17  shows  the  simple  arrangement  by  which 
workmen  are  enabled  to  enter  or  leave  the  cylinder, 
without  allowing  the  compressed  air  to  escape;  as  well 
as  the  general  principle  of  the  entire  process. 

L  L  is  a  separate  small  chamber,  the  air-lock,  which  is 
removed  when  a  new  length  of  pipe  is  to  be  added ;  aud  afterward 
replaced  and  firmly  bolted  on.  This  chamber  has  a  small  air-tight 
door  d,  by  which  it  can  be  entered  from  without ;  and  another,  o, 
opening  into  the  cylinder.  The  flaps,  t,  h.  of  both  doors,  open  in- 
ward, or  toward  the  cylinder.  This  chamber  also  has  two  stopcocks;  one,  a,  in  its  floor,  communi- 
cating with  the  cylinder;  and  one  e..  above,  communicating  with  the  open  air.  At  s  is  a  bent  tube, 
also  with  acock,  which  passes  air-tight  through  the  side  and  the  bottom  of  the  air-lock.  Through 
it  the  compressed  air  is  forced  into  the  cylinder,  by  an  air  force  pump  or  condenser:  and  through  it 
the  same  air  is  allowed  to  escape  at  a  later  period.  A  siphon  is  shown  at  77  n«.  A  drum  tv  is  used 
for  hoisting  theexcavated  material  from  the  bottom,  to  the  air-lock  :  its  axle  it  passes  air-tight  through 
stuffing-boxes  in  the  sides  of  the  lock  ;  the  hoisting  being  done  by  men  outside.  This  is  the  general 
arrangement  employed  by  Mr  W.  J.  McAlpine,  CE.  of  New  York,  at  Harlem  bridge ;  and  from  his 
description  of  it,  ours  has  been  condensed.  The  cylinders  were  there  6  ft  diam,  \%  ins  thick,  and  in 
lengths  of  9  ft,  bolted  together  through  inside  flanges  /.  as  the  sinking  went  on.  The  air-lock  is  6  ft 
diam,  by  nearly  6  ft  high  :  with  sides  of  boiler  iron  ;  and  top  and  bottom  of  cast  iron.  See  p  631. 

Now  suppose  the  cylinder  C  C  to  be  let  down,  and  steadied  in  position,  as  in  the  fig;  and  the  air- 
lock L  L  to  be  adjusted  on  top  of  it.  The  next  process  is  to  force  in  air  through  the  curved  tube  »  ; 
the  flap  t  of  the  lower  door  o.  and  the  cock  o,  being  previously  closed.  As  the  compressed  air  accu- 
mulates in  the  cylinder,  it  forces  out  the  water;  which  escape.*  partly  beneath  the  bottom  of  the  cyl- 
inder, and  partly  by  rising  through  the  siphon  nn,  aud  flowing  out  at  p.  The  door  o  being  already 
closed,  and  that'at  d  open,  the  air  in  the  air-lock  is  in  the  same  condition  as  that  outside;  so  that 
workmen  can  enter  it  readily.  Having  done  so,  thny  close  the  door  d.  and  the  cock  e:  and  open  the 
cock  a,  through  which  condensed  air  from  the  cylinder  rushes  upward,  soon  filling  the  air-lock. 
When  this  is  done,  the  flap  t  falls :  and  the  workmen  descend  through  the  door  o  by  :i  bidder,  or  by  a 
bucket  lowered  by  the  drum  tv,  to  the  bottom.  Here  they  loosen  and  excavate  the  material  as  deep 
as  they  can  ;  and",  filling  it  inio  a  bucket  or  bag.  they  signal  to  those  outside,  who  raise  it  to  the  air- 
lock. When  done,  they  ascend  to  the  air-lock,  close'the  door  o.  and  the  cock  o;  and  open  the  cock  e, 
through  which  the  condensed  air  in  the  lock  soon  escapes,  leaving  the  internal  air  the  same  as  that 
outside.  The  door  d  is  then  opened,  the  buckets  of  earth  me  removed,  and  the  men  go  out.  Finally 
the  cock  at  «  is  opened,  the  condensed  nir  in  the  cylinder  escapes  through  it  to  the  outside  air.  and 
the  cylinder  sinks  by  its  own  weight  into  the  cavity  and  loosened  soil  prepared  for  it  at  its  base,  and 
which  is  now  forced'np  into  the  cylinder  by  the  rush  of  the  returning  water.  The  process  is  then 
repeated.  The  sinking  will  often  vary  from  0  to  10  or  more  feet  at  oue  operation.  Until  depths  of 
40  or  50  ft.  most  men  can  endure  the  pressure  of  the  condensed  air  ;  but  as  the  depth  increases  this 
becomes  more  difficult,  and  positively  dangerous  to  life.  Cast  iron  cylinders  15  ft  diam  ;  aud  great 
caissons,  Fig  18,  have  been  thus  sunk ;  but  at  times  at  great  expeuse  aud  trouble. 

The  cylinder  should  be  guided  in  its  descent  by  a  strong  frame,  which 

may  be  supported  by  piles.  Otherwise  it  will  be  apt  to  tilt,  aud  thus  give  great  trouble  to  settle  it 
upon  its  exact  place.  Have  been  sunk  in  deep  water  by  divers  undermining  inside. 

Hollow  cylinders,  or  other  forms  of  brickwork  or  ma- 
sonry, with  a  strong  curb  or  open  ring  of  timber  or  iron  beneath  them,  may  be 
gradually  sunk  by  undermining  and  excavating  from  the  inside;  aud  form  very  stable  foundations. 
Under  water  this  may  be  done  by  properly  shaped  scoops,  with  or  without  the  aid  of  the  diving-bell, 
according  to  the  depth,  &c.  On  laud  it  will  often  be  the  most  economical  and  satisfactory  mode, 
especially  in  firm  soils.  The  descent  may  be  assisted  by  loading  them,  if,  as  sometimes  happens,  the 
friction  of  their  sides  against  the  earth  outside  prevents  tueir  siuKiug  by  their  owu  'weight:  A  brick 
cylinder,  46  ft  outer  diam,  walls  'A  ft  thick,  has  been  sunk  40  ft  iu  dry  sand  and  gravel,  without  uuy 
difficulty.  It  was  built  18  ft  high,  (on  a  wooden  curb  '21  ius  thics,)  aud  weighed  300  tons  before  tUe 
sinkiug  was  begun.  The  interior  earth  was  excavated  slowly,  so  that  the  siukiug  was  about  1  ft  per 
day  ;  the  walls  being  built  up  us  it  sank.  Tunnel  shafts  are  at  tunes  *o  sunk. 

On  the  Rhine  for  a  coal  sliaft,  a  brick  cylinder  25 1£  feet  diam  was  first  thus 
sunk  by  its  own  weight  76  ft  through  sand  and  gravel ;  then  an  interior  oue,  15  ft  diam,  was  sunk  iu 
the  same  way  to  the  depth  of  256  ft  below  the  surface ;  of  which  depth  all  the  180  ft  below  the  flrst 
cylinder  was  a  running  quicksand.  At  256  ft  frictiou  rendered  the  cylinder  immovable.  The  quick 
•and  was  removed  by  boriug ;  uo  pumping  was  done ;  but  the  water  was  permitted  to  keep  toe  cyl  full. 


328 


FOUNDATIONS. 


The  entire  foundation  for  a  large  pier  of  masonry  has  been  sunk  in  this  manner,  in  a  single  m&M ; 
a  sufficient  number  of  vertical  openings  being  left 'in  it  for  the  workmen  to  descend,  or  for  tools  to  be 
inserted  for  undermining.  This  is  generally  a  very  slow  and  tedious  operation,  especially  under 
water.  It  may  often  be  expedited  by  diving-bells  or  by  diving-dresses.  It  will  generally  be  better  to 
make  the  mass  wider  at  bottom  than  above  it,  so  as  to  diminish  friction  against  the  outside  earth. 
On  laud,  water  may  at  times  be  used  for  softening  the  bottom  earth.  By  keeping  the  interior  of  such 
hollow  masonry  dry,  It  may  even  be  built  downward  from  the  surface ;  by  undermining  only  a  por- 
tion of  its  circumference  at  a  time,  filling  said  portion  with  masonry,  and  then  removing  and  filling 
the  other  portion  ;  and  so  on  in  successive  stages  of  2  or  3  ft  downward  at  a  time.  This  mode  may  be 
adopted  also  when  friction  has  stopped  the  sinking  of  a  mass  by  its  own  weight  when  undermined. 

The  sand  pump  as  used  at  the  St  Louis  bridge  will  often  be  of  service  in  raid- 
ing sand  from  cylinders  while  being  sunk  in  water.  With  a  pump  pipe  of  3.5  ins  bore,  and  a  water 
jet  under  a  pressure  of  150  Ibs  per  sq  inch,  20  cub  yds  of  sand  per  hour  were  raised  125  feet.  A  jet  of 
air  has  also  been  successfully  used  in  the  same  way,  as  at  the  East  River,  N  Y,  suspension  bridge.  &c. 

Fascines.  On  marshy  or  wet  quicksand  bottoms,  foundations  may  be  laid  by 
first  depositing  large  area*  of  layers  of  fascines,  or  stout  twigs  and  small  branches, 
strongly  .tied  together  in  bundles  from  6  to  12  ft  long,  and  from  6  ins  to  2  ft  in  diam. 

The  layers  or  strata  of  bundles  should  cross  each  other.  A  kind  of  floating  raft  or  large  mattress 
is  first  made  of  those,  and  then  sunk  to  the  bottom  by  being  loaded  with  earth,  gravel,  stones,  &c. 
In  this  manner  the  abutments  and  piers  of  the  great  suspension  bridge  at  Kieff,  in  Russia,  with  spans 
of  440  ft,  were  founded  in  1852,  on  a  shifting  quicksand.  There  the  fascine  mattresses  extend  100  ft 
beyond  the  bases  of  the  masonry  which  rests  upon  them. 

Fascines  may  be  used  in  the  same  way  for  sustaining  railway  embankments,  &c,  over  marshy 
ground,  but  they  will  settle  considerably. 

Sand-piles.  We  have  already  alluded  to  the  use  of  sand  well  rammed  in  layers 
into  trenches  or  foundation  pits ;  but  it  may  also  be  used  in  soft  soils,  in  the  shape 
of  piles.  A  short  stout  wooden  pile  is  first  driven  5  to  10  feet  or  more,  according  to 
the  case.  It  is  then  drawn  out,  and  the  hole  is  filled  with  wet  sand  well  rammed. 
The  pile  is  then  again  driven  in  another  place,  and  the  process  repeated.  The  inter- 
vals may  be  from  I  to  3  ft  in  the  clear.  Platforms  may  be  used  on  these  piles  as  on 
wooden  ones.  If  the  sand  is  not  put  in  wet,  it  will  be  in  danger  of  afterward  sink- 
ing from  rain  or  spring  water.  Jn  this  case,  as  with  fascines,  it  is  well  to  test  the 
foundation  by  means  of  trial  loads.  Some  settlement  must  inevitably  take  place 
until  all  the  parts  come  to  a  full  bearing :  but  it  will  be  comparatively  trifling.  The 
same  occurs  in  every  large  work  to  some  extent ;  as  in  a  roof  or  arch  of  great  span, 
whether  of  wood,  iron,  or  masonry  ;  so  also  with  all  tall  piers,  walls,  &c,  &c.  Sandy 
foundations  under  water  should  be  surrounded  by  stout  well-driven  sheet-piling,  to 
prevent  the  enclosed  sand  from  running  out  in  case  the  outer  sand  is  washed  away  ; 
and  should  also  be  defended  by  a  deposit  of  random-stone.  See  Sand-piles,  p  636. 

On  bad  bottoms  under  water,  small  artificial  islands  of  good  soil  have 

been  deposited;  and  the  masonry  founded  upon  them.  Canal  locks  and  other  structures  may  at 
times  be  advantageously  founded  in  this  way  in  marshy  soils.  If  necessary,  a  depth  of  several  feet 
of  the  bad  soil  may  be  dredged  out  before  the  firmer  soil  is  deposited;  and  the  latter  may  be  weighted 
by  a  trial  load  to  test  its  stability. 

The  mode  of  laying  a  foundation  under  water,  by  building  the  masonry  upon  a  timber  platform 
above  water,  upheld  by  strong  screws,  and  lowered  into  the  water  as  the  work 
is  finished  in  the  open  air.  a  course  or  two  at  a  time,  has  of  late  been  much  employed  with  entire 
success,  in  large  bridge-piers  in  deep  water.  It  however  is  not  new.  It  was  suggested  more  than 
100  years  ago  by  Belidor. 

Piles  are  driven  6  to  10  ft  apart  around  the  space  to  be  occupied  by  the  pier ;  having  their  tops  con- 
nected by  heavy  timber  cap-pieces.  These  last  uphold  the  screws,  which  work  through  them.  The 
whole  is  braced  against  lateral  motion. 

The  shaded  part  of  Fig  18  shows  a  transverse  section  of  the  caisson  of  yellow- 
pine  timber  and  cement,  for  the  Brooklyn  tower  of  East  River  (N  Y) 
suspension  bridge,  of  1600  ft  clear  span.  It  is  168  ft  long  at  bottom,  and  102  ft  wide. 
A  longitudinal  section  resembles  the  transverse  one,  except  in  being  longer,  and  in 
showing  more  shafts  .1.  Of  these  there  are  6,  arranged  in  pairs,  for  expedition  and  as 
a  precaution  against  accident.  Namely,  two  water-shafts  J,  each  7  ft  by  6^  ft  across, 
for  removing  by  buckets  and  hoisting  apparatus,  the  material  excavated  beneath  the 

caisson ;  together  with  such 


8 


water  as  may  accumulate  at 
o  o;  two  air-shafts  of  21  ins 
diam,  through  which  air  is 
forced  from  above,  to  expel 
the  water  from  the  chamber 
C  S  S  D  below  the  caisson,  so 
as  to  allow  the  laborers  to 
work  there  at  undermining; 
the  expelled  water  escaping 
under  the  foot  C  D  of  the  cais- 
son, into  the  river ;  and  two 
supply  shafts  of  42  ins  diam, 
for  admitting  laborers,  tools, 

«fcc.    The  several  shafts  of  course  have  air-chambers  on  top,  on  the  saine  principle  as 

Fig  17,  to  prevent  the  escape  of  the  compressed  air  in  s  s. 


COST   OF    DREDGING.  329 

The  shafts  are  of  >4  Inch  boiler  iron.  The  foot  C  D,  nine  timbers  high,  is  continuous,  extending 
eutirely  around  the  caisson;  its  bottom  is  shod  with  cast  iron  ;  its  four  corners  are  strengthened  by 
wooden  knees  20  ft  long. 

From  the  bottom,  up  to  the  line  N,  N,  14  ft,  the  caisson  is  built  of  horizontal  layers  of  timbers  one 
foot  square;  the  layers  crossing  each  other  at  right  angles;  and  the  timbers  of  each  layer  touching 
each  other  well  forced  and  bolted  together;  and  all  the  joints  filled  with  pitch.  To  aid  in  preventing 
leakage,  the  nuts  and  heads  of  the  screws  have  India-rubber  washers ;  also  all  outside  seams,  as  well 
as  all  the  seams  of  the  layer  of  timbers  N,  N,  are  thoroughly  calked;  and  a  layer  of  tin,  enclosed 
between  two  layers  of  fejt,  is  placed  outside  of  each  outer  joint;  and  over  the  entire  top  of  the  layer 
next  helow  N,  N. 

When  the  caisson  was  built  up  to  N,  N,  on  land,  it  was  launched,  floated  into  position,  and  anchored ; 
after  which  were  added  for  sinking  it,  fifteen  courses  of  timbers  one  ft  square;  and  laid  one  ft  apart 
in  the  clear ;  with  the  intervals  filled  with  concrete.  The  top  course  A  B  is  of  solid  timber,  to  serve 
as  a  floor  for  supporting  machinery,  &c.  It  was  sunk  some  feet  below  the  very  bottom  of  the 
river,  in  order  to  avoid  the  teredo. 

Cribs  are  sunk  outside  of  the  caisson,  to  form  temporary  wharves  for  boats  carrving  away  excavated 
material  ;  and  for  vessels  bringing  stone,  &c. 

When  the  caisson  was  sunk,  and  the  water  forced  out  from  the  chamber  or  space  CSS  D. workmen 
began  to  excavate  uniformly  the  enclosed  area  of  river  bottom,  so  as  to  allow  the  caisson  to  descend 
slowly  untilit  reached  a  firm  substratum.  The  space  C  S  S  D,  as  well  as  the  shafts,  was  then  filled  up 
aolid  with  concrete  masonry.  A  coffer-dam  was  built  on  top  of  the  caisson;  and  in  it  the  regular 
masonrv  of  the  tower  was  started.  The  total  height  of  this  tower  including  the  caisson,  is  about  300 
ft.  For  full  details  see  report.  1873,  of  W.  A.  Roebling  the  chief  engineer. 

A  CLUMP  OF  PILES  WELL  DRIVEN;  and  then  enclosed  by  an  iron  cylinder  sunk  to  a 
firm  bearing,  and  filled  with  concrete,  is  an  excellent  foundation.  The  piles  may 
extend  to  the  top  of  the  cylinder,  and  thus  be  enclosed  in  the  concrete.  Such  ail 
arrangement  has  been  patented  by  S.  B.  Gushing,  C.  K.,  Providence,  R.  I.  The  cyl- 
inder and  concrete  serve  to  protect  the  piles  from  sea-worms,  and  from  decay  above 
low  water ;  and  are  not  intended  to  support  the  load  above  them. 

€ost  of  a  diving  dress,  with  air-pump  and  tubes,  about  $1000.  Alfred 
Hale  &  Co,  332  Washington  St,  Boston,  Mass. 

Two  men  can  work  the  air-pump  to  50  ft  depth. 

COST  OF  DEEDGING. 

DREDGING  is  generally  done  by  skilled  contractors,  who  own  the  requisite  machines, 
scows  or  lighters,  &c ;  and  who  make  it  a  specialty.  It  is  necessary  to  specify  whether 
the  dredged  material  is  to  be  measured  in  place  before  it  is  loosened;  or  after  being 
deposited  in  the  scow:  because  it  occupies  more  bulk  after  being  dredged.  It  was 
found,  in  the  extensive  dredgings  for  deepening  the  River  St  Lawrence  through  the 
Lake  of  St  Peter,  that  on  an  average  a  cub  yd  of  tolerably  stiff  mud  in  place,  makes 
1.4  yds  in  the  scow ;  or  1  in  the  scow,  makes  .715  in  place.  Also  stipulate  whether  the 
removal  of  bowlders,  sunken  trees,  &c,  is  to  constitute  an  extra.  These  often  require 
sawing  and  blasting  under  water.  The-cost  per  cub  yd  for  dredging  varies  much 
with  the  depth  of  water:  the  quantity  and  character  of  the  material :  thedist  to  which 
it  has  to  be  removed;  whether  it  can  be  at  once  discharged  from  the  machine  by 
means  of  projecting  side-shoots  or  slides ;  or  must  be  discharged  into  scows,  to  be  re- 
moved to  a  short  dist  by  poling,  or  to  a  greater  dist  by  steam  tugs ;  whether  it  can  be 
dropped  or  dumped  into  deep  water  by  means  of  flap  or  trap  doors  in  the  bottom  of 
the  hoppers  of  the  scows ;  or  must  be  shovelled  from  the  scows  into  shallow  water,  (at 
say  4  to  8  cts  per  yd ;)  or  upon  land,  (at  say  from  6  to  10  or  20  cts  for  the  shovelling 
alone,  or  shovelling  and  wheeling,  as  the  case  may  be ;)  whether  much  time  must  be 
consumed  in  moving  the  machine  forward  frequently,  as  when  the  excavation  is 
narrow,  and  of  but  little  depth;  as  in  deepening  a  canal,  &c;  whether  many  bowl- 
ders and  sunken  trees  are  to  be  lifted  ;  whether  interruptions  may  occur  from  waves 
in  storms;  whether  fuel  can  be  readily  obtained,  Ac,  &c.  These  considerations  may 
make  the  cost  per  cub  yd  in  one  c?ise  from  2  to  4  times  as  great  as  in  another.  The 
actual  cost  of  deepening  a  ship-channel  through  Lake  St  Peter,  to  IS  ft,  from  its  orig- 
inal depth  of  11  ft,  for  several  miles  through  moderately  stiff  mud,  was  14  cts  per 
cub  yd  in  place,  or  10  cts  in  the  scows;  including  removing  the  material  by  steam 
tugs  to  a  dist  of  about  V^a  mile,  and  dropping  it  into  deep  water.  This  includes  re- 
pairs of  plant  of  all  kinds,  but  no  profit.  It  was  a  favorable  case.  When  the  buckets 
work  in  deep  water  they  do  not  become  so  well  filled  as  when  the  water  is  shallower, 
because  they  have  a  more  vertical  movement,  and,  therefore,  do  not  scrape  along  as 
great  a  distance  of  the  bottom.  Hence  one  reason  why  deep  dredging  costs  more 
per  yard;  in  addition  to  having  to  bertifted  through  a  greater  height.  Perhaps  the 
following  table  is  tolerably  approximate  for  large  works  in  ordinary  mud,  sand,  or 
gravel ;  assuming  the  plant  to  have  been  paid  for  by  the  company ;  and  that  common 
labor  costs  $1  per  day. 


330 


COST   OF   DREDGING. 


Table  of  actual  cost  of  d redgi iig  on  a  large  scale;  inclnd» 
iiiS  dropping  tlie  material  into  scows,  alongside;  or  into 
side-shoots,  011  board.  Common  labor  $1  per  day.  Repairs 
of  plant  are  included  ;  but  no  pro  lit  to  contractor.  (Original.] 

Cts  per  Yard,  'Cts  per  Yard, 
iu  place.   j   iu  scow. 


Depth 

Cts  per  Yard, 

Cts  per  Yard, 

Depth 

iu  Ft. 

iu  place. 

IU  SCOW. 

iu  Ft. 

Less  than  10 

8.4 

6 

25  to  30 

10  to  15 

9.8 

7 

80  to  35 

15  to  20 

11.2 

8 

35  to  40 

20  to  25 

14.0 

10 

18.2 
25.2 
35.0 


13 

18 
25 


For  to  wing  of  the  scows  by  steam  tugs  to  a  dist  of  %  mik',  and  dropping  the  mud  into  deep  water,  add 

4  cts  per  yard  iu  the  scow  ;  for  >£  mile,  6  cts ;  for  %  mil.  .  b  cts  ;  for  1  mile,  10  cis     Add  profit  to  con- 
tractor.   On  a  small  scale  work  is  done  to  a  less  advantage;  and  a  corresponding  increase  must  be  made 
in  these  prices.    Also,  if  the  contractor  himself  furnishes  the  dredgers  and  plant,  a  still  further  addi- 
tion must  be  made.     It  is  evident  that  the  subject  admits  of  no  great  precision.    Small  jobs,  even  iu 
favorable  material,  but  iu  inconvenient  positions,  may  readily  cost  two  or  three  limes  as  much  per  yd 
as  the  above:  and  in  very  hard  material,  as  iu  ceme'iitt-cl  gravel  and  clay,  four  or  five  times  as  much 
for  the  dredging.     The  cost  of  towing,  however,  will  remain  as  before,  if  wages  are  the  same. 

At  present  Wages,  iSTe.  if  a  contractor  provides  all  the  plant  for  dredging  and  towing, 
the  total  cost,  including  his  profit,  will  be  about  3  time-  the  above  table. 

The  cost  of  dredgers,  tugs.  &c,  will  vary  of  course  with  their  capabilities,  strength  of  construction, 
style  of  finish,  whether  having  accommodations  for  the  men  to  live  on  board  or  not.  &c.  When  for 
use  in  salt  water,  the  bottoms  of  both  dredgers  and  scows  should  be  coppered,  to  protect  them  from  sea- 
worms;  and  if  occasionally  exposed  to  high  waves,  both  should  be  extra  strong.  The  most  powerful 
machines  on  the  St  Lawrence  cost  about  $15000  each  ;  and  removed  in  10  working  hours  on  an  average 
about  1800  cub  vds  iu  place,  or  25'20  in  the  scows.  Good  machines,  capable,  under  similar  circumstances, 
of  doing  as  much,  may,  however,  be  built  for  about  $25000  to  $30000  To  remove  this  quantity  to  a 
dist  of  %  tol  mile,  would  require  two  steam  tugs,  costing  about  $«000  to  $10000  each;  and  4  to  6  scows, 
(some  to  be  loading  while  otners  are  awav,)  holding  from  30  to  60  cub  yds  each  ;  and  costing  from  $800 
to  $1500  each  at  the  shop.  Scows  with  two  hoppers  are  best.  Such  a  dredger  would  require  at  least 
8  or  10  men,  including  captain,  engineer,  fireman,  and  cook.  Each  tug  4  or  5  men ;  and  each  scow  2 
men.  The  engineer  should  be  a  blacksmith  :  or  a  blacksmith  should  be  added.  In  certain  cases  a 
physician,  clerk,  assistant  engineer,  &c.  mav  be  needed. 

Dredgers  are  often  built  on  the  principle  of  the  Yankee  Excavator,  with  but  a  single  bucket  or  dip- 

Ser,  of  from  1  to  2  cub  yds  capacity.    Hull  about  25  by  60  ft.    Draft  3  ft.    Cylinder  about  7  or  8  ins 
iam;  15  to  18  inch  stroke;  ordinary  working  pressure  50  to  80  Ibs  per  sq  inch,  according  to  hardness 
of  material.     Cost  $8000  to  $12000.     Will  raise  as  an  average  days'  work  ^10  hours)  from  200  to  500 
yds  in  place,  or  280  to  700  in  the  scow,  according  to  the  depth,  nature  of  the  material,  Ac.     Require 

5  or  7  men  in  all  aboard,  including  cook.     Burn  J^  to  1  ton  of  coal  daily.     Tolerably  large  bowlders, 
and  sunken  logs,  can  be  raided  by  the  dipper.* 

When  the  material  is  hard  and  compacted,  the  buckets  of  dredgers  should  be  armed  with  strong 
steel  teeth  projecting  from  their  cutting  edge.  On  arriving  at  such  material,  every  alternate  bucket 
is  sometimes  unshipped.  By  arranging  the  buckets  so  as  to  dredge  a  few  feet  in  advance  of  the  hull, 
low  tongues  of  dry  land  may  be  cut  away  ;  the  machine  thus  digging  its  own  channel.  The  dailj 
work  iu  such  cases  will  not  average  half  as  much  as  in  wet  soil. 

On  small  operations,  dredgers  worked  by  two  or  more  horses, 

instead  of  by  steam,  will  answer  very  well  in  soft  material ;  or  e.ven  in  moderately  hard,  by  reducing 
the  size  and  number  of  the  buckets.  "  A  two-horse  machine  will  raise  from  50  to  100  yards  of  ordinary 
mud  in  place,  or  70  to  140  in  the  scow,  per  day,  at  from  12  to  15  ft  depth. 

Soft  material  in  small  quantity,  and  at  moderate  depth,  may  be  removed  by  the 
•low  and  expensive  mode  of  the  bag-scoop,  or  bag-spoon. 

This  is  simply  a  bag  B,  made  of  canvas  or  leather,  and  having  Its 
mouth  surrounded  by  au  oval  iron  ring,  the  lower  part  of  which  ia 
sharpened  to  form  :i  cutting  ed?e.  It  has  a  fixed  handle  h.  and  a 
swivel  handle  ».  One  man  pushes  the  bag  down  into  the  mud  by  ft, 
while  another  pulls  it  along  by  the  rope  g;  and  when  filled,  another 
raises  it  bv  the  rope  c.  and  empties  it.  If  the  bag  is  large,  a  wind- 
lass may  he  used  f  >r  raising  it.  The  men  may  work  from  a  scow  or 
raft  properlv  anchored.  Or  a  long-handled  metal  spoon,  shaped  like 
a  deeply-dished  hoe.  mav  be  used  by  only  one  man  ;  or  a  larger  spoon 
may  be  (ruined  by  a  man,  and  dragged  forward  and  backward  by  a 
horse  walking  in  a  circle  on  the  scow,  &c,  &c. 

The  weight  of  a  ciib  yd  <>f  wet  dredged  mud,  pure  sand,  or  gravel,  averages 
about  \YZ  tons;  say  111  Ths  per  cub  ft:  muddy  gravel,  full  l]/£  tons;  say  l'2-i  IDS  per 
cub  ft.  l'ur<;  sand  or  gravel  dredge  easily;  also  bed.s  of  shells  Wet  dredged  clay 
will  slide  down  a  shoot  inclined  at  from  1  to  ft.  to  1  to  3,  according  to  Us  freedom 
from  sand.  £c  ;  but  wet  sand  or  gravel  will  not  slide  down  even  ii  to  1,  without  a  free 
flow  of  water  to  ai-1  it;  otherwise  it  requires  much  pushing. 


•)f  The  writer  has  seen  cases  in  which  a  circular  saw  for  logs  in  deep  water,  would  have  been  a 
very  useful  addition  to  a  dredger.  It  should  be  worked  by  steam;  and  be  adjustable  to  different 
depths.  It  would  cost  but  about  $500. 

The  American  Dredging  C'O,  No.  10  8.  Delaware  Avenue,  Philada,  make  both 
dredgers  and  pile  drivers  of  many  patterns ;  and  contract  for  dredging  and  piling  on  any  scale. 


RETAINING-WALLS. 


331 


RETAINIM-WALLS, 


Art.  1.  We  here  speak  only  of  walls  sustaining  earth;  for  those  sustaining 
water,  see  Hydrostatics.  A  retaining -wall  is  one  for  sustaining  the  pres 
of  earth,  sand,  or  other  filling  or  backing,  deposited  behind  it  after  it  is  built ;  in 
distinction  to  a  face- wall,  which  is  a  similar  structure  for  preventing  the  fall  of 
earth  which  is  in  its  undisturbed  natural  position,  but  in  which  a  vert  or  inclined 
face  has  been  excavated.  The  earth  is  then  in  so  consolidated  a  condition  as  to  exert 
little  or  no  lateral  pres,  and  therefore  the  wall  may  generally  be  thinner  than  a 
retaining  one. 

This,  however,  will  depend  upon  the  nature  and 
position  of  the  strata  in  which  the  face  is  cut.  If 
the  strata  are  of  rock,  with  interposed  beds  of  clay, 
earth,  or  sand  ;  and  if  they  dip  or  incline  toward  the 
wall,  it  may  require  to  be  of  far  greater  thickness 
than  any  ordinary  retaining- wall ;  because  when  the 
thin  seams  of  earth  become  softened  by  infiltrating 
rain,  they  act  as  lubrics,  like  soap,  or  tallow,  to  fa- 
cilitate the  sliding  of  the  rock  strata ;  and  thus  bring 
an  enormous  pres  against  the  wall.  Or  the  rock  may 
be  set  in  motion  by  the  action  of  frost  upon  the  clay 
seams ;  or,  as  sometimes  occurs,  by  the  tremor  pro- 
duced by  passing  trains.  Even  if  there  be  no  rock, 
still  if  the  strata  of  soil  dip  toward  the  wall,  there 
will  always  be  danger  of  a  similar  result;  and  addi- 
tional precautions  mast  be  adopted,  especially  when 
the  strata  reach  to  a  much  greater  height  than  the 
wall.  A  vertical  wall  has  both  c  o 
and  d  »  vert. 

Experience,  rather  than  theo- 
ry, must  be  our  guide  in  the  building  of 
both  kinds  of  wall.  We  recommend  that 
the  hor  thickness  a  6,  Fig  1,  at  the  base  of  a 
vert  or  nearly  vert  retaining-wall  c  d  b  a, 

which  sustains  a  backing  of  either  s;md,  gravel,  or  earth,  level  with  its  top  c  d, 
as  in  the  fig,  should  not  be  less  than  the  following,  in  railroad  practice,  when  the 
foundations  are  not  more  than  about  three  feet  deep. 

When  the  backing?  is  deposited  loosely,  as  usual,  as  when 
dumped  from  carts,  cars,  &c. 

Wall  of  cut-stone,  or  of  Jirst-class  large  ranged  rubble, 

in  mortar a.b 35  of  its  entire  vert  height  d  b. 

;      good  common  xcabbled  mortar-rubble,  or  brick.  A  "  "       '• 

"      well-scabbled  dry  rubble 5  "  "  "       " 

With  good  masonry,  however,  we  may  take  the  height  d  8  instead  of  d  b,  and  then 
the  above  proportions  of  d  s  will  give  a  sufficient  thickness  at  the  ground-line  o  s. 
See  Table,  p  338. 

When  the  backing  is  somewhat  consolidated  in  hor  layers, 

each  of  these  thicknesses  may  be  reduced,  but  no  rule  can  be  given  for  this. 

The  offset  o  e,  in  front  of  the  wall,  is  not  included  in  these  thicknesses. 

When,  however,  the  backing  is  a  pure  clean  sand,  or  gravel,  we  should  use  only  the  full  dimen- 
sions; inasmuch  as  the  tremor,  caused  by  passing  traius,  would  neutralize  any  supposrd  advantage 
from  ramming  materials  so  devoid  of  cohesion.  Such  sand  may  be  rammed  with  much  advantage 
for  the  purpose  of  compacting  it  in  foundations;  but  a  diff  principle  is  involved  in  that  cuse.  When 
it  is  done  even  with  cohesive  earths,  with  a  view  of  saving  masonry  in  retaininp-walls.  it  is  probable 
that  the  expense  will  generally  be  found  quite  equal  to  that  of  the  masonry  saved.  See  Rem  4,  p  339. 

The  base  ah  in  Fig  1.  is  J^  of  the  height  bd.  In  the  foregoing  thicknesses  at  base,  the  back  d  b 
of  the  wall  is  supposed  to  be  vert;  and  the  face  ca  either  vert,  or  battered  (sloped  or  inclined  back- 
ward) to  an  extent  not  exceeding  about  1^  inches  to  a  foot;  which  limit  it  is  rarely  advisable  to  PX- 
ceed  in  practice,  owing  to  the  bnd  effect  of  rain,  &c.  upon  the  mortar  when  the  batter  is  great.  The 
base  of  a  vert  wall  need  not  in  fact  be  as  thick  as  one  with  a  battered  face ;  but  when  the  batter  does 
not  exceed  1.5  inches  to  a  foot,  the  diff  is  very  small.  See  Table,  Art  7  p  338. 

REM.  1.  A  mixture  of  sand,  or  earth,  with  a  larg-e  proportion 

OP  ROUND  BOWI.DKRS.  paving  pebbles,  Ac,  will  welch  considerably  more  thnn  the  material*  ordinarily 
used  for  backing;  and  will  exert  a  greater  pres  aeainst  the  wall :  the  thickness  of  which  should  be 
increased,  say  about  J^th  to  %th  part,  when  such  backing  has  to  be  used. 

RKM.  2.  The  wall  will  be  stronger  if  all  the  courses  of  masonry  be  laid 
with  an  inclination  inward,  as  at  oe.b-,  especially  if  of  dry  masonry, 
or  if  time  cannot  be  allowed  (as  it  always  should  be,  when  practicable)  for  the  mor- 
tar to  set  properly,  before  the  backing  is  deposited  behind  it.  The  object  of  inolin- 


332 


RETAINING-WALLS. 


ing  the  courses,  is  to  place  the  joints  more  nearly  at  right  angles  to  the  direction 
/P,  Figs  «.  7,  and  8,  of  the  pres  against  the  back  of  the  wall ;  and  thus  diminish 
the  tendency  of  the  stones  to  slide  on  one  another,  and  cause  the  wall  to  bulge.  See 
Art  19  of  Force  in  Rigid  Bodies.  When  the  courses  are  hor,  there  is  nothing  to  pre- 
vent this  sliding,  except  the  friction  of  the  stones,  one  upon  the  other,  when  of  dry 
masonry ;  or  friction  and  the  mortar,  when  the  last  is  used.  But  if,  as  is  frequently 
the  case,  (especially  in  thick  and  hastily  built  walls,)  this  has  not  had  time  to  harden 
properly,  it  will  oppose  but  little  resistance  to  sliding.  But  when  the  courses  are 
inclined,  they  cannot  slide,  without  at  the  same  time  being  lifted  up  the  inclined 
planes  formed  by  themselves.  In  retaining-walls,  as  in  the  abuts  of  important 
arches,  the  engineer  should  place  as  little  dependence  as  possible  upon  moriar;  but 
should  rely  more  upon  the  position  of  the  joints,  fur  stability. 

An  objection  to  this  inclining  of  the  joints  in  dry  (without  mortar)  walls,  is  that  rain-water,  falling 
on  the  battered  face,  is  thereby  carried  inward  to  the  earth  backing:  which  thus  becomes  soft,  and 
settles.  This  may  be  in  a  great  measure  obviated  by  laying  the  outer  or  face-courses  hor;  or  by 
using  mortar  for  a  depth  of  ouly  about  a  foot  from  the  face.  The  top  of  the  wall  should  be  protected 
by  a  coping  c  d,  Fig  1,  which  had  better  project  a  few  ins  in  front.  After  the  masonry  has  been 
built  up  to  the  surface  of  the  ground,  the  foundation  pit  should  be  filled  up;  and  it  is  well  to' con- 
solidate the  filling  by  ramming,  especially  in  front  of  the  wall. 

The  back  d  b  of  the  wall  should  be  left  rough.    In  brickwork  it 

would  be  well  to  let  every  third  or  fourth  course  project  an  inch  or  two.  This  increases  the  friction 
of  the  eurth  against  tlie  back,  and  thus  causes  the  resultant  of  the  forces  acting  behind  the  wall  to 
become  more  nearly  vert;  and  to  fall  farther  within  the  base,  giving  increased  stability.  It  also  con- 
duces to  strength  not  to  make  each  course  of  uniform  height  throughout  the  thickness  of  the  wall; 

three  courses.  By  this  means  the  whole  masonry  becomes  niore'efl'tctually  interlocked  or  bonded 
together  as  one  mass:  and  therefore  less  liable  to  bulge.  Very  thick  walls  may  consist  of  a  facing 
of  masonry,  and  a  backing  of  concrete. 

REM.  3.  It  is  the  pres  itself  of  the  earth  against  the  back,  that  creates  the  friction,  which  in  turn 
modifies  the  action  of  the  pres  ;  as  the  wt  or  pres  of  a  body  upon  an  inclined  plane  produces  friction 
between  the  body  and  the  plane,  sufficient,  perhaps,  to  prevent  tlie  body  from  sliding  down  it.  A  re- 
taining-wall  is  overthrown  by  being  made  to  revolve  around  its  outer  toe  or  edge  e  Fig  1.  as  n.  ful- 
crum, or  turning-point;  but 'in  order  thus  to  revolve,  its  back  must  first  plainly  rise;  and  in  doing 
so  must  rul)  against  the  backing,  and  thus  encounter  and  overcome  this  triction.  The 
friction  exists  the  same,  whether  the  wall  stands  firm  or  not;  as  in  the  case  of  the 
body  on  an  inclined  plane  ;  the  only  did'  is  that  in  one  case  it  prevents  motion  ;  and 
in  the  other  ouly  retards  it. 

Where  deep  freezing:  occurs  the  back  of  the  wall  should 

be  sloped  forwards  for  3  or  4  ft  below  its  top  as  at  c  o,  which  should  be  quite  smooth 
so  as  to  lessen  the  hold  of  the  frost  and  prevent  displacement. 

REM.  4.  When  the  wall  is  too  thin,  it  will  generally  fail 
by  bulging?  outward,  at  about  ^  of  its  height  above  the 
ground,  as  at  a,  in  Fig  -J.  A  slight  bulging  in  a  new  wall 
does  not  necessarily  prove  it  to  be  actually  unsafe.  It  is 
generally  due  to  the  newness  of  the  mortar,  and  to  the 
greater  pres  exerted  by  the  fresh  backing;  and  will  often 
cease  to  increase  after  a  few  months.  It  need  not  excite 
apprehension  if  it  does  not  exceed  ^  inch  for  each  foot  in 
thickness  at  a.  S^e  Remark  3.  Art  7,  p  339. 

>  The  young  engineer  need  not  in  practice  concern  himself  particularly  about  the  PRECIS- 
SP  GRAY  OF  HIS  BACKING,  or  about  the  ANGLE  OF  SLOPS:  at  which  it  will  stand  ;  for  the  material  which 
he  deposits  behind  his  wall  one  day,  may  be  dry  and  incoherent,  so  HS  to  slope  at  1%  to  1  ;  the  next 


he  deposits  behind  his  wall  one  day,  may  be  dry  and  incoherent,  so  HS  to  slope  at  1%  to  1  ;  the  next 
day  rain  may  convert  it  into  liquid  mud.  seeking  its  own  level,  like  water ;  the  next  it  may  be  ice. 
capable  of  sustaining  a  considerable  load,  as  a  vert  pillar. 

Moreover,  he  cannot  foretell  what  may  be  the  nature  of  his  backing;  for.  as  a  general  rule,  this- 
must  consist  of  whatever  the  adjacent  excavation  may  produce  from  time  to  time :  sand  to-day,  rock 
to-morrow,  &c.  Retaining-walls  are  therefore  usually  built  before  the  engineer  knows  the  character 
of  their  backing;  so  that  in  practice,  these  theoretical  considerations  have  comparatively  but  little 
weight.  Theory,  uncontrolled  by  observation  and  common  sense,  will  lead  to  great  errors  in  every 
department  of  engineering ;  but,  on  the  other  hand,  no  amount  of  experience  alone  will  compensate 
for  an  ignorance  of  theory.  The  two  must  go  hand-in-hand. 

Again,  the  settlement  of  the  backing  under  its  own  wt,  aided 
by  the  tremors  produced  by  heavy  trains  at  high  speed;  its  expansion  by  frost,  or 
by  the  infiltration  of  rain;  the  hydrostatic  pressure  arising  from  the  admission  of 
the  latter  through  cracks  produced  in  the  backing  during  long  droughts ;  as  well  as 
its  lubricating  action  upon  it,  (diminishing  its  friction,  and  giving  it  a  tendency  to 
slide,)  &c,  exert  at  times  quite  as  powerful  an  overturning  tendency  as  the  legitimate 
theoretical  pres  does.  The  action  of  these  agencies  is  gradual.  Careful  observation 
of  retaining-walls  year  after  year,  will  often  show  that  their  battered  faces  are  be- 
coming vertical.  Then  they  will  begin  to  incline  outward;  and  eventually  the  wall 
will  fail.  Theory  omits  loads  that  may  come  on  backing  increasing  its  pres. 


RETAINING- WALLS. 


333 


Assuming  the  theoretical  views  advanced  by  Professor  Moseley  to  be  correct  as 
theories,  the  thicknesses  which  we  have  recommended  in  Art  1,  for  mortar  walls, 
correspond  to  from  7  to  14  times;  and  for  dry  walls  about  10  to  20  times,  the  pros' 
assigned  by  him  ;  and  we  do  not  consider  ours  greater  than  experience  has  shown 
to  be  necessary.  See  Table  3.  Retaining-walls  designed  by  good  engineers,  but  in 
too  close  accordance  with  theory,  (which  assumes  that  a  resistance  equal  to  twice 
the  theoretical  pres  is  sufficient,)  have  failed;  and  the  inference  is  fair  that  many  of 
those  which  stand  have  too  small  a  coefficient  of  safety. 

The  fact  is,  (or  at  least  so  it  appears  to  us,)  there  mast  be  defects  in  the  theoretical  assumptions  of 
•ome  of  the  most  prominent  writers  who  give  practical  rules  on  this  subject.  Thus  Poncelet,  who 
certainly  is  at  their  head,  states  that  his  tables,  for  practical  use,  give  thicknesses  of  base  for  sus- 
taining 1  j8j.  times  the  theoretrcal  pres  ;  and  this  he  considers  amply  safe.  Yet,  for  a  vert  wall  of  cut 
granite,  his  base  for  sustaining  dry  sand  level  with  the  top,  as  in  Fig  1,  is  .35  of  the  vert  height- 
and  for  brick,  .45.  But  the  writer  found  that  when  not  subject  to  tremor,  a  wooden  model  of  a  vert 
wall  weighing  but  28  fi>s  per  cub  ft,  and  with  a  base  of  .35  of  its  height,  balanced  perfectly  dry  sand 
sloping  at  1%  to  1,  and  weighing  89  fts  per  cub  ft. 


:n 


TARIES  AS  THEIR  SPECIFIC  GRAVITIES  ;  and,  since  granite  weighs  about  1H5 
Ibs  per  cub  foot,  or  6  times  as  much  as  our  model,  it  follows,  we  conceive, 
that  a  wall  of  that  material,  with  a  base  of  .35  of  its  height,  must  have 
a  resistance  of  6  times  any  true  theoretical  pres,  instead  of  only  1.8 
times;  and  that  his  brick  wall  must  have  about  5  times  the  mere"  bal- 
ancing resistance.  Our  experiments  were  made  in  an  upper  room  of  a 
strongly  built  dwelling  ;  and  we  found  that  the  tremor  produced  by  pass- 
ing vehicles  in  the  street,  by  the  shutting  of  doors,  and  walking  about 
the  room,  sufficed  to  gradually  produce  leaning  in  walls  of  considerably 
more  than  twice  the  mere  balancing  stability  while  quiet :  and  it  appears 
to  us  that  the  injurious  effects  of  a  heavy  train  would  be  comparatively 
quite  as  great  upon  an  actual  retaining-wall,  supporting  so  incohesive 
a  material  as  dry  sand. 

Since,  therefore,  Poncelet's  wall  is  in  this  instance  sufficiently  stable 
for  practice,  it  seems  to  us  that  his  theory,  which  neglects  the  effect  of 
tremors,  &c,  must  be  defective.  He  also  gives  4  of  the  height  as  a  suf- 
ficiently safe  thickness  for  a  vert  granite  wall  supporting  stiff  earth;  but 
we  suspect  that  very  few  engineers  would  be  willing  to  trust  to  that  pro- 
portion, when,  as  usual,  the  earth  is  dumped  in  from  carts,  or  cars;  espe- 
cially during  a  rainy  period.  If  deposited,  and  consolidated  in  layers, 
theory  could  scarcely  assign  any  thickness  for  the  wall ;  for  the  backing  thus  becomes,  as  it  were,  » 
mass  of  unburnt  brick,  exerting  no  hor  thrust ;  and  requiring  nothing  but  protection  from  atmospheric 
influence,  to  insure  its  stability  without  any  retaining-vaill.  It  is  with  great  diffidence,  and  distrust 
in  our  opinions,  that  we  venture  to  express  doubts  respecting  the  assumptions  of  so  profound  an  in- 
vestigator and  writer  as  Poncelet;  and  we  do  so  only  with  the  hope  that  the  views  of  more  compe- 
tent persons  than  ourselves,  may  be  thereby  elicited.  Our  own  have  no  better  foundation  than  ex- 
periments with  wooden  and  brick  models,  by  ourselves  ;  combined  with  observation  of  actual  walls. 

Art.  3.  After  a  wall  a  b  c  o,  Fig  3,  with  a  vert  back,  has  been  proportioned  by 
our  rule  in  Art  1,  it  may  be  converted  into  one.  with  an  oflfsetted 

back,  as  a  i  n  o.  This  will  present  greater  resistance  to  overturning;  and  yet  con- 
tain no  more  material.  Thus,  through  the  center  t  of  the  back,  draw  any  line  i  n ; 
from  n  draw  n  .<?,  vert;  divide  i  s  into  any  even  number  of  equal  parts;  (in  the  fig 
there  are  4 ;)  and  divide  s  n,  into  one  more  equal  parts ;  (in  the  fig  there  are  5.)  From 
the  points  of  division  draw  hor,  and  vert  lines,  for  forming  the  offsets,  as  in  the  fig. 

In  the  offsetted  wall,  the  cen  of  grav  is  thrown  farther  back  from  the  toe  o,  than 
in  the  other,  thus  giving  it  increased  leverage  and  resistance;  but  within  ordinary 
practical  limits,  the  diff  is  very  small ;  and  since  the  triangle  of  supported  earth  is 
greater  than  when  the  back  is  vert,  its  pres  is  also  greater;  so  that  probably  no  ap- 
preciable advantage  attends  that  consideration.  The  increase  of  thickness 
near  the  base,  diminishes,  however,  the 
leverage  v  a,  Fig  8,  of  the  pres  /P,  of  the 
earth  against  the  back.  The  center  of  pressure  of 
this  pres  is  in  both  cases  at  %  the  vert  'height,  meas- 
ured from  the  bottom;  and  it  is  therefore  plain  that 
the  farther  back  from  the  front  it  is  applied,  the  shorter 
must  v  a  become.  Moreover,  in  the  offsetted  back,  the 
direction  of  the  pres  becomes  more  nearly  vert  than 
when  the  back  is  upright.  It  is  to  these  causes,  rather 
than  to  the  throwing  back  of  the  cent  of  grav,  that 
the  offsetted  wall  owes  its  increase  of  stability  over 
one  with  a  vert  back. 

Art.  4.  When,  as  in  Fig  4,  the  backing  is  higher  than  the 
wall,  and  slopes  away  from  its  inner  edge  d,  at  the  natural  slope  d  s,  of  1%  to  1,  we 
are  confident  that  the  following  thicknesses  at  base  will  at  least  be  found  sujficie.nt 

22 


334 


RETAINING-WALLS. 


for  vert  walls  with  sand.  They  are  deduced  from  the  experiments  just  alluded  to, 
and  are  but  rude  approximations,  with  no  scientific  basis.  We  should  not  have  in- 
serted them,  but  for  the  fact  that  we  know  of  no  others  for  this  case.  See  p  337. 

The  first  column  contains  the  vert  height  s  v,  of  the  earth,  as  compared  with  the 
vert  height  of  the  wall  ;  which  latter  is  assumed  to  be  1  ;  so  that  the  table  begins 
with  backing  of  the  same  height  as  the  wall,  as  in  Fig  1.  These  vert  walls  may  be 
changed  to  others,  with  battered  faces,  by  Art  8  ;  or  without  any  such  proceeding, 
their  faces  may  be  battered  to  any  extent  not  exceeding  1%  inches  to  a  foot,  or  1  in 


8,  without  sensibly  atfecting  their  stability,  without  increasing  the  base. 
TABLE   1.    (Original.) 


ill 

i 

||| 

Us 

Wall 
of 

Good 

Mortar 

Wall 

ll| 

Wall 
of 

Good 
Mortar 

Wall 

o£<2 

Cut  Stone, 

Rubble, 

of 

o3.§ 

Cut  Stone 

Rubble, 

of 

115 

in 
Mortar. 

Brick. 

good  dry 
Rubble.      | 

111 

in 
Mortar. 

or 
Brick. 

good  drv 
Rubble'. 

-s; 

IS! 

IP 

|f* 

£8-0 

1. 

.35 

.40 

.50 

2. 

.58 

.63 

.73 

1.1 

.42 

.47 

.57 

2.5 

.60 

.65 

.75 

1.2 

.46 

.51 

.61 

3. 

.62 

.67 

.77 

1.3 

.49 

.54 

.64 

4. 

.63 

.68 

.78 

1.4 

.51 

.56 

.66 

6. 

.64 

.69 

.79 

1.5 

.52 

.57 

.67 

9. 

.65 

.70 

.80 

1.6 

.54 

.59 

.69 

14. 

.66 

.71 

.81 

1.7 

.55 

.60 

.70 

25. 

1.8 

.56 

.61 

.71 

or  more 

.68 

.73 

.83 

Art.  5.  But  when  the  slope  n  rt  Fig  5,  of  1%  to  1,  starts  from  the  outer  edge  n 
of  the  wall,  greater  thickness  is  required.  Poncelet  gives  the  following  for  this 
case,  for  dry  sand. 

TABLE    2. 


Wall 

of 
Cut  Stone 

in 
Mortar. 


.35 

.393 

.439 

.485 

.532 

.579 

.617 

.645 


.707 


if 

«5 

Wall 

•55 

Wall 
of 

Wall 

of 

•°  '£ 

Cut  Stone 

of 

Brickwork. 

g"g 

in 

Brickwork. 

&  = 

Mortar. 

.452 

.498 
.548 

2.4 
3.0 
4.0 

.762 
.811 
.852 

1.02 
l.H 
1.18 

.604 

6.0 

.883 

1.25 

.665 

11.0 

.909 

1  28 

.726 

21.0 

.922 

1.31 

.778 

31.0 

.926 

l.:52 

.824 

Infinite. 

.934 

1.34 

.847 

.903 
.930 

When  the  earth  reaches  above  the  top  of  the  wall,  as  in  Figs  4  and  5,  the  wall  is  Surcharged  ; 

and  the  earth  that  is  above  the  top,  is  called  the  SURCHARGE.  When  the  surcharge  is  carefully  deposited 
above  the  wall,  so  as  to  slope  back  at  a  steeper  angle  than  1^  to  1,  as  say  at  1  to  1,  theory  does  not 
require  the  wall  to  be  as  thick.  Notwithstanding  Poncelet's  high  position,  the  writer  cannot  imagine 
that  the  base  of  a  brick  wall  need  be  so  great  as  1%  times  its  height  for  any  height  of  sand  whatever. 

Art.  6.    On  the  theory  of  re  tain  ing- walls.    Let  b  c  a  m,  Fig  6,  be 

such  a  wall,  upholding  backing  or  filling  csmg;  the  upper  surf  c  *  of  which  is 
hor,  and  level  with"  the  top  b  c  of  the  wall ;  and  let  ra  s  represent  the  nat  slope  of  the 
earth  which  composes  the  backing;  mg  being  hor. 

Abundant  experience  on  public  works  shows  that  this  slope,  whether  for  sand,  gravel,  or  earth, 
when  dry,  may  be  practically  taken  at  1%  to  1 ;  that  is,  1J$  hor.  to  1  of  vert  measurement;  which 
corresponds  to  an  angle  sm q  of  33°  41'  with  the  hor:  which  is  also  about  the  angle  at  which  bricks 
and  roughlv  dressed  masonrv  begin  to  slide  on  each  other.  This  angle,  however,  varies  considera- 


RETAINING- WALLS. 


335 


bly  ;  being  greatly  influenced  by  the  degree  of  dryness,  or  dampness,  of  the  material ;  so  that  mode- 
rately damp  sand  or  earth  will  stand  at  a  slope  of  1  to  1,  or  at  an  angle  of  45°.  Whatever  it  may  be, 
it  is  called  THE  ANGLE  OF  NAT  SLOPE  of  the  material  under  consideration.  In  theoretical  calculations 
for  walls,  it  is  safest  to  assume  (as  we  have  done  throughout)  that  the  backing  is  perfectly  dry,  since 


Fitf.7 


Its  pre«  is  then  greatest ;  unless  it  be  supposed  to  be  so  wet  as  to  possess  some  degree  of  fluidity.  The 
triangle  cms  of  earth  above  the  nat  slope  ma,  tends  to  slide  down  said  slope,  but  is  prevented  from 
BO  doing  by  the  wall. 

It  is  assumed  in  all  cases,  that  the  wall  is  secured  from  sliding  along  its  base,  see  Art  9;  that  it  is 
thick  enough  to  prevent  failure  by  bulging ;  and  that  it  will  fail  only  by  overturning,  by  rotating 
around  its  toe,  a,  as  a  fulcrum.  The  thickness  necessary  to  insure  safety  against  the  last  will  also  be 
sufficient  to  prevent  bulging.  Now  referring  only  to  Fig  6  with  a  vert  back,  if  the  angle  o  ma,  con- 
tained between  the  natural  slope  m  a,  and  a  vert  line  m  o,  drawn  from  the  inner  bottom  edge  m  of 
the  wall,  be  divided  by  a  line  m  t,  into  two  equal  angles,  o  m  t,  t  m  t,  then  the  angle  o  in  t  is  called 
THE  ANGLE,  and  m  t  THE  SLOPE,  op  MAXIMUM  PRESSURE.  The  triangular  prism  of  earth,  of  which 
o  m  t  is  a  section,  or  an  end  view,  is  called  THE  PKISM  OF  MAX  PRES  ;  because,  if  considered  as  a  wedge 
acting  against  the  back  of  the  wall,  it  would  produce  a  greater  pres  upon  it  than  wou»d  the  entire 
triangle  c  m  «  of  earth,  considered  as  a  single  wedge.  For  although  the  last  is  the  heaviest,  yet  it  is 
more  supported  by  the  earth  below  it.  Calculation  shows  that  if  we  consider  the  earth  o  m  a  to  be 
thus  div  into  wedges  by  any  line  m  t,  the  wedge  that  will  press  most  against  the  wall  is  that  formed 
when  m  t  divides  the  angle  o  m  t,  or  the  arc  o  i,  into  two  equal  parts. 

Since  mg  is  hor,  and  mo  vert,  the  two  form  an  angle  of  90°:  consequently  the  angle  of  max  pres 
is  plainly  found  by  taking  the  angle  smg  of  nat  slope  from  90°,  and  div  the  rem  by  2.  Thus  a  nat 

elope  of  IJi  to  1,  or  33°  41',  taken  from  90°,  leaves  56°  18';  and  J^ll?l.  —   28°  9'    the  COr- 

responding  angle  o  m  t  of  max  pres. 

For  ease  of  calculation,  only  one  foot  of  the  length  of  the  wall,  and  of  its  backing,  is  usually  con- 
sidered. The  number  of  cub  ft  of  wall,  or  of  backing,  is  then  equal  to  that  of  the  square  feet  in 
their  respective  profiles,  or  cross-sections. 

Now,  according  to  Moseley,  if  we  assume  the  particles  of  earth  composing  the 
backing  to  be  perfectly  dry,  and  devoid  of  cohesion,  (or  tendency  to  stick  to  each 
other,)  which  is  very  nearly  the  case  in  pure  sand;  and  if  we  suppose  the  wall  to  be 
suddenly  removed,  then  the  triangle  of  earth  cm£,  comprised  between  the  slope  mt 
of  max  pres,  and  the  vert  back  c  m  of  the  wall,  Fig  6,  would  slide  down,  under  the  in- 
fluence of  a  force  which  may  be  represented  by  y  P,  acting  in  a  direction  y  P,  at  right 
angles  to  the  face  c  m  of  the  triangle  of  earth  ;  (or  in  other  words,  at  right  angles 
to  the  back  of  the  vert  wall,)  its  center  of  force  being  at  P,  distant  %  way  between 
m  and  c,  measured  from  the  bottom ;  and  its  amount  equal  to  either  of  the  following: 

I  of  the  triangle  of  earth  c  m  t  X  o  t 
~  ~y*p"  vert  depth  o  m  ~~ '  or 

Wt  of  a  single  cub  . 
jyo  %9  P«rp  pres       ft  of  the  backing  > 

yP  =_  _ 

In  view  of  the  great  uncertainty  involved  in  the  matter  of  the  actual  pressure  of 
earth  against  retainiug-walls  in  practice  (see  Art  2,  p  332),  arid  in  order  to  furnish 
a  simple  rule  which,  although  entirely  unsupported  by  theory,  is  still  (in  the  writer's 
opinion)  sufficiently  approximate  for  ordinary  practical  purposes,  we  shall  assume 
that  No  1  of  the  two  foregoing  formulas  applies  near  enough  to  walls  with  in- 
clined backs  c  w,  also,  as  Figs  7  and  8,  (precisely  as  they  are  lettered,)  at  least 
until  the  back  of  the  wall  inclines  forward  as  much  as  tt  ins 
hor,  to  1  foot  vert,  or  at  an  angle  cmo  of  26°  34'.  What  follows  on 
retaiiiing-walls  will  involve  this  incorrect  assumption,  and 
must  be  regarded  merely  as  giving  safe  approximation. 

Some  appear  to  assume  this  perp  pres  to  be  the  only  one  acting  against  the  back 
of  the  wall;  and  hence  arrive  at  erroneous  practical  conclusions.  For  when,  in 
order  to  prevent  this  force  from  causing  the  triangle  of  earth  to  slide,  we  place  a 
retaining-wall  in  front  of  it,  then,  instead  of  motion,  the  force  will  produce  pres  of 
the  earth  against  the  wall ;  (see  Art  3,  of  Force,  p  445.)  But  in  producing  pres,  it 


r 
1  °>f 


336 


RETAINING- WALLS. 


C  0 


necessarily  produces  the  new  force  of  friction,  between  the  pressed  surfaces  of  the 
earth  and  wall.  That  is,  it'  a  wall  were  to  begin  to  overturn  around  its  toe  a  as  a 
fulcrum,  its  back  c  m  must  of  course  rise,  and  iu  so  doing  must  rub  against  the 
earth  filling  in  contact  with  it;  and  this  rubbing  would  evidently  act  to  impede  the 
overturning.  So  long  as  the  wall  does  not  move,  the  same  friction  assists  in  pre- 
venting overturning.  To  ascertain  the  amount  and  effect  of  this  friction,  let  ;/P,  Fig 
8,  represent  by  scale,  the  force  perp  to  the  back  c  m;  and  supposed  to  have  been  pre- 
viously calculated  by  the  foregoing  formula  No  1.  Make  the  angle  yPf  equal  to 
the  angle  of  wall  friction,*  draw '  yf  at  right  angles  to 
y  P,  or  parallel  to  m  c ;  make  P  x  equal  to  yf,  and  com- 

Elete  the  parallelogram  P  yfx.    Then  will  x  P  represent 
y  the  same  scale,  the  amount  of  the  friction 
ag-aiiist  the  back  of  the  wall.    Since  the  fric- 
tion acts  in  the  direction  of  the  back  c  m,  (see  end  of  Art 
62,  of  Force  in  Rigid  Bodies,)  it  may  be  considered  as 
acting  at  any  point  P,  in  that  line ;  (see  Art  18,  of  Force 
at  page  J5-.)     Hence  we  have  acting  at  P,  two  forces  ; 
namely,  the  perp  force  y  P,  and  the  friction  x  l>;  conse- 
quently, by  comp  and  res  of  force,  the  diag/P  of  the 
parallelogram  P  yfx,  if  measured  by  the  same  scale,  will 
give  us  the  amount  of  their  resultant ;  which  is  the 
approx  single  theoretical  force,  both  in 
amount  and  in  direction,  which  the  wall 
has  to  resist,  including  the  wall  friction. 
o          But  this  force, /P,  is  also  always  equal  to  the  perp 
H  O«   force  y  P,  mult  by  the  nat  sec  of  the  angle  y  Pf  of 
sa          the  wall  friction ;  (or  divided  by  its  nat  cosine)  and  of 
\  course  may  be  ascertained  thus : 

vt  of  triangle  v      .  v  nat  sec  of  angle  y  P  f  wt  of  ^  A  . 

Approx  theoreti    _          c  m  t          *  °  *  *        of  wall  friction  cmt*     l 


cal  pros  f  P 


vert  depth  o  r 


cos  y  P  f  X  o  m 


Or  finally,  if  it  is  assumed,  as  we  do  throughout,  that  the  earth  is  perfectly  dry  (in 
asmuch  as  its  pressure  is  then  the  greatest)  and  that  the  angles  of  nat  slope,  and 
of  wall  friction  are  then  each  33°  41'  or  1.5  to  1,  then  in  Figs  6,  7  and  8,  if  the  angle 
cm  n  between  the  back  c  m  and  the  vert  o  m  does  not  exceed  about  26°  34'  we  may 
assume 


Approx  theoretical 
pres  f  P 


=  wt  of  triangle  c  m  t  x  .643 


which  includes  the  action  of  the  friction  of  the  earth  against  the  back  of  the  wall. 

REM.  1.    When  the  back  of  the  wall  is  offset  ted  or  stepped,  as 

in  Fig  3,  instead  of  being  simply  battered,  as  in  Figs  7  and  8,  the  direction  of  the 
pres  of  the  earth  will  be  the  same  as  if  the  back  had  the  batter  i  n,  on  the  principle 
given  in  Art  34,  Fig  17,  of  Force  in  Rigid  Bodies,  p  464. 

REM.  2.  Now  to  find  both  the  overturning-  tendency  of  the 
earth,  and  the  resistance  of  the  wall  against  being  overturned  around  its  toe  a  as 
a  fulcrum,  first  find  the  ceu  of  grav  g  of  the  wall  (p  442),  and  through  it  draw  a 
vert  line  g  h.  Prolong /P  towards  rand  draw  a  v  perp  to  it.  By  any  scale  make 
«o  =  wt  of  wall,  and  si  =  calculated  pres /P.  Complete  the  parallelogram  sino, 
and  draw  its  diagonal  *  n,  which  will  be  the  resultant  of  the  pres/P  and  of  the  wt 
of  the  wall ;  and  should  for  safety  be  such  that  aj  be  not  less  than  about  one-fifth 
of  a  m,  even  with  best  masont-y  and  unyielding  soil.  Otherwise  the  great  pressure  so 
n^ar  the  toe  a  may  either  fracture  the  wall  or  compress  the  soil  near  that  point 
so  that  the  wall  will  lean  forward.  In  walls  built  by  our  rule,  Art  1,  or  by  table, 
p  338,  aj  will  be  more  than  one-fifth  of  a  m.  The  pres  /  P  if  mult  by  its  leverage 
a  v  will  give  the  moment  of  the  pres  about  a;  and  the  wt  of  the  wall  mult  by  its 
leverage  e  a  will  give  that  of  the  wall.  The  wall  is  safe  from  overturning  in  pro- 
portion as  its  moment  exceeds  that  of  the  pres.  It  is  assumed  to  be  safe  against 
sliding,  breaking,  or  settling  into  the  soil.  See  Art  15,  p  530. 


*This  aiigrle  of  wall  friction  is  that  at  which  a  plane  of  masonry  must 

b°  inclitip'1  to  the  horizontal  so  that  dry  sand  or  earth  would  slide  down  it.     It  is  about  the  same  as 
the  oat  slope,  or  33°  41',  or  1.5  to  1 ;  and  its  oat  secant  is  1.202,  and  its  nat  cos  .832. 


RKTAINJNG-WALLS. 


337 


Item.  4.  If  the  earth  slopes  downward  from  0,  as 
at  A  or  B,  instead  of  being  hor  as  in  Figs  6,  7,  8,  use  the  wt  of  the 
earth  c  m  n  instead  of  c  m  t,  m  n  being  the  slope  of  max  pressure. 
In  A  the  point  of  application  will  still  be  at  P  (at  one  third  of 
m  C)  as  in  6,  7,  8 ;  but  in  B  it  will  be  a  little  higher  as  explained 
below  for  Fig  9. 

Surcharged  walls  are  those  in  which  the  earth  backing 
extends  above  the  tops  of  the  walls. 

According  to  theory,  when  as  in  Fig  9,  there  is  a  surcharge 
v  c  k  of  backing,  sloping  away  from  c  at  its  natural  slope  c  v, 
the  max  pres  against  the  wall  is 
attained  when  the  earth  reaches  to 
the  level  of  </,  where  the  slope  rntd 
of  max  pres  intersects  the  face  of  the 
nat  slope  cv;  so  that  if  afterward  the 
earth  is  raised  to  v,  or  to  any  greater 
height,  no  additional  pres  is  thereby 
thrown  against  the  back  of  the  wall. 
So  also  if  the  earth  slopes  from  b,  or 
from  between  c  and  6,  except  that 
then  the  slope  m  d  of  max  pres  must 
extend  up  to  meet  this  other  slope. 

The  appro x i mate  amount 

of  the  oblique  pres,  when  the  wall  is 
surcharged,  (as  in  any  of  the  Figs  4, 
5,  9,)  may  be  found  on  the  same  prin- 
ciple as  when  the  earth  is  level  with 
its  top;  namely,  instead  of  the  trian- 
gle cmt  of  earth,  Figs  6,  7,  8,  9,  find 
the  wt  of  alt  the  earth  d  .<?  m  t,  Fig  4, 
d  in  t  r,  Fig  o,  or  c  d  m.  Fig  9  (if  the 
surcharge  reaches  to  d  or  v,  or  higher), 

included  between  the  slope  m  d  of  max  pres,  the  back  of  the  wall,  and  the  front 
slope;  omitting,  however,  any  which,  like  den,  Fig  5,  rests  on  the  top  of  the  wall 
(and  thus  adds  to  its  stability)  when  the  slope  starts  in  front  of  c.  Having  found 
this  weight,  then  for  dry  hacking  the 


Fig  9 


y 


Jib 


a 


9aW  } 


ap,ro*m 


««•«  «» 


•  X  -6  13, 


including  the  action  of  the  friction  of  the  earth  against  the  hack  of  the  wall  ;  near 
enough  (in  the  writers  opinion)  for  practical  purposes  in  so  uncertain  a  matter; 
but  essentially  empirical. 

The  direction  of  the  pressure  thus  found  will  be  the  same  rs  when 
the  earth  is  level  with  the  top  be;  namely,  as  in  Figs  6  and  7.  first  d'aw  a  line,  as 
P  y.  perp  to  the  hack  c  in  whether  vert  or  inclined.  Then  draw  another  line,  as  P/, 
making  the  angle  t/P/=  the  angle  of  wall  friction,  which  we  all  along  assume  to 
be  3o°  41',  or  1.5  to  1.  But  it  is  of  the  utmost  importance  to  bear  in  mind  that  ils 
point  of  application  will  not  always  he  at  P  (one-third  of  the  height  of  the 
wall  above  m  )  as  heretofore;  for  in  all  cnses  it  will  be  at  that  point  P,  or  at  some 
higher  one  as  h,  where  the  hack  is  cut  by  a  line  /  P  or  el<,  Fig  I),  drawn  from  the 
cen  of  grav  of  the  sustained  earth  (omitting  any  that  rests  immediately  on  the  top 
be),  and  parallel  to  the  slope  md  of  max  pres:  and  such  a  line  will  strike  at  one- 
third  the  height  of  the  wail  only  when  ihe  sustained  earth  'cm  or  tic  m  form.**  a 
complete  triangle.  one  of  whose  angles  is  at  the  inner  top  edge  c  of  the  wall. 
In  all  other  cases  said  line  for  a  surcharge  will  strike  above  P. 

This  change  in  the  position  of  the  point  of  application,  serves  to  expl.Mn  the  !*reat  increase  of  thick. 
ness  required  to  meet  thefirtt  small  surcharges  of  earth  above  ttie  ton  of  (he  wall.  These  raise  the 
point  of  application  considerably  more  than  snh^eiiuent  ones  do.  Thus,  in  Fie  SI.  suppose  the  sur- 
charge to  reacli  only  to  n  r:  then  the  line  Ji  e.  drawn  throuch  the  cen  of  grav  r,r  the  earth  mnr  c  m, 
and  parallel  torn  d.  will  plainly  intersect  the  hack  c.  m  :>t  a  higher  point  h,  tlum  one  I.  P  drawn  through 
the  cen  of  grav  of  the  trianirle  of  earth  d  r.  m,  or  of  t  c  m:  which  latter  line  intersects  it  at  P,  at  one- 
third  of  it<  heisrht  from  the  ha-e  m  n.  So  that  with  the  small  surcharge,  we  have  not  only  the  pres 
of  a  greater  quantity  of  parth  than  t  c.  m.  hut  a  greater  leverage  is  given  to  K  tor  overthiowing  the 
wail.  But  when  'he  snrohavir"  i>s  made  to  reach  d.  then  the  earth  d  c  m.  being  like  t  c  m  a  triangle, 
the  line  through  it--  ren  r.f  grav  \v  II  strike  the  hick'-  m  at  P,  at  the  same  height  a*  that  through  t  cm; 
or  lower  down,  and  with  a  shorter  leverage,  thau  that  through  mnr  cm.  The  difi  is  still  greater 
when  the  surcharge  starts  at  6. 


338 


RETAINING-WALLS. 


Art.  7.  On  page  331,  Fig  1,  we  recommend  that  the  base  o  a  at  the  ground- 
line  of  well  built  vortical  walls  should  not  be  less  than  .35,  or  .4,  or  .6  of  the 
height  ds  above  said  line,  depending  on  the  kind  of  masonry.  But  a  wall  with  a 
battered  (inclined)  front  or  face  as  found  by  Art  8,  p  3  >9  (by  which  the  following 
table  was  prepared),  will  be  as  strong,  and  at  the  same  time  contain  less  masonry 
than  a  vert  wall,  although  the  battered  one  will  have  the  thickest  base  os. 

Table  3,  of  thicknesses  at  base  o  s9  Fig?  1,  and  at  top  c  </,  of 
walls  with  battered  faces,  so  as  to  be  as  strong  as  vertical 
<»aes  which  contain  more  masonry. 

For  the  ctib  yds  of  masonry  above  o  s  per  foot  run  of  wall,  mult  the 
.square  of  the  vert  height  ds  by  the  number  in  the  column  of  cub  yds.  Then 
add  the  foundation  masonry  below  o  s.  See  also  Table,  p  341.  Also  study  Kerns  1 
and  2,  Art  8,  p  339. 

(Original.) 


All  the  walls  below  have  the  same  strength 
as  a  vert  one  whose  base  os,  fig  1—  .35 
of  its  htdn. 

All  the  walls  below  have  the 
same  strength  as  a  vert  oue 
whose  base  o  s,  fig  1=.4 
of  its  ht  ds. 

All  the  walls  below  have  the 
same   strength   as    a   vert 
one  whose  base  os,  fig  1  = 
.5  of  its  ht  ds. 

Cut  stone. 

Mortar  rubble. 

Dry  rubble. 

Batter,  in 
ins  to  a  ft. 

Rase,  in 
pts  of 
ht. 

Top,   in 
pts  of 
ht. 

C  yds  per 
ft  run. 

Base,  in 
pts  of 
ht. 

Top,     in 
pts  of 
ht. 

C  yds  per 
ft  run. 

Base,  in 
pts  of 
ht. 

Top,    in 
pts  of 
ht. 

C  yds  per 
ft  run. 

0 

.350 

.350 

.01296 

.400 

.400 

.01482 

.500 

.500 

.01852 

*A 

.352 

.310 

.01226 

.401 

.359 

.01407 

.501 

.459 

.01778 

i 

.355 

.270 

.01158 

.403 

.320 

.01339 

.503 

.420 

.01709 

V/2 

.359 

.234 

.01098 

.408 

.283 

.01280 

.506 

.381 

.01643 

2 

.364 

.197 

.010^9 

.413 

.246 

.01220 

.510 

.343 

.01580 

2^ 

.371 

.163 

.00989 

.419 

.210 

.01165 

.516 

.308 

.01526 

3 

.379 

.129 

.00941 

.425 

.17o 

.01111 

.522 

.272 

.01470 

&A 

.389 

.096 

.00898 

.435 

.143 

.01070 

.528 

.236 

.01415 

4 

.400 

.066 

.00863 

.445 

.110 

.01028 

.537 

.204 

.01372 

5 

.425 

.007 

.00800 

.468 

.051 

.00961 

.555 

.138 

.01283 

Triangle 

.429 

.000 

.00794 

.490 

.OnO 

.00907 

.612 

.000 

.01133 

Moselev  and  others  quote  Gadroy,  for  a  DRY  SAND  SLOPING  AT  21°.  It  would  be  better  to  cease  from 
circulating  such  evident  mistakes.  Dry  sand  will  stand  at  no  less  angle  for  a  savant  than  for  any- 
body else.  For  practical  purposes,  we  may  say  that  dry  sand,  gravel  and  earths,  slope  at  33°  41'  or 
1?^  to  1 ;  as  abundant  experience  on  railroad  embkts  proves.  Poncelet  gives  tables  for  walls  to  sup- 
port dry  earth  sloping  at  1  to  1.  or  45°:  but  as  we  do  not  believe  in  the  existence  of  such  earth,  we 
omit  such  tables.  Sand,  gravel,  and  earths  may  be  moistened  to  diff  degrees,  so  as  to  stand  at  any 
angle  between  hor  and  vert;  and  by  moistening  and  ramming,  the  earths  may  be  con  verted  into  com- 
p;ict  masses,  exerting  little  or  no  pres;  and  may  even  so  continue  after  they  become  dry  ;  being  then, 
in  fact,  a  kind  of  air-dried  brick.  It  is  sometimes  difficult  to  know  whether  earth  or  sand  is  perfectly 
dry  or  not;  and  an  exceedingly  small  degree  of  moisture  will  cause  them  to  stand  at  1  to  1,  in  small 
heaps,  such  as  hare  probably  been  observed  by  the  authorities  on  the  subject.  The  writer  found  that 
fine  sand  from  the  sea  shore,  and  under  cover,  would  stand  at  1%  to  1  during  warm  dry  weather,  and 
at  L  to  I  when  the  air  was  damp.  Yet  no  diff  whatever  in  its  degree  of  moisture  was  perceptible  to 
the  feeling.  Its  susceptibility  to  dampness  was  of  course  owing  to  salt.  A  few  handfuls  of  dry  earth 
may  perhaps  be  coquetted  into  standing  at  1  to  1  on  a  table ;  but  so  far  as  our  observation  extend-;, 
when  it  is  dumped  in  large  quantities  from  carts  and  wheelbarrows,  its  slope  is  about  1>^  to  1 ;  sind 
this  we  consider  the  proper  one  to  be  used  in  practical  calculations,  where  safety  is  the  consideration 
of  paramount  importance. 

The  less  the  nat  slope,  the  greater  Is  the  pres :  and  sinc^  the 
slope  is  least  when  the  backing  is  perfectly  dry,  (omitting  of  course  its  condition 
when  so  absolutely  \u?i  as  to  become  partially  fluid,)  we  have,  on  the  score  of  safety, 
confined  our  tables  to  dry  backing.  As  stated  in  Art  1,  we  cannot  recommend  dimen- 
sions less  than  those  there  given,  when  we  consider  the  rough  treatment  to  which 
masonry  is  exposed  on  public  works. 

In  carrying:  a  road  along;  dang-erons  precipices,  we  should 
rather  be  tempted  at  times  to  make  thicker  walls.  We  imagine,  for  instance,  that 
the  centrifugal  force  of  a  heavy  train,  whirling  around  a  sharp  curve,  convex  on  the 
dangerous  side,  should  not  be  overlooked  in  designing  walls  for  such  localities.  This 
force  is  hor;  and  is  applied  near  the  top  of  the  wall :  and,  consequently,  its  leverage 
may  be  considered  as  equal  to  the  height:  whereas  the  theoretical  pres  of  the  earth 
is  oblique  :  and  is  applied  at  ^  of  the  height  from  the  bottom  ;  so  that  its  leverage 
about  the  toe  of  the  wall  is  very  short.  Moreover,  the  simple  weiffht  of  the  train,  pro- 
duces pres  against  the  wall ;  as  well  as  that  of  the  backing.  All' such  considerations 
are  omitted  by  theorists.  The  dangerous  pres  caused  by  tremors,  Ac,  cannot  be 


RETAINING- WALLS. 


339 


assumed  to  be  applied  at  %  of  the  height  from  the  bottom ;  nor  indeed,  can  it  be 
calculated  at  all. 

REM.  2.  Wharf  walls  are  an  instance  where  the  thickness  should  be  increased, 
notwithstanding  that  the  pres  of  the  water  in  front  helps  to  sustain  them.  The  earth 
behind  such  walls,  is  not  only  liable  to  be  very  heavily  loaded  when  vessels  are  dis- 
charging; but  is  apt  to  become  saturated  with  water,  especially  below  low-water 
level;  and  thus  to  exert  a  very  great  pres  against  the  walls.  Moreover,  the  water 
gets  under  the  wall;  and  by  its  upward  pressure  virtually  reduces  its  weight,  and 
consequently  its  stability.  The  same  cause  of  course  diminishes  the  friction  of  the 
wall  upon  its  base.  Such  walls  are,  therefore,  very  liable  to  slide,  if  the  foundation 
is  smooth,  and  horizontal ;  and  have  done  so  even  when  the  foundation  had  a  con- 
siderable inclination  backward,  as  in  Fig  1.  See  Art  9. 

REM.  3.  A  retaining-wall  is  usually  in  greater  danger  for  a  few  months  after  its  completion,  than 
after  time  has  been  allowed  for  the  mortar  to  harden  perfectly  ;  and  for  the  backing  to  settle.  When 
there  are  suspicions  of  the  safety  of  a  new  wall,  it  would  be  well  to  place  strong  temporary  shores 
against  it,  at  about  %  to  %  of  its  height  above  ground.  In  some  cases,  permanent  buttresses  of 
masonrv  may  be  built  for  the  purpose.  They  should  be  well  bonded  into  the  wall. 

REM.  4.  The  pres  of  the  earth  backing  will  be  much  reduced,  if  the  first  few  feet  of  its  height  be 
made  up  in  thin  hor  layers,  to  be  consolidated  by  being  used  by  the  masons  instead  of  scaffolding;  as 
snown  at  A,  Fig  1.  Frequently  this  can  be  done  without  inconvenience  ;  and  at  very  trifling  cost. 

Art.  8.  To  change  a  vert  retaining-wall,  into  one  with  a 
battered  face,  which  shall  present  an  equal  resistance 
against  overturning;  although  requiring  less  masonry. 

This  is  sometimes  termed  a  transformation  of  profile.    (Original.) 

Let  a  b  o  i,  Fig  10,  be  the  vert  wall.    Mult  its  bas« 
o  t,  by  1.225 ;  (1.22475  is  nearer ;)  the  prod  will  be  the  -i  i 

base  o  e,  of  a  triangular  wall  b  o  e,  possessing  the     _J£. a  s        ft 

same  stability;  and  yet  not  requiring  much  more 
than  half  the  masonry  of  the  vert  one.  See  Rem  1. 
This  being  done,  suppose  a  wall  to  be  desired  with  a 
face  batter,  of  say  3  ins  to  a  ft ;  or  1  in  4.  From  the 
point  n,  where  the  face  of  the  triangular  wall  inter- 
sects that  of  the  vert  one.  step  off  vert  any  4  short 
equal  spaces ;  and  from  the  upper  one  m,  step  off  one 
space  hor,  to  v.  Through  v  and  n  draw  the  dotted 
line  s  t,  which  evidently  will  batter  1  in  4.  Then  is 
b  s  t  o  approximately  the  reqd  wall ;  but  a  little 
.  thicker  than  necessary.  To  reduce  it,  from  t  draw 
the  dotted  line  t  b.  Mark  the  point  c,  where  this 
line  intersects  the  face  a  i.  of  the  vert  wall;  and 
through  c  draw  d  I,  parallel  to  s  t.  Then  Is  b  d  I  o 
the  reqd  wall.  Our  fig  is  drawn  in  an  exaggerated 
manner,  so  as  to  avoid  confusion  in  the  lines.  The 
base  o  e  of  the  triangular  wall,  would  not  in  reality 
be  near  so  great  as  it  is  represented. 


FiglO 


i  I  t 


It  will  be  observed  that  as  the  base  increases,  the  quantity  of  masonry  diminishes. 

REM.  1.  The  battered  wall  will  in  fact  be  safer  than  the  vert 
one,  and  for  this  reason ;  although  the  wall  after  being  transformed  from  a  vert 
one.  to  one  with  a  battered  face,  (and  consequently  with  a  greater  base,)  has  the  same  moment  of  sta- 
bility as  before  ;  and  although  the  amount  of  pres  of  the  earth  against  it.  also  remains  unchanged, 
still  the  new  wall  is  better  able  to  resist  this  pres  than  before.  In  other  words,  the  moment  or  pu- 
dency of  the  pres  to  upset  the  wall,  has  become  Jess.  For  let  a  ?>  m  »i,  Fig  11,  represent  a  vertical 
wall :  and  fo  the  amount  and  direction  of  pres  behind  it.  Now,  the  leverage  with  which  this  pres 
tends  to  overturn  t.he  wall  around  its  toe  m,  is  the  dist  m  8, 
measured  from  the  toe  or  fulcrum  m,  and  at  right  angles  to 
the  direction  fo  s  c  of  the  pres  ;  and  this  leverage  mult  by 
the  force  /  o,  gives  the  overturning  tendency  or  moment  of 
naid  force.  See  "Moments  and  leverage,"  p  473.  Again, 
let  any.  represent  a  triangular  wall  of  the  same  stability 
as  the  other,  as  found  by  our  rule.  Here  we  still  have  the 
same  »mount  fo.  and  direction  fo  s  c.  of  pres  force  against 
the  wall;  but  it  now  acts  to  overturn  the  wall  any 
around  the  toe  y :  and  then-fore,  with  the  reduced  leverage 
y  ~.  Consequently,  its  overturning  tendency  is  less  than 
before.  Therefore,  in  ordinary  language,  we  may  say  that 
the  wall  is  stronger  than  before,  although  its  moment  of 
stability,  or  standing  tendency,  has  in  itself  undergone  no 
change.  If  the  pres/o  against  the  vert  back  were  hor,  as 
in  the  case  of  water,  then  its  leverage  would  evidently  be 
the  same  in  both  walls;  and  the  proportion  between  the 
overturning  moment  of  the  pres,  and  the  moments  of 
stabilitv  of  the  two  walls,  would  be  constant.  P  528. 

REM.  2.  In  attempting  to  reduce  the  masonry  by  adopt- 
ing a  wall,  o  b  e.  Fig  10.  of  a  triangular  section  :  or  of  one 
nearly  approaching  a  triangle,  special  attention  should  be 
given  to  the  quality  of  the  masonry  near  the  thin  toe  e; 
which  will  otherwise  be  apt  to  crack,  or  fail  under  the  preg. 


340 


RETAINING- WALLS. 


Eg  13 


MOREOVER,  WHEN  COMMON  MORTAR  is  useis.  WITHOUT  AN  ADMIXTURE  OP  CEMENT,  which  It  never 
should  be,  in  retaiuiug-walls,whereduraOility  is  au  object,  a  great  batter  is  objec- 
tionable; inasmuch  as  the  rain,  combined  with  frost,  &c,  soon  destroys  the  uior- 
tar.  In  such  cases,  therefore,  the  batter  suould  not  exceed  1  orl^jiustoa  ft;  and 
even  then,  at  least  the  poiiuiug  of  tne  joints,  and  a  few  feet  in  height  of  both 
the  upper  and  the  lower  courses  of  masonry,  should  be  doue  with  cement,  or 
cement-mortar.  We  have  observed  a  most  marked  diffiu  the  corrosion  of  the  mor- 
tar, where,  in  the  same  walls,  with  the  same  exposure,  one  portion  has  been  built 
with  a  vert  face ;  and  another  with  a  batter  of  but  1>$  inch  to  a  foot.  Common 
mortar  will  never  set  properly,  and  continue  firm,  when  it  is  exposed  to  mois- 
ture from  the  earth.  This  is  very  observable  near  the  tops  and  bottoms  of 
abuts,  retaining- walls.  <tc ;  the  lime-mortar  at  those  parts  will  generally  be 
found  to  be  rendered  entirely  worthless.  A  profile  somewhat  like  Fig  12,  may 
at  times  prove  serviceable,  instead  of  the  triangular.  This  is  the  form  of  the 
Gothic  buttress  ;  which  probably  had  its  origin  in  the  cause  just  spoken  of. 

Art.  9.    A  retaining-wall   may  slide,  without 
T?U  <•   1Q        losing  its  verticality;  and,  indeed,  without  any  danger 
J:  10    I  >>       of  being  overturned.     This  is  very  apt  to  occur  if  it  is  built  upon 
a  hor  wooden  platform ;    or  upon  a  level  surf  of  rock,  or  clay, 
without  other  means  than  more  friction  to  prevent  sliding.    This  may  be  obviated 
by  inclining  the  base,  as  in  Fig  I ;  by  founding  the  wall  at  such  a  depth  as  to  pro- 
vide a  proper  resistance  from  the  soil  in  .front;  or  in  case  of  a  platform,  by  securing 
one  or  more  lines  of  strong  beams  to  its  upper  surf,  across  the  direction  in  which 
sliding  would  take  place.    On  wet  clay,  friction  may  be  as  low  as  from  .2  to 
y±  the  weight  of  tne  wall ;  on  dry  earth,  it  is  about  ^  to  % ;  and  on  sand  or  gravel, 
about  %  to  %.    The  friction  of  masonry  on  a  wooden 
fcj          platform,  is  about  -6^  of  the  wt,  if  dry ;  and  %  if  wet. 

__~|  Counterforts,  shown  in  plan  at  c  c  c,  Fig  13.  consist  in 
an  increase  of  the  thickness  of  the  wall,  at  its  back,  at  regular  inter- 
vals of  its  length.  Wo  conceive  them  to  be  but  little  better  than  a 
waste  of  masonry.  When  a  wall  of  this  kind  fails,  it  almost  in- 
variably separates  from  its  counterforts  :  to  which  it  is  connected 
merely  by  the  adhesion  of  the  mortar ;  and  to  a  slight  extent,  by  the 
bonding  of  the  masonry.  The  table  in  Art  7  shows  that  a  very  small  addition  to  the  base  of  a  wall,  is 
attended  by  a  great  increase  of  its  strength;  we  therefore  think  that  the  masonry  of  counterforts 
would  be  much  better,  and  more  cheaply  employed  in  giving  the  wall  an  additional  thickness,  along 
its  entire  length;  and  for  the  lower  third  of  its  heignt.  Counterforts  are  very  generally  used  ic 
retaining-walls  by  European  engineers;  but  rarely,  if  ever,  by  Americans. 

Buttresses  are  like  counterforts,  except  that  they  are  placed  in  front  of  a  wall  instead  of  be- 
hind it;  and  that  their  pronle  is  generally  triangular,  or  nearly  so.  They  greatly  increase  its  strength; 
but.  being  unsightly,  are  seldom  used,  except  as  a  remedy  when  a  wall  is  seen  to  be  failing. 

Laitd-tieS,  or  long  rods  of  iron,  have  been  employed  as  a  makeshift  for  upholding  weak  re-  . 
taining-wa]ls.     Extending  through  the  wall  from  its  face,  the  laud  ends  are  connected  with  anchors 
of  masonry,  cast-iron  or  wooden  posts;  the  whole  being  at  some  dist  below  the  surface. 

Retaining  walls  With  curved  profiles  are  mentioned  here  merely  to  cau- 
tion the  young  engineer  against  building  them.  Although  sanctioned  by  the  practice  of  some  high, 
authorities,  they  really  possess  no  merit  sufficient  to  compensate  for  the  additional  expense  and  trou- 
ble of  their  construction. 

Art.  1O.  Among  military  men,  a  retaining-wall  is  called  a  revetment.  When  the 
»arth  is  level  with  the  top,  a  SCarp  revetment;  when  above  it.  a  counterscarp 
revetment,  or  a  demi-revetment.  When  the  face  of  the  wall  is  battered,  a  sloping ;  and  when  the  back 
is  battered,  a  countersloping  revetment.  The  batter  is  called  the  till  IIS. 

Art.  11.  Our  own  experiments  lead  us  to  believe  that  the  pressure 
against  a  wall,  Fig  9,  from  dry  sand,  ck,  level  with  the  top,  is  not  at  all  dimin- 
ished by  reducing  the  width  cs  of  the  sand,  until  it  becomes  about  one-sixth  of 
the  dist  c«,  pertaining  to  the  angle  cms  of  natural  slope.  With  less  than  that 
limit,  the  pres  was  plainly  diminished.  It  would  therefore  seem  to  be  dangerous 
to  make  a  retaining-wall  thin,  merely  because  the  backing  does  not  entend  to  t.* 

Table  4.  of  contents  in  cub  yards  for  each  foot  in  length 
of  retaining-walls,  with  a  thickness  at  base  equal  to  .4  of  the  vert  height. 
Face  batter,  \%  inches  to  a  foot ;  or  J^th  of  the  height.  Back  either  vert,  or  stepped 
according  to  the  rule  in  Art  3,  Fig  3.  The  strength  is  very  nearly  equal  to  that  of 
avert  wall  with  a  base  of  .4  its  height.  See  table  p  338.  Experience  has  proved 
that  such  walls,  when  composed  of  well-scabbled  mortar  rubble,  are  safe  under  all 
ordinary  circumstances  for  earth  level  with  the  top.  Steps  or  offsets,  oe,  at  foot, 
Fig  1,  are  not  here  included. 


*'"  A  singular  fact  which  the  writer  believes  he  was  the  first  to  notice  in  1859,  We  cannot  under- 
stand how  correct  results  are  to  be  expected  from  experiments  like  those  of  Gen.  Pasley,  and  others, 
who  confined  their  bucking  in  a  box.  placing  their  walls  in  front  of  it.  The  friction  of  the  backing 
against  the  sides  of  the  box  must  diminish  the  pres  against  the  wall,  and  thus  lead  to  adopting  too 
slight  a  thickness.  We  conceive  that  an  experimental  wall  should  diminish  in  height  and  thickness, 
each  way  from  its  central  portion,  (preserving,  however,  the  same  proportion  between  the  two,> 
until  it  terminates  in  a  point  at  each  end.  Ours  were  made  in  this  way. 


STONE    BRIDGES. 


341 


TABLE  4.    (Original.) 


Ht. 

Cub. 

Ht. 

Cub. 

Ht. 

Cub. 

Ht. 

Cub. 

Ht. 

Cub. 

Ht. 

Cub. 

Ft. 

Yds. 

Ft. 

Yds. 

Ft. 

Yds. 

Ft. 

Yds. 

Ft. 

Yds. 

Ft. 

Yd*. 

1 

.013 

1034 

1.38 

20 

5.00 

29« 

10.9 

48 

28.8 

74 

68.5 

M 

.028 

11 

1.51 

X 

5.25 

30 

11.3 

49 

30.0 

76 

722 

2 

.050 

% 

1.G5 

21 

5.51 

31 

12.0 

50 

31.3 

78 

761 

H 

.078 

12 

1.80 

K 

5.78 

32 

12.8 

51 

325 

80 

80.0 

3 

.113 

X 

1.95 

22 

6.05 

33 

13.6 

52 

33.8 

82 

84.1 

X 

.153 

13 

2.11 

J4 

6.33 

34 

14.5 

53 

35.1 

84 

88.4 

4 

.200 

X 

2.23 

23 

661 

35 

15.3 

54 

36.5 

86 

92.5 

« 

.253 

14 

2.45 

1A 

6.90 

86 

16.2 

55 

37.8 

88 

96.8 

5 

.313 

H 

J.63 

21 

7.20 

37 

17.1 

56 

39.2 

90 

101.3 

H 

.378 

15 

2.81 

J4 

7.50 

38 

18.1 

57 

40.6 

92 

105.8 

6 

.450 

X 

3.00 

7.81 

39 

190 

58 

42.1 

94 

1105 

X 

.528 

16 

3.  '21) 

>* 

8.13 

40 

20.0 

59 

43.5 

96 

115.2 

7 

.613 

Vi 

3.40 

26 

8.45 

41 

21.0 

60 

45.0 

98 

120.1 

X  1  ."03 

17 

3.61 

X 

8.78 

42 

22.1 

62 

48.1 

100 

125.0 

8 

.800 

M 

3.83 

27 

9.12 

43 

23.1 

64 

51.2 

102 

130.1 

X 

.903 

18 

4.05 

^ 

945 

44 

24.2 

66 

54.5 

104 

135.2 

9 

1.01 

H 

4.28 

28 

9.80 

45 

25.3 

68 

57.8 

106 

140.5 

K 

1.13 

19 

4.51 

J4   102 

46 

26.5 

70 

61.3 

10 

1.25 

X 

4.75 

29    1  10.5 

47 

27.6 

72 

648 

See  preceding  footnote. 


STONE  BEIDGES, 


Art.  1.  In  an  arch  st  s,  Fig  1,  the  dist  eo  is  called  its  span ;  t'a  its  rise;  t  its 
Crown  ;  its  lower  boundary  line,  *ao,  its  soffit,  or  intrados  ;  the  upper  one, 
rtr,  its  back,  or  extrados.  The  terms  soffit  and  back  are  also  applied  to  the 
entire  lower  and  upper  curved  surfaces  of  the  whole  arch.  The  ends  of  an  arch,  or 
the  showing  areas  comprised  between  its  intrados  and  extrados,  are  its  faces ;  thurf 
the  area  st  s  a  is  a  face.  The  inclined  surfaces  or  joints,  re,  r<>,  upon  which  the  feet 
of  the  arch  rest,  or  from  which  the  arch  springs,  are  the  skewbacks.  Lines 
level  with  e  and  o,  at  right  angles  to  the  faces  of  the  arch,  and  forming  the  lower 
edges  of  its  fret,  (see  nn,  Fig  2^,)  are  the  spria- in-  lines,  or  springs.  The 
blocks  of  which  the  arch  itself  is  composed,  are  the  arcth-stones,  or  vonssoirs. 
The  center  one,  la,  is  the  keystone;  and  the  lowest  ones,  ss.  the  springers. 
The  term  archblock  might  be  substituted  for  voussoir,  and  like  it  would  apply  to 
brick  or  other  material,  as  well  as  to  stone.  The  parts  tr,tr,  are  the  ll aim  dies  ; 
and  the  spaces  /  r  /,  t  r  b,  above  these,  are  the  spandrels.  The  material  deposited 
in  these  spaces  is  the  spandrel  filling; ;  it  is  sometimes  earth,  sometimes  ma- 
soriry ;  or  partly  of  each,  as  in  Fig  1. 

In  large  arches.it  often  consists  of  several  parallel  SPANDREL-WALLS.  II,  Fig  2**j,  running  lengthwise 
of  the  rondway,  or  astraddle  of  the  arch.  They  are  covered  at  top  either  by  small  arches  from  wall  to 
wall,  or  by  fla't  stones,  for  supporting  the  material  of  the  roadway.  They  are  also  at  times  connected 
together  by  vert  cross- walls  at  intervals,  for  steadying  them  laterally,  as  at  tt.  Fig  2^.  The  parts 
gpen,  gpon,  Figl,  are  the  ABUTMENTS  of  the  arch;  en,  on,  the  'faces ;  gp,  gp.  the  backs;  and 
p  n,  p  n,  the  bases  of  the  abuts.  The  bases  are  usually  widened  by  feet,  steps,  or  offsets,  d  d.  for  dis- 
tributing the  wt  of  the  bridge  over  a  greater  area  of  foundation  ;  thus  diminishing  the  danger  of  set- 
tlement. The  distance  t  a  in  any  arch-stone,  is  called  its  depth. 

The  only  arches  in  common 
use  ibr  bridges,  are  the  circular, 
(often  called  segmental);  and 
the  elliptic. 

Art.  2.  To  find  the 
depth  of  keystone  for 
cut-stone  arches, 
whether  circular  or 
elliptic.*  (Original.) 

Find  the  rad  eo,  Fig  1,  which 
will  touch  the  arch  at  o,  a,  and 
e.  Add  together  this  rad,  and 
half  the  span  o  e.  Take  the  sq 
rt  of  the  sum.  Div  this  sq  rt 
by  4.  To  the  quot  add  fa  of  a 
ft.  Or  by  formula, 

*  Inasmuch  as  the  rules  which  we  give  for  arches  and  abuts  are  entirely  ordinal  and  novel,  it  may 
uot  be  arnias  to  state  that  they  are  not  altogether  empirical ;  bat  are  based  upon  accurate  drawing* 


342 


STONE    BRIDGES. 


Depth  of  key  __   (v '  Rad  +  half  span\     ,     o /Vw 
in  feet  \  4  / 

For  second-class  work,  this  depth  may  be  increased  about  i/£th  part;  or 
for  brick  or  fair  rubble,  about  ^tli.  See  table  of  Keystones,  p  345. 

In  large  arches  it  is  advisable  to  increase  the  depth  of  the  archstones  toward  the 
springs  ;  but  when  the  span  is  as  small  as  about  60  to  80  or  100  feet,  this  is  not  at  all 
HKWxxary  if  the  stone  is  good;  although  the  arch  will  be  stronger  if  it  is  done.  In 
practice  this  increase,  even  in  the  largest  spans,  does  not  exceed  from  %  to  1A  tny 
depth  of  the  key ;  although  theory  would  require  much  more  in  arches  of  great  rise. 
See  Kx  2,  p  46S,  also  Ex  6,  p  478,  of  Force  in  Rigid  Bodies. 

REM.  To  find  the  rad  c  o,  whether  the  arch  be  circular  or  elliptic.  Square 
half  the  span  P,  o.  Square  the  whole  rise  i  a.  Add  these  squares  together:  div  the 
sum  by  twice  the  rise  i  a.  Or  it  may  be  found  near  enough  for  this  purpose  by  the 
dividers,  from  a  small  arch  drawn  to  a  scale. 


Amount  of  pressure  sustained  by  archstones.    IF  t>r«  jges  of  the 

same  width  of  roadway,  if  all  the  other  parts  bore  to  each  other  the  same  proportion  t*s  t  je  spans,  the 
total  pres,  as  well  as  the  pret  per  »q  ft,  in  arches  of  the  same  proportionate  rise,  wojld  increase  as 
the  squares  of  the  spans.  But  in  practice  the  depth  of  the  archstones  increases  much  less  rapidly 
than  the  span  ;  while  the  thickness  of  the  roadway  material,  and  the  extraneous  load  per  sq  ft,  remain 
the  same  whether  the  span  be  great  or  small.  Hence  the  pressures  increase  less  rapidly  than  the 
squares  of  the  spans.  Thus  in  two  bridges  of  the  same  width,  but  with  spans  of  100  and  200  ft,  with 
depths  of  archstones  taken  from  our  table  page  345,  and  uniform  from  key  to  spring;  supposed  to  be 
filled  up  solid  with  masonry  of  160  tbs  per  cub  ft,  to  a  level  of  about  15  inches  above  the  crown,  (in- 
cluding the  stone  paving  of  the  roadway);  with  an  extraneous  load  of  100  B>s  per  sq  ft;  and  the 
leverages  y  m,  y  o,  Fig  52,  p  479,  measured  from  the  center  y  of  the  skewback  ;  the  pressures  will  be 
approximately  as  follows : 


Span  1OO  ft. 


AT  KEY. 
For  1  ft  in 
width  of 
its  entire 
depth. 

Per  sq  ft. 

AT  81 

For  1  ft  in 
width  of 
its  entire 
depth. 

•RING. 
Per  sq  ft. 

AT 
For  1  ft  in 
width  of 
its  entire 
depth. 

KEY. 
Per  sq  ft. 

AT  SP 
For  1  ft  in 
width  of 
its  entire 
depth. 

RING. 
Per  sq  ft. 

Rise. 

w 

1 

X 

X 

Tons. 

ox 

MX 

31  14 
25 
18 

Tons. 
13*£ 
12k 
11 
9 
6% 

Tons. 
58 

57 

57^ 

61^ 
67^ 

Tons. 
18^ 
19 
20 

mi 

25 

Tons. 
126 
112 
97 

80  \i 
67  fc 

S£ 

•UK 

24>$ 
21 

l£H 

Tons. 
179 

181 
188 
207 
230 

Tons. 
42 
44 

47J4 
54  y4 
61^ 

Span  200  ft. 


Here  it  is  seen  that  the  entire  pros  of  the  200  ft  span,  with  different  rises,  averages  but  little  more 
than  three  times  that  of  the  100  ft  one  ;  while  the  pres  per  sq  ft  of  each  stone  averages  but  nbout  2^ 
times  that  of  the  100  ft  one.  It  will  also  be  seen  that  with  the  same  span,  the  pres  at  the  key  becomes 
less,  while  that  at  the  spring  becomes  greater,  as  the  rise  increases.  Also  that  when  the  archstones 
are  of  uniform  depth,  the  pres  at  either  spring  of  a  semicircular  arch  is  about  4  times  as  great  as  at 
the  key  ;  whereas  when  the  rise  is  but  %  of  the  span,  the  pres  at  spring  averages  but  about  %  greater 
than  at  the  key.  These  proportions  vary  somewhat  in  different  spans. 

The  greater  pres  per  sq  ft  at  the  springs  may  be  reduced  by  increasing  the  depth  of  the  archstones 
towards  the  springs.  This  however  is  not  necessary  in  moderato  spans,  inasmuch  as  good  stone  will 
be  safe  even  under  this  greater  pres. 

By  USillg:  parallel  spandrel  Walls,  see  Fig  2^,  p  346,  or  by  partly  filling 
with  earth  instead  of  masonry,  the  pres  on  the  archstones  may  be  diminished,  say,  as  a  rough  average, 
about  -i-  part. 

and  calculations  made  by  the  writer,  of  lines  of  pres,  Ac.  of  arches  from  1  to  300  ft  span,  and  of  every 


STONE   BRIDGES. 


343 


Table  1.    Of  some  existing-  arches,  with  both  their  actual,  and  their 

calculated  depths  (by  our  rule)  of  keystone.  Where  the  two  depths  are  given  in  the  last  column,  the 
smallest  is  for  first  class  cut-stone,  and  the  largest  for  good  rubble,  or  brick.  Those  also  which  are 
not  specified  are  of  first-class  cut-stone.  C  stands  for  circular,  E  for  elliptic.  For  2d  class  work,  add 
about  %th  part;  and  for  brick,  or  fair  rubble,  about  J4th. 


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O 

Monocacy  Aqueduct,  Chesapeake  &  Ohio  Canal... 
Llaurwast,  Wales.  Turnpike  
Avon  Viaduct.  England.  Brick  in  cement  

Tonoloway  Culvert,  under  Cbes  &  Ohio  Canal.  I 

Philadelphia  &  Reading  R  R  
"  
Edinburg  &  Dalkeith  R  R.  Scotland  

*  The  width  is  not  given  in  any  account  that  we  have  seen,  but  probably  it  was  not  more  than  5  or  6 
ft.  It.  was  tested  by  a  distributed  load  of  360  tons,  and  by  a  weight  of  5  tons  falling  18  ins ;  and  was 
uninjured  thereby."  Total  settlement  before  and  after  being  loaded,  about  1^  ins.  The  depth  of  arch- 
atones  was  increased  gradually  from  2.67  ft  at  key,  to  3.6  ft  at  springs.  Mortar  joints  full  ^  iunh 
thick.  Rise  about  y^-th  of  the  span. 


344 


STONE    BRIDGES. 


The  arch  on  the  BOURBONNAIS  RAILWAY,  is  probably  the  boldest;*  and  THE  CABIN  JOHN  ARCH,  by 
Oapt,  now  Gen'l  M.  0.  Meigs,  US  Army,  the  grandest  stone  one  in  existence.  PONT-Y  PRYDD,  in 
Wales,  is  a  common  road  bridge,  of  very  rude  construction  ;  with  a  dangerously  steep  roadway.  It 
was  built  entirely  of  rubble,  in  mortar,  by  a  common  country  mason,  in  1750;  and  is  still  in  perfect 
condition.  Only  the  outer,  or  showing  arch-stones,  are  2.5  ft  deep  ;  and  that  depth  is  made  up  of  two 
stones.  The  inner  arch-stones  are  but.  1.5  ft  deep  ;  and  but  from  6  to  9  inches  thick.  The  stone  quar- 
ried with  tolerably  fair  natural  beds;  and  received  little  or  no  dressing  in  addition.  The  bridge  is  a 
fine  example  of  that  ignorance  which  often  passes  for  boldness.  PONT  NAPOLEON  carries  a  railroad 
across  the  Seine  at  Paris.  The  arches  are  of  the  uniform  depth  of  4  ft,  from  crown  to  spring.  They 
are  composed  chiefly  of  sm>ill  rough  quarry  chips,  or  spawls ;  well  washed,  to  free  them  from  dirt 
and  dust;  and  then  thoroughly  bedded  in  good  cement;  and  grouted  with  the  same.  It  is  in  fact  an 
arch  of  cement-concrete.  The'  PONT  DE  ALMA,  near  it,  and  built  in  the  same  way,  has  elliptic  arches 
of  from  126  to  141  ft  span  ;  with  rises  of  i  the  span.  Key  4.9  ft.  These  two  bridges,  considering  tb« 
want  of  precedent  in  this  kind  of  construction,  on  so  large  a  scale,  must  be  regarded  as  verv  bold; 
and  as  reflecting  the  highest  credit  for  practical  science,  upon  their  engineers,  Darcel  and  Couche. 
Some  trouble  arose  from  the  unequal  contraction  of  the  different  thicknesses  of  cement.  They  show 
what  may  be  readily  accomplished  in  arches  of  moderate  spans,  by  means  of  small  stone,  and  good 
hydraulic  cement  when  large  stone  fit  for  arches  is  not  procurable.  In  Pont  Napoleon  the  depth  of 
arch  is  barely  what  our  rule  gives  for  second  class  cut-stone. 

REM.  Our  engineers  are  usually  toosparing-  of  cement.  Itshould 
be  freely  used,  not  only  in  the  arches  themselves,  and  in  the  masonry  above  them,  as  a 
protection  from  rain-soakage ;  but  in  abuts,  wing-walls,  retaining-walls,  and  all  other 
important  masonry  exposed  to  dampness.  The  entire  backs  of  important  brick  arches 
should  be  covered  with  a  layer  of  good  cement,  about  an  inch  thick.  The  want  of  it 
can  be  seen  throughout  most  of  our  public  works.  The  common  mortar  will  be 
found  to  be  decayed,  and  falling  down  from  the  soffits  of  arches;  and  from  the  joints 
of  masonry  generally,  within  from  3  to  6  ft  of  the  surface  of  the  ground.  The  mois- 
ture rises  by  capillary  attraction,  to  that  dist  above  the  surf  of  the  nat  soil ;  or 
descends  to  it  from  the  artificial  surf  of  embankments,  &c ;  therefore,  cement-mortar 
should  be  employed  in  those  portions  at  least.  The  mortar  in  the  faces  of  battered 
walls,  even  when  the  batter  is  but  1  to  !*/£  inches  per  foot,  is  far  more  injured  by  rain 
and  exposure,  than  in  vert  ones ;  and  should  therefore  be  of  the  best  quality.  See 
MORTAR,  Ac. 

We  have,  however,  seen  a  quite  free  percolation  of  surface  water  through  brick 
arches  of  nearly  3  ft  in  depth,  even  when  cement  was  freely  used.  In  aqueduct 
bridges,  we  believe  that  cement  has  not  been  found  to  prevent  leaks,  whether  the 
arches  were  of  brick,  or  even  of  cut-stone.  May  not  this  be  the  effect  of  cracks 
produced  by  settlement  of  the  arch;  or  by  contraction  and  expansion  under  atmos- 
pheric influence?  Cement  at  any  rate  prevents  the  joints  from  crumbling. 

Art.  3.  The  keystones  for  lararo  elliptic  arches  by  the  best  en- 
gineers, appear  generally  to  be  about  %  part  deeper  than  our  rule  requires ;  or  than 
is  considered  necessary  for  circular  ones  of  the  same  span  and  rise.  We  do  not.  how- 
ever, see  any  sufficient  reason  for  this.  The  elliptic  arch,  with  its  spandrel  filling. 


Jl ^^-TT^^i -It  lms  slightly  less  wt;  and  that  wt  ha* 

a  trifle  less  leverage  than  in  a  circulai 
one;  and  consequently  it  exerts  less 
pres  both  at  the  key,  and  at  the  skew- 
back.  See  London,  Gloucester,  and 
Waterloo  bridges,  in  the  preceding 
table.  See  upper  footnote  p  349. 

RKM.  Young  engineers  arc  apt  to  affect  shallow  arch-stones;  but  it  would  be  far  better  to  adop 
the  opposite  course:  for  not  only  do  deep  ones  make  a  more  stable  structure,  but  a  thin  arch  is  ai 
unsightly  an  object  as  too  slender  a  column.  According  to  our  own  taste, 'arch-stones  fullv  X  .ieepe: 
than  our  rule  gives  for  first-class  cut  stone,  are  greatly  to  be  preferred  when  appearance  is' consulted 
Especially  when  an  arch  is  of  rough  rubble,  which  costs  about  the  same  whether  it  is  built  up  ai 
arch,  or  as  spandrel  filling,  it  is  mere  folly  to  make  the  arches  shallow.  Stability  and  durability 
ohould  be  the  objects  aimed  at;  and  when  they  can  be  attained  even  to  excess,  without  increased  cost 
it  is  best  to  do  so. 

Rem.  Brick  arches,  from  their  great  number  of  joints  are  apt  to  settle 
much  more  than  cut  stone  ont-s  when  the  centers  are  removed,  and  thereby  tc 
derange  the  shape  of  the  arch,  and  at  times,  without  due  care,  even  to  endanger 
its  safety,  especially  if  it  b;3  large  and  flat.  When  the  span  exceeds  about  oO  to  3£ 
ft,  and  particularly  if  flat,  use  only  brick  of  superior  quality  in  good  cement 
mortar.  With  even  best  materials  and  work  we  adviso  the  young  engineer  not 
to  attempt  brick  arches  for  railroad  bridges  of  greater  spans  than  about  the  fol- 
lowing. Considerably  larger  ones  than  some  of  them  h:w<>  been  built,  and  have 
stood ;  but  their  coefs  of  safety  are  not  in  all  cases  satisfactory.  In  this  table  the 
rise  is  in  parts  of  the  span.  For  more  on  brick  arches,  see  Art  10,  p  674. 


STONE    BRIDGES. 


345 


Table  2.  Depths  of  keystones  for  arches  of  first-class  cut  stone, 
by  Art  2.  For  second  class  add  full  one-eighth  part ;  and  for  superior  brick  one- 
fourth  to  one-third  part,  if  the  span  exceeds  about  15  or  20  ft. 

Rise,  in  parts  of  the  span. 


Original. 


SPAN. 
Feet. 

1 

i      I 

t 

i 

i 

* 

TV 

Key.  Ft. 

Key.  Ft. 

Key.  Ft. 

Key.  Ft. 

Key.  Ft. 

Key.  Ft. 

Key.  Ft. 

2 

.55 

.56 

.58 

.60 

.61 

.64 

.68 

4 

.70 

.72 

.74 

.76 

.79 

.83 

.88 

6 

.81 

.83 

.86 

.89 

.92 

.97 

1.03 

8 

.91 

.93 

.96 

1.00 

1.03 

1.09 

1.16 

10 

.99 

KOI 

1.04 

1.07 

1.11 

1.18 

1.26 

15 

1.17 

1.19 

1.22 

1.26 

1.30 

1.40 

1.50 

20 

1.32 

1.35 

1.3S 

1.43 

1  48 

1.59 

1.70 

25 

1.45 

1.4H 

1.53 

1.58 

1.64 

1.76 

1.88 

30 

1.57 

1.60 

1.65 

1.71 

1.78 

1.91 

2.04 

35 

1  68 

1.70 

1.76 

1.83 

1.90 

2.04 

219 

40 

1.78 

1.81 

1.88 

1.95 

2.03 

2.18 

2.33 

50 

1.97 

200 

2.08. 

2.16 

2.25 

2.41 

2.58 

60 

2.14 

2.18 

2.26 

2.35 

2.44 

2.62 

2.80 

80 

2.44 

2.49 

2.58 

2.68 

2.78 

298 

3.18 

100 

2.70 

2.75 

2.86 

2.97 

3.09 

3.32 

3.55 

120 

2.94 

2.99 

3.10 

3.22 

8.35                  3  61 

3.88 

140 

3.16 

3.21 

3.33 

3.46 

3.60                  3.87 

4.15 

160 

3.36 

3.44 

3.58 

3.72 

3.87 

4.17 

180 

3.56 

3.63 

3.75 

3.90 

4.06 

4.38 

200 

3.74 

3.81 

3.95 

4.12 

4.29 

220 

3.91 

4.00 

4.13 

4.30 

4.48 

240 

4.07 

4.15 

4.30 

4.48 

260 

4.23 

4.31 

4.47 

4.66 

280 

4.38 

4.46 

4.63 

300 

4.53 

4.62 

4.80 

Art.  4.  To  proportion  the  abuts  for  an  arch  of  stone  or 
brick,  whether  circular  or  elliptic.  (Original.) 

The  writer  ventures  to  offer  the  following  rule,  in  the  belief  that  it  will  be  found 
to  combine  the  requirements  of  theory  with  those  of  economy  and  ease  of  applica- 
tion, to  perhaps  as  great  an  extent  as  is  attainable  in  an  endeavor  to  reduce  so  com- 
plicated a  subject,  to  a  simple  and  reliable  working?  rule  for  prac- 
tical bridge-builders.  This  is  all  that  he  claims  for  it/Notwithstanding  its 
simplicity,  it  is  the  result  of  much  labor  on  his  part.  It  applies  equally  to  the  smallest 
culvert,  and  to  the  largest  bridge ;  whatever  may  be  the  proportions  of  span  and  rise  ; 
and  to  any  height  of  abut  whatever.  It  applies  also  to  all  the  usual  methods  of  filling 
above  the  arch  ;  whether  with  solid  masonry  to  the  level  vf,  Fig  2,  of  the  top  of  the 
arch;  or  entirely  with  earth  ;  or  partly  with  each,  as  represented  in  the  fig:  or  with 
parallel  spandrel-walls  extending  to  the  back  of  the  abut,  as  in  Fig  2}^.  Although 
the  stability  of  an  abut  cannot  remain  precisely  the  same  under  all  these  conditions, 
yet  the  diff  of  thickness  which  would  follow  from  a  strict  investigation  of  each  par- 
ticular case,  is  not  sufficient  to  warrant  us  in  embarrassing  a  rule  intended  for  popu- 
lar use.  by  a  multitude  of  exceptions  and  modifications  which  would  defeat  the  very 
object  for  which  it  was  designed.  We  shall  not  touch  upon  the  theory  of  arches, 
except  in  the  way  of  incidental  allusion  to  it.  Theories  for  arches,  and  their  abuts, 
omit  all  consideration  of  passing  loads;  and  consequently  are  entirely  inapplicable 
in  practice  when,  as  is  frequently  the  case,  (especially  in  railroad  bridges  of  moderate 
spans,)  the  load  bears  a  large  ratio  to  the  wt  of  the  arch  itself.  Hence  the  theoretical 
line  of  thrust  has  no  place  in  such  cases.  Our  rule  is  intended  for  common  practice : 
and  we  conceive  that  no  error  of  practical  importance  will  attend  its  application  to 
any  case  whatever ;  whether  the  arch  be  circular  or  elliptic. 

It  gives  a  thickness  of  abut,  which,  without  any  backing 
of  earth  behind  it,  is  safe  in  itself,  and  in  all  cases,  against 
the  pres.  when  the  bridge  is  unloaded.  Moreover,  in  very  large  arches, 
in  which  the  greatest  load  likely  to  come  upon  them  in  practice  is  small  in  comparison 
with  the  wt  of  the  arch  itself,  and  the  filling  above  it,  our  abuts  would  also  be  safe 
from  the  loaded  bridge,  without  any  dependence  upon  the  earth  behind  them  ;  but 
as  the  arches  become  less,  and  consequently  the  wt  of  the  load  becomes  greater  in 
proportion  to  that  of  the  arch,  and  of  the  filling  above  it,  we  must  depend  more  and 
more  upon  the  resistance  of  the  earth  behind  the  abuts,  in  order  to  avoid  the  neces- 
sity of  giving  the  latter  an  extravagant  thickness.  It  will  therefore  be  understood 
throughout,  that  our  rules  suppose  that  after  the  bridge  is  finished,  earth  will  be  deposited 
behvtid  the  abuts,  and  to  the  height  of  the  roadway,  as  usual;  except  when  parallel 
spandrel  walls  are  used. 


346 


STONE   BRIDGES. 


To  proportion  the  abutments  of  a  stone  bridge. 

RULE.  Find,  as  in  Art  2,  the  rad  c  o,  Fig  2,  in  ft,  which  will  touch  the  arch  at  o, 
a,  and  e.  Div  this  rad  by  5.  To  the  quot  add  y1^  of  the  rise,  and  2  ft.  The  sum  will 
be  the  thickness  on  or  ey  of  each  abut  at  the  springing  line,  or  the  level  from  which 
the  arch  starts  for  any  abut  whose  height  os  does  not  exceed  1%  times  the  base  sp. 
If  of  rough  rubble  add  6  ins  to  insure  full  thickness  in  every  part. 


Thicks  o  n  of  abut  at  spring 

in  ft,  when  the  height  o  s 

does  not  exceed  \%  times  the 

base  8  p 


Rad  in  ft 
6 


rise  in  ft 
10 


. 

*     J 


Mark  the  points  n  and  y  thus  ascertained.  Next,  from  the  center  t,  of  the  span 
or  chord  e  o,  lay  off  i  />,  equal  to  ^  part  of  the  span.  Join  a  h ;  and  through  n,  and 
parallel  to  a//,  draw  the  indefinite  line  gnp  of  the  abut.  Do  the  same  with  the 
other  abut.  Make  y  m  and  ng  each  equal  to  half  the  entire  height  i  t  of  the  arch ; 
and  from  g  draw  a  straight  line  g  x,  touching  the  back  of  the  arch  as  high  up  as  pos- 
sible ;  or  still  better,  as  shown  at  t  m,  with  a  rad  d  t  or  d  TO,  (to  be  found  by  trial,) 
describe  an  arc  t  m.  Then  gx  or  t  m,  will  be  the  top  of  the  masonry  filling  above  the 
arch.* 


Fi 


;u 


Now  find  by  trial  the  point  s,  Fig  2,  at  which  the  thickness  sp  is  equal  to  two- 


*  EXCEPT   WHEN    THE   RISE    IS 
BUT    ABOUT  4-  OF    THE    SPAN,  OB 

LESS;  in  which  case  carry  the 
masonry  up  solid  to  the  level 
vtf,  of  the  top  of  the  arch.  Or 
if  the  arch  is  a  large  one,  ex- 
ceeding say  about  60  ft  span  ; 
and  especially  if  its  rise  is 
greater  than  about  -^  of  its 
span,  it  is  better  to  economize 
masoqry  by  the  use  of  parallel 
interior  spandrel-walls,  1  1.  Fig 
2^.  carried  up  to  vtf.  Fig  2. 
Indeed,  such  interior  walls  may 
often  be  advantageously  intro- 
duced in  much  smaller  arches. 
When  high,  they  are  steadied 
by  occasional  cross-  walls,  as  tt, 
Pig  2^.  Their  feet  should  be 


.      n  Fig  2«  the  dark  part 
the  second  cross*  wall,  similar  to  tt. 


STONE    BRIDGES.  347 

thirds  of  the  corresponding  vert  height  o  s,  and  draw  up.  Then  will  the  thickness 
on  or  ey  be  that  at  the  springing  line  of  the  given  circular  or  elliptic  arch  of  any 
rise  and  span;  and  the  lineg'j?  will  be  the  back  of  the  abut;  provided  its  height  o"s 
does  not  exceed  1%  times  sp;  or  in  other  words,  provided  xp  is  not  less  than  %  of 
o  s.  In  practice,  sp  will  rarely  exceed  this  limit;  and  only  in  arches  of  considerable 
rise.  But  if  it  should,  as  for  instance  at  o  7,  then  make  the  base  q u  equal  to  sp,  added 
to  one-fourth  of  the  additional  height  sg\  and  draw  the  back  uw,  parallel  to  g p  ; 
and  extending  to  the  same  height,  &c,  as  in  Fig  2.  If,  however,  this  addition  of  % 
of  *•  q  should  in  any  case  give  a  base  q  u,  less  than  one-half  the  total  height  o  q,  (which 
will  very  rarely  happen  in  practice,)  then  make  qu  equal  to  half  said  total  height ; 
drawing  the  back  parallel  to  gp,  and  extending  it  to  tne  same  height  as  before.  The 
additional  thicknesses  thus  found  below  sp,  have  reference  rather  to  the  pres  of  the 
earth  behind  the  abut,  than  to  the  thrust  of  the  arch.  In  a  very  high  abut,  the  inner 
line  g  p  would  give  a  thickness  too  slight  to  sustain  this  earth  safely. 

When  the  height  o b,  Fig  2,  of  the  abut  is  less  than  the  thickness  on  at  spring,  a 
small  saving  of  masonry  (not  worth  attending  to,  except  in  large  flat  arches)  may  be 
effected  by  reducing  the  thickness  of  the  abut  throughout,  thus:  Make  ok  equal  to 
on,  and  draw  kl.  Make  <>z  equal  to  3^  of  on,  and  draw  Iz.  Then,  for  any  height 
oh  of  abut  less  than  on,  draw  h  r,  terminating  in  I  z.  This  bv  will  be  sufficient  base, 
if  the  foundations  are  firm.  The  back  of  the  abut  will  be  drawn  upward  from  v, 
parallel  to  g  />,  and  terminating  at  the  same  height  as  g  or  w. 

REM.  1.  All  the  abuts  thus  found  will  (with  the  provisions  in  Art  6)  be  safe, 
without  any  dependence  upon  the  wing-walls ;  no  matter  how  high  the  embkt  may 
extend  above  the  top  of  the  arch.  If  the  bridge  is  narrow,  and  the  inner  faces  of 
the  wing-walls  are  consequently  brought  so  near  together  as  to  afford  material  as- 
sistance to  the  abuts,  the  latter  may  be  made  thinner;  but  to  what  extent,  must 
depend  upon  the  judgment  of  the  engineer. 

We,  however,  caution  the  young  practitioner  to  be  careful  how  he  adopts  dimensions  less  than  those 
given  by  our  rule.  There  are  certain  practical  considerations,  such  as  carelessness  of  workmanship; 
newness  of  the  mortar;  danger  of  undue  strains  when  removing  the  centers;  liability  of  derange- 
ment during  the  process  of  depositing  the  earth  behind  the  abuts,  and  over  the  arch  ;  &c,  which  must 
not  be  overlooked;  although  it  is  impossible  to  reduce  them  to  calculation. 

Whenever  it  can  be  done,  the  centers  should  remain  in  place  until  the  embkt  is  finished;  and  for 
some  time  afterward,  to  allow  the  mortar  to  set  well. 

RKM.  '2.  A  good  deal  of  liberty  is  sometimes  taken,  in  reducing  the  quantity  of  masonry  above  the 
springing  line  of  arches  of  considerable  rise,  and  of  moderate  spans.  When  care  is  taken  to  leave 
the  centers  standing  until  the  earth  filling  is  completed  above  the  arch,  and  behind  its  abuts,  so  that 
it  may  not  be  deranged  by  accident  during  that  operation ;  and  when  good  cement  is  used  instead  of 
common  mortar,  such  experiments  may  be  tried  with  comparative  safety  ;  especially  with  culvert 
arches,  in  which  the  depth  of  arch-stones  is  great  in  proportion  to  the  span.  They  must,  however,  be 
left  to  the  judgment  of  the  engineer  in  charge  ;  as  no  specific  rules  can  be  laid  down  for  them.  They 
can  hardly  be  regarded  as  legitimate  practice,  and  we  cannot  recommend  them.  We  have  known 
nearly  semicircular  arches,  of  30  to  40  ft  span,  to  be  thus  built  successfully,  with  scarcely  a  particle 
of  masonry  above  the  springs  to  back  them.  Such  arches,  however,  are  apt  to  fall,  if  at  any  future 
period  the  earth  filling  is  removed,  without  taking  the  precaution  to  first  build  a  center  or  some  other 
support  for  them.  Even  when  the  embkt  can  be  finished  before  the  centers  are  removed,  we  cannot 
recommend  (and  that  only  in  small  spans)  to  do  less  than  to  make  ng.  Fig  2,  equal  to  X  of  the  total 
height  it  of  the  arch  ;  and  from  g  so  found,  to  draw  a  straight  line  touching  the  back  of  the  arch  as 
high  up  as  possible. 

REM.  3.  We  have  said  nothing  about  battering  the  faces  of  the  abuts, 

because  in  the  crossing  of  streams,  the  batter  either  diminishes  the  water-way;  or 
requires  a  greater  span  of  arch.  Such  a  batter,  however,  to  the  extent  of  from  % 
to  1^  ins  to  a  ft,  is  useful,  like  the  offsets,  for  distributing  the  wt  of  the  structure, 
and  its  embkt,  over  a  greater  area  of  foundation ;  especially  when  the  last  is  not 
naturally  very  firm  ;  or  when  the  embkt  extends  to  a  considerable  height  above  the 
arch.  In  our  tables,  Nos  3  and  5,  of  approximate  quantities  of  masonry  in  semi- 
circular bridges  of  from  2  to  50  ft  span,  the  faces  are  supposed  to  be  vert. 

Art.  5.  Abutment-piers.  When  a  bridge  consists  of  several  arches,  sus- 
tained by  piers  of  only  the  usual  thickness,  if  one  arch  should  by  accident  of  flood, 
or  otherwise,  be  destroyed,  the  adjacent  ones  would  overturn  the  piers;  and  arch 
after  arch  would  then  fall.  To  prevent  this,  it  is  usual  in  important  bridges  to  make 
some  of  the  piers  sufficiently  thick  to  resist  the  pres  of  the  adjacent  arches,  in  case 
of  such  an  accident ;  and  thus  preserve  at  least  a  portion  of  the  bridge  from  ruin. 
Such  are  called  abutment-piers. 

Our  formula  of  —  _j_  - ---  -j-  2  ft,  for  the  thickness  at  spring;  with  the  back  battering  as  before, 

at  the  rate  of  ?X-  of  the  span  to  the  rise ;  face  vert ;  will  of  itself  (without  any  modification  for  great 
heights')  give  a  perfectly  safe  abut-pier,  for  any  unloaded  bridge  ;  and  to  any  height  whatever  ;  due 
regard  being  had,  however,  to  the  consideration  alluded  to  in  the  next  Art.  Thus,  for  an  abut-pier 
as  high  as  o  q,  Fig  2 ;  or  of  any  greater  height ;  it  is  only  necessary  first  to  find  the  thickness  o  n  at 
spring  as  before  ;  and  then  draw  the  battered  back  gnp:  extending  it  down  to  the  base  at  B  ;  with- 
out adding  %  of  the  additional  height  «  q.  This  addition  is  made  in  the  ease  of  abuts,  that  they 


348 


BTONE   BRIDGES. 


0 


may  be  secure  from  the  pres  of  the  earth  behind  them ;  as  well  as  from  the  pres  of  the  arch ;  a  con. 

sideratiou  which  does  not  apply  to  abut-piers ;  in  which  only  tlie  pres  of  the  arch  is  to  be  resisted. 

But  although  the  abut-pier  tlius  found  by  our  formula,  would  be  abundantly 
safe,  yet  its  shape  a  b  c  o.  Fig  3,  is  inadmissible.  In  practice  it  would  be 
changed  to  one  somewhat  like  that  shown  by  the  dotted  lines  ;  having  an  equal 
degree  of  batter  on  both  faces.  This  of  course  requires  more  masonry,  with 
but  little  increase  of  stability  ;  but  that  cannot  be  avoided. 

WHEN  AN  ABUT-PIER  is  BUILT  IN  DEEP  WATER,  or  in  a  shallow  stream  sub- 
ject to  high  freshets,  care  must  be  taken  that  water  cannot  find  its  way  under 
the  pier,  and  thus  produce  an  upward  pres,  which  will  either  diminish,  or 
entirely  counteract  its  efficiency  as  an  abut.  See  Remark  2,  Art  4,  of  Hy- 
dros t  a  lies,  p  525. 

Art.  6.    Inclination  of  the  course's  of  masonry 
below  the  spring's  of  an  arch.    Although  our  fore- 
going rule  gives  a  thickness  of  abut  which  cannot  be  overturned, 
or  upset,  by  the  pres  of  the  arch,  yet  if  the  arch  be  of  large  span, 
and  small  rise,  its  great  hor  thrust  may  produce  a  sliding  out- 
ward of  the  masonry  near  the  level  of  the  springs,  if  the  stones 
are  laid  in  hor  courses;  especially  if  the  mortar  has  not  set  well. 
This  danger,  it  is  true,  could  be  avoided  by  conflning  the  courses  together 
by  iron  bolts  and  cramps  ;  or  by  increasing  considerably  the  thickness  of  the 
abuts  ;  but  the  expense  of  doing  either  of  these,  leads  to  the  cheaper  expedient 
of  inclining  the  masonry,  as  shown  between  o  and  n,  Pig  4 ;  the  courses  near  o 
being  steeper;  and  gradually  becoming  less  steep  near  n. 
By  this  process  the  arch  is  virtually  prolonged  into  the  body 
of  the  abut,  so  far  that  when  the  inclination  of  the  lower 
masonry  ceases,  as  at  n,  the  direction  of  the  theoretical 
line  of  thrust,  or  of  pres  of  the  arch   (rudely  represented 
by  the  dotted  curved  line  o  n)  is  nearly  at  right  angles  to 
the  joints  of  the  hor  masonry  below  n;  and  consequently, 
said  thrust  is  unable  to  produce  sliding  at  that  point.  Be- 
tween o  and  n,  the  line  of  pres  is  everywhere  so  nearly  at 
right  angles  to  the  variously  inclined  joints,  as  to  preclude 
the  possibility  of  sliding  in  that  interval  also.  See  Art  63 
of  Force   in   Rigid   Bodies.*   The  abut  being  thus  safe 
throughout  from  both  overturning  and  sliding,  can  fail 
only  from  defective  foundations;  or  from  the  inferiority 
of  the  stone  of  which  it  ia  built;  and  which,  if  soft,  may 
be  crushed. 

This  inclination  of  the  masonry  is  as  neces- 
sary in  an  elliptic  arch,  Fig  4%,  as  in  a  circular 
one.  The  direction  of  the  line  of 

thrust  naouniinan  elliptic  arch,  differs  but  slightly  from  that  in  a 
circular  one  of  the  same  span  and  rise.    The  amount  of  thrust  is  also  nearly  the 

same  in  both ;  consequently, 
the  same  precautions  for  re- 
sisting it,  must  be  adopted 
in  each.  The  dotted  line 
n  a  o  u  m,  in  Fig  4^,  gives  a 
tolerable  general  idea  of  the 
theoretical  line  of  thrust  in 
an  elliptic  arch,  when  the 
filling  above  the  arch,  and 
abuts,  is  either  as  repre- 
sented in  the  fig;  or  when 
it  is  wholly  of  earth,  or 


*  This  curved  line  of  pressures  is  found  in  the  manner  directed  at  Rem  1,  p  492.  and  at  Fig  25,  p 
531.  Rankine,  Moseley,  and  others  call  it  the  line  of  resistance,  and  ap- 
ply line  of  pressures  to  another  line  which  need  not  be  introduced  in  a  practical  consideration  of 
abutments,  walls,  dams,  &c.  They  however  call  any  given  point  in  our  line  a  center  of  pressure, 
because  at  any  part  of  the  height  of  the  abut  such  point  shows  where  all  the  pressure  or  thrust  may 
for  many  purposes  be  assumed  to  be  concentrated.  The  perversion  of  common  technical  terms  is 
reprehensible.  Said  other  line  had  better  been  called  the  line  of  resultants.  We,  both  here,  and  on 
p  493,  531,  and  elsewhere,  call  the  one  to  which  we  refer  the  line  of  pressure, 
or  of  thrust,  simply  because  bridge  masons  have  no  idea  of  any  other  line 
curving  through  an  abut,  and  inasmuch  as  the  pressure  or  thrust  is  greatest  in  that  line  they  very 
properly  so  term  it.  Again,  we  have  said  above  and  elsewhere  that  the  bed-joints  should  be  nearly 
perp  to  this  line  of  thrust.  Theory  properly  requires  them  to  be  at  right 
angles  to  the  resultants  which  cut  the  bed-joints  at  this  line  of 

thrust.  Still  we  know  that  on  account  of  the  friction  of  masonry  we  may  with  perfect  safety  vary  as 
much  as  about  30°  from  a  right  angle  to  these  resultants,  without  depending  at  all  on  the  strength 
of  the  mortar ;  see  Art  63,  p  487  ;  and  in  using  our  rule  of  thumb  on  the  next  page  for  drawing  in- 
clined bed-joints,  we  shall  always  be  far  within  the  limit  of  30°;  and  therefore  fully  safe  from  sliding. 
Oar  rules  do  not  call  for  this  line,  nor  for  anything  more  than  the  spaa,  rise,  and  radius  of  the  arch. 


STONE    BRIDGES.  349 

wholly  of  solid  masonry;  although  both  its  amount,  and  its  direction,  will  differ 
somewhat  according  as  one  or  the  other  of  these  modes  of  filling  is  adopted.* 

The  elliptic  form  is  plainly  unfavorable  for  uniting  the  arch-stones  with  the  inclined  masonry  near 
the  springs,  so  as  to  receive  "the  thrust  properly  ;  or  about  at  right  angles  to  its  res  .Itant.  In  ordi- 
nary cases  this  difficulty  may  be  overcome  by  making  the  joints  of  only  the  outside  or  showing  arch- 
•tones  to  conform  to  the  elliptic  curve;  as  between  e  and  a;  while  the  joints  of  the  inner  or  hidden 
cues,  may  have  the  directions  shown  between  g  and  u,  nearly  at  right  angles  to  the  line  of  thrust.  It 
will  rarely  happen,  however,  that  the  young  engineer  will  have  to  construct  elliptic  arches  of  suffi- 
cient magnitude  to  require  either  this,  or  any  equivalent  expedient.  For  spans  less  than  50  ft,  with 
rises  not  less  than  about  ^  of  the  span,  nothing  of  the  kind  is  actually  necessary,  if  the  mortar  is 
good,  and  has  time  to  harden. 

In  order  to  incline  the  masonry  of  any  abut  with  sufficient  accuracy,  it  would 
be  necessary  first  to  trace  the  curved  line  of  pres  of  the  given  arch,  as  directed  in 
Art  72  of  Force  in  Rigid  Bodies,  so  as  to  arrange  the  bed  joints  about  at  right  angles 
to  it  at  every  point  of  its  course;  but  we  offer  the  following  process  as  sufficing  for  all 
ordinary  practical  purposes;  while  its  simplicity  places  it  within  the  reach  of  the  com- 
mon mason.  In  actual  bridges  the  direction  of  tne  actual  thrust  changes  as  the  load 
is  passing;  therefore,  in  practice  no  given  degree  of  Inclination  of  the  abut  masonry 
can  conform  to  it  precisely  during  the  entire  passage.  Consequently,  any  excess  of 
refinement  in  this  particular,  becomes  simply  ridiculous ;  especially  in  small  spans. 

Rule  for  inclining:  the  beds  of  the  masonry  in  the  abnts. 

Add  together  the  rad  cm,  Fig  4;  and  the  span  of  the  arch.  Div  the  sum  by  5.  To 
the  quot  add  3  ft.  Make  o  t,  on  the  rad,  equal  to  the  last  sum.  Then  is  t  a  central 
point,  toward  which  to  draw  the  directions  of  the  beds,  as  in  the  fig.  Draw  t  s  hor, 
and  from  t  as  a  center,  describe  the  arc  oy\o  being  the  center  of  the  uniform  depth 
of  the  arch.  From  y  lay  off  on  the  arc  the  dist  y  n,  equal  to  one-sixth  part  of  t  y ; 
draw  t  n  a.  It  will  never  be  nec'ssary  to  C  "line  the  masonry  below  this  t  n  a. 
Neither  need  the  inclination  extend  entirely  to  the  face  m  i  of  the  abut;  but  may 
stop  at  e,  about  half-way  between  i  and  n.  From  e  upward,  the  inclination  may 
extend  forward  to  the  line  e  m.  See  Foot-note  p  348. 


*  It  will  be  seen  that  in  consequence  of  the  more  sudden  curvature  of  an  elliptic  arch,  near  the 
parts  a  and  «.  the  dotted  line  of  thrust  iu  Fig  4J^  may  pass  entirely  out  from  the  arch-stones  at  those 
points,  and  enter  again  below.  This  causes  a  tendency  in  the  parts  of  the  arch  in  that  vicinity  to 
rise  ;  and  thus  permit  the  portions  near  the  crown  o,  to  descend.  If  the  arch  in  Fig  4%  were  a  seg- 
ment of  a  circle,  the  direction  of  the  line  of  thrust  would  become  but  little  changed;  and  the  arch 
would  consequently  coincide  with  it  more  nearly  than  in  the  elliptic  one;  and  hence  would  be  more 
stable  than  this  last.  If  the  arch  is  a  full  ".emicircle,  with  stones  of  ordinary  depth,  the  line  of 
thrust  wil!  pass  out  of  it,  and  into  it  again,  as  in  the  elliptic  one ;  producing  the  same  deranging  ten- 
dency. See  Rem  3,  p  493,  of  Force  In  Rigid  Bodies.  So  long  as  theintrados  is  elliptic,  or  a  full  semi- 
circle, with  keystones  of  the  ordinary  proportions,  no  increase  in  the  depth  of  the  arch-stones  from 
the  key  toward  the  springs,  will  prevent  the  line  of  thrust  from  thus  falling  below  the  arch  :  or  even 
materially  change  its  direction  in  an  actual  bridge.  Thus,  if  in  Fig  4J^,  the  depth  of  the  arch- stones 
were  increased  toward  the  springs,  even  as  much  as  is  shown  by  the  dotted  line  w  s,  the  line  of  thrust 
in  the  finished  bridge  would  still  be  along  the  dotted  line  o  an;  the  greater  depth  serving  merely  to 
distribute  the  thrust  over  a  greater  area.  If,  instead  of  solid  masonry,  a  hollow  vault  were  constructed 
at  w  to  support  the  roadway,  the  line  of  thrust  at  a  would  rise  a  little ;  and  by  thus  building  some 
parts  of  the  spandrels  hollow,  and  other  parts  solid,  much  may  be  done  in  the  way  of  keeping  tbsline 
of  thrust  within  the  arch-stones.  The  same  could  be  accomplished  by  greatly  increasing  the  depth 
of  all  the  arch-stones;  for  iu  that  case  a  line  parallel  to  o  an,  and  passing  through  the  center  of  the 
depth  of  the  new  keystone,  as  well  as  that  of  the  new  springer,  would  everywhere  fall  within  thearch- 
Btones ;  although  not  to  the  same  extent  in  all  of  them.  This  may,  in  fact,  be  the  reason  for  the  great 
depth  of  key  in  elliptic  arches,  referred  to  in  Article  3.  In  cases  where  appearances  may  be  sac- 
rificed to  strength,  we  might  first  find  (by  Rem  3,  p  493,  of  Force  in  Rigid  Bodies,)  the  line  of  thrust  or 
pressure  of  a  required  arch ;  and  if,  as  in  Fig  4^  above,  U  does  not  fall  well  within  the  arch-stones, 
we  mav  arrange  the  latter  so  as  to  coincide  with  it.  Sucft  a  change  in  the  arch-stones  would  of  course 
in  turn  slightly  change  the  direction  of  the  line  of  thrust;  but  in  ordinary  practice  this  might  be 
neglected.  As  a  general  rule,  the  change  in  the  shape  of  the  arch  would  be  detected  by  but  few  per- 
sons; nor  indeed,  does  it  follow  that  the  new  curve  must  be  at  all  ungraceful. 

As  a  segment  arch  becomes  more  flat,  the  line  of  thrust  coincides  more  nearly  with  the  arch-stones. 
The  writer  draws  the  following  inference,  from  the  results  with  many  of  his  own  diagrams.  Front 
the  center  c,  Fig  1^.  from  which  any  segmental  arch  whose  rise  does  not  exceed  .4  of  its  span,  IN 
described,  draw  two  lines  c  o,  c  o.  forming  angles  of  25°  with  c  a.  Then  supposing  the  depth  of  the 
keystone  t,o  be  according  to  our  rule,  make  the  depths  of  the  arch- stones  between  o  and  «,  everywhere 
equal  to  1^  times  that  of  the  key.  From. the  key  increase  the  depth  gradually  to  o  and  o.  Then  will 
the  theoretical  line  of  thrust  of  the  arch  (supposed  to  be  filled  up  to  the  level  of  n  n)  everywhere  fall 
within  the  actual  arch-stones ;  nowhere  tpproachine;  their  extrados  or  intrados  nearer  than  about  % 
of  the  depth  of  the  arch  at  the  same  point ;  and  this  will  be  the  case  whether  the  filling  be  either  earth, 
or  solid  masonry :  or  of  parallel  spandrel  walls  :  or  partly  of  masonry,  and  partly  of  earth ;  both  dis- 
posed as  indicated  by  the  line  «.  t,  or  i,  &c;  although  both  the  direction  and  intensity  of  the  line  of 
thrust  will  of  course  vary  slightly  under  these  different  conditions.  This  appears  to  the  writer  to 
be  an  important  fact  in  practice,  as  regards  segmenta!  arches  of  great  spans. 

t  The  feet  of  both  elliptic  and  semicircular  arches  are  always  made  hor;  bat  it  is  plain  from  lig 
4}£.  that  this  practice  is  at  variance  with  correct  principles  of  stability  in  the  case  of  the  ellipse.  It 
is  the  same  in  the  semicircle.  In  ordinary  bridges  of  the  latter  form,  the  vert  pres,  or  weight  resting 
on  each  skewback.  is  (roughly  speaking)  usually  about  from  3%  to  4  times  the  hor  pres  on  the  same  ; 
and  the  total  pres  is  about  4  times  as  great  as  the  pres  on  the  keystone.  Therefore,  theoretically,  the 
skewback  should  usually  be  about  4  times  as  deep  as  the  keystone  ;  and  Us  bed,  instead  of  being  hor, 
•bould  be  inclined  at  the  rate  of  ahont  1  vert  to  4  hor.  See  Example  2,  p  468,  of  Force  in  Rigid  Bodiei. 

23 


350 


STONE    BRIDGES. 


S     0    C 


When  the  arch  is  flat,  this  inclination  may  become  so  steep,  especially  in  the  upper  parts,  that 
struts,  or  shores  of  some  kind,  must  be  used  for  preventing  the  ma- 
sonry from  sliding  down,  until  the  completion  of  the  arch  secures  it 
from  doing  so.  The  hor  courses  between  the  face  TO  i,  and  the  line 
o  e,  will  aid  somewhat  in  this  respect. 

The  method  thus  described,  should  be  applied  to  all  very  largt 
arches  whose  rise  is  %,  or  less,  of  the  span.  As  before  remarked,  it 
is  not  actually  necessary  in  arches  not  exceeding  about  50  ft  span, 
and  not  flatter  than  ?-  of  the  span.  Indeed,  if  the  earth  filling  can 
be  deposited  before  the  centers  are  removed,  these  limits  may  be  con- 
siderably extended  without  danger.  Still,  since  a  certain  degree  of 
inclination  is  attended  with  very  little  trouble  or  expense,  we  would 
recommend  for  even  such  arches,  a  process  somewhat  like  the  follow- 
ing: From  half  the  span  take  the  rise.  Div  the  rem  by  3.  Make  o  t, 
Fig  5,  equal  to  the  quot.  Draw  t  n,  and  o  m,  hor.  Div  the  angle 
t  o  m  into  two  equal  parts,  by  the  line  o  a.  Incline  tl.e  masonry  so 
as  to  be  parallel  to  o  a,  as  far  down  as  t  n.  The  inclined  course* 
may  extend  out  to  the  face  o  t,  or  not,  at  pleasure. 

REM.  1.  The  necessity  for  inclining  the  courses  becomes  greater  as  the  span  in- 
creases; because  the  pres  from  arches  of  the  same  transverse  width,  but  otherwise 
of  the  same  proportions,  and  of  diff  spans,  increases  nearly  aa  the  squares  of  the 
spans;  whereas  the  thickness  of  the  abut  at  the  springs  increases  only  nearly  in  the 
same  proportion  as  the  span  itself.  Hence,  in  the  larger  span,  the  abut  at  spring 
presents  a  less  proportion  of  mortar-joint,  and  of  friction,  to  prevent  sliding,  than 
in  the  smaller  one.  In  practice,  the  pres  of  arches  does  not  increase  quite  as  mpidly 
as  the  squares  of  the  spans,  because  the  depth  of  the  arch-stones  does  not  increase 
as  fast  as  the  spans;  thus  reducing  the  relative  wt  of  the  large  one  considerably. 

REM.  2.    To  find  the  length  (ah,  Fig  7) 
.    ,  from  face  to  face  of  a  culvert.     From 

1     .11.   ct  the  height  A  t  of  the  embkt,  take  the  above  ground  height  n  a 

of  the  culvert;  the  rem  will  be  the  height  ho  of  the  embkt 
above  the  culvert.  Then  the  reqd  length  a  ft  is  plainly  equal 
to  the  top  width  id  of  the  embkt,  added  to  the  two  dists  as, 

cfc,  which  correspond  to  its  steepness  of  side-slopes.     Thus,  if 

Jl  t  ^.       the  side-slope  is,  as  usual,  1^  to  1,  then  as  and  c  b  will  each  be 

equal  to  1%  times  o  ft ;  or  the  two  together  will  be  3  times  o  h. 

So  that  if  the  width  i  d  is  1 4  ft,  and  A  o  5  ft,  the  length  a  b  will  be 

14  -f  (5  X  3)  =  14  +  15  =  29  ft. 

Art.  7.  The  following  tables,  3,  4,  and  5,  of  quantities,  will 
be  found  useful  for  expediting  preliminary  estimates ;  for  which  purpose  chiefly  they 
are  intended  ;  hence  no  pains  have  been  taken  to  make  them  scrupulously  correct, 
but  rather  a  little  in  excess  of  the  truth.  The  first  column  of  Table  3  contains  the 

total  vert  height  o  c,  Fig  6,  from  the 
crown  n  of  a  semicircular  arch,  to 
the  foundation  or  base  gm  of  its 
abut.  The  other  columns  give  ap- 
proximately the  number  of  cub  yds 
contained  in  each  running  foot,  or 
foot  in  length  of  the  culvert,  or 
bridge,  measured  from  end  to  end 
(face  to  face)  of  the  arch  proper; 
and  including  only  the  arch  and  its 
abuts,  as  shown  in  Fig  1 ;  or  in  the 
half  section  opmgy  in  Fig  6;  in- 
cluding footings  to  the  abuts,  but 
omitting  the  wing-walls  (w  n),  and 
the  spandrel-walls  (*),  Figs  6  and 

At  the  foot  of  each  column  is  the  approximate  content  in  cub  yds  of  the  two 
spandrel- walls  by  themselves;  one  over  each  face  of  the  arch. 

These  spandrel- walls  are  calculated  on  the  supposition  that  their  thickness  at  base,  at  their  junc- 
tion with  the  wing- walls,  where  their  height  iso  greatest,  is  equal  to  -^  of  their  height  at  that  point : 
except  where  that  proportion  gives  a  less  thickness  at  top  than  2J^  ft;  and  that  they  extend  2  ft  (o  a) 
above  the  top  o  of  the  arch.  At  the  top  of  the  arch,  they  are  all  supposed  to  be  2%  f't  thick  at  top; 
that  being  assumed  to  be  about  the  least  thickness  admissible  in  a  rubble  wall  in  such  a  position. 
Both  the  back  and  the  face  are  supposed  to  be  vert.  The  contents  of  these  spandrel-walls  will  vary 
somewhat,  however,  even  in  the  same  span,  with  the  height  of  the  abut  and  the  arrangement  of  the 
•wings.  They,  however,  constitute  so  small  a  proportion  of  the  entire  contents  given  in  Table  5.  that 
this  consideration  may  be  neglected  in  preliminary  estimates.  They  are  so  firmly  bonded  into  the 
masonry  of  the  wings  at  their  highest  points,  arid  so  strongly  connected  by  mortar  with  the  backing 
of  the  arch  at  their  bases,  that  they  require  no  greater  thickness  however  high  the  emb  may  be. 

The  contents  of  the  four  wing-walls,  of  which  nj w  b,  Fig  6,  is  one, 
will  be  found  in  a  table  (No.  4)  immediately  following  that  for  the  body  of  the  cul- 
Vert.  We  have  also  added  a  table  (No.  5)  for  complete  semicircular  culverts  «f 
vari ous  lengths,  including  their  spandrel  and  wing  walla. 


\Fie-.6 


STONE   BRIDGES. 


351 


REM.  1.  Although  the  thickness  of  wing-walls  increases  in  all  parts  with  their 
height,  they  are  not  made  to  show  thicker  at  nj  than  at  ti,  Fig  6 ;  but  (as  seen  in  the 
fig)  are  ofisetted  at  their  back  t  n,  a  little  below  their  slanting  upper  surf  ij,  so  as 
to  give  a  uniform  width  for  the  steps  or  flagstones,  as  the  case  may  be,  with  which 
they  are  covered.  In  the  fig  the  covering  is  supposed  to  be  of  flagstones  ;  but  steps 
are  preferable,  being  less  liable  to  derangement.  To  prevent  the  flagstones  from 
sliding  down  the  inclined  plane  jt,  the  lower  stone  i  should  be  deep  and  large,  and 
laid  with  a  hor  bed.  The  flags  are  sometimes  cramped  together  with  iron,  and  bolted 
down  to  the  wall.  Steps  require  nothing  of  that  kind,  as  seen  at  s,  Fig  11,  p  355. 

REM.  2.  The  tables  show  the  inexpediency  of  too  much  con- 
tracting the  width  of  water-way,  with  a  view  to  economy,  by  adopting 
a  small  span  of  arch,  when  a  culvert  of  greater  span  can  be  made,  of  the  same  total 
height. 

For  the  wings  must  be  the  same,  whether  the  span  be  great  or  small,  provided  the  total  height  it 
the  same  fn  both  cases  ;  and  since  the  wings  constitute  a  large  proportion  of  the  entire  quantity  of 
masonry,  in  culverts  of  ordinary  length,  the  span  itself,  within  moderate  limits,  has  comparatively 
little  effect  upon  it.  Thus,  the  total  masonry  in  a  semicircular  culvert  of  3  ft  span,  8  ft  total  height, 
and  60  ft  long  between  the  faces  of  the  arch,  is,  by  Table  5,  151^6  cub  yds  ;  while  that  of  a  5  ft  span, 
of  the  same  height  and  length,  is  152.4.  A  semicircular  bridge  of  25  ft  span,  24  ft  total  height,  and 
40  ft  between  the  faces  of  the  arch,  contains  1031  cub  yds ;  while  one  of  35  ft  span,  of  the  same  height 
and  length,  contains  1134  yds ;  so  that  in  this  case  we  may  add  nearly  50  per  cent  to  the  water-way, 
by  increasing  the  masonry  of  the  bridge  but  y^jth  part. 

REM.  3.  Partly  for  the  same  reason,  and  partly  because  the  cnlverts  for  a 
double-track  road  are  not  twice  as  long;  as  those  for  a  single- 
track  one,  the  quantity  of  culvert  masonry  for  the  former  will  not  average  more 
than  about  from  %  to  %  part  more  than  that  for  the  latter;  so  that  it  frequently 
becomes  expedient  to  finish  the  culverts  at  once  to  the  full  length  required  for  a 
double  track,  although  the  embkts  may  at  first  be  made  wide  enough  for  only  a 
single  one,  with  the  intention  of  increasing  them  at  a  future  time  for  a  double  one. 

Thus,  the  average  size  of  culverts  for  a  single  track  may  be  roughly  taken  at  6  ft  span,  30  ft  lon^ 
from  face  to  face,  and  10  ft  total  height;  and  such  a  one  contains,  by  Table  5,  140  cub  yds.  For  a 
double  track,  it  would  require  to  be  about  12  feet  longer;  and  we  see  by  Table  3  that  this  will  add 
2.67  X  12  ~  32  cub  yds  ;  making  a  total  of  172  yds  instead  of  140  ;  thus  adding  rather  >2ss  than  y^ 
part.  When  the  culverts  are  under  very  high  embkts,  and  consequently  much  longer,  the  addition 
for  a  double  track  becomes  comparatively  quite  trifling. 

Table  3,  of  approximate  numbers  of  cnb  yds  of  masonry 
per  foot  run,  contained  in  the  arches  and  abutments  only,  as 

shown  in  Fig  1  (omitting  wings,  and  the  spandrel-walls  over  the  faces  of  the  arches) 
of  semicircular  culverts  and  bridges,  of  from  2  to  50  ft  span,  and  of  different  total 
heights,  ht,  Fig  1,  or  o  c,  Fig  6.  It  will  be  seen  that  in  many  cases,  a  bridge  of  larger 
span  contains  less  masonry  than  one  of  smaller  span,  when  their  total  heights  are  th* 
same.  There  is  a  liberal  allowance  for  footings  or  offsets  at  the  bases  of  the  abut* 
TABLE  3.  (Original.) 


Total 
Height. 

Span 
2ft. 

Span 
3ft. 

Span 

4ft. 

Span 
5ft. 

Span 
6ft. 

Span 
8ft. 

Span 
10ft. 

Span 
12ft. 

Span 
15ft. 

Feet. 
2 

Cub.   y. 
42 

Cub.  y. 

Cub.  y. 

Cub.  y. 

Cub.  y. 

Cub.  y. 

Cub.   y. 

Cub.  y. 

Cub.  y. 

3 

60 

63 

67 

4 

79 

83 

87 

92 

.97 

5 

99 

1  04 

1.08 

1  15 

1.21 

g 

1  28 

1  28 

1  28 

1  37 

1.46 

1  58 

1.69 

7 
8 
9 
10 
11 

1.62 
2.01 
2.45 
2.94 

1.59 
1.96 
2.38 
2.85 
3  38 

1.55 
1.91 
2.31 
2.76 
3  26 

1.64 
1.95 
2.29 
2.72 
3.19 

1.72 
1.99 
2.27 
2.67 
3  12 

1  85 
2.13 
2.42 
2.77 
3.16 

1.97 
2.26 
2.56 

2.87 
3.19 

2J2 
2.38 
2.65 
2.93 
3.23 

"3.62" 
3.34 
3  67 

12 

3.98 

3.82 

3.72 

3.62 

3.57 

3.52 

3.55 

4.01 

13 

4  42 

4.29 

4.17 

4.10 

4.02 

2.86 

4.36 

14 

5.08 

4.90 

4.77 

4.67 

4.57 

4.41 

4.72 

15 

5.57 

5.42 

5.30 

5.17 

5.01 

5.09 

16 

6  30 

6.12 

5.97 

5.82 

5.56 

5.69 

17 

6  87 

6  70 

6  52 

6.26 

6.34 

18 

7  69 

7.48 

7.27 

7.01 

7.04 

19 

8  32 

8.07 

7.71 

7.69 

20 

9  20 

8  92 

8.56 

8.49 

21 

9.82 

9.46 

9.34 

22 

10.8 

10.3 

10.2 

23 

11.3 

11.1 

24 

12.3 

12.1 

25 

13.2 

26 

14.2 

Contents  of  the  two  spandrel- walls,  over  the  two  ends  of  the  arch,  in  cub  yds. 
I      2.9      I     3.7      |      4.4      |      5.2      |     5.8      |      7.9     |     9.8     I      12.      I     16. 


352 


STONE    BRIDGES. 


TABLE  3.    (Continued.) 


Total 
Height. 

Spaa 
20  ft. 

Spaa 
25ft. 

Total 
Height. 

Span 
35  ft. 

Total 
Height. 

Span 
50ft. 

Feet. 
12 

Cub.  v. 
4.60 

Cub.  y. 

Feet. 
20 

Cub.  y. 
10.5 

Feet. 

27 

Cub.  y. 
18.0 

13 

4.98 

21 

11.0 

28 

18.7 

14 

5.37 

6.10 

22 

11.6 

29 

19.4 

15 

5.77 

6.41 

23 

12.2 

30 

20.1 

16 

6.18 

6.76 

24 

12.7 

31 

20.9 

17 

6.60 

7.16 

25 

13.3 

32 

21.6 

18 

7.03 

7.61 

26 

13.8 

33 

22.4 

19 

7.47 

8.10 

27 

14.5 

34 

23.1 

20 

8.12 

8.60 

28 

15.1 

35 

23.9 

21 

8.82 

9.02 

29 

15.7 

36 

24.7 

22 

9.57 

9.72 

30 

16.3 

37 

25.5 

23 

10.4 

10.4 

31 

17.0 

38 

26.3 

24 

11-3 

11.1 

32 

181 

39 

27.1 

25 

12.2 

12.1 

33 

19.2 

40 

28.0 

26 

13.1 

13.0 

34 

20.4 

41 

28.8 

27 

14.1 

14.0 

35 

21.7 

42 

30.0 

28 

15.2 

15.0 

36 

23.0 

43 

31.5 

29 

i6.3 

16.1 

37 

243 

44 

33.0 

30 

17.4 

17.2 

38 

25.7 

45 

34.6 

31 

18.6 

18.4 

39 

27.2 

46 

36.3 

32 

19.9 

19.6 

40 

28.7 

47 

38.1 

SH 

21  2 

20.9 

41 

30.2 

48 

39.8 

5* 

22.6 

22.2 

42 

31.8 

49 

41.6 

35 

24.0 

23.6 

43 

33.5 

50 

43.6 

36 

25.4 

25.0 

44 

35.2 

51 

45.5 

37 

26.9 

26.5 

45 

36.9 

52 

47.4 

38 

285 

28.0 

46 

38.7 

53 

49.4 

39 

30.1 

29.5 

47 

40.6 

54 

51.6 

40 

31.7 

31.2 

48 

42.5 

55 

53.7 

41 

32  8 

49 

44.4 

56 

55.9 

42 

34.5 

50 

46.4 

57 

58.1 

43 

36.3 

58 

60.4 

44 

38.1 

59 

62.7 

45 

40.0 

60 

65.1 

Contents  of  the  two  spandrel-walls,  over  th«  two  ends  of  the  arch,  in  cub  yds. 
28.          |     42.          II  |      85.        II  |    195. 


Art.  8.  The  following*  table  of  contents  of  wing-walls,  or  wings,  will, 
like  the  preceding  one,  be  useful  in  making  preliminary  estimates.  The  wings 
no,  no,  shown  in  plan  at  Fig  8,  are  supposed  to  form  an  angle  aoc,  of  120°,  with  the 
face,  or  end  o  o  of  the  culvert.  Their  outer  or  small  ends  n  n,  are  all  assumed  to  be  of 
the  dimensions  shown  on  a  larger  scale  at  E.  Thickness  at  base  at  every  part  equal 
to  T^y  of  the  height  of  the  wall  at  said  part;  except  when  that  proportion  becomes 
too  small  to  allow  the  width  or  thickness  at  top  to  be  2.5  ft ;  in  which  case  it  is  en- 
larged at  such  parts  sufficiently  for  that  purpose.  See  Remark  2.  This  happens  only 


2.5 


{m 


TL 


3.125 


xTIL 


when  the  height  m  m,  Fig  E,  of  the  wing,  becomes  less  than  9  ft.  Batter  of  face,  1% 
ins  to  a  ft ;  or  1  in  8.  Back  vert ;  but  offsetted,  if  necessary,  for  a  short  dist  below 
the  top,  so  as  to  give  a  uniform  showing  top  thickness  of  2%  ft.  The  masonry  is 
supposed  to  be  good  well-scabbled  mortar  rubble.  The  height  given  in  the  first 
column  is  the  greatest  one;  or  that  at  oo,  (or  wj,  Fig  6,)  where  the  wing  joins  the 
face  of  the  culvert.  In  the  table  no  allowance  is  made  for  footings  (offsets  or  steps) 
at  the  base  of  the  wings  J  as  these  are  frequently  omitted  in  wings  on  good  founda- 


STONE    BRIDGES. 


353 


tiona.    In  taking  out  quantities  from  the  table,  bear  in  mind  that  the  height  of  the 
wings  is  usually  a  little  greater  than  that  of  the  culvert  itself. 

Table  4,  of  approximate  contents,  in  cub  yds,  of  the  four 
wing- walls  of  a  culvert,  or  bridge.    (Original.) 

The  heights  are  taken  where  greatest;  as  atj  w,  Fig  6 


Height 

Length 
of 

Cub.  vds. 
in 

Height 
of 

Length 
of 

Cub.  yds. 
in 

wing. 

Feet. 

Feet. 

Feet. 

Feet. 

6 

1.73 

4.04 

30 

43.S 

818 

7 

3.46 

8.85 

32 

46.8 

997 

8 

5.20 

14.6 

34 

50.3 

1192 

9 

6.U3 

21.5 

36 

53.7 

1414 

10 

8.66 

30.2 

38 

57.2 

1661 

11 

10.4 

40.9 

40 

60.7 

1928 

12 

12.1 

53.7 

42 

64.2 

2220 

14 

15.6 

85.2 

44 

67.6 

2552 

16 

19.1 

128 

46 

71.1 

2912 

18 

2'2.5 

183 

48 

74.6 

3306 

20 

26.0 

247 

50 

78.0 

3741 

22 

29.5 

329 

55 

86.7 

4942 

24 

32.9 

426 

60 

95.3 

6404 

26 

36.4 

541 

65 

104 

8131 

28 

39.8 

672 

70 

113 

10155 

To  reduce  cub  yds  to  perches  of  25  cub  ft,  mult  by  1.080. 
To  reduce  perches  to  cub  yds,  mult  by  .926,  or  div  by  1.08. 


The  contents  for  heights  intermediate  of  those  in  the  table  may  be  found  approximately  by  simple 
proportion. 

RKM.  1.  It  is  not  recommended  to  actually  prolong  all  wings  until  their  dimen- 
sions become  as  small  as  shown  at  E,  in  Fig  8.  In  large  ones  it  will  generally  be 
more  economical  to  increase  their  end  height  m  m,  a  lew  feet.  The  contents,  how- 
ever, may  be  readily  found  by  the  table  in  that  case  also.  Thus  suppose  the  height 
«»f  the  wings  at  one  end  to  be  30  ft,  and  at  the  other  end  8  ft;  we  have  only  to  sub- 
tract the  tabular  content  for  S  ft  high,  from  that  for  30  ft  high.  Thus,  818  — 14.6  = 
803.4  cub  yds  required  content. 

HEM.  2.  It  might  be  supposed  that  inasmuch  as  the  wings  of  arches  often  have  to 
sustain  the  pressure  from  embankments  reaching  far  above  their  tops,  they  should, 
like  ordinary  retaining-walls,  be  made  much  thicker  in  that  case.  But  the  fact  that 
they  derive  great  additional  stability  from  being  united  at  their  high  ends  to  the 
body  of  the  bridge  or  culvert,  renders  such  increase  unnecessary  when  pioportioned 
by  our  rule ;  no  matter  how  far  the  earth  may  extend  above  them ;  as  shown  by 
abundant  experience. 

Relying  upon  this  aid.  we  may  indeed,  when  the  earth  does  not  extend  above  the  top.  reduce  the 

the  wings,  instead  of  heinc  splayed  or  flared  out,  as  at  on.  on.  merely  form  straight  prolongations 
of  the  abutments  of  the  arch,  as  shown  by  the  dotted  lines  at  og  w.  In  this  case  the  pressure  of  the 
earth  against  the  wings  is  less  thnn  when  they  are  splayed.  We  have  known  the  thickness  at  o 
to  be  reduced  in  such  cases  to  rnther  less  than  %  tne  height,  when  the  wings  were  15  ft  high,  and 
the  height  of  the  embankment  above  their  tops  16  feet  in  one  case,  and  36  ft  in  another.  In  another 
instance,  similar  wings  25^  ft  high,  and  with  29  ft  of  embankment  above  their  top.  had  their  bases 
at  o  rather  less  than  -^  of  the  height.  In  all  these  cases,  the  uniform  thickness  at  top  was  2.5  feet; 
backs  vertical.  We  mention  them  because  this  particular  subject  does  not  seem  to  be  reducible  to 
any  practical  rule.  The  last  wall  appears  to  us  to  be  too  thin  ;  especially  if  the  earth  is  not  deposited 
in  layers:  and  after  allowing  the  mortar  full  time  to  set.  The  labor,  however,  required  in  compact- 
ing the  earth  carefully  in  layers,  may  cost  more  than  is  thereby  saved  in  the  masonry.  The  young 
practitioner  must  bear  this  in  mind  when  he  wishes  to  economize  masonry  by  such  means :  and  also 
that  the  thin  wall  may  bulge,  or  fail  entirely,  if  the  earth  backing  is  deposited  while  the  mortar  i* 
imperfectly  set. 


354 


8TONE   BRIDGES. 


Table  5.  Approximate  contents  in  cubic  yards,  of  com- 
plete semicircular  culverts  and  bridges  of  from  2  to  50  feet 
span;  including  the  2  spandrel  walls ;  and  the  4  wings ;  all  proportioned  by  the 
foregoing  directions:  and  taken  from  the  two  preceding  tables.  The  height  in  the 
second  column,  is  from  the  top  of  the  keystone  to  the  bottom  of  the  foundation.  The 
wings  are  calculated  as  being  2  ft  higher  than  this,  including  the  thickness  of  the 
coping.  The  wings  are  frequently  carried  only  to  the  height  of  the  top  of  the  arch; 
thus  saving  a  good  deal  of  masonry.  Table  4,  of  wings  alone,  will  serve  to  make  tha 
proper  deduction  in  this  case. 

The  several  lengths  are  from  end  to  end,  or  from  face  to  face,  of  the  arch  proper. 
The  contents  for  intermediate  lengths  maybe  found  exactly;  arid  those  for  inter- 
mediate heights,  quite  approximately,  by  simple  proportion.  In  this  table,  as  in 
No.  3,  it  will  be  observed  that  when  the  heights  are  Lite  same  in  both  casts,  a  larger 
span  frequently  contains  less  masonry  than  a  smaller  one.  A  semicircular  culvert 
or  bridge  contains  less  masonry  than  a  flatter  one,  when  the  total  height  is  the  same 
in  both  cases;  therefore,  the  first  is  the  most  economical  as  regards  cost;  but  it  does 
uot  afford  as  much  area  of  water-way  ;  or  width  of  headway. 

(Original.) 


a 
1 

of 

^ 

~Sb 
'£ 

33 

2- 

JS 

tj 
J= 

Length. 
30  Ft. 

as 

|3 

«U 

« 

1 

f£ 
g 

&J 
*>=* 

SB 

JS  ^ 

•£,&« 

§| 

ta56" 
g§ 

j=  « 
«>&« 
§3 

J=  w 

&*< 

C  g 

Ft. 

2 

Ft. 
5 
6 
7 
8 
10 

CubY. 
27 
37 
49 
63 
101 

Cub  Y. 

32 
43 
57 
7» 
116 

CubY. 
42 
56 
73 
93 
145 

Cub.Y. 
52 
69 
89 
113 
175 

Cub.Y. 
72 
94 
122 
153 
234 

Cub.Y. 
92 
120 
154 
193 
291 

Cub.Y. 
112 
146 
187 
233 
351 

Cub.Y. 
132 
171 
219 
273 
410 

Cub.Y. 
152 
197 
251 
313 
469 

Cub.Y. 
172 
222 
284 
353 
527 

Cub.Y. 
192 

248 
316 
393 
586 

Cub.Y. 
212 
274 
349 
433 
645 

3 

5 
6 

7 
8 
10 
12 

28 
38 
49 
63 
101 
149 

34 
44 
57 
73 
115 
169 

44 
57 
73 
93 
143 
208 

54 
70 

89 
112 
172 

248 

75 
95 
121 
152 

229 
328 

96 
121 
153 
191 

286 
407 

117 
146 

184 
230 
343 

487 

138 
172 
216 
269 
400 
567 

158 
198 
247 
308 
457 
646 

179 
223 
280 
348 
514 
726 

200 
249 
312 
387 
571 
806 

221 
275 
343 
426 
628 
885 

4 

5 
6 

7 
8 
10 
12 
14 

30 
38 
49 
63 
100 
147 
209 

35 
45 
57. 
73 
114 
166 
234 

46 
58 
73 
92 
141 
204 
285 

57 
70 
88 
111 
169 
243 
336 

78 
96 
119 
149 
224 
319 
437 

100 
122 
150 
188 
279 
395 
539 

122 
147 
181 
226 
335 
472 
641 

143 
173 
212 
264 
390 
548 
742 

165 
198 
243 
302 
445 
625 
844 

186 
224 
274 
340 
500 
701 
945 

208 
250 
305 
379 
555 
777 
1047 

229 
275 
336 
417 
611 
854 
1149 

5 

6 
7 
8 
10 
12 
14 

41 
52 
65 
100 
146 
207 

47 
60 
75 
114 
165 
231 

61 
76 
94 
141 

202 
280 

75 
93 
114 
168 
239 
329 

102 
125 
153 
223 
314 
427 

130 
158 
192 

277 
388 
525 

157 
191 
231 
331 
463 
623 

184 
224 
270 
386 
537 
721 

212 
257 
309 
440 
611 
819 

239 
289 
348 
495 
686 
917 

267 
322 
387 
549 
760 
1015 

294 
355 

426 
603 
835 
1113 

6 

7 
8 
10 
12 
14 
16 

53 
66 
100 
146 
206 
281 

62 
76 
113 
164 
219 
311 

79 
96 
140 
200 
277 
373 

96 
116 
167 
236 
325 
434 

131 
156 
220 
308 
420 
556 

165 
196 
274 
381 
516 
679 

200 
236 
327 
453 
611 
801 

234 
276 
380 
526 
706 
923 

268 
316 
434 
598 
802 
1046 

303 
356 

487 
670 
897 
1168 

337 
396 
541 
743 
993 
1291 

372 
436 
594 
815 
1088 
1413 

8 

7 
8 
10 
12 
14 
16 
18 

57 
70 

lot 

147 
206 
.  281 
367 

67 

81 
118 
165 
230 
310 
405 

85 
102 
145 

200 
276 
370 

480 

104 
124 
173 
236 
323 
430 
554 

141 
166 
228 
308 
416 
549 
704 

178 
209 
284 
379 
510 
669 
854 

215 
251 
339 
450 
603 
788 
1003 

252 
294 
395 
522 
696 
908 
1153 

289 
337 
450 
593 
790 
1027 
1302 

326 
379 
505 
664 
883 
1146 
1452 

363 
422 
561 
736 
977 
1266 
1602 

400 
464 
616 
807 
1070 
1385 
1/51 

10 

8 
10 
12 
14 
1(5 
18 

74 
107 
148 
207 
280 
366 

85 
121 
166 
229 
309 
402 

108 
150 
201 
275 
368 
475 

131 
179 
236 
321 

426 

548 

176 
236 
306 
412 
542 
693 

221 
294 
377 
504 
659 
839 

266 
351 

447 
595 
775 
984 

311 
408 
518 
686 
801 
1129 

357 
466 

588 
778 
1008 
1275 

402 
523 
658 
869 
1124 
1420 

447 
581 
729 
961 
1241 
1565 

492 
638  ' 
799 
1052 
1357 
1711 

12 

10 
12 
14 

16 
18 
20 

110 
151 
206 
279 
364 
470 

125 
168 
228 
306 
399 
512 

154 
204 
272 
362 
469 
598 

183 
239 
317 
418 
540 
684 

242 
310 
405 
529 
680 
855 

301 
381 
493 
640 
820 
1026 

359 
452 
581 
751 
960 
1197 

418 
523 
669 
862 
1100 
1368 

476 
594 
758 
974 
1241 
1540 

535 
665 
846 
1085 
1381 
1711 

594 
736 
934 
1196 
1521 
1882 

652 
807 
1022 
1307 
1661 
2053 

STONE    BRIDGES. 


355 


Table  5  — (Continued.)    (Original.) 


a 

SS 

Wi&< 

«.&, 

«,£ 

Sfc, 

s.£ 

*fc 

*fe 

w^ 

0. 

OQ 

•§ 

g£ 

gS 

gg 

g$ 

s« 

|a 

£§ 

£S 

gS 

£8 

|g 

BO 
«g 

jo 

£J 

Ft. 

Ft. 

Cub.Y. 

Cub.Y. 

Cub.  Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

12 

162 

182 

222 

262 

842 

422 

502 

583 

663 

743 

823 

903 

14 

215 

239 

286 

333 

427 

522 

616 

711 

805 

899 

994 

1088 

1  ^ 

Id 

285 

313 

370 

427 

541 

654 

768 

882 

996 

1110 

1223 

1337 

18 

369 

404 

474 

545 

686 

826 

967 

1108 

1249 

1390 

1530 

1671 

20 

473 

515 

600 

685 

855 

1024 

1194 

1364 

1534 

1704 

1873 

2043 

22 

595 

646 

748 

850 

1054 

1258 

1462 

1666 

1870 

2074 

2278 

2482 

14 

237 

264 

317 

371 

478 

586 

693 

801 

908 

1015 

1123 

1230 

1C, 

304 

335 

397 

458 

582 

706 

829 

953 

1076 

1200 

1324 

1447 

9n 

18 

381 

416 

486 

556 

697 

838 

97S 

1119 

1259 

1400 

1541 

1681 

20 

479 

520 

601 

682 

844 

1007 

1169 

1332 

1494 

1656 

1819 

1981 

22 

598 

646 

741 

837 

1028 

1220 

1411 

1603 

1794 

1985 

2177 

2368 

24 

739 

795 

908 

1021 

1247 

1473 

1699 

1925 

2151 

2377 

2603 

2829 

16 

327 

360 

428 

496 

631 

766 

901 

1036 

1172 

1307 

1442 

1577 

18 

403 

441 

517 

594 

746 

898 

1050 

1202 

1355 

1507 

1659 

1811 

20 

500 

543 

629 

715 

887 

1059 

1231 

1403 

1575 

1747 

1919 

2091 

25 

22 

614 

663 

760 

857 

1051 

1246 

1440 

1635 

1829 

2023 

2218 

2412 

24 

751 

807 

919 

1031 

1255 

1479 

1703 

1927 

2151 

2375 

2599 

2823 

26 

909 

974 

1104 

1234 

1494 

1754 

2014 

2274 

2534 

2794 

3054 

3314 

28 

1085 

1160 

1310 

1460 

1760 

2060 

2360 

2660 

2960 

3260 

3560 

3860 

22 

685 

743 

859 

975 

1207 

1439 

1671 

1903 

2135 

2367 

2599 

2831 

24 

817 

880 

1007 

1134 

1388 

1642 

1896 

2150 

2404 

2658 

2912 

3166 

26 

969 

1033 

1181 

1309 

1585 

1861 

2137 

2413 

2689 

2965 

3241 

3517 

35 

28 

1130 

1205 

1356 

1507 

1809 

2111 

2413 

2715 

3017 

3319 

3621 

3923 

30 

1327 

1408 

1571 

1734 

2060 

2386 

2712 

3038 

3364 

3690 

4016 

4342 

83 

1549 

1639 

1820 

2001 

2363 

2725 

3087 

3449 

3811 

4173 

4535 

4897 

35 

1946 

2054 

2271 

2488 

2922 

3356 

3790 

4224 

4658 

5092 

5526 

5960 

30 

1494 

1594 

1795 

1996 

2398 

2&00 

3202 

3604 

4006 

4408 

4810 

5212 

32 

1711 

1819 

2035 

2251 

2683 

3115 

3547 

3979 

4411 

4843 

5275 

5707 

34 

1956 

2071 

2302 

2533 

2995 

3457 

3919 

4381 

4843 

5305 

5767 

6229 

^0 

36 
38 

2228 
2519 

2350 
2650 

2597 
2913 

2844 
3J76 

3338 

3832 

4326 

4820 

5314 

5808 

6302 

6796 

40 

2835 

2975 

3255 

3535 

4095 

4655 

5215 

5775 

6335 

6895 

7455 

8015 

42 

3197 

3347 

3647 

3947 

4547 

5147 

5747 

6347 

6947 

7547 

8147 

8747 

45 

3818 

3991 

4337 

4683 

5375 

6067 

6759 

7451 

8143 

8835 

9527 

10219 

50 

5063 

5281 

5717 

6153 

7025 

7897 

8769 

9641 

10513 

11385 

12257 

13129 

Art.  9.  Especial  pains  should  be  taken  to  secure  an  unyielding-  foun- 
dation for  culverts  and  drains  under  high  einbkts)  otherwise 
the  superincumbent  weight,  especially  under  the  middle  of  the  embkt,  may  squeeze 
them  into  the  soil  below,  if  soft  or  marshy;  and  thus  diminish  the  area  of  water- 
way, or  at  least  cause  an  ugly  settlement  at  the  midlength  of  the  culvert.  Also,  in 
soft  ground,  the  embkt  ma,y  press  the  side  walls  closer  together,  narrowing  the 
channel.  This  may  be  prevented  by  an  inverted  arch,  or  a  bed  of  masonry,  between 
the  walls.  A  stratum  from  3  to  6  ft  thick,  of  gravel,  sand,  or  stone  broken  to  turn- 
pike size,  will  generally  give  a  sufficient  foundation  for  culverts  in  treacherous 
marshy  ground  ;  or  quicksand,  with  but  a  moderate  height  of  embkt.  It  should  ex- 
tend a  few  feet  beyond  the  masonry  in  every  direction,  and  should  be  rammed  ;  the 
sand  or  gravel  being  thoroughly  wet,  if  possible,  to  assist  the  consolidation.  Piling 
will  sometimes  be  necessary.  If  the  masonry  is  built  upon  timber  platforms,  or  a 
smooth  surface  of  rock,  care  must  be  taken  to  prevent  it  from  sliding,  from  the  prea 
of  the  earth  behind  it.  This  same  pres  may  even  overthrow  the  piles,  if  they  are 
not  properly  secured  against  it. 
Art  1O.  I>r;»  MIS. 
Brains  of  the  dimen- 
sions in  Fig  11,  con- 
tain 1  perch,  of  25 
cub  ft  ;  or  .926  of  a 
cub  yd,  per  ft  run. 


They  are  frequently 
built  of  dry  scabbled 
rubble,  and  paved  with 
spawls.  When  there  ia 
much  wash  through 
them,  with  a  consider- 
able slope,  it  is  better  to 
continue  the  foundation 


356 


STONE    BRIDGES. 


solid  clear  across.  This  is  often  done  without  those  causes,  inasmuch  as  the  additional  masonry  ia  & 
mere  trifle:  and  the  excavation  of  a  siugle  broad  foundation-pit  is  less  troublesome  than  that  of  two 
narrow  ones.  A  deep  flag-stone /at  the  entrance,  and  others  at  short  dists  of  the  length,  may  be  in- 
troduced in  both  drains  and  culverts,  to  protect  from  undermining. 

These  drains  extend  under  the  entire  width  of  the  embkt,  from  toe  to  toe;  and  may  terminate  in 
steps,  as  in  the  side  view  at  S.  They  are  of  course  better  when  built  with  mortar,  with  an  admixture 
of  cement  to  prevent  the  water  when  full  from  leaking  into  and  softening  the  embankment. 

Sometimes  two  or  three  such  drains  may  be  placed  parallel  to  each  other,  instead  of  a  culvert. 
When  two  are  so  placed,  they  contain  only  1>6  times  the  masonry  of  one  ;  still  their  use  will  generally 
involve  no  saving  of  masonry  over  a  culvert.  A  man  can  crawfthrough  Fig  11  to  clean  it. 

Art.  11.  The  drainage  of  the  roadways  of  stone  bridges  of  several 
arches,  is  generally  effected  by  means  of  open  gutters,  which  descend  slightly  from 
the  crown  of  the  arches,  each  way,  until  they  reach  to  near  the  ends  of  the  re 
spective  spans. 

There  they  discharge  into  vertical  iron  pipes  built  into  the  masonry.  The  upper  ends  of  the 
pipes  should  be  covered  by  gratings.  When  inconvenience  would  result  from  the  water  falling  upon 
persons  passing  under  the  arches,  these  pipes  may  be  carried  down  the  entire  height  of  the  piers; 
but  when  such  is  not  the  case,  they  may  extend  only  to  the  soffit,  or  under  face  of  the  arch ;  allowing 
the  water  to  fall  freely  through  the  air  from  that  height. 

Table  6,  of  approximate  contents,  in  cub  yds,  of  a  solid 
pier  of  masonry,  6  ft  by  22  ft  on  top;  and  battering  1  inch  to  a  ft  on  each  of 
its  4  faces.  The  contents  of  masonry  of  such  forms  must  be  calculated  by  the  prismoidal  formula ; 
and  not  by  taking  the  length  and  breadth  of  the  pier  at  half  its  height  as  an  average  length  and 
breadth,  as  is  sometimes  done.  This  incorrect  method  would  give  only  6492  cub  yds  as  the  content 
of  the  pier  200  ft  high;  instead  7178  yds,  its  true  content.  High  piers  may  for  economy.be  built  hol- 


low, with  or  without  interior  cross-walls  for  strengthening  them,  as  the  case  may  require;  and  the 
batter  is  generally  reduced  to  %  inch  or  less  to  a  foot.     Hollow  piers  require  good  well-  bedded  ma- 
*°arJ'                                                                  (Original.) 

Ht. 

Ft. 

Lgth 
at 
baso. 

Bdth 
at 

base. 

Cubic 
yards 

Ht. 

Ft. 

Lgth 
at 
base. 

Bdth 
at 
base. 

Cubic- 
yards 

Ht. 
Ft, 

Lgth 
at 
base. 

Bdth 
at 
base. 

Cubic 
yards. 

6 

23. 

7. 

32.5 

52 

30.67 

14.67 

537 

128 

43.33 

27.33 

2759 

7 

.17 

.17 

38.6 

54 

31. 

15. 

570 

130 

.67 

.67 

2848 

8 

.33 

.33 

44.9 

56 

.33 

.33 

605 

132 

44. 

28. 

2940 

9 

23.5 

7.5 

51.3 

58 

.67 

.67 

641 

134 

.33 

.33 

3032 

10 

.67 

.67 

58. 

60 

32. 

16. 

679 

136 

.67 

.67 

3126 

11 

.83 

.83 

64.8 

62 

.33 

.33 

717 

138 

45. 

29. 

3222 

12 

24. 

8. 

71.7 

64 

.67 

.67 

757 

140 

.33 

.33 

3320 

13 

.17 

.17 

79. 

66 

33. 

17. 

798 

142 

.67 

.67 

34-20 

14 

.33 

33 

86.4 

68 

.33 

.33 

840 

144 

46. 

SO. 

3521 

15 

24.5 

8.5 

94. 

70 

.67 

.67 

884 

146 

.33 

.33 

3623 

»      16 

.67 

.67 

102 

72 

34. 

18. 

928 

148 

.67 

.67 

3728 

17 

.83 

.83 

110 

74 

.33 

.33 

973 

150 

47. 

31. 

3835 

18 

25. 

9. 

118 

76 

.67 

.67 

1021 

152 

.33 

.33 

3944 

19 

.17 

.17 

127 

78 

35. 

19. 

1070 

154 

.67 

.67 

4056 

20 

.33 

.33 

135 

80 

.as 

.33 

1120 

156 

48. 

32. 

4168 

21 

25.5 

9.5 

144 

82 

.67 

.67 

1171 

158 

.33 

.33 

4-284 

22 

.67 

.67 

153 

84 

36. 

20. 

1224 

160 

.67 

.67 

4402 

23 

.83 

.83 

163 

86 

,38 

.33 

1278 

162 

49. 

33. 

4520 

21 

26. 

10. 

172 

88 

.67 

.67 

1334 

164 

.33 

.33 

4640 

25 

.17 

.17 

182 

90 

37. 

21. 

1392 

166 

.67 

.67 

4763 

26 

.33 

.33 

192 

92 

.33 

.33 

1451 

168 

50. 

34. 

4887 

27 

26.5 

10.5 

202 

94 

.67 

.67 

1510 

170 

.33 

.33 

5014 

28 

.67 

.67 

212 

96 

38. 

22. 

1569 

172 

.67 

.67 

5143 

29 

.83 

.83 

223 

98 

.33 

.33 

1631 

174 

51. 

35. 

5275 

30 

27. 

11. 

234 

100 

.67 

.67 

1695 

176 

.33 

.33 

5409 

31 

.17 

.17 

245 

102 

39. 

23. 

1761 

178 

.67 

.67 

5545 

32 

.33 

.33 

256 

104 

.33 

.33 

1829 

180 

52. 

36. 

5680 

33 

27.5 

11.5 

•268 

106 

.67 

.67 

1899 

182 

.33 

.33 

6820 

34 

.67 

.67 

280 

108 

40. 

24. 

1968 

184 

.67 

.67 

5962 

35 

.83 

.83 

292 

110 

.33 

.33 

2041 

186 

53. 

37. 

6106 

36 

28. 

12. 

304 

112 

.67 

.67 

2115 

188 

,?3 

.33 

625? 

38 

.33 

33 

329 

114 

41. 

25. 

2191 

190 

.67 

.67 

640C 

40 

.67 

67 

356 

116 

.33 

.33 

2269 

192 

54. 

38. 

6552 

42 

29. 

13. 

383 

118 

.67 

.67 

2346 

194 

.33 

.33 

6704 

44 

.33 

.33 

411 

120 

42. 

26. 

2424 

196 

.67 

.67 

6859 

46 

.67 

.67 

441 

122 

.33 

.33 

2504 

198 

55. 

39. 

7016 

48 

30. 

14. 

472 

124 

.67 

.67 

2587 

200 

.33 

.o3 

7178 

50 

.33 

.33 

504 

126 

43. 

27. 

2672 

202 

.67 

.67 

7339 

BOARD    MEASURE. 


357 


BOAED  MEASUKE. 


Remark  on  following-  table.  The  table  extends  to  12  ins  by  24  ins,  but 
it  is  easy  to  find  for  greater  sizes  ;  thus,  for  example,  the  board  measure  in  a  piece  of  19  by  22,  will 
be  twice  that  <  i  a  piece  of  19  by  11,  or  17.42  X  2  =  34.84  ft  board  meas  ;  or  that  of  19*4  by  22,  will  be 
that  of  10H  by  22  added  to  that  of  9  by  22,  or  18.79  +  16.50  =  35.29.  A  foot  of  board  meas  is  equal  to 
1  foot  square  and  1  inch  thick,  or  to  144  cub  ins.  Hence  1  cab  ft  =  12  fi  board  meas. 


a  . 

Feet  of  Board  Measure  conta  ned  in  one  running  foot  of  Scantlings 

a 

31 

1000  ft  board  measure  =  83tf  cub  ft. 

*3 

11 

THICKNESS  2N"  INCHES. 

Is 

fen 

BM 

^ 

1 

1M 

1x4 

l?i 

2 

234 

2x4 

2?* 

3 

r 

Ft  Bd.M. 

Ft.  Bd.M. 

Ft.  Bd.M. 

Ft.Bd.M.  Ft.  Bd.M. 

Ft.  Bd.M. 

Ft.  Bd.M. 

Ft.  Bd.M. 

Ft.  Bd.M. 

x4 

.0208 

.0260 

.0313 

.0365 

.0417 

.0469 

.0521 

.0573 

.0625 

y 

IX 

.0417 

.0521 

.0625 

.0729 

.0833 

.0938 

.1042 

.1146 

.1250 

?x 

XX 

.0625 

.0781 

.0938 

.1094 

.1250 

.1406 

.1563 

.1719 

.1875 

3 

1. 

.0833 

.1042 

.1250 

.1458 

.1667 

.1875 

.2083 

.2292 

.2500 

i 

.1042 

.1302 

.1563 

.1823 

.2083 

.2344 

.2604 

.2865 

.3125 

VX 

.1250 

.1563 

.1875 

.2188 

.2500 

.2813 

.3125 

.3438 

.13750 

£t 

XX 

.1458 

.1823 

.2187 

.2552 

.2917 

.3281 

.3646 

.4010 

.4375 

I? 

2. 

.1667 

.2083 

.2500 

.2917 

.3333 

.3750 

.4166 

.4583 

.5000 

2. 

.1875 

.2344 

.2813 

.3281 

.3750 

.4219 

.4688 

.5156 

.5625 

IX 

.2083 

.2604 

.3125 

.3646 

.4167 

.4688 

.5208 

.5729 

.6250 

?5 

ax 

.2292 

.2865 

.3438 

.4010 

.4583 

.5156 

.5729 

.6302 

.6875 

9? 

3. 

.2500 

.3125 

.3750 

.4375 

.5000 

.5625 

.6250 

.6875 

.7500 

3. 

.2708 

.3385 

.4063 

.4739 

.5416 

.6094 

.6771 

.7448 

.8125 

.2917 

.3646 

.4375 

.5104 

.5833 

.6563 

.7292 

.8021 

.8750 

«^ 

.3125 

.3906 

.4689 

.5469 

.6250 

.7031 

.7813 

.8594 

.9375 

y 

4. 

.3333 

.4167 

.5000 

.5833 

.6667 

.7500 

.8333 

.9167 

1.000 

4. 

.3542 

.4427 

.5312 

.6198 

.7083 

.7969 

.8854 

.9740 

1.063 

VX 

.3750 

.4688 

.5625 

.6503 

.7500 

.8438 

.9375 

1.031 

1.125 

x4 

IX- 

.3958 

.4948 

.5938 

.6927 

.7917 

.8906 

.9896 

1.086 

1.188 

1 

5. 

.4167 

.5208 

.6250 

.7292 

.8333 

.9375 

1.042 

1.146 

1.250 

5 

.4375 

.5469 

.6563 

.7656 

.8750 

.9844 

1.094 

1.203 

1.313 

IX 

.4583 

.5729 

.6875 

.8020 

.9167 

1.031 

1.146 

1.260 

1.375 

ix 

XX 

.4792 

.5990 

.7188 

.9583 

1.078 

1.198 

1.318 

1  .4:;8 

• 

6. 

.5000 

.6250 

.7500 

'.8750 

1.000 

1.125 

1.250 

1.375 

1.500 

6.* 

1£ 

.5208 

.6510 

.7813 

.9115 

1.042 

1.172 

1.302 

1.432 

1.563 

J4 

.5417 

.6771 

.8125 

.9479 

1.083 

1.219 

1.354 

1.490 

1.625 

VX 

XX 

.5625 

.7031 

.8438 

.9844 

1.125 

1.266 

1.406 

1.547 

1.688 

y 

7. 

.5833 

.7292 

.8750 

1.021 

1.167 

1.312 

1.458 

1.604 

1.750 

7. 

.6042 

.7552 

.9063 

1.057 

1.208 

1.359 

1  510 

1.661 

1.813 

J4 

.6250 

.7813 

.9375 

1.094 

1.250 

1.406 

1.563 

1.719 

1.875 

IX 

xx 

.6458 

.8073 

.9688 

.130 

1  .292 

1.453 

1.615 

1.776 

1.938 

8 

8. 

.6667 

.8333 

1.000 

.167 

1.333 

1.500 

1.667 

1.833 

2.000 

8 

.6875 

.8594 

1.031 

.203 

1.375 

1.547 

1.719 

1.891 

2.063 

x4 

.7083 

.8854 

1.063 

.240 

1.417 

1694 

1.771 

1.948 

2.125 

• 

xtf 

.7292 

.9114 

1.094 

.276 

1.458 

1.641 

1.823 

2.005 

2.188 

1 

9. 

.7500 

.9375 

1.125 

.313 

1.500 

1.688 

1.875 

2.062 

2.250 

0 

IX 

.7708 

.9635 

1.156 

.349 

1.542 

1.734 

1.927 

2.120 

2.313 

J4 

.7917 

.9895 

1.188 

.385 

1.583 

1.781 

1.979 

2.177 

2.375 

ix 

xtf 

.8125 

1.016 

1.219 

.422 

1.625 

1.828 

2.031 

2.234 

2.438 

8 

10. 

.8333 

1.042 

1.250 

.458 

1.667 

1.875 

2.083 

2.292 

2.500 

10. 

M 

.8542 

1.068 

1.281 

.495 

1.708 

1.922 

2.135 

2.349 

2.563 

x4 

.8750 

1.094 

1.313 

.531 

1.750 

1.969 

2.188 

2.406 

2.JJ26 

xi 

9i 

.8958 

1.120 

1.344 

.568 

1.792 

2.016 

2.240 

2.463 

2.688 

11. 

.9167 

1.146 

1.375 

.604 

1.833 

2.063 

2.292 

2.521 

2.750 

lit 

M 

.9375 

1.172 

1.406 

1.641 

1.875 

2.109 

2.344 

2.578 

2.813 

x4 

.9583 

1.198 

1  438 

1.677 

1.917 

2.156 

2.396 

2  635 

2.875 

i£ 

xi 

.9792 

1.224 

1  469 

1.714 

1958 

2.203 

2.448 

2.693 

2.938 

H 

12. 

1.000 

1.250 

1.500 

1.750 

2.000 

2.250 

2.500 

2.750 

3.000 

12 

x4 

1.042 

1.302 

1.563 

1.823 

2.083 

2.344 

2.604 

2.865 

3.125 

V* 

T3. 

1.083 

1.354 

1.625 

1.896 

2.167 

2.438 

2.708 

2.979 

3250 

13 

J4 

1.125 

1.406 

1.688 

1.969 

2.250 

2.531 

2.813 

3.094 

3.375 

x4 

14. 

1.167 

1.458 

1.750 

2.042 

2333 

2.625 

2.917 

3.208 

3.500 

14 

x4 

1.208 

1.510 

1.813 

2.115 

2.417 

2.719 

3.021 

3.322 

3.625 

x4 

15. 

1.250 

1.563 

1.875 

2.188 

2.500 

2.813 

3.125 

3.438 

3.750 

15 

1.292 

1.615 

1.938 

2.260 

2.583 

2.906 

3.229 

3.552 

3.875 

x4 

16. 

1.333 

1.667 

2.000 

2.333 

2.667 

3.000 

3.333 

3.667 

4.000 

16 

x4 

1.375 

1.719 

2.063 

2.406 

2.750 

3.094 

3.438 

3.781 

4.125 

>4 

17. 

1.417 

1.771 

2.125 

2.479 

2.833 

3.188 

3.542 

3.896 

4.250 

17 

1.458 

1.823 

2.187 

2.552 

2.917 

3.2H1 

3.646 

4.010 

4.375 

18. 

1.500 

1.875 

2.250 

2.625 

3.000 

3.375 

3.750 

4.125 

4.500 

18. 

19. 

1.583 

1.979 

2.375 

2.771 

3.167 

3563 

3.958 

4.354 

4.750 

19. 

20. 

1.667 

2.083 

2.500 

2.917 

3.333 

3.750 

4.167 

4.583 

5.000 

20. 

21. 

1.750 

2.188 

2.625 

3.063 

8.500 

3.9?,8 

4.375 

4812 

5250 

21. 

22. 

1.833 

2.292 

2.750 

3.208 

3.667 

4.125 

4.583 

5.042 

5.500 

22. 

23. 

1.917 

2.396 

2.875 

3.354 

3.833 

4.313 

4.792 

5.270 

5.750 

23. 

24. 

2.000 

2.500 

3.000 

3.500 

4.000 

4.500 

5.000 

5.500 

6.000 

24. 

358 


BOARD    MEASURE. 


Table  of  Board  Measure  — (Continued.) 


P"  . 

51 

Feet  of  Board  Measure  conta  ned  in  one  runn  ng  foot  of  Scantlings 
of  different  dimensions.     (Original.). 

jj 

S3 

THICKNESS  IN  INCHES. 

j>M 

3X           3M 

3«              4 

4J4            4>4            4%              5 

*£ 

Ft.Bd  M.    Ft.BdM. 

Ft.Bd.M.    Ft.Bd.M. 

Ft.Bd.M 

Ft.Bd.M. 

FtBd.M.    Ft.  lid.  M 

Ft.Bd.M. 

. 

34 

.0677 

.0729 

.0781 

.0833 

.0885 

.0938 

.0990         .1042 

.1094 

34 

iz 

.1354 

.1457 

.1562 

.1667 

.1770 

.1875 

.1979         .2083 

.2188 

y>, 

9i 

.2031 

.2187 

.2344 

.2500 

.2656 

.2813 

.2969    !      .3125 

.3281 

% 

1. 

.2708 

.2917 

.3125 

.3333 

.3542 

.3750 

.3958 

.4167 

.4375 

i. 

.3385 

.3646 

.3906         .4167 

.4427 

.4688 

.4948 

.5208 

.5469 

34 

IX 

.4063 

.4375 

.4688         .5000 

.5313 

.5625 

.5938 

.6250 

.6563 

y% 

ax 

.4740 

.5104 

.5469         .5833 

.6198 

.6563 

.6927 

.7292 

.7656 

% 

2. 

.5417 

.5833 

.6250 

.6667 

.7083 

.7500 

.7917 

•8333 

.8750 

2. 

i/ 

.6094 

.6563 

.7031 

.7500 

.7969 

.8438 

.8906 

.9375 

.9844 

% 

jv 

.6771 

.7292 

.7813 

.8333 

.8854 

.9375 

.9896 

1.042 

1.094 

% 

3X 

.7448 

.8021 

.8594 

.9167 

.9740 

1.031 

1.089 

1.146 

1.203 

% 

3. 

.8125 

.8750 

.9375 

1.000 

1.062 

1.125 

1.188 

1.250 

1.313 

3. 

.8802 

.9479 

1.016 

1.083 

1.151 

1.219 

1.286 

1.354 

1.422 

34 

i^ 

.9179 

1  .021 

1.094 

1.167 

1.240 

1.313 

1.385 

1.458 

1.531 

H 

1.016 

1.094 

1.172 

1.250 

1.327 

1.406 

1.484 

1.563 

1.641 

% 

1083 

1.167 

1.250 

1.333 

1.416 

1.500 

1.583 

1.667 

1.750 

4. 

k 

1.151 

1  240 

1.328 

1.417 

1.504 

1.594 

1.682 

1.771 

1.859 

34 

| 

1.219 

1.313 

1.406 

1.500 

1.593 

1.688 

1.781 

1.875 

1.969 

y* 

1.286 

1.384 

1.484 

1.583 

1.681 

1.781 

1.880 

1  979 

2.078 

5.* 

1.354 

1.457 

1.566 

1.666 

1.770 

1.875 

1.979 

2.083 

2188 

5. 

34 

1.422 

1.530 

1.644 

1.750 

1.858 

1.969 

2.078 

2.188 

2297 

34 

y* 

1.490 

1.603 

1.722 

1.833 

1.947 

2  063 

2.177 

2.292 

2.406 

34 

% 

1.557 

1.676 

1.800 

1.917 

2.035 

2.156 

2.276 

2.396 

2.516 

% 

6. 

1.625 

1.750 

1.875 

2.000 

2.125 

2  250 

2.375 

2.500 

2.625 

6. 

34 

1.693 

1.823 

1.953 

2.083 

2.214         2.344 

2.474 

2604 

2.734 

34 

y* 

1.760 

1.896 

2.031 

2.167 

2.302     !    2.438 

2.573 

2.708 

2.843 

y% 

% 

1.828 

1.969 

2.109 

2.250 

2.391 

2.531 

2.672 

2.813 

2953 

% 

7. 

1.896 

2.042 

2.188 

2.333 

2.479 

2.625 

2.771 

2.917 

3.063 

7. 

v 

1  .9(54 

2.115 

2.266 

2.416 

2.568 

2719 

2.870 

3021 

3.172 

34 

J4 

2.031 

2  187 

2.344 

2.500 

2656 

2813 

2.969 

3.125 

3.281 

y* 

2.099 

2  260 

2422 

2.583 

2.745 

2.906 

3.068 

3.229 

3391 

8. 

2.167 

2333 

2.500 

2.667 

2.833 

3000 

3.167 

3.333 

3.500 

8. 

34 

2.234 

2.406 

2.578 

2.750 

2  922 

3.094 

3.266 

3.438 

3.609 

X 

34 

2.302 

2.479 

2656 

2.833 

3.010 

3.188 

3.365 

3.542 

3.718 

x 

?i 

2.370 

2.552 

2  734 

2.916 

3.099 

3.281 

3.464 

3.646          3.828 

9. 

2.438 

2.625 

2.813 

3.000 

3.187 

3.375 

3.563 

3.750          3.938 

9.4 

34 

2.505 

2.698     i    2.891 

3.083 

3.276 

3.469 

3.661 

3.854 

4.047 

34 

y^ 

2.573 

2.771         2.969 

3.167 

3.365 

3.563 

3.760 

3.958 

4.156 

H 

2.641 

2.844 

3.047 

3.250 

3.453 

3.656 

3.859 

4.063          4.266 

H 

10. 

2.708 

2.917 

3.125 

3.333 

3.542 

3.750 

3.958 

4.167          4.375 

10. 

34         2.776 

2.990 

3.203 

3.416 

3.630 

3.844 

4.057 

4.271           4.484 

x4" 

Ji          2.844 

3.063 

3.281         3.500 

3.719 

3.938 

4.156 

4.375     !     4.594 

\<, 

?4          2.911 

3.135 

3.359         3.583 

3.807 

4.031 

4.255 

4.479          4.703 

3^ 

11. 

2.979 

8.208 

3.438     i    3.668 

3.896 

4.125 

4.354 

4.583          4.813 

11. 

i/. 

3.047 

3.281 

3.516 

3.750 

3.984 

4.219 

4.453 

4.688          4.922 

34 

% 

3.115 

3.354 

3.594 

3.833 

4.073 

4.313 

4.552 

4.792          5.031 

y* 

3.182 

3.427 

3.672 

3.916 

4.161 

4.406 

4.651 

4.896          5.141     |         % 

12. 

3.250 

3.500 

3.750 

4.000 

4.250 

4.500 

4.750 

5.000 

5.250 

12. 

u 

3.385 

3.646 

3.906 

4.167 

4.427 

4.688 

4.948 

5.208 

5.469 

13. 

3.521 

3.792 

4.063 

4.333 

4.604 

4.875 

5.146 

5.417          5.688 

13." 

V<; 

3.656 

3.938 

4.219 

4.500 

4.781 

5.063 

5.344 

5.625          5.906 

H 

14. 

3.792 

4.083 

4.375 

4.667 

4.958 

5.250 

5.542 

5.833      !     6.125 

14. 

j£ 

3.927 

4.229 

4.531 

4.833 

5.135 

5.438 

5.740 

(5.042          6.344 

Jl3 

15. 

4.063         4.375 

4.688 

5.000 

5.313 

5.6'25 

5.938 

6.250      :     6.563 

15. 

4 

4.198         4.521 

4.844 

5.166 

5.490 

5.813 

6.135 

6.458      :     6.781 

% 

16. 

4.333         4.667 

5.000 

5.333 

5.667 

6.000 

6.333 

6.667      ;     7.000 

16. 

j£ 

4.469 

4.813 

5.156 

5.500 

5.844 

6.188 

6.531 

6.875     i     7.219 

^ 

17. 

4.604 

4.958 

5.313 

5.667 

6.021 

6.375 

6.729 

7.083 

7.438 

17. 

4.740 

5.104 

5.469 

5.833 

6.198 

6.563 

6.927 

7.292 

7.656 

18." 

4.875 

5.250 

5.625 

6.000 

6.375 

6.750 

7.125 

7.500 

7.875 

18. 

19. 

5.146 

5.542 

5.938 

6.333 

6.729 

7.125 

7.521 

7.917 

8.313 

19. 

20. 

5.417 

5.833 

6.250 

6.667 

7.083 

7.500 

7.917 

8.333 

8.750 

20. 

21. 

5.688 

6.125 

6.563 

7.000 

7.438 

7.875 

8.313 

8.750 

9.188 

21. 

22. 

5.958 

6.417 

6.875 

7.333 

7.792 

8.250 

8.708 

9.167 

9.625 

22. 

23. 

6.229 

6.708 

7.188 

7.667 

8.145 

8.625 

9.104 

9.583 

10.06 

23. 

34! 

6.500 

7.000 

7.500 

8.000 

8.500 

9.000 

9.500 

10.00 

10.50 

24. 

Creosote  or  flead  oil  is  by  far  the  best  preservative  for  timber  yet  known. 
It  is  certainly  effective  against  sea  worms  for  at  least  25  years,  if  thoroughly  applied.  It  is  a  colorless 
oily  fluid  distilled  chiefly  from  coal  tar.  Its  peculiar  antiseptic  quality  is  due  to  its  carbolic  and 
cresylic  acids  ;  but  it  also  acts  by  filling  the  pores  of  the  wood  so  as  to  exclude  air  and  moisture.  It 
weighs  about  8.8  Ibs  per  U.  S.  gkllon.  Boils  at  500°  to  600°  Fah.  From  .5  to  1  gallon,  or  say  4.5  to 
9  Ibs  (or  10  to  12  Ibs,  if  exposed  to  sea  worms)  are  required  per  cub  ft,  depen  ing  on  the  nature  of  the 
timber  It  is  used  at  a  temp  of  200°  to  300°  F,  and  under  a  pressure  of  150  to  200  Ibs  per  *q  inch,  for 


BOARD    MEASURE. 


359 


Table  of  Board  Measure  — (Continued.) 


.s* 
si 

Feet  of  Board  Measure  conta  ned  in  one  running  foot  of  Scantlings 
of  different  dimensions.     (Original.) 

**  OQ 

n 

I1 

THICKNESS    IN 

INCHES. 

2s 

£H 

5& 

5% 

6 

6J4 

6M 

6% 

7 

7*4 

'*     j  

Ft.Bd.M. 

Ft.Bd.M. 

Ft.Bd.M. 

Ft.Bd.M. 

Ft.Bd.M- 

Ft.Bd.M. 

FuBd.M. 

FtLd.m. 

rt.Ld.M. 

K 

.1146 

.1198 

.1250 

.1302 

.1354 

.1406 

.1458 

.1510 

.1563 

y*. 

14 

.2292 

.2396 

.2500 

.2604 

.2708 

.2813 

.2917 

.3021 

.3125 

8 

H 

.3438 

.3594 

.3750 

.3906 

.4063 

.4219 

.4375 

.4531 

.4688 

H 

I 

.4583 

.4792 

.5000 

.5208 

.5417 

.5625 

.5833 

.6012 

.6250 

i. 

X 

.5729 

.5990 

.6250 

.6510 

.6771 

.7031 

.7292 

.7552 

.7813 

y* 

M 

.6875 

.7188 

.7500 

.7812 

.8125 

.8438 

.8750 

.9062 

.9375 

5 

X 

.8021 

.8385 

.8750 

.9115 

.9479 

.9844 

1.020 

1.057 

1.094 

H 

2 

.9167 

.9583 

1.000 

1.042 

1.083 

1.125 

1.167 

1.208 

1.X50 

2. 

K 

1.031 

1.078 

1.125 

1.172 

1.219 

1.266 

1.313 

1.359 

1.406 

X 

1.146 

1.198 

1.250 

1.302 

1.354 

1.406        1  458 

1.510 

'  1.5G3 

J4 

K 

1.260 

1.318 

1.375 

1.432 

1.490 

1.547 

1.604 

1.661 

1.719 

% 

3. 

1.375 

1.438 

1.500 

1.562 

1.625 

1.688 

1.750 

1.813 

I.fe75 

3. 

y± 

1.490 

1.557 

1  .625 

1JJ93 

1.760 

1.828 

1.896 

l.%4 

2.0ol 

X 

1.604 

1.677 

1.750 

1.823 

1.896 

1,969 

2.042 

2.115 

2.188 

H 

H 

1.719 

1.797 

1.875 

1.953 

2.031 

2.109 

2.188 

2  266 

2.344 

H 

4. 

1.833 

1.917 

2.000 

2.083 

2.167 

2.250 

2.333 

2.417 

2500 

4. 

Vi 

1.948 

2.036 

2.125 

2.214 

2.302 

2.391 

2.479 

2.568        2.«56 

X 

9t 

2.063     !    2.156 

2.250 

2.344 

2.438 

2.531 

2625 

2.719 

2.813 

X 

H 

2.177 

2.276 

2.375 

2.474 

2.573 

2.672 

2.771 

2.870 

2.!!G9 

H 

5 

2.292 

2.396 

2.500 

2.604 

2.708 

2.813 

2.<J17 

8.021 

3.125 

b 

y< 

2.406 

2.516 

2.625 

2.734 

2844 

2.953 

3.063 

3.172 

3.  '281 

u 

2.521 

2.635 

2.750 

2.865 

2.979 

3.094 

3.208 

3.323 

3.438 

y 

?* 

2.635 

2.755 

2.875 

2.995 

3.115 

3.234 

3.354 

3.474 

3.594 

H 

6. 

2.750 

2.875 

3.000 

3.125 

3.250 

3.375 

3.500 

3.625 

3750 

6. 

2.865 

2.995 

3.125 

3.255 

3.385 

3.516 

3.646 

3.776 

3  i!OG 

y± 

^ 

2.979 

3.115 

3.250 

3.385 

3.521 

3.656 

3.792 

3.927 

4.063 

% 

?i 

3.094 

3.234 

3.375 

3.516 

3.656 

3.71)7 

3.938 

4.078 

4.219 

H 

7. 

3.208 

3.354 

3.500 

3.646 

3.792 

3.938 

4.083 

4.229 

4.375 

7. 

K 

3.323 

3.474 

3.625 

3.776 

3.927 

4.078 

4.229 

4.380 

4.531 

y*. 

J4 

3.438 

3.594 

3.750 

3.906 

4.063 

4.219 

4.375 

4.531 

4.688 

H 

5i 

3.552 

3.714 

3.875 

4.036 

4.1<J8 

4.359 

4.521 

4.682 

4.844 

% 

8. 

3.667 

3.833 

4.000 

4.167 

4.333 

4.500 

4.667 

4.833 

5.000 

8. 

y* 

3.781 

3.953 

4.125 

4.297 

4.469 

4.641 

4.813 

4.984 

5.156 

y± 

% 

3.896 

4.073 

4.250 

4.427 

4.604 

4.781 

4.5)57 

5.135 

5  313 

% 

% 

4.010 

4.193 

4.375 

4.557 

4.740 

4.922 

5.103 

5.286 

5.469 

H 

9. 

4.125 

4.313 

4.500 

4.687 

4.875 

5.063 

5.249 

5.438 

5C25 

4.240 

4.432 

4.625 

4.818 

5.010 

5.203 

5.395 

5.589 

5.781 

i 

L/ 

4.354 

4.552 

4.750 

4.948 

5.146 

5.344 

5.541 

5740 

5.JJS8 

ji 

a^ 

4.469 

4.672 

4.875 

5.078 

5.281 

5.484 

5.687 

5.891 

6.0U 

H 

10. 

4.583 

4.792 

5.000 

5.208 

5.417 

5.625 

5.833 

6.042 

6.250 

10. 

K 

4.698 

4.911 

5.125 

5.339 

5.552 

5.766 

5.979 

6.193 

6.406 

H 

H 

4.813 

5.031 

5.250 

5.469 

5.688 

5.906 

6.125 

6.344 

6.563 

% 

M 

4.927 

5.151 

5.375 

5.599 

5.823 

6.047 

6.271 

6.495 

6.71!) 

Jl 

11 

5.012 

5.271 

5  500 

5.729 

5.958 

6.188 

6.417 

6.646 

6.876 

11 

5.156 

5.391 

5.625 

5.859 

6.094 

6.328 

6.563 

6.797 

7.(i:u 

iz 

5.271 

5.510 

5.750 

5990 

6.229 

6.469 

6.708 

6.948 

7.188 

•^ 

& 

5.385 

5.630 

5.875 

6.120 

6.365 

«.609 

6.854 

7.099 

7.:;44 

H 

12. 

5.500 

5.750 

6.000 

6.250 

6.500 

6.750 

7.000 

7.250 

7.500 

I* 

Ji 

5.729 

5.990 

6.250 

6.510 

6.771 

7.031 

7.292 

7.552 

7.813 

K 

13. 

5.958 

6.229 

6.500 

6.771 

7.042 

7.313 

7.583 

7.854 

8.125 

13. 

& 

6.188 

6.469 

6.750 

7.031 

7.313 

7.594 

7.875 

8.156 

8.438 

y 

H. 

6.417 

6.708 

7.000 

7.292 

7.583 

7.875 

8.167 

8.458 

8.750 

14. 

>i 

6.646 

6.948 

7.250 

7.552 

7.854 

8.156 

8.458 

8.760 

9.063 

X 

15. 

6.8/5 

7.188 

7.500 

7.812 

8.125 

8.438 

8.750 

9.063 

9.375 

15. 

J* 

7.104 

7.427 

7.750 

8.073 

8.396 

8.719 

9.042 

9.365 

9.688 

H 

16. 

7.333 

7.6G7 

8.000 

8.333 

8.667 

9.000 

9.333 

9.667 

10.00 

16. 

^ 

7.563 

7.906 

8.250 

8.594 

8.938 

9.281 

9.625 

9.969 

10.31 

X 

17. 

7.792 

8.146 

8.500 

8.854 

9.208 

9.563 

9.917 

10.27 

10.63 

17. 

-  X 

8.021 

8.385 

8.750 

9.115 

9.479 

9.844 

10.21 

10.57 

10.94 

H 

18. 

8.250 

8.625 

9.000 

9.375 

9.750 

10.13 

10.50 

10.88 

11.25 

18. 

19. 

8.708 

9.104 

9.500 

9.896 

10.29 

10.69 

11.08 

11.48 

11.88 

19. 

20. 

9.167 

9.583 

10.00 

10.42         10.83 

11.25 

11.67 

12.08 

12.50 

20. 

21. 

9.625 

10.06 

10.50 

10.94       111.38 

1181 

12.25 

12.69 

13.13 

21. 

22. 

10.08 

10.54 

11.00 

11.46       111.92 

12.38 

12.83 

13.29 

13.75 

22. 

23. 

10.54 

11.02 

11.50 

11.98       112.46 

12.94 

13.42 

13.90 

14.38 

23. 

24. 

11.00 

11.50 

12.00 

12.50         13.00 

13.50 

14.00 

14.50 

15.00 

24. 

1 

I 

slightly  more  brittle  and  inflammable.  In  hot  weather  it  exudes  to  some  extent,  and  discolors  the 
timber.  It  preserves  spikes  driven  into  the  timber.  Its  peculiar  smell  excludes  the  timber  from 
dwellings.  Ite  cost  is  12  to  15  cts  per  gallon  ;  and  that  of  the  process,  including  oil.  will  range  from 
12  to  30  cts  per  cub  ft  of  timber,  according  to  the  quantity  of  oil,  &o;  ordinarily  perhaps  about  15 
to  20  cts.  At  times  it  may  be  well  for  economy  to  first  reduce  the  timbers  to  their  intended  final 
dimensions.  The  most  perfect  process  is  that  patented  by  I'rof.  Charles  A.  Seely,  of  New  York, 
under  whosp  patent  Mr  W.  T.  Pelton,  N  Y,  contracts  to  furnish  apparatus  and  oil,  and  apply  the 
process;  which  i*  called  Seelyizing,  or  oarboliziug.  It  may  be  applied  to  either  green  01  seasoned 


360 


BOARD    MEASURE. 


Table  of  Board  Measure  —  (Continued.) 


li 

Feet  of  Board  Measure  contained  in  one  runn  ng  foot  of  Scantlings 
of  different  dimensions.      (Original.) 

a 

2§ 

|>H 

THICKNESS  IN  INCHES. 

•SH 

79< 

8 

8x* 

8H 

8% 

9 

9M 

9H 

m 

£M 

Ft.Bd.M.    Ft.Bd.M. 

FtBd.M. 

Ft.Bd.M 

FtBd.M 

Ft.Bd.M 

Ft.Bd.M 

FtBd.M.  JFt.Bd.M. 

' 

J4 

.1615 

.1667 

.1719 

.1771 

.182: 

.1875 

.1927 

.1979  !       .2031 

^ 

^6 

.3229 

.3333 

.3438 

.3542 

.3646 

.3750 

.3854 

.3958 

.4063 

X 

X 

.4844 

.5000 

.5156 

.5313 

.5467 

.5625 

.5781 

.5938 

.6094 

1. 

.6458 

.6667 

.6875 

.7083 

.7292 

.7500 

.7708 

.7917 

.8125 

i. 

% 

.8073 

.8333 

.8594 

.8854 

.9115 

.9375 

.9635 

.9896 

1.016 

S 

.9688 

1.000 

1.031 

1.063 

1.094 

1.125 

1.156 

1.188 

1.219 

y* 

5i 

1.130 

1.167 

1.203 

1  240 

1.276 

1.313 

1.349 

1.385 

1.422 

H 

2. 

1.292 

1.333 

1.375 

1.417 

1.458 

1.500 

1.542 

1.583 

1.625 

2. 

xi 

1.453, 

1.500 

1.547 

1.594 

1.641 

1.688 

1.734 

1.781 

1.828 

M 

S 

1.615 

1.667 

1.719 

1.771 

1.822 

1.875 

1.927 

1.979 

2.031 

/^J 

X 

1.776 

1.833 

1.891 

1.948 

2.005 

2.063 

.120 

2.177 

2.234 

2 

3. 

1.938 

2.000 

2.063 

2.125 

2.188 

2.250 

.313 

2.375 

2.438 

3. 

i^[ 

2.099 

2.  167 

2.234 

2.302 

2.370 

2.438 

.505 

2.573 

2.641 

Ji 

K 

2.260 

2.333 

2.406 

2.4*9 

2.552 

2.625 

.698 

2.771 

2.844 

% 

H 

2.422 

2.500 

2.578 

2.656 

2.734 

2.813 

.891 

2.969 

3.047 

X 

4. 

2.583 

2.667 

2.750 

2.833 

2.917 

3.000 

3.083 

3.167 

3.250 

4. 

j£ 

2.745 

2.833 

2.922 

3.010 

3.099 

3.188 

3.276 

3.365 

3.453 

ix 

Ji 

2.906 

3.000 

8.094 

8.188 

3.281 

3.875 

3.469 

3.563 

3.656 

L£ 

K 

3.068 

3.167 

3.266 

3!365 

3.464 

3.563 

3.661 

3.760 

3.859 

H 

5. 

3.229 

3.333 

8.438 

3.542 

3.646 

3.750 

3.854 

3.958 

4.063 

5. 

M 

3.391 

3.500 

3.609 

3.719 

3.828 

3.938 

4.017 

4.156 

4.266 

24 

Q 

3.552 

3.667 

3.781 

3.896 

4.010 

4.125 

4.240 

4.354 

4.469 

$6 

K 

3.714 

3.833 

3.953 

4.073 

4.193 

4.313 

4.432 

4.552 

4.67'2 

H 

6. 

3.875 

4.000 

4.125 

4.250 

4.375 

4.500 

4.625 

4.750 

4.875 

6. 

ix 

4.036 

4.167 

4.297 

4.427 

4.557 

4.688 

4.818 

4.948 

5.078 

X 

^ 

4.198 

4.333 

4.469 

4.604 

4.740 

4.875 

5.010 

5.146 

5.281 

j£ 

4.359 

4.500 

4.641 

4.781 

4.922 

5.063 

5.203 

5.344 

5.484 

% 

7. 

4.521 

4.667 

4.813 

4.958 

5.104 

5.250 

5.360 

5.542 

5.688 

7. 

U 

4.682 

4.833 

4.984 

5.135 

5.286 

5.438 

5.590 

5.740 

5.891 

J4 

H 

4.844 

5.000 

5.156 

5.313 

5.469 

5.625 

5.782 

5.938 

6.094 

J3 

5.005 

5.167 

5.328 

5.490 

5.651 

5.813 

5.975 

6.135 

6.297 

H 

8.* 

5.167 

5.333 

5.500 

5.667 

5.833 

6.000 

6.167 

6.333 

6.500 

8. 

i/ 

5.328 

5.500 

5.672 

5.844 

6.016 

6.188 

6.359 

6.531 

6.703 

IX 

M 

5.490 

5.667 

5.844 

6.021 

6.198 

6.375 

6.552 

6.729 

6.906 

y* 

?i 

5.651 

5.833 

6.016 

6.198 

6.380 

6.563 

6.745 

6.927 

7.109 

x 

9. 

5.818 

6.000 

6.188 

6.375 

6.563 

6.750 

6.938 

7.125 

7.313 

9. 

^ 

5.974 

6.167 

6.359 

6.552 

6.745 

6.938 

7.130 

7.323 

7.516 

/4 

Q 

6.135 

6.333 

6.531 

6.729 

6.927 

7.125 

7.323 

7.521 

7.719 

c 

% 

6.297 

6.500 

6.703 

6.906 

7.109 

7.313 

7.516 

7.719 

7.922 

ix 

10. 

6.458 

6.667 

6.875 

7.083 

7.292 

7.500 

7.708 

7.917 

8.125 

10. 

ix 

6.620 

6.833 

7.047 

7.260 

7.474 

7.688 

7.901 

8.115 

8.328 

ix 

}*        ..781 

7.000 

7.219 

7.438 

7.656 

7.875 

8.094 

8.313 

8.531 

:u 

%           343 

7.167 

7.391 

7.615 

7.839 

8.063 

8.286 

8.510 

8.734 

H 

11. 

7.104 

7.333 

7.563 

7.792 

8.021 

8.250 

8.479 

8.708 

8.938 

11. 

7.266 

7.500 

7.735 

7.969 

8.203 

8.438 

8.672 

8.906 

9.141 

S 

7.427 

7.667 

7.906 

8.146 

8.386 

8.625 

8.865 

9.104 

9.344 

j£ 

2i 

7.589 

7.833 

8.078 

8.323 

8.568 

8.813 

9.057 

9.302 

9.547 

9i 

12. 

7.750 

8.000 

8.250 

8.500 

8.750 

9.000 

9.250 

9.500 

9.750 

12. 

Ji 

8.073 

8.333 

8.594 

8.854 

9.115 

9.375 

9.635 

9.896 

10.16 

% 

13. 

8.396 

8.666 

8.938 

9.208 

9.479 

9.750 

10.02 

10.29 

10.56 

13. 

8.719 

9.000 

9.281 

9.563 

9.844 

10.13 

10.41 

10.69 

10.97 

Js 

14. 

9.042 

9.333 

9.625 

9.917 

10.21 

10.50 

10.79 

11.08 

11.38 

14. 

9.365 

9.666 

9.969 

10.27 

10.57 

10.88 

11.18 

11.48 

11.78 

% 

15. 

9.688       10.000 

0.31 

10.63 

10.94 

11.25 

11.56 

1.88 

12.19 

15. 

J^ 

10.01 

10.33 

0.66 

10.98 

1.30 

11.63 

11.95 

2.27 

12.59 

M 

16. 

10.33 

10.67 

1.00 

11.33 

1.67 

12.00 

12.33 

2.67 

13.00 

16. 

Ji 

10.66 

11.00 

1.34 

11.69 

2.03 

12.38 

12.72 

3.06 

13.41 

w 

17. 

0.98 

11.33 

1.69 

12.04 

2.40 

12.75 

13.10 

3.46 

13.81 

17. 

11.30 

11.66 

2.03 

12.40 

2.76 

13.13 

13.49 

3.85 

14.22 

\i 

18. 

1.63 

12.00 

2.38 

12.75 

3.13 

13.50 

13.88 

4.25 

14.63 

18. 

19. 

2.27 

12.67 

3.06 

13.46 

3.85 

14.25 

14.65 

15.04 

15.44 

19. 

20. 

2.92 

13.33 

3.75 

14.17 

4.58 

15.00 

15.42 

15.83 

16.25 

20. 

21. 

3.56 

14.00 

4.44 

14.88 

5.31 

15.75 

16.19 

16.63 

17.06 

21. 

22. 

4.21 

14.66 

5.13 

15.58 

16.04 

16.50 

16.96 

17.42 

17.88 

23. 

4.85 

15.33 

15.81 

16.29 

16.77 

17.25 

17.73 

18.21 

18.69         23. 

24. 

15.50 

16.00 

16.50 

17.00 

17.50 

18.00 

18.50 

19.00 

19.50         24. 

I 

timber.     See  admirable  paper  on  this  subject  by  Col.  T.  J.  Cram,  U  S  Engs,  Jour  Franklin  Inst, 
July.  Aug.  Sept  1873.     Aiso  footnote,  p  414;  also  p  362.  at  top. 

If  the  sap  of  yreen  timber  be  prevented  from  escaping  at  fhe  ends  of 
the  sticks,  as  in  the  case  of  girders,  &c,  enclosed  airtight  In  brickwork  or  masonrr.  its  fermentation 
will  produce  dry  rot.  The  painting  or  varnishing  of  green  timber  conduces  to 
the  same  end.  In  a  free  circulation  of  dry  air  timber  will  endure  for  centuries,  if  not  attacked  by 
worms.  Alternate  exposure  to  water  and  air  produces  wet  rot* 


BOARD    MEASURE. 


361 


Table  of  Board  Measure— (Continued.) 


-I 

Feet  of  Board  Measure  contained  in  one  running  foot  of  Scantlings 
of  different  dimensions.     (Original.) 

a  . 

ij 

•So 

THICKNESS  IN  INCHES. 

£M 

10 

10* 

10* 

10% 

11 

11* 

11* 

11* 

12 

; 

Ft.  Bd.M. 

Ft.  Bd.M. 

FtJBd.M. 

Ft.  Bd.M. 

Ft.Bd.M.jFt.Bd.M. 

Ft.  Bd.M. 

Ft.  Bd.M. 

Ft.  Bd.M. 

,, 

.2083 

.2135 

.2188 

.2240 

.2292 

.2344 

.2396 

.2448 

.2500 

* 

u 

.4167 

.427  1 

.4375 

.4479 

.4583 

.4688 

.4792 

.4896 

.5000 

IX 

if 

.6250 
.8333 

.6406 

.8542 

.6563 
.8750 

.6719 

.8958 

.6875 
.9167 

.7031 
.9375 

.7188 
.9583 

.7344 
.9792 

.7500 
1.000 

i* 

1.042 

1.068 

1.094 

1.120 

1.146 

1.172 

1.198 

1.224 

1.250 

?x 

1.250 

1.821 

1.313 

1.344 

1.375 

.406 

1.438 

1.469 

1.500 

* 

•X 

1.458 

1.495 

1.531 

1.568 

1.604 

.641 

1.677 

1.714 

1.750 

% 

2. 

1.667 

1.708 

1.750 

1.792 

1.833 

.875 

1.917 

1.958 

2.000 

2. 

1.875 

1.922 

1.969 

2.016 

2.063 

.109 

2.156 

2.203 

2.250 

* 

\f 

2.083 

2.135 

2.188 

2.240 

2.292 

.344 

2.396 

2.448 

2.500 

* 

s 

2.292 

2.349 

2.406 

2.464 

2.521 

.578 

2.635 

2.693 

2.750 

9i 

3. 

2.500 

2.563 

2.6'25 

2.688 

2.750 

2.813 

2.875 

2.938 

3.000 

3. 

2.708 

2.776 

2.844 

2.911 

2.979 

3.047 

3.115 

3.182 

3.250 

\6 

2.917 

2.990 

3.063 

3.135 

3.208 

3.281 

3.354 

3.427 

3.500 

* 

H 

3.125 

3.203 

3.281 

3.359 

3.438 

3.516 

3.594 

3.672 

3.750 

H 

4. 

3.333 

3.417 

3.500 

3.583 

3.667 

3.750 

3.833 

3.917 

4.000 

4. 

3.542 

3.630 

3.719 

3.807 

3.896 

3.984 

4.073 

4.161 

4.250 

X 

?x 

3.750 

3.844 

3.938 

4.031 

4.125 

4.219 

4.313 

4.406 

4.500 

»x 

3.958 

4.057 

4.156 

4.255 

4.354 

4.453 

4.552 

4.651 

4.750 

% 

5. 

4.167 

4.271 

4.375 

4.479 

4.583 

4.688 

4.791 

4.896 

5.000 

5. 

4.375 

4.484 

4.594 

4.703 

4.813 

4.922 

5.031 

5.141 

5.250 

?x 

4.583 

4.698 

4.813 

4.927 

5.042 

5.156 

5.270 

5.385 

5500 

* 

•  X 

4.792 

4.911 

5.031 

5.151 

5.271 

5.391 

5.510 

5.630 

5.750 

H 

6. 

5.000 

5.125 

5.250 

5.375 

5.500 

5.625 

5.750 

5.875 

6.000 

6. 

5.208 

5.339 

5.469 

5.599 

5.729 

5.859 

5.990 

6.120  1     6.250 

Hi 

5.417 

5.552 

5.688 

5.823 

5.958 

6.094 

6.229 

6.365        6.500 

* 

S 

6.625 

5.766 

5.906 

6.047 

6.188 

6.328 

6.469 

6.609 

6.750 

9i 

f. 

5.8!<3 

5.979 

6.125 

6.271 

6.417 

6.563 

6.708 

6.854 

7.000 

7. 

IX 

3.042 

6.193 

d.344 

6.495 

6.646 

6.797 

6.948 

7.099 

7.250 

M 

6.250 

<».406 

6.563 

6.719 

6.875 

7.031 

7.188 

7.344 

7.500 

* 

8 

S.458 

6620 

S.781 

6.943 

7.104 

7.266 

7.427 

7.589 

7.750 

K 

a.4 

6.667 

6.833 

7.000 

7.167 

7.333 

7500 

7.667 

7.833 

8.000 

8 

ii 

6.875 

f.047 

1.219 

7.391 

7.563 

7.734 

7.906 

8.078 

8.250 

x4 

It 

7.083 

7.260 

7.438 

7.615 

7.792 

7.969 

8.H6 

8-323 

8.500 

7.292 

7.656 

r.839 

8.021 

8.203 

8.385 

8.568 

8.750 

5i 

*.* 

7.500 

7'.388 

7.875 

8.063 

9  .'250 

8.438 

8.625 

8.813 

9.000 

9. 

7.708 

7.901 

8.094 

3.286 

8.479 

8.672 

8.865 

9.057 

9.250 

* 

1£ 

7.917 

8.115 

8.313 

8.510 

8.709 

.  3.906 

9.104 

9.302 

9.500 

i£ 

«x 

8.125 

8.323 

8.531 

8.734 

9.141 

9.344 

9.547 

9.750 

x5 

TO. 

8.333 

8.542 

8.750 

8.958 

9'.167 

9.375 

9.583 

9.792 

10.00 

10. 

8.542 

8.755 

8.969 

9.182 

9.396 

9.609 

9.823 

10.04 

10.25 

y± 

* 

8.750 

8.969 

9.188 

9.406 

9.625 

9.844 

1C.  06 

10.28 

10.50 

3 

^ 

8.958 

9.182 

9.406 

9.630 

9.854 

10.08 

10.30 

10.53 

10.75 

Jf- 

9.167 

9.396 

9.625 

9.854 

10.08 

031 

10.54 

10.77 

11.00 

a. 

9.375 

9.609 

9.844 

1008 

10.31     |     0.55 

10.78 

11.02 

11.25 

* 

* 

9.583 

9.823 

10.06 

10.30 

10.54     1     0.78 

11.02 

11.26 

11.50 

* 

* 

9.792 

10.04 

10.28 

10.53 

10.77 

1.02 

11.26 

11.51 

11.75 

H 

12. 

10.00 

10.25 

10.50 

10.75 

11.00 

1.25 

11.50 

11.75 

12.00 

12. 

10.42 

10.68 

10.94 

11.20 

11.46 

1.72 

11.98 

12.24 

12.50 

* 

13. 

10.83 

11.10 

11.38 

11.65 

11.92 

2.19 

12.46 

12.73 

13.00 

13. 

11.25 

11.53 

11.81 

12.09 

12.38 

2.66 

12.94 

13  22 

13.50 

14. 

11.67 

11.96 

12.25 

12.54 

12.83 

3.13 

13.42 

13.71 

14.00 

14. 

12.08 

12.39 

12.69 

12.99 

13.29 

3.59 

13.90 

14.20 

14.50 

15. 

12.50 

12.81 

13.13 

13.44 

13.75 

4.06 

14.38 

14.69 

15.00 

15. 

12.92 

1324 

13.56 

13.89 

14.21 

4.53 

14.85 

1518 

15.50 

16. 

13.33 

13.67 

14.00 

14.33 

14.67 

15.00 

15.33 

15.67 

1600 

16. 

13.75 

14.09 

.  14.44 

14.78 

15.13 

5.47 

15.81 

16.16 

16.50 

17. 

14.17 

1452 

14.88 

15.23 

15.58 

15.94 

16.29 

16.65 

17.00- 

17. 

M 

14.58 

14.95 

15.31 

15.77 

16.04 

16.41 

16.77 

17.14 

17.50 

w 

re 

15.00 

15.38 

15.75 

16.13 

16.50 

16  88 

17.25 

17.63 

18.00 

18. 

19. 

15.83 

16.23 

16.63 

17.02 

17.42 

17.81 

18.21 

18.60 

1900 

19. 

20. 

16.67 

17.08 

17.50 

17.92 

18.33 

18.75 

19.17 

19.58 

20.00 

20. 

21. 

17.50 

17.94 

18.38 

18.81 

19.25 

19.69 

20.13 

20.56 

21.00 

21. 

22. 

18.33 

18.79 

19.25 

19.71 

20.17 

20.63 

21.08 

21.54 

22.00 

22. 

23. 

19.17 

19.65 

20.13 

20.60 

21.08 

21.56 

22.04 

22.52 

23.00 

23. 

24. 

20.00 

20.50 

21.00 

21.50 

22.00 

^2.50 

23.00 

23.50 

24.00 

24. 

Price  of  lumber,  Phila,  1873:  Spruce  joists,  $28  to  $30  per  1000  ft  board 

meas.     Hemlock  joists,  $20  to  $23.     Yellow  pine  floor  boards,  $40  to  $60.     White  pine  boards,  $30  to 
$70,  according  to  quality,  degree  of  seasoning,  &c.     Sawed  W  pine  timbers  $40  to  $50.     Heart  Y  pine 

$50  to  $55.    Hemlock  $20  to  $25.    In  188O  prices  range  from  |  to  J  less. 


362 


WEIGHT   OF   CAST    IRON. 


The  durability  of  timber  in  situations  either  dry  or  merely  damp  (not 
wet),  is  said  to  be  increased  by  soaking  for  a  week  or  two  in  a  solution  of  either 
aalt  or  quicklime,  in  water.  See  also  "  Creosote,"  p  358. 

TABLE  OF  WEIGHT  OF  CAST  IRON.* 

The  weight  of  a  pattern  of  perfectly  dry  white  pine,  if  mult 
by  20,  will  give  approximately  the  wt  of  the  casting.  If  well  seasoned,  but  still  not 
perfectly  dry,  mult  by  19,  or  by  18. 

Assuming  450  Bbs  to  a  cub  ft,  a  pound  contains  3.8400  cubic  inches ;  a  ton  5  cub  ft ; 
and  a  cubic  inch  weighs  .2604  ft)s. 


1  Thickness 
or  Diameter 
1  in  Inches. 

Thick- 
ness or 
Diam. 
indeci- 
mals  of 
a  foot. 

Wt.  of  a 
Square 
Foot. 
Lbs. 

Wt.  of  a 
Square 
bar.  1  ft. 
long. 
Lbs. 

Wt.  of  a 
Round 
bar,  I  ft. 

Wt.  of 

Balls. 
Lbs. 

t 

1  Thickness 
or  Diameter 
in  Inches. 

Thick- 
ness or 
Diam. 
in  deci- 
mals of 
a  foot. 

Wt.  of  a 

Square 
Foot. 
Lbs. 

Wt.  of  a 
Square 
bar.  1  ft. 
long. 
Lbs. 

Wt.  of  a 
Round 
bar,  1  ft. 
long. 
Lbs. 

Wt.  of 
Balls. 
Lbs. 
t 

1-32    .0026 

1.173 

.003 

.002 

3^ 

.2604 

117.3 

30.52 

23.97 

4.162 

1-16     .0052 

2.344 

.012 

.010 

M 

.2708 

121.8 

33.01 

25.93 

4.681 

3-32 

.0078 

3.516 

.027 

.021 

.0001 

% 

.2813 

126.5 

35.60 

27.95 

5.243 

M 

.0104 

4.687 

.048 

.038 

.0003 

% 

.2917 

131.2 

38.28 

30.07 

5.846 

5-32 

.0130 

5.861 

.076 

.060 

.0005 

% 

.3021 

135.9 

41.07 

32.25 

6.498 

3-16 

.0156 

7.032 

.110 

.086 

.0009 

H 

.3125 

140.6 

43.95 

34.51 

7.193 

7-32 

.0182 

8.203 

.150 

.118 

.0014 

.3229 

145.3 

46.93 

36.85 

7.934 

x4 

.0208 

9.375 

.195 

.154 

.0021 

4. 

.3333 

150.0 

50.01 

39.27 

8.726 

9-32 

.0234 

10.54 

.247 

.194 

.0030 

xi 

.3438 

154.7 

53.18 

41.77 

9.572 

5-16 

.0260 

11.73 

.305 

.240 

.0042 

M 

.3542 

159.3 

56.46 

44.33 

10.47 

11-32 

.0287 

12.89 

.370 

.290 

.0056 

9i 

.3646 

164.0 

59.82 

46.99 

11.42 

.0313 

14.06 

.440 

.346 

.0072 

.3750 

168.7 

63.33 

49.71 

12.43 

13-32 

.0339 

15.24 

.516 

.400 

.0092 

xi 

.3854 

173.4 

66  >6 

52,52 

13.49 

7-16 

.0365 

16.41 

.598 

.470 

.0114 

% 

.3958 

178.1 

70.52 

55.39 

14.62 

15  32  1   .0391 

17.56 

.687 

.540 

.0140 

y* 

.4063 

182.8 

74.28 

58.34 

15.81 

H        .0417 

18.75 

.781 

.610 

.0170 

5. 

.4167 

187.5 

78.12 

61.37 

17.05 

916    .0469 

21.10 

.989 

.777 

.0243 

xi 

.4271 

192.2 

82.10 

64.47 

18.35 

K 

.0521 

23.44 

1.221 

.959 

.0334 

V* 

.4375 

196.9 

86.14 

67.65 

19.73 

n-re 

.0573 

25.79 

1.478 

1.161 

.0444 

.4479 

201.6 

90.29 

70.52 

21.18 

.0625 

28.12 

1.758 

1.381 

.0575 

xl2 

.4583 

206.2 

94.54 

74.26 

22.68 

13*16 

.0677 

30.47 

2.064 

1.621 

.0732 

% 

.4688 

210.9 

98.89 

77.66 

24.27 

% 

.0729 

32.81 

2.393 

1.880 

.0913 

% 

.4792 

215.6 

103.3 

81.16 

25.93 

15-16 

.0781 

35.16 

2.747 

2.158 

.1124 

% 

.4896 

220.3 

107.9 

84.72 

27.41 

1. 

.0833 

37.50 

3.125 

2.455 

.1363 

6. 

.5000 

225.0 

112.5 

88.36 

29.44 

1-16 

.0885 

39.84 

3.528 

2.771 

.1636 

M 

.5208 

234.4 

122.1 

95.89 

33.28 

.0938 

42.19 

3.955 

3.107 

.1942 

xl2 

.5417 

243.8 

132.0 

103.7 

37.44 

3-16 

.0990 

44.53 

4.407 

3.461 

.2284 

% 

.5625 

253.1 

142.4 

111.9 

41.94 

\i 

.1042 

46.87 

4.883 

3.835 

.2664 

7. 

.5833 

262.5 

153.2 

120.2 

46.77 

f-16 

.1094 

49.22 

5.384 

4.229 

.3084 

X 

.6042 

271.9 

164.2 

129.0 

51.97 

N 

.1146 

51.57 

5.909 

4.640 

.3546 

.6250 

281.3 

175.8 

138.1 

57.54 

7-16 

.1198 

53.91 

6.461 

5.073 

.4058 

% 

.6458 

290.7 

187.7 

147.4 

63.47 

.1250 

56.26 

7.033 

5.523 

.4603 

8. 

.6667 

300.0 

200.1 

157.0 

69.82 

9-16 

.1302 

58.60 

7.632 

5.993 

.5204 

M 

.6875 

309.4 

212.7 

167.0 

76.58 

.1354 

60.94 

8.253 

6.484 

.5852 

J£ 

.7083 

318.8 

225.8 

177.3 

83.74 

11-16 

.1406 

63.28 

8.900 

6.991 

.6555 

9£ 

.7292 

328.2 

239.3 

187.9 

91.35 

ax 

.1458 

65.63 

9.572 

7.518 

.7310 

9. 

.7500 

337.4 

253.1 

198.8 

99.42 

13-16 

.1510 

67.97 

10.27 

8.064 

.8122 

.7708 

346.8 

267.4 

210.0 

107.9 

% 

.1563 

70.32 

10.99 

8.630 

.8991 

.7917 

356.2 

282.1 

221.5 

116.8 

15-16 

.1615 

72.66 

11.73 

9.215 

.9920 

H 

.8125 

365.6 

297.0 

233.3 

126.3 

3. 

.1667 

75.01 

12.50 

9.821 

1.073 

10. 

.8333 

375.0 

312.5 

245.5 

136.3 

.1771 

79.70 

14.11 

11.09 

1.308 

.8542 

384.4 

328.4 

257.8 

146.8 

IX 

.1875 

84.40 

15.83 

12.43 

1.554 

M 

.8750 

393.7 

344.5 

270.6 

157.9 

% 

.1979 

89.07 

17.63 

13.85 

1.827 

% 

.8958 

403.1 

361.2 

283.7 

169.3 

ix 

.2083 

93.75 

19.54 

15.34 

2.131 

11. 

.9167 

412.5 

378.2 

297.0 

181.5 

fcx 

.2188 

98.44 

21.54 

16.56 

2.467 

.9375 

421.9 

395.5 

310.6 

194.2 

ax 

.2292 

103.2 

23.64 

18.56 

2.835 

v 

.9583 

431.2 

413.3 

324.6 

207.3 

tx 

.2396 

107.8 

25.84 

20.29 

3.241 

H 

.9792 

440.6 

431.4 

338.8 

219.2 

3. 

.2500 

112.6 

28.13 

22.10 

3.682 

12. 

1  Foot. 

450. 

450. 

353.4 

235.6 

t  Wts  of  balls   are  as  the  cubes  of  their  diams. 

To  find  the  weight  of  a  spherical  shell.    From  the  weight  of  a  ball 
which  has  the  outer  diam  of  the  shell,  take  the  wt  of  one  which  has  its  inner  diam. 

*  For  Copper,  mult  by  1.2 ;  Ltad,  mult  by  1.6;  Brass,  add  1-7 th ;  Zinc,  mult 

by  .97.     All  approximate. 


WEIGHT   OF   CAST-IRON   PIPES. 


363 


WEIGHT  OF  CAST-IRON  PIPES  per  running  foot, 

Assuming  the  weight  of  cast-iron  at  460  ftn  per  cub  ft,  or  .2604  ft)  per  cub  inch.  No 
allowance  is  here  made  for  the  spigot  and  faucet-joints  used  in  water-pipes  As 
the^e  are  now  commonly  made,  (see  Hydraulics,  Fig  38,)  they  add  to  the  weight  of 
each  length  or  section  of  pipe  of  any  size,  about  as  much  as  that  of  8  inches  in 
length  of  the  plain  pipe  as  given  in  the  table.  See  Hydraulics,  Art  16;  also  next 
table.  For  lead-pipe  mult  by  l.G ;  copper,  mult  by  1.2 ;  brass,  add  l-7th ; 
welded  iron,  mult  by  1.0667,  or  add  one  fifteenth  part. 


iii 

s^s 

THICKNESS    OP    PIPE    IN   INCHES. 

§S.S 

H     \      % 

K 

x 

« 

K 

1 

1* 

191 

IX    !     1H 

\H    \      2 

Wtin 

Wiia 

Wtia 

Wtiu 

Wt  in 

Wt  in 

Wt  in 

Wt  in 

Wt  in 

Wt  in 

Wt  in  Wt  in 

Wt  in 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

LJbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

1. 

3.07 

5.07 

7.38 

9.93 

12.9 

16.2 

19.7 

23.5 

27.7 

32.1 

36.9 

47.4 

59.1 

3.89 

6.00 

8.61 

11.5 

14.8 

183 

22  2 

26.3 

30.8 

35.5 

40.6 

51.7 

64.0 

% 

4.30 

6.92 

9.84 

13.1 

16.6 

20.5 

24.6 

29.1 

33.8 

38.9 

44.3 

56.0 

68.f 

% 

4  .92 

7.84 

11.1 

14.6 

18.5 

22.6 

27.1 

31.8 

36.9 

42.3 

48.0 

60.3 

73.8 

2. 

5.53 

8.76 

12.3 

16.2 

20.3 

24.8 

29.5 

34.6 

40.0 

45.7 

51.7 

64.6 

78.7 

6.15 

9.69 

13.5 

17.7 

222 

26.9 

32.0 

37.4 

43.1 

49.0 

55.4 

68.9 

86.7 

i^ 

6.76 

10.6 

14.8 

19.2 

24.0 

29.1 

34.5 

40.1 

46.1 

52.4 

59.1 

73.2 

88.6 

H 

7.37 

11.5 

16.0 

20.8 

25.9 

31.2 

36.9 

42.9 

49.2 

55.8 

62  7 

77.5 

93.5 

3 

7.93 

12.5 

17.2 

22  3 

27.7 

33.4 

39.4 

457 

52.3 

59.2 

66  4 

81.8 

98.4 

8.60 

13.4 

18.5 

23.8 

29.5 

35.5 

41.8 

48.4 

554 

62.3 

70.1 

86.1 

103. 

\6 

9.21 

14.3 

19.7 

25.4 

31.4 

37.7 

44.3 

51.2 

58.4 

65.9 

73.8      90.4 

108. 

% 

9.83 

15.2 

20.9 

26.19 

33.2 

398 

46.8 

54.0 

61.5 

69.3 

77.5;     94.7 

113. 

4. 

10.3 

16.1 

22.2 

28.5 

35.1 

42.0 

49.2 

56.7 

64.6 

72.7 

81.2      99.0 

118. 

y* 

11.1 

17.1 

23.4 

30.0 

369 

44.1 

51.7 

59.5 

67.7 

76.1 

84.9;    103. 

123. 

X 

11.7 

18.0 

21.6 

31.5 

38.8 

46.3 

54.1 

62.3 

70.7 

79.5 

88.61    108. 

1'28. 

x 

12.3 

18.9 

25.8 

33.1 

40.6 

48.5 

56.6 

65.0 

73.8 

83.9 

92.3     112. 

133. 

5. 

12.9 

19.8 

27.1 

34.6 

42.5 

50.6 

59.1 

67.8 

76.9 

87.2 

9«.0     116. 

138. 

% 

13.5 

20.8 

28.3 

36.1 

44.3 

52.8 
54  9 

61.5 

70.6 
73  3 

80.0 
83.0 

90.6 
94.0 

99.6 
103 

121. 
125. 

143. 
148. 

% 

14^8 

22.6 

30  8 

39.2 

48.0 

57^1 

66.4 

7e!i 

86.1 

97^4 

107.' 

129! 

153] 

6. 

15.4 

23.5 

32.0 

40.8 

49.8 

59.2 

68.9 

78.9 

89.2 

99.8 

111. 

134. 

158. 

X 

16.6 

25.4 

34.5 

43.8 

53.5 

63.5 

73.8 

84.4 

95.3 

107. 

118. 

142. 

167. 

7. 

17.8 

27.2 

36.9 

46.9 

57.2 

67.8 

78.7 

89.4 

102. 

113. 

126. 

151. 

177. 

X 

19.1 

29.1 

39.4 

50.0 

60.9 

72.1 

83.7 

955 

108. 

120. 

133. 

159. 

187. 

8. 

20.3 

30.9 

41.8 

53.1 

64.6 

76.4 

88.6 

101. 

114. 

127. 

140. 

168. 

197. 

« 

21.5 

32.8 

44.3 

56.1 

68.3 

80.7 

93.5 

107. 

120. 

134. 

148. 

177. 

207. 

9. 

22.8 

34.6 

46.8 

59.2 

72.0 

85.1 

98.4 

112. 

126. 

140. 

155. 

185. 

217. 

X 

240 

36.4 

49.2 

62.3 

75.7 

89.3 

103. 

118. 

132. 

147- 

163. 

194. 

226. 

10. 

25.1 

38.3 

51.7 

65.3 

79.4 

93.6 

108. 

123. 

138. 

154. 

170. 

202. 

235. 

h 

23.4 

401 

54.1 

68.4 

83.0 

97.9 

113.2 

129. 

145. 

161. 

177. 

211. 

245. 

11. 

27.6 

42.0 

566 

71.5 

86.7 

102. 

118. 

134. 

151. 

168. 

185. 

220. 

255. 

X 

28.8 

43.8 

59.1 

746 

90.4 

107. 

123. 

140. 

157. 

174. 

192. 

228. 

265. 

12. 

30.0 

45.7 

61.5 

77.7 

94.1 

111. 

128. 

145. 

163. 

181. 

199. 

237. 

275. 

13. 

32.5 

49.4 

66.4 

83.8 

102. 

120. 

138. 

156. 

175. 

195. 

214. 

254. 

294. 

14. 

35.0 

5H.1 

71.4 

89.4 

109. 

128. 

148. 

168. 

188. 

208. 

229. 

271. 

314. 

15. 

37.4 

56.7 

76.3 

96.1 

116. 

137. 

158. 

179. 

200. 

222. 

244. 

289. 

334. 

16. 

39.1 

60.4 

81.2 

102. 

124. 

145. 

167. 

190. 

212. 

285. 

258. 

306. 

353. 

17. 

42.3 

61.1 

«6.1 

108. 

131. 

154. 

177. 

201. 

225. 

249. 

273. 

323. 

373. 

18. 

44.8 

67.8 

91.0 

115. 

139. 

163. 

187. 

212. 

237. 

262. 

288. 

340. 

393. 

19. 

47.3 

71.5 

96.0 

121. 

146. 

171. 

197. 

223. 

249. 

276. 

303. 

357. 

412. 

20. 

49.7 

75.2 

101. 

127. 

153. 

180. 

207. 

234. 

261. 

289. 

317. 

375. 

432. 

21. 

52.2 

78.9 

106. 

133. 

161. 

188. 

217. 

245. 

274. 

303. 

332. 

392. 

452. 

22. 

54.6 

82.6 

111. 

139. 

168. 

196. 

227. 

256. 

286. 

316. 

347. 

409. 

471. 

23. 

57.1 

86.3 

116. 

145. 

175. 

208. 

236. 

267. 

298. 

330. 

362. 

426. 

491. 

24. 

59.6 

89.9 

121. 

152. 

183. 

214. 

246. 

278. 

311. 

343. 

375. 

441. 

oil. 

25. 

62.0 

95.6 

126. 

158. 

190. 

223. 

256. 

289. 

323. 

357. 

391. 

461. 

53t. 

26. 

61.5 

97.3 

131. 

164. 

198. 

231. 

266. 

300. 

335. 

370. 

406. 

478. 

550. 

27. 

66.9 

101. 

135. 

170. 

205. 

240. 

276. 

311. 

348. 

384. 

421. 

495. 

570. 

28. 

69.4 

5. 

140. 

176. 

Ill 

249. 

286. 

323. 

360. 

397. 

436. 

512. 

590. 

29. 

71.8 

9. 

145. 

182. 

220. 

257. 

295. 

334. 

372. 

411. 

450. 

530. 

609. 

30. 

74.2 

2. 

150. 

188. 

227. 

266. 

305. 

345. 

384. 

424. 

465. 

547. 

629. 

31. 

76.7 

6. 

155. 

195. 

234. 

275. 

315. 

356. 

397. 

438. 

480. 

564. 

649. 

32. 

79.1 

0. 

160. 

201. 

242. 

283. 

325. 

367. 

409. 

451. 

495. 

581. 

668. 

33. 

81.6 

Si 

165. 

207. 

249. 

292. 

335. 

378. 

421. 

465. 

509. 

698. 

688. 

34. 

84.1 

7. 

170. 

213. 

257. 

300. 

345. 

389. 

434. 

479. 

524. 

616. 

708. 

35. 

86.5 

I. 

175. 

219. 

264. 

309. 

354. 

400. 

446. 

492. 

539. 

633. 

726. 

36. 

sa.o 

4. 

180. 

225. 

271. 

318. 

364. 

411. 

458. 

506. 

554. 

650. 

746. 

42. 

104. 

6. 

210. 

262. 

315. 

370. 

423. 

478. 

532. 

588. 

644. 

753. 

864. 

48. 

119. 

8. 

239. 

298. 

359. 

422. 

482. 

544. 

605. 

669. 

733. 

856. 

982. 

For  warming  buildings  by  steam  it  usually  suffices  to  allow  1  sq  ft 
of  cast  or  wrought  pipe  surface  for  each  120  cub  ft  of  space  to  be  warmed  ;  and  1  cub 
ft  of  boiler  for  each  2000  cub  ft  of  such  space. 


364 


WEIGHT    OF    CAST-IRON    WATER-PIPES. 


WEIGHT  OF  CAST-IROX  WATER-PIPES, 

As  used  in  Phila,  and  tested  by  hydraulic  press  before  delivery,  to  an  internal 
pres  of  300  ft>s  per  sq  inch  This  table  includes  spigots  and  faucets,  or  bells.  The 
pipes  are  required  to  be  made  of  remeltcd  strong  tough  gray  pig  iron,  such  as  may 
be  readily  drilled  and  chipped ;  and  all  of  more  than  4  ins  diam  to  be  cast  verti- 
cally, with  the  bell  end  down.  Deviations  of  5  per  cent  above  or  below  the  theo- 
retical weights,  are  allowed  for  irregularities  in  casting,  which  it  seems  impossi- 
ble to  avoid. 


Diam. 

Length. 

Weight. 

Thickness 
of  body. 

Thickness 
of  bell. 

Thickness 
ofleadjoint. 

Ins. 

Ft.       Ins. 

Lbs. 

Ins. 

Ins. 

Ins. 

3 

9 

135 

A 

K 

K 

4 

9 

190 

% 

H 

A 

i 

12        3 

220 

H 

it 

* 

• 
8 

12        3^ 
12        3J4 

370 
500 

I 

| 

% 

% 

10 

12        3^ 

640 

« 

1 

% 

12 

12        3}$ 

850 

ft 

% 

A 

16 

12        3tf 

12GO 

M 

i 

$ 

20 

12        3>* 

1815 

H 

>ft 

A 

Price,  Phila,  1880,  about  $56  to  $67  per  ton ;  or  say  2^  to  3  cents  per  ft);  de- 
pending on  size,  and  quantity  ordered.  Elbows,  connections,  &c.  about  6  to  15  per  ct  more.  la 
ordering  anything  by  the  ton,  be  careful  to  specify  the  number  of  fts  (2240).  This  prevents  mis- 
understandings. 

Approximate  average  prices*iii  cents  per  Ib,  currency,  of 
iron  and  steel  in  Philada,  in  1880.  Iron  bars  of  ordinary  sizes  (%  inch  to 
2  ins  diam  or  square,)  best  Norway,  5^;  Swedish,  5;  American  common,  round  or 
square,  3^  to  3%;  refined,  3%  to  4;  increasingly  degrees,  for  both  larger  and 
smaller  sizes,  to  about  40  or  50  per  ct  more  for  bars  of  either  %  inch  or  of  4  inches 
diam  or  square.  Extra  refined,  1  ct  per  ft)  more.  Hoop  iron.  4%  for  \%  inch 
or  more  wide;  to  7  cts  for  very  thin  %  inch  wide.  Sheet  iron,  common,  Nos 
10  to  16,  4% ;  Nos  16  to  25,  7 ;  best,  about  50  per  cent  more.  Best  Russian,  15  cts. 
Plates,  common,  4^;  best,  6.  Angle,  4.  T,  4%.  Rolled  I  beams,  4% 
to  5.  Plioeiiix  columns,  5  to  5%.  Rails,  2%  to  3.  Pig  iron,  common, 
1%  to  1%;  best,  2.  Cast  iron  house  fronts,  fitted  and  put  up,  6%  to  7. 
Ordinary  castings,  2^  to  3%  cents.  Brass  castings,  20  to  22  cts. 

Steel  bars,  ordinary  sizes,  common,  9^  to  10^;  best,  13  to  14;  machinery,  9. 
Very  small,  or  large  sizes,  up  to  40  or  50  per  ct  more.  Sheet  steel,  common,  8 
to  10;  best,  13  to  14.  Plates,  homogeneous,  8^.  Steel  rails,  3*4  to  3%. 
Cast  steel,  ordinary,  7  to  9  cts ;  tire,  5  to  6 ;  spring,  5 ;  tool,  12  to  13  cts. 
Weight  per  foot  run  of  welded  wrought -iron  tubes,  usually  in 
lengths  of  18  ft.  Other  sizes  and  lengths  made  to  order. 


Inner 
Diam. 

Outer 
Diam. 

Ths. 

Wt.  per 
Foot. 

Inner 
Diam. 

Outer 
Diam. 

Ths. 

Wt.  per 
Foot. 

Inner 
Diam. 

Outer 
Diam. 

Ths. 

Wt.  per 
Foot. 

Ins. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Ins. 

Lbs. 

.270 

.405 

.068 

.243 

.928 

1.660 

.366 

5.065 

2.331 

2.855 

.262 

7.126 

.320 

.540 

.110 

.505 

.995 

1.315 

.160 

1.976 

2.468 

2.876 

.204 

5.773 

.320 

.840 

.260 

1.613 

1.048 

1.315 

.134 

1.670 

3.067 

3.500 

.217 

7.547 

.364 

.540 

.088 

.422 

1.108 

1.900 

.396 

6.370 

3.548 

4.000 

.226 

9.055 

.437 

.675 

.119 

.708 

1.294 

1.660 

.183 

2.891 

4.026 

4.500 

.237 

10.728 

.478 

1.050 

.286 

2.338 

1.380 

1.660 

.140 

2.258 

4.508 

5.000 

.247 

12.492 

.494 

.675 

.091 

.561 

1.459 

2.375 

.458 

9.390 

5.045 

5.563 

.259 

14.564 

.580 

.840 

.130 

.987 

1.504 

1.900 

.198 

3.604 

6.065 

6.625 

.280 

18.767 

.623 

.840 

.109 

.845 

1.611 

1.900 

.145 

2.694 

7.023 

7.625 

.301 

23.410 

.675 

1.315 

.320 

3.405 

1.807 

2.855 

.524 

13.656 

7.982 

8.625 

.322 

28.348 

.764 

1.050 

.143 

1.387 

1.917 

2.375 

.229 

5.400 

9.001 

9.688 

.344 

34.077 

.824 

1.050 

.113 

1.126 

2.067 

2.375 

.154 

3.667 

10.019 

10.750 

.366 

40.641 

Prices  of  welded  iron  tubes.  Philada,  1880,  approx;  2  ins  diam,  7U 

cts  per  ft  ;  3  ins  diam,  8  cts ;  6  ins  diam,  9  cts  :  8  ins  diam,  1 1  ^  cts.     Made  on  a  large  scale  by  W.  C 
Allison  ft  Co,  Junction  Car  Works,  32d  and  Walnut  Sts,  Philada, 
*  In  1882  all  about  20  per  ct  lower. 


365 


The  foregoing  well  known  firm  of  W.  C.  Allison  Jk  Co  are  builders  of  R. 
R.  cars  of  all  kinds;  and  furnish  all  the  appliances  for  roofs,  buildings,  and  bridges 
of  iron ;  and  furnish  railroad  supplies,  &c,  &c. 

Seamless  I>rawn  Brass  ami  Copper  Tubes  of  the  American 

Tube  Works,  Boston,  Mass.     Approximate  sizes  and  weights.* 
Although  tliis  is  the  Go's  own  table,  it  is  not  consistent  in  itself;  for  where  two 
thicknesses  are  given,  the  wts  in  some  cases  correspond  with  the  least,  and  in  others 
with  the  greatest ;  and  there  is  no  way  to  discriminate  between  them. 


Outer 
Diam. 
Ins. 

•sr- 

Ths. 
Inch. 

Wt.  i 
per 
Brs. 

n  Ibs. 
ft. 
Cop. 

Outer 
Oiam. 
IDS. 

LRth. 

Ft. 

Ths. 
In. 

Wt.  i 
per 
Brs. 

u  fts. 
ft. 
Cop. 

Outer 
Diam. 
Ins. 

•*?• 

Ths. 
lu. 

Wt.  i 
per 
Brs. 

nibs, 
ft. 
Cop. 

N 

10 

.049 

.375 

.375 

.083 

.095 

K 

12 

.058 

.500 

.500 

\% 

13 

to 

1.75 

1.80 

2K 

12 

to 

3. 

3.13 

Yt 

10 

.058 

.625 

.625 

.109 

.120 

1 

10 

.065 

.750 

.750 

1» 

12 

1.88 

1.94 

2« 

12 

3.13 

3.25 

IK 

10 

.065 

.875 

.875 

2 

15 

2.2 

2.25 

3 

12 

3.33 

3.5 

.083 

2K 

13 

ik 

2.25 

2.38 

3K 

10 

< 

3.5 

3.63 

IX 

15 

to 

1.25 

1.25 

«J 

14 

«« 

2.38 

2.50 

3k' 

10 

< 

3.88 

4.13 

.109 

'2H 

13 

<< 

2.50 

2.67 

3^ 

10 

• 

4.25 

4.38 

IH 

10 

" 

1.375 

1.375 

.095 

4 

10 

5. 

5.25 

w 

14 

" 

1.50 

1.60 

ZX 

13 

to 

2.75 

3. 

5 

10 

M 

7. 

8. 

IK 

12 

" 

J.631 

1.70 

.120 

These  tubes  are  furnished  either  plain,  or  tinned  with  pure  tin.  The  Co  also  make  the  connections, 
elbows,  bends,  &c.  of  the  same  metals,  usually  required  in  plutnbincr.  The  brass  tubes  have  been 
used  in  Boston,  &c,  as  a  substitute  for  lead  service-pipes  in  dwellings  for  several  years  ;  and  nre  much 
approved  of.  They  have  also  been  introduced  in  the  City  Hospital,  and  in  some  of  the  hotels. 


irices,  either  brass  or  copper,  per  foot:  Diam  %,  25  cts;  %, 

—  ,  ,., .  _jh.  60  cts ;  1^,  70  cts ;  1%,  80  cts.     If  tinned,  add  about  one-seventh  part  to  th 

prices.     Liberal  discounts  on  large  sales.     1880. 

24 


*  Appro x  pri 

s  ;  %,  45  cts  :  1  inch. 


366 


WEIGHT   OF    WROUGHT    IRON    AND   STEEL. 


Table  of  Weight  of  WROUGHT  IRON*  and  STEEL. 

Wrought  iron  is  here  taken  at  485  fts  per  cub  f t ;  or  a  sp  gr  of  7.77.  Very  pure 
soft  wrought  iron  weighs  from  4&8  to  492  fts  per  cubic  foot.  Light  weight  indicates 
impurities,  and  weakness.  Steel  weighs  about  49 J  Ibs  per  cubic  toot ;  therefore,  for 
steel,  an  addition  must  be  made  to  tlie  tabular  amounts,  of 
about  1  pound  in  1OO  Ibs. 

At  485  Ibs  per  cub  ft,  a  cubic  inch  weighs  .28067  ft ;  a  ft  contains  3.5629  cub  ins ; 
and  a  ton,  4.6186  cub  ft ;  and  this  is  about  the  average  of  hammered  iron  The  usual 
assumption  is  480  Ibs  per  cub  ft;  which  is  nearer  the  average  of  ordinary  mild  iron. 
At  480  fts,  a  cubic  inch  weighs  .2778  of  a  ft);  a  ft)  contains  3.6  cub  ins  ;  a  ton,  4  6bb7 
cub  ft ;  a  rod  of  1  sq  inch  area,  weighs  10  fts  per  yard ;  or  3%  fts  per  foot,  exactly. 


hiekness 
Diameter 

i  Inches. 

Thick- 
ness or 
Diani. 
in  deci- 
mals of 

Wt.  of  a 

Square 
Foot. 

Wt.  of  a 
Square 
bar,  1  ft. 
long. 

Wt.  of  a 
Round 
bar,  1  ft. 

long. 

Wt.  of 
Balls. 

ill 
•*?  = 

Thick- 
ness or 
Diam. 
in  deci- 
mals of 

Wt.  of  a 
Square 
Foot. 

Wt.  of  a 
Square 
bar.  1  ft. 
long. 

Wt.  of  a 
Round 
bar.  1  ft. 
long. 

Wt.  of 
Ball*. 

Kfc« 

a  foot. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

~  o'~ 

a  foot. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

1-32 

.0026 

1.263 

.0033 

.0026 

*x 

.2604 

126.3 

32.89 

25.83 

4.484 

1-16 

.0052 

2.526 

.0132 

.0104 

.2708 

131.4 

3557 

27.94 

5.045 

3-32 

.0078 

3.789 

.0296 

.0233 

.0001 

?'8 

.2813 

1364 

38.37 

30.13 

5.649 

X 

.0104 

5.052 

.0526 

.0414 

.0003 

X 

.2917 

141.5 

41.26 

32.41 

6.301 

5-32 

.0130 

6.315 

.0823 

.0646 

.0005 

% 

.3021 

146.5 

44.26 

34.76 

7.000 

3-16 

0156 

7.578 

.1184 

.0930 

.0009 

K 

.3125 

151.6 

47.37 

37.20 

7.750 

7-32 

.0182 

8.841 

.1612 

.1266 

.0015 

% 

.3229 

156.6 

50.57 

39.72 

8.550 

X 

.0208 

10.10 

.2105 

.1653 

.0023 

4. 

.3333 

161.7 

53.89 

42.33 

9.405 

932 

.0234 

11.37 

.2665 

.2093 

.0033 

K 

.3438 

166.7 

57.31 

45.01 

10.32 

5-16 

.0260 

13.63 

.3290 

.2583 

.0045 

tt 

.3542 

171.8 

60.84 

47.78 

11.28 

11-32 

.0287 

13.89 

.3980 

.3126 

.0060 

% 

.3046 

176.8 

64.47 

50.63 

12.31 

M 

.0313 

15.16 

.4736 

.3720 

.0078 

X 

.3750 

181.9 

68.20 

53.57 

13.39 

13-32 

.0339 

16.42 

.5558 

.4365 

.0098 

>A 

.3854 

Ifc6.9 

72.05 

56.59 

14.54 

7-16 

.0365 

17.68 

.6446 

.5063 

.0123 

H 

.3958 

192.0 

75.S-.9 

5t).«9 

15.75 

15-32 

0391 

18.95 

.7400 

.581:} 

.0151 

% 

.4063 

1H7.0 

80.05 

62.87 

17.03 

M 

.0417 

20.21 

.8420 

.6613 

.0184 

5. 

.4167 

202.1 

84.20 

(if,.  13 

18.37 

9-16 

.0469 

22.73 

1.066 

.8370 

.0262 

# 

.4271 

207.1 

88.47 

69.48 

19.78 

% 

.0521 

25.26 

1.316 

1.033 

.0359 

k 

.4375 

212.2 

92.83 

72.91 

21.26 

11-16 

.0573 

27.79 

1.592 

1.250 

.0478 

S/8 

.4479 

217.2 

97.31 

76.43 

22.82 

% 

.0625 

3031 

1.895 

1.488 

.0620 

% 

.4583 

222.3 

101.9 

80.02 

24.45 

13-16 

.0677 

32.84 

2.223 

1.746 

.0788 

H 

.4688 

227.3 

10fi.6 

83.70 

26.16 

X 

.0729 

35.37 

2.579 

2.025 

.0985 

H 

.4792 

232.4 

111.4 

87.46 

27.94 

15-16 

.0781 

37.89 

2.960  " 

2.325 

.1211 

% 

.4816 

237.5 

116.3 

91.31 

29.80 

1. 

.0833 

40.42 

3.368 

2.645 

.1470 

6. 

.5000 

242.5 

121.3 

95.23 

31.74 

1-16 

.0885 

42.94 

3.803 

2.986 

.1763 

H 

.51:08 

252.6 

131.6 

103.3 

35.88 

K 

.0938 

45.47 

4.263 

3.348 

.2093 

% 

.5417 

262.7 

142.3 

111.8 

40.36 

3-16 

.0990 

48.00 

4.750 

3.730 

.2461 

K 

.5625 

272.8 

153.5 

S0.5' 

45.19 

M 

.1042 

50.52 

5.263 

4.133 

.2870 

7. 

.58."3 

282.9 

165.0 

2f!.6 

50.40 

5*16 

.1094 

5305 

5.802 

4.557 

.3323 

g 

.6042 

293.0 

177.0 

3Ji.O 

56.00 

% 

.1146 

55.57 

6.368 

5.001 

.3820 

8 

.6-;:o 

303.1 

Ih9.5 

48.1 

62.00 

7-16 

.1198 

58.10 

6.9IJO 

5.466 

.4365 

x 

.64f,8 

313.2 

V02.3 

5«.8 

68.40 

X 

.1250 

60.63 

7.578 

5.952 

.4960 

8. 

.6fi07 

823.3 

215.6 

C9.3 

75.24 

9-16 

.1302 

63.15 

8.223 

6.458 

.5606 

M 

.6875 

333.4 

229.3 

80.1 

82.52 

N 

.1354 

65.68 

8.893 

6.985 

.6306 

/*> 

.70a3 

343.5 

243.4 

91.1 

90.25 

11-16 

.1406 

68.20 

9.591 

7.533 

.7062 

% 

.7292 

353.6 

247.9 

202.5 

98.45 

% 

.1458 

70.73 

10.31 

8.101 

.7876 

9. 

.7500 

363.8 

272.8 

214.3 

107.1 

13-16 

.1510 

73.26 

11.07 

8.690 

.8750 

M 

.7708 

373.9 

288.2 

226.3 

116.3 

y» 

.1563 

75.78 

11.84 

9.300 

.9688 

N 

.7917 

384.0 

304.0 

238.7 

126.0 

15-16 

.1615 

78.31 

12.64 

9.930 

1.069 

X 

.8125 

394.1 

320.2 

251.5 

136.2 

2. 

.1667 

80.83 

13.47 

10.58 

1.176 

10. 

.8333 

404.2 

336.8 

264.5 

146.9 

x 

.1771 

85.89 

15.21 

11.95 

1.410 

X 

.8542 

414.3 

353.9 

277.9 

158.2 

.1875 

90.94 

17.05 

13.39 

1.674 

H 

.8750 

424.4 

371.3 

291.6 

170.1 

*/ 

.1979 

9599 

19.00 

14.92 

1.969 

% 

.8958 

434.5 

389.2 

305.7 

182.6 

$4 

.2083 

101.0 

21.05 

16.53 

2.296 

11. 

.9167 

444.6 

407.5 

320.1 

195.6 

% 

.2188 

106.1 

23.21 

18.23 

2.658 

.9375 

454.7 

426.3 

334.8 

209/2 

H 

.2292 

111.2 

25.47 

20.01 

3.056 

M 

.9583 

464.8 

445.4 

349.8 

223.5 

% 

.2396 

116.2 

27.84 

21.87 

3492 

H 

.9792 

474.9 

465.0 

365.2 

238.4 

3. 

.2500 

121.3 

30.31 

23.81 

3.968 

12. 

1  Foot. 

485. 

485. 

380.9 

253.9 

To  find  the  weight  of  a  spherical  shell.    From  the  weight  of  a  ball 
which  has  the  outer  diam  of  the  shell,  take  the  wt  of  one  which  has  its  inner  diam. 

*  For  Copper,  add  l-7th  part.    Lead,  mult  by  1.47.    Brass,  mult  by  1.06. 

Zinc,  mult  by  ,9.    Tin,  mult  by  .95.    All  approximate. 


WEIGHT   OF   SHEET    METALS. 


367 


Weight  of  one  square  foot  of  Rolled,  or  Sheet  Iron,  Steel, 
Copper,  or  Brass  Plates.     (From  llaswell.) 


Thickness  by  the  Birmingham  gauge  for 
iron  wire,  sheet  iron,  and  steel. 

For  rolled  lead,  multiply  copper  by  1.3  ;  and 
for  zinc,  multiply  wrought  iron  by  the  deci- 
mal .9. 


Thickness  by  the  American  gauge. 

For  rolled  lead,  multiply  copper  by  1.3;  aud 
for  zinc,  multiply  wrought  irou  by  the  decimal  .9. 
Silver  is  >j  heavier  thau  steel. 


*l 

fco 

Thick- 
ness. 

Iron. 

Steel. 

Copper. 

Brass. 

o£ 

ii> 

Thick- 
ness. 

Iron. 

Steel. 

Copper. 

Brass. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Ins. 

LOB. 

Lbs. 

Lbs. 

Lbs. 

0000 

.454 

18.35 

18.54 

20.5662 

19.4312 

0000 

.46 

18.63 

18.87 

20.838 

19.688 

000 

.425 

17.18 

17.35 

19.2525 

18.19 

000 

.40964 

16.58 

16.80 

18.5567 

17.5328 

00 

.38 

15.36 

15.51 

17.214 

16.264 

00  .3648 

14.77 

14.96 

16.5254 

15.6134 

0 

.34 

13.74 

13.87 

15  402 

14.552 

0  .32486 

13.15 

13.32 

14.7162 

13.904 

1 

.3 

12.13 

12.25 

13.59 

12.84 

.2893 

11.70 

11.86 

13.1053 

12382 

2 

.284 

11.48 

11.59 

12.8652 

12.1552 

2 

.25763 

10.43 

1057 

11.6706 

11.0268 

3 

.259 

10.47 

10.57 

11.7327 

11.0852 

3 

.22942 

9.291 

9.415 

10.3927 

9.8192 

4 

.238 

9.619 

9.715 

10.7814 

10.1864 

4 

.20431 

8.273 

8.384 

9.2552 

8.7445 

5 

.22 

8.892 

8.981 

9.966 

9.416 

5 

.18194 

7.366 

7.462 

8.2419 

7.i87 

6 

.203 

8.205 

8.287 

9.1959 

8.6884 

6 

.16202 

6.561 

6.648 

7.3395 

6.9345 

7 

.18 

7.275 

7.348 

8154 

7.704 

7 

.14428 

5.842 

5.920 

6.5359 

6.1752 

8 

.165 

6.669 

6.736 

7.4745 

7.062 

8 

.12849 

5.203 

5.272 

5.8206 

5.4994 

9 

.148 

5.981 

6041 

6.7044 

6.3344 

9 

.11443 

4.633 

4.695 

5.1837 

4.8976 

10 

.134 

5.416 

5.470 

6.0702 

5.7352 

10 

.10189 

4.125  I  4.180 

4.6156 

4.3609 

11 

.12 

4.850 

4.899 

5.436 

5.136 

11 

.090742 

3.672  I   3.723 

4.1106 

3.8838 

12 

.109 

4.405 

4.449 

4.9377 

4.6652 

12 

.08080h 

3.272    3.315 

3.6606 

3.4586 

13 

.095 

3.840 

3.878 

4.3035 

4.066 

13 

.071961 

2.916 

2.952 

3.2598 

30799 

14 

.083 

3.355 

3.388 

3.7599 

3.5524 

14 

.064084 

2.592 

2.629 

2.903 

2  7428 

15 

.072 

2.910 

2.939 

3.2616 

3.0816 

15 

.05706* 

2.311 

2.341 

2.5852 

2.4425 

16 

.065 

2.627 

2.653 

2.9445 

2.782 

l(i 

.05082 

2.052 

2.085 

2.3021 

2.1751 

17 

.058 

2.344 

2.367 

2.6274 

2.4824 

17 

.045257 

1.825 

1857 

2.0501 

1.937 

18 

.049 

1.980 

1.999 

2.2197 

2.0972 

18 

.040303 

1.681 

1.653 

1.8257 

1.725 

19 

.042 

1.697 

1.714 

1.9026 

1.7976 

19 

.03589 

1.452 

1.468 

1.6258 

1.5361 

20 

.035 

1.415 

1.429 

1.5855 

1.498 

20 

.031961 

1.293 

1.311 

1.4478 

1.3679 

21 

.032 

1.293 

1.305 

1.4496 

1.369H 

21 

.028462 

1.152 

1.166 

1.2893 

1.2182 

22 

.028 

1.132 

1.143 

1.2684 

1.1984 

22 

.025847 

1.026 

1.040 

1.1482 

1.0849 

23 

.025 

1.010 

1.020 

1.1325 

1.07 

23 

.022571 

.913 

.9-25 

1.0225 

.96604 

24 

.022 

.8892 

.8981 

.9966 

.9413 

24 

.0201 

.814 

.824 

.91053 

.86028 

25 

.02 

.8083 

.8164 

.906 

.856 

25 

.0179 

.724 

.734 

.81087 

.76612 

26 

.018 

.7225 

.7348. 

.8154 

.7704 

26 

.01594 

.644 

.653 

.72208 

.68223 

27 

.016 

.6467 

.6532 

.7428 

.6848 

271.014195 

.574 

.582 

.64303 

.60755 

28  i  .014 

.5658 

.5715 

.6342 

.5992 

28 

.012641 

.511 

.518 

.57264 

.54103 

291  .013 

.5254 

.5307 

.5889 

.5564 

29 

.011257 

.455 

.471 

.50994   .4818 

30,  .012 

.4850 

.4899 

.5436 

.5136 

30 

.010025 

.405 

.410 

.45413   .42907 

31  i  .010 

.4042 

.4082 

.453 

.428 

311.008928 

.360 

.366 

.404441  .38212 

32   .009 

.3638 

.3674 

.4077 

.3852 

32L00795 

.321  '   .326 

.36014!  .3402B 

33  j  .008 

.3233 

.3265 

.3624 

.3424 

33 

.00708 

.286    .290 

.32072   .30302 

34   .007 

.2829 

.2857 

.3171 

.2996 

34 

.006304 

.254 

.258 

.28557 

.26981 

35 

.005 

.2021 

.2041 

.2265 

.214 

35 

.005614 

.226 

.230 

.25431 

.24028 

36 

.001 

.1617 

.1633 

.1812 

.1712 

36  .005 

.202 

.205 

.2265 

.214 

37  .004  453 

.180 

.182 

.20172 

.19059 

38  .003965 

.159 

.162 

.17961 

.1697 

39  .003531 

.142 

.144 

.15995 

.1511H 

40  .003144 

.127 

.128 

.14242 

.13456 

Approximate  prices  in  1873  for  copper  or  brass  sheets,  Nos  0  to  25,  from 
42  to  48  cts  per  fl>.  Ingots,  30  to  35  cts.  Castings,  35  to  45  cts.  Sheet  iron,  com- 
mon, Nos  10  to  16,  514 ;  Nos  16  to  25,  8  cts ;  best,  about  50  per  ct  more.  Sheet  steel, 
common,  8  to  10  cts  ;  best,  15  to  17  cts. 


The  first  two  cols  vary  considerably  in  different  books.    Ours  are  from  a  table  prepared  by  an  Kng- 
Hsh  maker  of  the  g;iuge«  themselves.     See  "  Gauge,"  in  Tomliuson's  Cyclopedia. 

No  trade  Stupidity  is  more  thoroughly  senseless  than  the  adherence  to  the  various 
Birmingham,  Lancashire,  &c,  gauges;  instead  of  at  once  denoting  the  thickness  and  diameter  of 
sheets,  wire,  &c,  by  the  parts  of  an  inch  ;  as  has  long  been  suggested.  Thus,  No.  ^,  or  No.  -^  wire, 
or  sheet-metal  of  any  kind,  should  be  understood  to  mean  ^  or  J^  of  an  inch  diaru,  or  thickness.  To 
avoid  mistakes,  which  are  very  apt  to  occur  from  the  number  of  gauges  in  use  :  and  from  the  absurd 
practice  of  applying  the  same  No.  to  different  thicknesses  of  different  metals,  in  different  towns,  it  is 
best  to  ignore  them  all ;  and  in  giving  orders,  to  define  the  diameter  of  wire,  and  the  thickness  of 
sheet-metal,  by  parts  of  an  inch.  Or  the  weight  per  hundred  ft  for  wire:  or  per  «q  ft  for  sheets. 
may  be  employed.  We  believe  that  the  foregoing  Birmingham  gauge  applies  to  zinc,  copper,  brass. 


368 


WEIGHT   OF   WIRE. 


and  lead;  although  It  ia  generally  stated  to  be  for  iron  and  steel  only.  Another  Birmingham  gauge 
Is  used  for  sheet  brass,  gold,  silver,  and  some  other  metals;  but  we  have  ueverseen  it  stated  what 
those  others  are.  There  are  different  gauges  even  for  wire  to  be  used  for  different  purposes;  and 
various  firms  have  gauges  of  their  own ;  not  even  according  among  themselves. 

The  American  gauge  differs  from  all  others,  and  is  equally  senseless.  We  have  Stubs  (England) 
gauge:  but  as  Mr.  Stubs  makes  various  English  gauges,  the  term  by  itselj  means  nothing.  Gene- 
rally, however,  in  our  machine  shops,  it  applies  to  the  Birmingham  gauge  of  the  preceding  table. 

Weig-ht  of  one  foot  in  length  of  Wire,  of  Iron,  Steel,  Copper, 
or  Brass.     (From  Haswell.) 


)iameters  by  the  Birmingham  gauge  for 
iron  >vire,  sheet  iron,  and  steel. 

Diameters  by  the  American  gauge. 

|    a 

Iron. 

Steel. 

Copper. 

Brass. 

»i 

1 

Iron. 

Steel. 

Copper. 

Brass. 

C5       5 

^o 

Q 

Ins. 

j 
Lbs.      j      Lbs.     { 

Lbs. 

Lbs. 

Ins. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

>00   .454 

.546207 

.55136 

.623913 

.589286 

)000 

.46 

.56074 

.56603 

.640513 

.605176 

K)0   .425 

.4786 

16 

.48317 

I 

.5 

46' 

52 

.516407 

000 

.40964 

.444683 

.448879 

.507946 

.479908 

00    .38 

.3826 

i 

.38627 

.4 

37C 

99 

.41284 

00 

.3648 

.352659 

.355986 

.40283 

.380666 

0   .34 

.3063 

I 

.30923 

.3 

491 

21 

.3305 

0 

.32486 

.279665 

.282303 

.319451 

.301816 

1    .3 

.2385 

.24075 

.2 

72- 

3 

.25731 

1 

.2893 

.221789 

.223891 

.253342 

.239353 

2    .284 

.2137 

58 

.21575 

> 

.2 

441 

46 

.230596 

2 

.25763 

.175888 

177548 

.200911 

.189818 

3    .259 

.1777 

35 

.17944 

I 

.2 

D3C 

54 

.19178)5 

3 

.22942 

.13948 

.140796 

.159323 

.150522 

4    .238 

.1501 

)7 

.15152 

J 

.1 

71 

61 

.161945 

4 

.20431 

.110616 

.11166 

126353 

.119376 

5    .22 

.1282 

.12947 

.1 

46: 

07 

.138376 

5 

.18194 

08772 

.088548 

.1002 

.094666 

6   .203 

.1092 

4 

.11023 

I 

.1 

241 

4 

.117817 

6 

.16202 

.069565 

.070221 

.079462 

.075075 

7    .18 

.0858 

.08666 

r 

.0 

J8C 

75 

.092632 

7 

.14428 

055165 

.055685 

.063013 

.059545 

8   .165 

.0721 

?6 

.07282 

r 

.0 

324 

1 

.077836 

8 

.12849 

.043751 

.044li»4 

.049976 

.047219 

9    .148 

.0580 

16 

.05859, 

.0 

03 

.062624 

9 

.11443 

.034699 

.035026 

.039636 

.037437 

10    .134 

.0475* 

S3 

.04803 

i 

.0 

->4; 

53 

.051336 

10 

.10189 

.027512 

.027772 

.031426 

.029687 

11    .12 

.038K 

.03852 

.0 

IK 

89 

04117 

11 

.090742 

.02182 

.022026 

.024924 

.023549 

12    .109 

.0314* 

5 

.03178 

> 

.0 

64 

.033968 

12 

.080808 

.017304 

.017468 

.019766 

.018676 

13    .095 

.0239 

6 

.02414 

j 

.0 

273 

19 

.025802 

13 

.071%! 

.013722 

.013851 

.015674 

.014809 

14    .083 

.0182, 

6 

.01842* 

$ 

.0 

20? 

53 

.019696 

14 

.064084 

010886 

.010989 

.012435 

.011746 

15     072 

.0137;. 

8 

.01386' 

.0 

I5f 

92 

.014821 

15 

.057061- 

008631 

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16    .065 

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6 

.01130} 

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27 

S9 

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17    .058 

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5 

.00899J 

.0 

01 

83 

.009618 

17 

.045257 

005427 

.005478 

.006199 

.005857 

18    .049 

.0083C 

3 

.00642; 

.0 

172 

68 

.006864 

18 

.04030: 

004304 

.004344 

.004916 

.004645 

19    .0*2 

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5 

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4 

.005043 

19 

.03589 

003413 

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20    .035 

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6 

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08 

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24    .022 

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14 

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25     02 

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12 

11 

.001144 

25 

.0179 

0008491 

.0008571 

.0009699 

.  00091  6S 

26   .018 

.00085 

86 

.00086S 

7   I.OC 

09 

307 

.0009263 

26 

.01594 

0006734 

.0006797 

.0007692 

.0007267 

27    .016 

.00067 

84 

.000684 

8   I.OC 

07 

749 

.0007319 

27 

.01419c 

000534      .0005391 

.0006099 

.0005763 

28    .014 

.00051 

94 

.000524 

3 

.01 

or. 

933 

.0005604 

28 

.012641 

0004235    .0004275 

.0004837 

.000457 

29    .013 

.00044 

79 

.000452 

1 

.Of 

i).') 

16 

.0004832 

29 

.011257 

0003358    .0003389 

.0003835 

.0003624 

30   .012 

.00038 

16 

.000385 

2 

.oc 

104 

J59 

.0004117 

30 

.010025 

0002663 

.0002688 

.0003042 

.0002874 

31    .01 

.00026 

5 

.000267 

5 

.00 

327 

.0002859 

31 

.008928 

0002113    .0002132 

.0002413 

.000228 

32    .009 

.00021 

47 

.000216 

7 

.01 

02 

152 

.0002316 

32 

.00795 

0001675    .0001691 

.0001913 

.0001808 

33    .008 

.00016 

96 

.000171 

2 

.01 

01 

W 

.000183 

33 

.00708 

0001328    .0001341 

.0001517 

.0001434 

34   .007 

.00012 

99 

.000131 

1 

.on 

01 

183 

.0001401 

34 

006304 

0001053   ;.  0001063 

.0001204 

.0001137 

35    .005 

.00006 

625 

.000066 

ss 

.00 

00 

f568 

.00007148 

35 

.005614 

00008366  .00008445  .0000956 

.00009015 

36   .004 

.00004 

24 

.000042 

8 

.00 

00 

1843 

.00004574 

36  .005 

00006625  .00006687  .0000757 

.0000715 

37 

.00445." 

00005255  .00005304  .00006003 

.00005671 

lirmingham  g>aus?e  for  sheet 

39  .'  003531 

00004166  .00004205  .00004758    uuuu**sfo 
00003305    00003336  .00003775  i  .00003566 

Brass,    i 

Silver, 

Go 

If 

1,   and  all 

40  .003144 

0000262   ;.C0002644i  .00002992;  .00002827 

metals  except  iron  and  steel? 

'                                    ' 

1 

Price  of  brass  and  copper  wire, 

approximate  for  1880.  Alison  ia  Brass 

£    ^ 

| 

1 

2    Zi 

1 

| 

«fc  Copper  €o.,  Nos  19  &  21  Cliff  St, 
N  YorK     For  100  ^^  nv  rnf^v^ 

Inch 
.004     ' 
.005     8 

Inch 
.015 
.016 

13 
14 

Inch 
.036   19 
.041   20 

Inc 
.06 
.06 

•25 
26 

Inch 
.095 
.103 

j  Inch 
31    .133 
32   .143 

Nos  0  to  25  Coppe 
"    "  "   "  Highl 
"    "  "   "  Low  b 

r                  46  ct< 

>  per  ft). 

arass  36  " 
rass  40  " 

.008     J 

.019 

15 

.047    '21 

.07 

11 

.113 

33    .145 

.010    1C 

.024 

16 

.051    22 

.07 

'K 

.120 

34    .148 

.012    1] 

.029 

17 

.057   23 

.077 

29 

.124 

35   .158 

.013    IV 

.034 

18 

.061    24 

.082 

30 

.126 

36  .167 

IRON    WIRE. 


369 


Table  of  Iron  wire  made  by  the  Trenton  Iron  Co;  Trenton,  N.  J. 
Charles  Hewitt  Esq,  President. 

The  numbers  of  the  wires  in  the  first  column  are  those  of  the  Trenton  Iron 
Co ;  and  correspond  to  somewhat  smaller  diaius  than  those  ot  the  Birmingham  gauge. 


No.  by 
Tr.  Ir. 
CosG. 

Diam. 
las. 

Area. 
Sq.  ius. 

No.  to  a 
sq.  in.  of 
metal. 

Wt.  of 
100  yds. 
Lbs. 

Breakg 
pull. 

Lbs. 

No.  by 
Tr.Ir. 
CosG. 

Diam. 

lus. 

Area. 
Sq.  ins. 

No.  to  a 
sq.in.of 
metal. 

Wt.  of 
100yds. 
Lbs. 

Breakg 
pull. 
Lbs. 

0 

.305 

.07306 

13.69 

73.96 

5926 

0 

.130 

.01327 

75.36 

13.43 

1233 

1 

.285 

.06379 

15.68 

64.58 

5226 

1 

.1175 

.01084 

92.25 

10.97 

1010 

2 

.265 

.05515 

18.13 

55.83 

4570 

2 

.105 

.00866 

115.5 

8.767 

810 

3 

.245 

.04714 

21.21 

47.72 

3948 

3 

.0925 

.00672 

148.8 

6.804 

631 

4 

.225 

.03976 

25.15 

40.25 

3374 

4 

.080 

.00503 

198.8 

5.092 

474 

5 

.205 

.03301 

30.29 

33.42 

2839 

0 

.070 

.00385 

259.7 

3.897 

372 

6 

.190 

.02835 

35.27 

28.70 

2476 

6 

.061 

.00292 

342.5 

2.956 

292 

7 

.175 

.02405 

41.58 

24.35 

2136 

7 

.0525 

.00216 

462.9 

2.187 

222 

8 

.160 

.02011 

49.73 

20.36 

1813 

8 

.045 

.00159 

629.0 

1.610 

169 

S 

.145 

.01651 

60.57 

16.71 

1507 

20 

.033 

.00086 

1162.8 

.870 

96 

The  wire  in  this  table  is  supposed  to  be  hard,  bright,  or  unannealed. 
Annealing  renders  wire  more  pliable,  but  less  elastic;  and  reduces  its  strength  per- 
haps 20  or  2o  per  cent.  The  strength**  in  the  last  column  are  at  the 

rate  of  81000  ft>s  per  sq  inch  for  No.  0 ;  9WOO  for  No  10;  and  111000  for  No.  20:  and 
are  say  from  15  to  *25  per  ct  greater  than  those  of  ordinary  market  wire.  The  wires 
in  the  table  are  all  made  of  the  same  specially  prepared  iron  ;  and  the  increased 
strength  per  sq  in  of  the  smaller  ones  is  due  to  their  more  frequent  passage  through 
the  draw-plates.  The  choice  Swedish  and  Norway  irons  would  not  yield  wire  of 
much  greater  strength  ;  while  the  cost  would  be  about  60  per  ct  more. 

Hard  steel  wire  averages  about  twice  the  strength  of  iron  wire. 

Unannealed  or  hard  brass  wire  has  about  ^ths  the  strengths  of  the  above 
table,  and  about  ^  more  weight.  If  annealed,  only  full  half  the  strength. 

Hard  copper  wire  may  be  taken  at  %  of  the  tabular  strengths,  and  full 
•^  more  weight. 

To  find  approximately  the  number  of  straight  wires  that 
can  be  got  into  a  cable  of  given  diameter. 

Divide  the  diameter  of  the  cable  in  inches,  by  the  diameter  of  a  wire  in  inches. 
Square  the  quotient.  Multiply  said  square  by  the  decimal  .77.  The  result  will  be 
correct  within  about  4  or  5  per  cent  at  most,  in  u  cylindrical  cable. 

The  solidity,  or  metal  area  of  all  the  wires  in  a  cable,,  will  be 
to  the  area  of  the  cable  itself,  about  as  1  to  1.3.  In  other  words,  the  area  of  the 
voids  is  nearly  %  that  of  the  cable;  while  that  of  the  wires  is  fully  %  that  of  the 
cable.  All  approximate. 


Prices  approx  for  1880. 


I  ron  W  i  re  of  quality  of  the  above  table. 

Best  Cast  Steel  wire. 


Number  of  wire  by  Trenton  Iron  Co's  gauge. 
0  to  9     |  10  to  13  |  14  to  17   |   18  to  20  |      23      |      28 


Approximate  price  iu  cents  per  pound. 

8         I      10         I      12^      I      14      I      16 
I      21J4      I      26^      I      30}$      I     60     | 


Tinned  wire  is  about  one  third  higher  price. 

The  prices  of  annealed  and  unannealed  wire  are  the  same. 

The  Co  make  other  wire  of  specified  quality  to  order. 

BUCKLED  PLATES 

of  plate  iron  are  usually  3  or  4  ft  square,  from  1-20  to  38  ins  thick,  with  a  flat  rim  about  2  ins  wide 
all  around,  with  rivet  or  bolt  holes  for  holding  the  plate  firmly  down  to  its  intended  place.  The  rest 
of  the  plate  is  stamped  into  the  form  of  a  kind  of  groined  arch  rising  from  1  to  2  ins  in  the  center. 
They  are  verv  strong,  and  are  used  for  the  floors  of  tire-proof  buildings,  and  of  city  iron  bridges, 
covered  with  asphalt  or  stone  paving.  &c.  One  of  3  ft  sq,  .25  inch  thick,  c  .rved,  1.75  ins.  nnd  with 
a  2  inch  rim  well  bolted  down  on  all  sides,  required  a  quiet,  equally  distributed  load  of  18  tons  to  crush 
it.  When  unbolted  the  strength  is  only  half  as  great. 

Table  of  safe,  quiet,  uniformly  distributed  loads  (.25  of  the 

ultimate  ones)  for  buckled  iron  plates  3  ft  square,  arched  1.75  ins,  and  well  bolted  down  on  all  sides. 
Also  approx  wt  iu  Ibs  per  sq  yd. 


B.W.G. 

lus. 

Wt. 

Tons. 

H>s. 

Ins. 

Wt. 

Tons 

fts. 

Ins. 

Wt.  Tons. 

fts. 

18 

.018 

17 

.27 

604 

X 

45 

1.0 

2240 

5-16 

113 

6.2 

13b88 

16 

.066 

24 

.43 

96  i 

3-16 

68 

2.5 

5600 

% 

135 

9.0 

20160 

12   !  .107 

39 

.64 

1433 

y* 

90 

4.5 

10080 

Soft  puddled  steel  will  bear  nearly  twice  as  much. 


370 


GALVANIZED    IRON. 


Weights  in  IDS  per  sq  ft,  of  galvanized  sheet  iron,  both  flat 
and  corrugated.  The  Nos  and  thicknesses  are  those  of  the  iron  before  it  is 
galvanized ;  but  the  weights  refer  to  'the  galvd  sheets.  When  a  flat  sheet  (the  ordi- 
nary size  of  which  is  from  '2  to  "2%  ft  in  width,  by  6  to  8  ft  in  length)  is  converted 
into  a  corrugated  one,  with  corrugations  5  ins  wide  from  center  to  center,  and  about 
an  inch  deep  (the  common  sizes,)  its  width  is  thereby  reduced  about  y^th  part,  or 
from  30  to  27  ins;  and  consequently  the  weight  per  sq  ft  of  area  covered  is  increased 
about  ^th  part.  When  the  corrugated  sheets  are  laid  upon  a  roof,  the  overlapping 
of  about  2%  ins  along  their  sides,  and  of  4  ins  aiorg  their  ends,  diminishes  the 
covered  area  about  ^th  part  more ;  making  their  weight  per  sq  ft  of  roof  about  %th 
part  greater  than  before.  Or  the  weight  of  corrugated  iron  per  sq  ft,  in  place  on  a 
roof,  is  about  ^  greater  than  that  of  the  flat  sheets  of  above  sizes  of  which  it  is  made. 

GALVANIZED  IRON. 

Weights  per  square  foot. 


No.  by 
Birming- 
ham wire 

Thick- 
ness in 
inches. 

Flat. 
Lbs. 

Corru- 
gated. 
Lbs. 

Corrug 
on  roof. 
Lbs. 

No.  by 
Birming- 
ham wire 

Thick- 
ness in 
inches. 

Flat. 
Lbs. 

Corru- 
gated. 
Lbs. 

Corrug. 
on  roof. 
Lbs. 

gauge. 

gauge. 

30 

.012 

.806 

.896 

1.08 

21 

.032 

l.«8 

1.81 

2.17 

29 

.013 

.857 

.952 

1.14 

20 

.035 

1.75 

1.94 

2.33 

28 

.014 

.897 

.997 

1.20 

19 

.042 

2.03 

2.26 

2.71 

27 

.016 

.978 

1.09 

1.30 

18 

.049 

2.32 

2.58 

3.09 

26 

.018 

1.06 

1.18 

1.41 

17 

.058 

2.68 

2.98 

3.57 

25 

.020 

1.14 

1.27 

152 

16 

.065 

2.96 

3.29 

3.95 

24 

.022 

1.22 

1.36 

1.62 

15 

.072 

3.25 

3.61 

4.33 

23 

.025 

134 

1.49 

1.79 

14 

.083 

3.69 

4.10 

4.92 

22 

.028 

1.46 

1.62 

1.95 

13 

.095 

418 

4.64 

5.57 

N 

os.  18  to  23  are  commonly  used  for  covering  roofs. 

The  galvanizing  is  simply  a  thin  film  of  zinc  on  both  sides  of  the 

sheet,  as  in  what  is  known  as  "  tinned  plates,"  or  "  tin  ;  "  which  are  in  reality  sheet  iron  similarly 
coated  with  tin.  Zinc,  like  tin,  resists  corrosion  from  ordinary  atmospheric  influences,  much  better 
than  iron  ;  and  hence  the  use  of  these  metals  as  a  protection  to  the  iron.  A  well  galvanized  roof, 
of  a  good  pitch,  will  suffer  but  little  from  5  to  6  years'  exposure  without  being  painted.  It  will  then 
take  paint  readily,  and  should  be  painted.  It  is  better,  however,  always  to  paint  tin  ones  at  once. 

Paint  does  not  adhere  well  to  new  zinc,  and  this  is  the  principal 

reason  why  new  galvanized  roofs  are  not  painted:  but  this  may  be  remedied  by  first  brushing  the 
zinc  over  with  the  following:  One  part  of  chloride  of  copper,  1  part  nitrate  of  copper,  1  part  of  sal- 
ammoniac.  Dissolve  in  64  parts  of  water.  Then  add  1  part  of  commercial  hydrochloric  acid.  When 
brushed  with  this  solution,  the  zinc  turns  black  ;  dries  within  12  to  24  hours,  and  may  then  be  painted. 
Paint  of  some  mineral  oxide  of  a  brown  color  is  generally  used :  one  coat  being  applied  to  both 
sides  in  the  shop  ;  and  the  other  after  being  put  on  the  roof.  Repainting  every  3  or  4  years  will  suffice 
afterward.  Ungalvanized  iron  (called  BLACK  IRON,  for  distinction)  is  also  very  enduring  for  roofs,  if 
well  painted  every  1  or  2  years.  The  chief  advantage  of  galvanized  roofing  is  that  it  does  not  require 
painting  so  often  as  the  black.  The  galvanizing  adds  about  %  of  a  ft  per  square  foot  of  surface,  or 
about  %  ft  per  sq  ft  of  sheet  as  coated  on  both  sides:  without  regard  to  the  thickness  of  the  sheet. 
On  this  principle  the  above  table  has  been  prepared.  Those  in  text- books  generally,  are  incorrect  as 
regards  the  thinner  sheets.  Paint  for  roofs  should  not  have  much  dryer.  See  Painting,  p  512. 

The   sulphurous  fumes  from  coal  are  very  corrosive  of 

EITHER  OALVANIZED  OR  BLACK  IRON  ;  as  may  be  seen  in  shops,  railroad  bridges,  or  engine  houses, 
roofed  with  either;  if  efficient  means  are  not  provided  for  carrying  off  the  smoke:  and  the  same  with 
other  metals.  THB  ACID  OP  OAK  TIMBER  is  said  to  destroy  the  zinc  of  galvanized  iron.  See  Tin  and 
Zinc.  Flat  iron  is  usually  nailed  upon  a  sheeting  of  boards  ;  but  the  strength  of  corrugated  iron 
obviates  the  necessity  for  this,  and  enables  it  to  stretch  5  or  6  ft  from  purlin  to  purlin,  without  inter- 
mediate  support.  The  corrugated  sheets  are  riveted  together  on  the  roof,  by  rivets  of  galvanized 
wire  about  X  inch  thick,  300  to  a  pound,  well  driven  (so  as  to  exclude  rain)' at  intervals  of  3  or  4 
inches,  all  around  the  edges.  The  rivet-holes  are  first  punched  by  machinery,  so  as  to  insure  coinci- 
dence in  the  several  sheets:  and  the  rivets  are  driven  hy  two  men,  one  above,  and  one  beneath  the 
roof.  For  black  iron,  ungalvanized  nails,  boiled  in  linseed  oil  as  a  partial  preservative  from  rust,  are 
commonly  used;  as  also  in  shingling  or  slating.  Galvanized  ones,  however,  would  be  better  in  nil 
the*e  cases;  or  even  copper  ones  for  slating  because  good  slate  endures  much  longer  than  either 
shingles  or  iron,  and  therefore  it  becomes  true  economy  to  use  durable  metals  for  fastening  It.  In 
none  of  these  cases,  however,  are  the  nails  fully  exposed  to  the  weather. 

The  sheets  of  flat  iron  are  put  together  by  overlapping  and 

FOLDING  THE  KDOE8.  much  the  same  as  shown  by  the  fig  page  378,  head  Tin  :  the  joints  which  run 
up  and  down  the  roof  being  the  same  as  at  s  a,  and  the  horizontal  ones  as  at  t  t; 
except  that  inasmuch  as  these  are  not  soldered  in  the  iron  sheets,  the  joint  is  made 
about  5i  to  1  inch  wide,  instead  of  %  inch,  the  better  to  provide  against  leaking. 
Cleats  are  used  as  in  tin,  with  2  nails  to  a  cleat.  The  iron  plates  are  best  laid  on 
sheeting  boards ;  but  in  sheds,  &c,  are  sometimes  laid  directly  on  rafters,  not  more 
thao  about  18  ins  apart  in  the  plear ;  the  plates  being  allowed  to  sag  a  little  between 


CORRUGATED    IRON. 


371 


Ihe  rafters,  so  as  to  form  shallow  gutters.  In  such  cases  it  is  well  to  bevel  off  the  tops  of  the  rafters 
•lightly,  as  in  this  fig. 

A  serious  objection  to  iron  as  a  roof  covering,  is  its  rapid  con- 
densation of  Atmospheric  moisture ;  which  falls  from  the  irou  in  drops  like  rain,  and  may  do  injury 
to  ceilings,  Hours,  or  articles  in  the  apartments  immediately  beneath  the  roof.  Painting  does  not 
appreciably  diminish  this ;  it  may,  however,  be  obviated  by  plastering,  as  shown  at  11,  of  Figs  21}^, 
of  Trusses,  page  268.  Corrugated  iron  uiakea  an  excellent  permanent  street  or  other  awning. 

Pri€e  Of  Corrugated  iron  in  ISSO,  7  to  9  cts  per  ft  ;  or  12  to  U  cts  if  galvanized, 
for  Noa.  14  to  26.  Does  not  require  sheets  of  best  iron,  lor  roofs. 

CORRIJOATED  IRON. 

Experiments  by  the  writer,  on  the  strength  of  corrugated 

iron. 
First.     A  sheet  d  d,  of  No.  16  iron,  (— -  j 

(about  y1^  inch  thick,)  27  ins  wide,  by  4  ft  long, 
with  live  complete  «  orrugations  of  5  ins  by  1  inch, 
was  laid  on  supports  o  ft  9  ins  apart.  A  block  of 
wood  c,  9  ins  wide,  by  7  ins  thick,  and  30  ins  long, 
was  placed  across  the  center,  and  gradually  loaded 
with  castings  weighing  ItiuO  Ibs. 

This  caused  a  deflection  at  the  center  of  precisely  J$  an 
inch.  On  the  removal  of  the  load  after  an  hour,  no  perma- 
nent set  was  appreciable.  The  severity  of  the  test  was  pur- 
posely increased  by  applying  the  several  castings  very 
roughly,  jolting  the  whole  as  much  as  possible.*  The  sus- 
pended area  of  the  sheet  was  8.44  sq  ft ;  and  since  the  actual  center  load  of  1600  Its  is  about  equiva- 
lent to  3000  fts  equally  distributed,  it  amounts  to  — —  =  355  fts  per  sq  ft  distributed.  But  3000  Ibs 

distributed  would  produce  a  deflection  of  but  about  full  %  of  an  inch.  Again,  355  Ibs  per  sq  ft 
is  about  4  times  the  weight  of  the  greatest  crowd  that  could  well  congregate  upon  a  floor.  Conse- 
quently tliis  iron,  at  3'  9"  span,  is  safe  in  practice  for  any  ordinary  crowd.  Moreover,  such  a  crowd 
would  produce  a  center  deflection  of  only  the  J^th  part  of  J£  of  an  inch  ;  or  yV  of  an  inch  ;  or  TT^-K 
of  the  clear  span  ;  which  is  but  two-thirds  of  Tredgold's  limit  of  -T^-Q  of  the  span.  See  Art  26  of 
Strength  of  Materials,  p  196. 

In  one  experiment  the  ends  of  the  sheets  rested  upon  supports  dressed  so  as  to  present  undulations 
corresponding  tolerably  closely  with  the  shape  of  the  corrugations;  but  in  the  other  the  supports 
were  flat,  and  each  end  of  the  sheet  rested  only  upon  the  lower  points  of  the  corrugations.  No  ap- 
preciable difference  was  observed  in  the  results 

Second.  An  arch  of  tfo.  18  (^ 
inch)  iron,  corrugated  like  the  foregoing, 
but  the  depth  of  corrugation  increased  to 
1*4  ins  by  the  process  of  arching  the  sheet ; 
clear  span  6  ft  1  inch  ;  rise  10  ins ;  breadth  27 
ins.  (of  which,  however,  only  25  ins  bore 
against  the  abutments  ) 

Each  foot  o  of  the  arch  abutted  upon  a  casting  .7', 
the  inner  portion  t  of  which  was  undulated  on  top,  to 
correspond  with  the  corrugations  of  the  arch,  which 
rested  upon  it.  At  j/.  (one-fourth  of  the  span.)  two 
wooden  blocks  were  placed,  occupying  a  width  of  9 
inches,  and  extending  across  the  arch  :  on  them  was 
piled  a  load.  1.  of  castings,  to  the  extent  of  4480  Ibs, 
or  2  tons.  Under  this  load  the  arch  descended  about 
H  an  inch  at  y,  becoming  flatter  along  that  side,  and 

slightly  more  curved  upward  along  the  unloaded  side  n.  Two  similar  blocks  were  then  placed  at  n, 
and  two  tons  of  load,  a,  were  piled  upon  them,  in  addition  to  the  2  tons  at  I ;  making  a  total  of  8960 
fts.or4tons.  Thi«  brought  the  arnh  more  nearly  back  to  its  original  shape;  but  still  slightly 
straightened  at  both  n  and  y.  and  a  little  more  curved  in  the  center.  The  load  was  then  increased  to 
10000  fts.  and  left  standing  for  several  days.  Two  iron  ties,  each  %  by  l?i,  which  were  used  for  pre- 
venting the  abutment  castings  j  from  spreading,  were  found  to  have  stretched  nearly  %  of  an  inch. 
Additional  ones  were  inserted,  and  the  load  increased  to  a  total  of  6  tons,  or  134-10  Ibs;  parts  of  it  on 
«  and  1.  and  part  in  the  shape  of  long  broad  bars  of  iron  at  the  center  of  the  arch,  below  the  loads  * 
and  I.  and  betwpen  n  and  y.  So  far  as  could  be  judged  by  eye.  the  shape  of  the  arch  was  now  almost 
perfect.  The  loads  s  and  \  did  not  touch  each  other.  After  standing  more  than  a  week,  the  load 
was  accidentally  overturned,  crippling  the  arch.  The  load  was  equal  to  about  1000  fts  per  sq  ft  of 

the  arch.  Such  arches  have  since  come  into  common  use  instead  of  brick,  for 
fireproof  floors. 

Curved  roofs  of  25  to  3O  ft  span,  rising  about  i^  span,  may  be  made 
of  ordinary  corrugated  iron  of  Nos  16  to  13,  riveted  as  usual ;  and  having  no  acces- 
sories except  tie-rods  a  few  feet  apart;  continuous  angle-iron  skewbacks;  and  thin 
vertical  rods  to  prevent  the  ties  from  sagging. 

»  Without,  however,  allowing  the  deflection  to  exoeed  the  ^  Inch ;  which  was  effected  by  means  of 
»  stop  under  the  sheet. 


372 


WEIGHT    OF    FLAT    IRON. 


Weight  of  1  ft  in  length  of  FLAT  ROLLED  IKON,  at  4SO  Ibs  per 
cubic  foot.  For  cast  iron,  deduct  TV  part;  for  steel,  add  ^g-;  for  copper,  add 
j\  for  cast  brass,  add  j1^;  for  lead,  add  ]^;  for  zinc,  deduct  -j^. 


-  a 

THICKNESS  IN  INCHES. 

£.2 

1-16 

H 

3-16    !      U 

5-16 

H 

7-16 

«      !      K 

H 

K 

1 

1. 

.2033 

.4166 

.6250 

.8333 

1.012 

1.250 

1.458 

1.666 

2.083 

2.500 

2.916 

3.333 

% 

.2344 

.4888 

.7033 

.9375 

1.172 

1.406 

1.640 

1.875 

2.344 

2.812 

3.280 

3.75 

y\ 

.2603 

.5210 

.7810 

1.042 

1  303 

1.563 

1.823 

2.083 

2605 

3.125 

3.646 

4.166 

X 

.2335 

.5730 

.8595 

1.146 

1.432 

1.719 

2.006 

2.292 

2.864 

3.438 

4.012 

4.583 

% 

.3125 

.6250 

.9375 

1.250 

1.562 

1.875 

2.188 

2.500 

3.125 

3.750 

4.375 

5.000 

% 

.3385 

.6771 

1.015 

1.354 

1.692 

2.031 

2.370 

2.708 

3.384 

4.062 

4.740 

5.416 

H 

Mi* 

.7292 

1.094 

1.458 

1.823 

2.188 

2.550 

2.916 

3.646 

4.375      5.105 

5.»33 

% 

3906 

.7812 

1.172 

1.562 

1.953 

2.344 

2.735 

3.125 

3.906 

4.688      5.470 

625 

2. 

.4136 

.8333 

1.25 

1.666 

2.083 

2.500 

2.916 

3.333 

4.166 

5.000 

5.833 

6.666 

X 

.4427 

.8855 

1.328 

1.771 

2.214 

2.6,>6 

3.098 

3.542 

4.428 

5.312 

6.196 

7.083 

.4888 

.9375 

1.406 

1.875 

2.344 

2.812 

3.281 

3.750 

4.688 

5.624 

6.562 

7.500 

% 

Ml* 

.9895 

1.484 

1.979 

2.474 

2.968 

3.463 

3.958 

4.948 

5.936 

6.926 

7.916 

X 

.5210 

1.012 

1.562 

2.083 

2.605 

3.125 

3.646 

4.166 

5.210 

6.250 

7.291 

8.333 

% 

.5470 

1.094 

1641 

2.187 

2.735 

3.282 

3.829 

4.375 

5.470 

6.564 

7.658 

8.750 

% 

.5730 

1.146 

1.719 

2.292 

2.865 

3.438 

4.011 

4.583 

5.730 

6.876 

8.022 

9.166 

% 

.5390 

1.198 

1.797 

2.396 

2.995 

3.594 

4.193 

4.792 

5.990 

7.188 

8.386 

9.583 

3. 

.625 

1.250 

1.875 

2500 

3.125 

3.750 

4.375 

5.000 

6.250 

7.500 

8.750 

10.00 

M 

.6515 

1.303 

1954 

2.605 

3.257 

3.908 

4.560 

5.210 

6.514 

7.816 

9.120 

10.42 

y* 

.6770 

1.354 

2.031 

2.7')8 

3.385 

4.062 

4.739 

5.416 

6.770 

8.124 

9.478 

10.83 

% 

.7031 

1.406 

2.109 

2.812 

3.516 

4.218 

4.921 

5.625 

7.032 

8.4o6 

9.842 

11.25 

x 

.7291 

1.458 

2.188 

2.916 

3.646 

4.375 

5.105 

5.833 

7.291 

8.750 

10.21 

11.66 

X 

.7555 

1.511 

2.266 

3.021 

3.777 

4.533 

5.288 

6.042 

7.554 

9.066 

10.58 

12.08 

% 

.7812 

1.562 

2.343 

3.125 

3.906 

4.686 

5.468 

6.25 

7.812 

9.372 

10.94 

12.50 

% 

.8070 

1.614 

2.421 

3.229 

4.035 

4.842 

5.65 

6.458 

8.070 

9.684 

11.30 

12.92 

A. 

.8333 

1.666 

2.500 

3.333 

4.166 

5.000 

5.833 

6.666 

8.333 

10.00 

11.66 

13.23 

x 

.8595 

1.719 

2.578 

3.438 

4.297 

5.156 

6.016 

6.875  j     8.594 

10.31 

12.03 

13.75 

H 

.8355 

1.771 

2.656 

3.542 

4.427 

5.312 

6.198 

7.083       8.854 

10.62 

12.40 

14.16 

% 

.9115 

1.823 

2.734 

3.616 

4.557 

5.468 

6.380 

7.291 

9.114 

10.94 

12.76 

14.58 

X 

.9375 

1.875 

2.812 

3.750 

4.687 

5.624 

6.562 

7.500 

9.374 

11.25 

13.12 

15.00 

X 

.9336 

1.927 

2.891 

3.854 

4.818 

5.782 

6.745 

7.708 

9.636 

11.56 

13.49 

15.42 

H 

.9895 

1.979 

2.968 

3.958 

4.917 

5.936 

6.926 

7.917 

9.894 

11.87 

13.85 

15.83 

X 

1.016 

2.031 

3.048 

4062 

5.080 

6.096 

7.112 

8.125 

10.16 

12.19 

14.22 

16.25 

6. 

1.0*2 

2.083 

3.125 

4.166 

5.210 

6.25 

7.291 

8333 

10.42 

12.50 

14.58 

16.66 

X 

1.088 

2.136 

3.204 

4.271 

5.340 

6408 

7.476 

8.542 

10.68 

12.81 

14.95 

17.08 

X 

1.094 

2.188 

3.282 

4.375 

5.470 

6.564 

7.658 

8.750 

10.94 

13.13 

15.31 

17.50 

K 

1.120 

2.240 

3.360 

4.479 

5.600 

6.720 

7.840 

8.958 

11.20 

13.44 

15.68 

17.92 

X 

1.146 

2.292 

3.438 

4.584 

5.730 

6.876 

8.022 

9.167 

11.46 

13.75 

16.04 

18.33 

% 

1.172 

2.344 

3.516 

4.687 

5.860 

7.032 

8.204 

9.375 

11.72 

14.06 

16.40 

18.75 

H 

1.198 

2396 

3.594 

4.791 

5.990 

7.188 

8.386 

9.583 

11.98 

14.37 

16.77 

19.16 

X 

1.224 

2.448 

3.672 

4.896 

6.120 

7.344 

8.568 

9.792 

12.24 

14.69 

17.13 

19.58 

6. 

1.250 

2.500 

3.750 

5.000 

6.250 

7.500 

8.750 

10.00 

12.50 

15.00 

17.50 

20.00 

H 

1.276 

2.552 

3.828 

5.104 

6.380 

7.656 

8.932 

10.21 

]2.76 

15.31 

17.86 

20.42 

X 

1.302 

2.604 

3.906 

5.208 

6.510 

7.812 

9.114 

10.42 

13.02 

15.62 

18.23 

20.83 

% 

1.328 

2.657 

3.984 

5.313 

6  610 

7.968 

9.297 

10.63 

13.28 

15.93 

18.59 

21.25 

« 

1.354 

2.708 

4.063 

5.417 

6.770 

8.126 

9.480 

10.83 

13.54 

16.25 

18.96 

21.66 

% 

1.381 

2.761 

4.143 

5521 

6.906 

8.286 

9.668 

11.04 

13.81 

16.57 

19.33 

22.08 

H 

1.406 

2.813 

4.218 

5.625 

7.030 

8.436 

9.843 

11.25 

14.06 

16.87 

19.69 

22.50 

% 

1.432 

2.864 

4.296 

5.729 

7.160 

8.592 

10.02 

11.46 

14.32 

17.18 

20.04 

22.92 

1. 

1.458 

2.916 

4.375 

5.833 

7.291 

8.750 

10.20 

11.66 

14.58 

17.50 

20.42 

23.33 

X 

1.484 

2.939 

4.452 

5.938 

7.420 

8.904 

10.39 

11.87 

14.84 

17.81 

20.78 

23.75 

M 

1.511 

3.021 

4.533 

6.042 

7.555 

9.066 

10.58 

12.08 

15.11 

18.13 

21.16 

24.16 

% 

1.536 

3.073 

4.608 

6.146 

7.680 

9.216 

10.75 

12.29 

15.36 

18.43 

21.50 

24.58 

H 

1.562 

3.125 

4.686 

6.250 

7.810 

9.372 

10.93 

1250 

15.62 

18.74 

21.86 

25.00 

H 

1.588 

3.177 

4.764 

6.354 

7.940 

9.528 

11.12 

12.71 

15.88 

19.05 

22.24 

25.42 

H 

1.615 

3.229 

4.845 

6.458 

8.075 

9.690 

11.31 

12.92 

16.15 

19.38 

22.62 

25.83 

x 

1.641 

3.281 

4.923 

6.562 

8.205 

9.846 

11.48 

13.13 

16.41 

19.69 

22.96 

26.25 

8. 

1.666 

3.333 

5.000 

6.666 

8.333 

10.00 

11.66 

13.33 

16.66 

20.00 

23.33 

26.66 

H 

1.693 

3.386 

5.079 

6.771 

8.455 

10.15 

11.85 

13.54 

6.91 

20.30 

23.70 

27.08 

1.719 

3.438 

5.157 

6.875 

8.595 

10.31 

12.03 

13.75 

7.19 

20.61 

24.06 

27.50 

% 

1.745 

3.489 

5.235 

6.979 

8.725 

10.47 

12.21 

13.96 

7.45 

20.94 

24.42 

27.92 

X 

1.771 

3.542 

5.313 

7.083 

8.855 

10.63 

12.40 

14.17 

7.71 

21.26 

24.80 

28.33 

1.797 

3.594 

5.391 

7.188 

8.985 

10.78 

12.58 

14.37 

7.97 

21.56 

25.16 

28.75 

$i 

1.823 

3.646 

5.469 

7.292 

9.115 

10.94 

12.76 

14.58 

8.23 

21.88 

25.52 

29.17 

% 

1.849 

3.698 

5.547 

7.3% 

9.245 

11.09 

12.94 

14.79 

8.49 

22.18 

25.88 

29.58 

9. 

1.875 

3.750 

5.625 

7.500 

9.375 

11.25 

13.12 

15.00 

8.75 

22.50 

26.24 

30.00 

X 

1.901 

3.802 

5.703 

7.604 

9.505 

11.41 

13.31 

15.21 

9.00 

22.81 

26.62 

30.42 

N 

1.927 

3.&H 

5.781 

7.708 

9.635 

11.56 

13.49 

15.42 

9.27 

23.12 

26.98 

30.83 

% 

1.953 

3.906 

5.859 

7.812 

9.765 

11.72 

13.67 

15.62 

9.53 

23.44 

27.34 

31.25 

^ 

1.979 

.958 

5.937 

7.916 

9.895 

11.87 

13.85 

15.84 

9.79 

23.74 

27.70 

31.67 

H 

2.005 

.010 

6.015 

8.021 

10.02 

12.03 

14.04 

16.04 

20.04 

24.06 

28.08 

32.08 

?* 

2.031 

.062 

6.093 

8.125 

10.16 

12.18 

14.21 

16.25 

20.32 

24.36 

28.42 

32.50 

M 

2.057 

.114 

6.171 

8.229 

10.29 

12.34 

14.40 

16.46 

20.58 

24.68 

28.80 

32.92 

LO. 

2.083 

.166 

6.250 

8.333 

10.41 

12.50 

14.58 

16.66 

20.82 

25.00 

29.16 

33.33 

H 

2.109 

4.219 

6.327 

8.438 

10.55 

12.65 

14.76 

16.87 

21.10 

25.30 

29.52 

33.75 

M 

2.135 

4.270 

6.405 

8.541 

10.67 

12.81 

14.94 

17.08 

21.34 

25.62 

29.88 

34.17 

WEIGHT    OF    ANGLE    AND   T   IRON. 


373 


Weight  of  1  ft  in  length  of  FLAT  ROLLED  IROX,  at  480  IDS 
per  cubic  foot  —  (Continued.) 


A  A 

•S  2 
£.2 

1-16 

« 

3-16 

T 

H 

HICK 

5-16 

NESS 
% 

IN  INCHES 

7-16    j      fc 

N 

H 

% 

j 

10% 

2.162 

.323 

6.486 

8.646 

10.81 

12.97 

15.13  1    17.29 

2  .62 

25.94 

30.26 

34.58 

y* 

2.188 

.375 

6.564 

8.750 

10.94 

13.13 

15.31 

17.50 

2  .88 

26.26 

30.62 

35.00 

% 

2.214 

.427 

6.642 

8.854 

11.07 

13.28 

15.50 

17.71 

2  .14 

26.56 

31.00 

35.42 

% 

2.239 

.479 

6.717 

8.958 

11.20 

13.43 

15.67 

17.92 

2  .40 

26.86   I    31.34 

35.83 

% 

2.?66 

.531 

6.798 

9.062 

11.33 

13.59 

15.86 

18.12 

2  .66 

27.18 

31.72 

36.25 

11. 

2.291 

.583 

6873 

9.166 

11.46 

13.75 

16.04 

18.33 

2  .90 

27.50 

32.08 

36.66 

K 

2.318 

.636 

6.954 

9.271 

1.59 

13.91 

16.22 

18.54 

23.18 

27.82 

32.44 

37.08 

54 

2.344 

.688 

7.032 

9.375 

1.72 

14.06 

16.40 

18.75 

23.44 

28.12 

32.80 

37.50 

X 

2.370 

.740 

7.110 

9.479 

1.85 

14.22 

16.59 

18.96 

23.70 

28.44 

33.18 

37.92 

X 

2.395 

.791 

7.185 

9.582 

1.97 

14.37 

16.76 

19.16 

23.94 

28.74 

33.52 

38.33 

^ 

2.422 

.844 

7.266 

9.688 

2.11 

14.53 

16.95 

19.37 

24.22 

29.06 

33.90 

38.75 

% 

2.-148 

.896 

7.344 

9.792 

2.24 

14.68 

17.13 

19.58 

24.48 

29.36 

34.26 

39.16 

K 

2.474 

.948 

7.422 

9.896 

2.37 

14.84 

17.32 

19.79 

24.74 

29.68 

34.64 

39.58 

12. 

2.500 

5.000 

7.500 

10.00 

2.50 

15.00 

17.50 

20.00 

25.00 

30.00 

35.00 

40.00 

Plate-iron  washers.    Standard  sizes.    Dianas  of  washers  and  bolt-holes 

in  ins.     Approx  thickness  by  Birmingham  wire  gauge,  p  367.     Approx  number  in  one  ft). 


Diams. 

Ths.  I      No. 

Diams. 

Ths. 

No. 

Diams. 

Ths. 

No. 

Ji 

X 

18 

543 

IM 

H 

14 

50 

2M 

15-16 

9 

8.7 

% 

5-16 

16 

228 

m 

9-16 

12 

30 

2H 

1    1-16 

9 

6.3 

IU 

5-16 

16 

147 

m 

K 

12 

25.7 

3! 

1  YA. 

9 

4.7 

t/ 

X 

16 

123 

1% 

11.16 

10 

17 

3 

1  % 

9 

3.7 

1 

7-16 

14 

70 

2 

13-16 

10 

10.7 

3^ 

1  H 

9 

3.0 

Price  in  Philada,  1880,  about  16  to  12  cts  per  ft>  for  diams  from  1.5  to  3  inches. 
Rolled  Star  Iron.    Standard  sizes.    Carnegie  Bros,  Pittsburgh. 

The  thickness  is  that  at  center  of  straight  part  of  one  of  the  four  arms,  in  ins.     Rolled  in  lengths 
of  20  to  25  ft.    Area  in  sq  ins.     Wt  in  tts  per  ft  run. 


Ins. 

Ths. 

Area. 

Wt. 

Ins. 

Ths. 

Area. 

Wt. 

4X4 

h 

3.6 

12 

2.5  X  2.5 

% 

1.58 

5.25 

S.5  X  3.5 

X 

3.3 

10 

2  X  2 

% 

.80 

2.75 

3X3 

X                24 

8 

1.5  X  1.5 

9-32 

.70 

2.30 

T  and  Angle  Iron.    Standard  sizes.*    Wt  per  ft.    Prices,  p  364. 


Flanges 
in  ins. 

Thicks. 
in  ins. 

Weight 
in  B)s. 

Flanges 
iuins. 

Thicks, 
in  ins. 

Weight 
inlbs. 

Flanges 
in  ins. 

Thicks, 
in  ins. 

Weight 

in  Ibs. 

y\  x  % 

X 

0.6 

2      X  2J4 

% 

5 

3VS  X  4 

^ 

12. 

%  x    yt 

3-16 

1. 

2v/  x  2V 

5-16 

4.5 

39f  X  4 

12.5 

1      X  1 
1     X  11A 

3-16 
3-16 

1.2 
1.4 

2       X  2% 
2%  X  2« 

5-16* 

5.3 
5. 

4      X  4 
4X4 

1 

9.6 
12.7 

\y±  x  \y\ 

2 

6. 

4      X  4 

19. 

y 

2.2 

2!/2  X  2^ 

% 

6.5 

4      X  4 

1 

25. 

\$L  X  \1A 

3-16 

1.8 

2>4  X3 

K 

6.7 

4      X5 

M 

4.3 

\y^  x  1H 

2.3 

3      X  3 

7.3 

5      X  3 

2.7 

14 

2.8 

N 

7.7 

5      X  6 

M 

7.7 

1J4  X  2 

H 

2.8 

3      X3H 

» 

10.4 

5^  X  3 

4. 

m  x  2 

1%  X  2 

5-16 

3.4 
3. 

3      X  4 
3      X  4 

B 

8.5 
11.3 

jj*xJ8 

K 

4.4 
9 

2      X  2 

H 

3.3 

3«  X  3^ 

S 

8.5 

6      X  4J^ 

5^ 

1 

2      X  2 

3.9 

^ 

10.8 

22 

2X2 

% 

4.7 

3^X3M 

7-16 

9.7 

THE  DIMENSIONS  IN  THIS  TABLE  ARE  FROM 

OUT  TO  OUT  EACH  WAY,  as  a  a,  o  c.  When  these,  and 
thethicknessof,  measured  at  the  center  of  each  arm, 
is  the  same  in  M.  N,  and  S.  there  will  scarcely  be  an 
appreciable  difference  in  the  weights  of  the  three. 
The  dimensions  in  the  table  are  MARKKT  SIZES;  that 
is,  they  can  be  bought  ready  made.  The  young  en- 
gineer should  bear  this  in  mind  in  such  matters 
when  designing;  and  not  introduce  sizes  that  have 
to  be  specially  made;  for  this  requires  the  manufac- 
turer to  prepare  a* new  set  of  rolls;  the  expense  of 
which  would  not  be  warranted  by  an  order  less  than 
some  thousands  o"  dollars. 


374 


BOLTS,    NUTS,    WASHERS. 


Weight  of  heads  and  lints  of  iron  bolts.*  The  weights  include  both 
a  head,  and  a  nut,  which  are  supposed  to  be  neatly  finished  off.  Rough  machine- 
made  ones  frequently  weigh  from  J^  to  %  more.  Wt  of  head  about  =  wt  of  nut. 

Diameter  of  bolt,  in  inches. 

K|K|X|K|K|X|     i    |  ix  |  **  |  m  |  «  |   «*  |   » 

Weight  of  an  hexagonal  head  and  nut  in  Ibs. 
.017    I     .057    I     .128    I     .267    I      .43    I      .73     I     1.10    I  2.14  I    3.78    I     5.6     I    8.75    I     17.    I     28.8 


.021 


.069 


.164 


Weight  of  a  square  head  and  nut  in  Ibs. 
.320    I      .55    I       .88    I    1.31    I    2.56    I    4.42  I      7.0 


10.5 


21. 


36.4 


Au  hexagonal  head  and  nut  together,  weigh  about  as  much  as  5  diams  of  length  of  the  bolt ;  or  a* 
6  diams,  if  they  are  square.     In  machine-made  bolts  up  to  2  ft  long,  the  head  is  in  one  piece  with  tlie 
shank ;  but  in  very  long  ones,  as  in  web-members  for  trusses,  it  is  simply  a  nut, 
like  the  other.   Blacksmiths  make  a  head  by  welding  on  a  ring ;  or  by  upsetting. 
The  thickness  y  t,  of  a  nut  N ;  or  m  g,  of  a  head,  II. 

should  be  at  least  equal  to  the  diam  co,  of  the  screw.  And  the  width  a  b, 
between  two  parallel  sides  of  the  hexagon,  is  1>^  times  the  same  diam.  Some 
machinists  add  %  inch  to  this  width,  for  all  diams.  If  square,  each  side  is 
made  equal  to  1^  diams  of  the  screw.  The  following  are  the  usual  nunibers 
of  screw-threads  cut  per  inch  of  length  at  the  ends  of  bolts.  The  depth  to 
which  the  threads  are  cut  is  a  little  less  than  their  pitch,  or  distance  apart 
from  center  to  center. 


I       | 


These  may  be  termed  the  American  stand- 
_  ard  dimensions.  "With  them  a  bolt  will  not  yield  by 
tripping  off  its  threads;  but  will  generally  break  off  just  below  the  nut,  where  the 
iminution  of  metal  by  the  cutting  of  the  threads  commences.  To  allow  for  this 
drtual  reduction  of  diam,  we  must  first  calculate  what  diam  is  necessary  for  sustain- 
ng  the  load,  or  strain,  which  will  come  upon  it;  and  then  add  enough  to  cover  this 
reduction.  That  is,  we  must  add  twice  the  depth  to  which  the  threads  are  cut.  For 
this  purpose,  it  will  suffice  to  increase  the  neat  calculated  diam  about  %  part,  for 
bolts  from  1  to  2%  ins  diam  ;  i  for  3  ins  diam ;  Y&  f°r  6  ins  diam.* 

This  of  course  involves  a  considerable  increase  in  the  quantity  of  iron  ;  but  is  necessary  when  the 
bolt  is  required  to  fit  close  in  the  hole  through  which  it  passes.     But  where  this  is  not  the  case,  the 
screw  end  may  be  UPSET,  or  made  so  much  thicker  than  the 
other  part  of  the  shank,  that  the  threads  may  be  cut  into  it, 
without  injury  to  the  strength  of  the  bolt;  as  in  the  next  fig. 

THE  FOLLOWING  TABLE  GIVES  THE  DIAM  FOR  EITHER  CASE. 

In  carpentry,  as  well  as  in  ties  for  masonry,  washers,  ww, 
of  either  cast  or  wrought  iron,  are  placed  between  the  timber, 
or  stone,  and  the  head  and  nut;  in  order  to  distribute  the 
pressure  over  a  greater  surface,  and  thus  prevent  crushing ; 
especially  in  timber. 

When  much  strained  against  wood, 

the  side  of  a  square  wrought-iron  washer  ;  or  the  diam  ww  of 
a  circular  one,  should  not  be  less  than  4  diams  of  the  screw, 
as  in  the  fis? ;  and  its  thickness,  t  w,  %  diam  at  least. 

Two  such  square  washers  will  together  weigh  as  much  as  18 
diams  in  length  of  a  round  rort  of  the  same  diam  as  the  screw  : 
or  with  a  square  head  and  nut.  as  much  as  24  diams.  Two 

of  same  diam  as  screw;  or  with  a  square  head  and  nut.  as 
much  as  20  diams.  Cast  iron  washers,  being  more  apt  to  split 
under  heavy  strains,  may  be  made  about  twice  as  thick  aa 
wrought  ones.  "When  the  strain  is  very  great,  the  diam  of 

*Price  of  nuts  in  Philada,  1880,  well  finished,  square  ones  of  1.5  to  3  ins  on 

a  side,  14  to  12  cts  per  ft :  lare^r  si/.^s.  12  to  10  ct->.     Hexagon  ones  about  25  per  ct  more.     Rough 
bolts  and  nuts  6  to  7  cts.     In  1883  »U  .ibout  30  per  ot  lower. 


BOLT8,    NUTS,    WASHERS. 


375 


.  22.0  ' 
.  35.0  • 


the  washer  may  be  5  or  6  times  that  of  the  screw ;  and  its  thickness  equal  to  diam  ;  but  4  diams  will  • 
suffice  for  most  practical  purposes,  or  even  2,5  when  there  ia  but  little  strain,  and  the  thickness  may 
then  be  but  .1  or  .2  diam  of  bolt.        See  p  373  for  such. 

When  the  screw  end  of  the  bolt  is  upset  as  at  «  e,  for  a  length  8  e  of  6  diams  of  the  shank  g  y.  we 
must  add  to  t  ,e  weight  of  the  entire  length  of  the  bolt,  an  allowance  sufficient  to  cover  the  addi- 
tional thickness.  This  allowance  will  equal  3J^  diains  of  length  of  the  body  or  shank  e  g  of  the  bolt. 
But  when  thus  upset,  the  head,  nut,  and  washers  must  be  enlarged  to  suit  the  thicker  screw.  For 
bolts  of  from  1  to  3  ins  diam.  the  total  allowance  must  be  increased  to  the  weight  of  a  length  equal  to 
35  diams  y  g  of  the  shank,  or  body  of  the  bolt,  supposing  the  washer.s  to  be  round  ;  and  the  bead  and 
nut  square.  When  a  neat  finish  is  required,  the  upper  edges  of  the  washers  are  shaped  as  shown  by 
the  dotted  lines  near  w  w,  and  frequently  the  upset  length  «  e  need  not  exceed  2  or  3  diams;  both  of 
which  slightly  diminish  the  foregoing  allowance. 

When  the  bolt  is  not  to  be  much  strained,  or  when  the  timber  is  hard,  the  washers 
may  be  but  3  diams  of  the  screw  in  width,  or  diam  ;  and  about  .4  of  a  diam  in  thick- 
ness; and  this  will  reduce  their  weight  fully  one-half.  For  ease  of  reference  we 
recapitulate  the  allowances  advisable  to  be  made  in  preliminary  estimates;  as  meas- 
ured by  the  weight  of  a  rod  of  the  same  diam  as  the  shank  eg\  the  length  of  the  rod 
}>eiug  given  in  its  own  diameters. 

BOLT  NOT  UPSET.   Square  head  and  nut  together 6  diams. 

Hexagon"     "      "        "          6      ' 

2  round  washers  :  %  diam  thick;  4  diams  in  diam;  wrought 14      " 

Square  head,  and  nut;  and  2  round  washers 20      " 

BOLT  UPSET Upsetting 3.5  " 

Square  head  and  nut  together 9.5  " 

Hexagon  "      "      "         "        7.9  " 

2  round  washers ;  %  diam  thick:  4  diams  of  screw  in  diam.... 

Upsetting;  squarehead;  and  nut;  and  2  round  washers 

If  the  washers  are  of  the  smaller  dimensions  given  above,  deduct  half  their  weight. 

Lock-nut  washers. 

When  bolts  are  subjected  to  much  rough  jolting,  as  at  rail-joints,  &c,  the  nuts  are 
liable  to  unscrew  themselves  in  time.  On  railroads  this  is  a  source  of  great  annoy- 
ance. SHAW'S  LOCK-NUT  WASHER  is  intended  to  prevent  this.  It  is  a  simple  circular 
washer  made  of  steel ;  with  a  slit  s  s  cut  through  it,  leaving  sharp  edges.  On  one  side, 
a,  of  the  slit,  the  metal  is  pressed  upward  about  %  inch  ;  and  that  on  the  other  side,  c, 
downward,  the  same  distance;  so  that  a  perspective  view  would  be  somewhat  as  at*. 
Now,  when  the  nut  is  screwed  down  over  the  washer,  in  the  direction  of  the  arrow, 
the  slit  offers  no  obstruction ;  but  if  the  nut  afterward  tends  to  unscrew  itself,  the 
sharp  upper  edge  of  the  slit,  along  a,  presents  friction  against  the  bottom  of  the  nut, 
sufficient,  it  is  said,  to  prevent  it  from  so  doing.  ANOTHER  DEVICB  is  to  cut  at  the 
end  of  the  screw  a  few  threads  of  a  screw  of  less  diam  than  the  main  one,  and  in  the 
opposite  direction.  The  nut  is  then  screwed  upon  the  larger  diam  ;  and  after  it  the 
lock-nut  is  screwed  in  the  other  direction  upon  the  smaller  diam,  until  it  comes 
into  contact  with  the  main  nut.  At  Figs  16  and  17  of  rail-joints,  other  methods 
will  be  seen,  p  396. 

THE  BILLINGS  LOCK-NUT  WASHER  is  also  said  to  be  effective.  It  is  a  thin 
hollow  cup  of  tempered  spring  steel,  n  n,  made  by  pressure,  from  a  flat  circular 
piece  previously  heated.  It  is  made  by  the  Wharton  Safety  Switch  Co,  of 
Philada.  Diam  2J4  ins;  height  %  inch;  thickness  %  inch;  weight  3}£  ounces.* 

Table   of  diameters,  weights,  and    approximate   breaking: 
strains,  for  round  rolled  iron  bolts,  ties,  or  bars;  assuming  the 

breaking  strain  per  square  inch  of  average  quality  of  rolled  iron  to  be  as  follows:  Up  to  1  inch 
square,  or  1  inch  diam,  20  tons,  or  44800  fcs  ;  from  1  to  2  ins  sq  or  diam,  19  tons  :  2  to  3  ins,  18  tons  ; 
3  to  4  ins,  17  tons ;  4  to  5  ins,  16  tons  ;  5  to  6  ins,  15  tons.  The  first  4  columns  of  the  table  are  to 
be  used  when  the  screw  end  of  the  bolt  is  enlarged  or  upset,  so  that  the  shank  or  body  of  the  bolt 
shall  not  be  weakened  by  the  cutting  of  the  screw  threads.  But  when  the  shank  is  so  weakened,  the 
diam  and  weight  of  the  bolt  must  be  taken  from  the  last  2  cols. 

Rem.  But  it  is  very  important  to  know  that  a  long  upset  rod  is  no 
stronger  than  one  not  upset,  against  slowly  applied  loads  or  strains.  Both  will 
then  break  at  about  midlength,  under  equal  pulls.  Therefore  in  such  cases  the 
col  of  greatest  diams  in  the  table  should  be  used. 

EXAMPLE  1.  To  find  the  diam  of  a  bolt,  that  shall  just  break  under  a  strain,  or  a  load  of  52.5  tons, 
we  see  by  the  table  and  opposite  52.5  ton*,  that  it  will  be  1%  ins  if  the  screw  end  is  enlarged,  and  2.3 
ins  if  it  is  not.  In  the  first  case,  the  weight  of  the  bolt  will  be  9.3  B>s  per  foot  run  ;  and  in  the  second, 
13.8  fts. 

EXAMPLE  2.  What  must  be  the  diam  in  order  to  sustain  a  strain  of  52.5  tons,  with  a  safety  of  3? 
Here  52.5  X  3  ~  157.5  tons.  In  the  table,  the  nearest  we  find  to  157.5  tons,  is  163.6;  opposite  which 
we  find  the  diam  3f^  ins.  A  diam  a  trifle  less  than  this  will  break  under  a  strain  of  157.5  tons ;  and 
consequently  will  have  a  safety  of  3  for  52.5  tons. 

The  breaks:  strains  in  this  table  will  also  answer  for  square 

BARS,  by  merely  increasing  them  ^  part;  for  a  round  bar  has  very  approximately  4  of  the  strength 
of  a  square  one  whose  side  is  equal  to  the  diam  of  the  round  one  ;  or  the  square  one  has  1 H  times  the 
Strength  of  the  round  one;  or,  more  correctly,  as  1  to  .7854.  For  the  Strength  Of 
COPPER  bolts,  multiply  the  tabular  ones  by  the  decimal  .8 ;  and  for  their  weight,  increase, 
that  of  iron  ones  4-  part.  Heads,  nuts,  and  washers,  are  not  included  in  the  table. 


L  me 

Ji^rWn 


*  Price  in  1880,  $5  per  100. 


376 


WEIGHT    OF    METALS. 


WEIOHT  AJTD  STRENGTH  OF  IRON  BOLTS.    (Original.) 
For  square  ones  or  for  copper  see  preceding  paragraph. 


Ends  enlarged,  or  upset. 

Ends  not 
enlarged. 

Ends  enlarged,  or  upset. 

Ends  not 
enlarged. 

Diam. 

Weight 

Break- 

Break- 

Diam. 

Weight 

Diam. 

Weight 

Break- 

Break- 

Diam. 

Weight 

of 

per  foot 

ing 

ing 

of 

per  foot 

of 

per  foot 

ing 

ing 

of 

per  foot 

thank 

run. 

strain. 

strain. 

•ihank 

run. 

shank 

run. 

strain. 

strain. 

shank 

run. 

Ins. 

Pds. 

Tons. 

Pds. 

Ins. 

Pds. 

Ins. 

Pds. 

Tons. 

Pds. 

lus. 

Pds. 

1A 

.oiu 

.245 

549 

1% 

8.10 

45.7 

10236S 

2.14 

12.0 

3-16 

.093 

.553 

1239 

13-16 

8.69 

49.0 

1097GO 

2.22 

12.9 

% 

.165 

.983 

2202 

.35 

.321 

y& 

9.30 

52.5 

117C03 

2.30 

13.8 

5-16 

.258 

1.53 

34-27 

.43 

.452 

15-16 

9.93 

56.0 

125440 

2.38 

14.7 

3/s 

.372 

2.21 

4950 

.50 

.654 

2. 

10.6 

59.7 

133728 

2.45 

15.7 

7  16 

.506 

3.00 

6720 

.58 

.897 

i/£ 

12.0 

63.8 

14291-2 

2  59 

17.5 

1A 

.661 

3.93 

8803 

.66 

1.14 

/4 

13.4 

71.6 

160384 

273 

19.5 

y-16 

.837 

4.97 

11133 

.73 

1.41 

3Z 

14.9 

79.7 

178528 

2.88 

21.6 

% 

1.03 

6.14 

13754 

.80 

1.67 

\/ 

16.5 

88.4 

198016 

3.02 

23.9 

11-16 

1.25 

7.42 

16621 

>8 

2.03 

oZ 

18.2 

97.4 

218176 

3.16 

26.1 

% 

1.49 

8.83 

19779 

.96 

2.41 

3? 

20.0 

106.9 

239456 

3.30 

28.5 

i3-u> 

1.75 

10.4 

2329  fi 

1.04 

281 

H 

21.9 

116.8 

261632 

3.45 

31.1 

% 

2.03 

12.0 

26880 

1  12 

3.26 

3 

23.8 

127.2 

284928 

3.60 

33.9 

15-16 

2.33 

138 

3031-2 

1.-20 

3.77 

1A 

27.9 

141.0 

315840 

3.86 

39.1 

lin. 

265 

15.7 

35163 

1.27 

4.27 

2 

32.4 

16X6 

366464 

4.12 

44.4 

1-16 

295 

16.8 

37632 

1.35 

4.77 

H 

37.2 

187.7 

420448 

4.41 

51.0 

l/8 

3.35 

IS.9 

42330 

1.42 

5.28 

4 

42.3 

•213.6 

478464 

4.70 

57.8 

3-16 

373 

21.1 

47  -2Q  i 

1.49 

5.81 

\/ 

47.8 

227.0 

508480 

498 

65.2 

1A 

4.13 

23.3 

5219-2 

1.n5 

6.39 

I/ 

536 

l?54.5 

570080 

5.25 

72.9 

5-16 

4.56 

25.7 

57568 

1.64 

70i 

/4 

59.7 

283.5 

635040 

5.53 

80.5 

% 

500 

28.2 

63168 

1.72 

7.74 

5. 

66.1 

314.2 

703808 

5.80 

88.1 

7-16 

5.47 

30.8 

6899-2 

i.*o 

8.48 

1A 

72.9 

324.7 

72732  - 

6.08 

97.0 

Y* 

5.95 

33.6 

7526 

1.87 

9.20 

g 

80.0 

356.4 

798336 

6.36 

106. 

9-16 

6.46 

36.4 

81536 

1.94 

988 

% 

87.5 

389.5 

8724SO 

6.63 

116. 

% 

699 

39.4 

88250 

2.00 

10.6 

6 

95.2 

424.1 

94998  \- 

6.90 

126. 

11-16 

7.53 

42.5 

95200 

•2.07   ill.  3 

See  Rein,  p  375. 

ROLLED  LEAD,  COPPER,  and  BRASS:  Sheets  and  Bars. 


Thickness 
or 

LEAD. 

COPPER. 

BRASS. 

Thickness 
or 

Diameter, 

Diameter, 

or  side, 

Sheets, 

Square 

Round 

Sheets, 

Square 

Round 

Sheets, 

Square 

Round 

or  side, 

in 

per 

Jars; 

Bars; 

per 

Bars; 

Bars; 

per 

liars  ; 

Bars; 

in 

Inches. 

Square 

IFoot 

1  Foot 

Square 

1  Foot 

1  Foot 

Square 

1  Foot 

1  Foot 

Inches. 

Foot. 

long. 

long. 

Foot. 

long. 

long. 

Foot. 

long. 

long. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

1-32 

1.86 

.005 

.004 

1.44 

.004 

.003 

1.36 

.004 

.003 

1-32 

1-16 

3.72 

.019 

.015 

2.89 

.015 

.012 

2.71 

.014 

.011 

1-16 

3-32 

5.58 

.044 

.034 

433 

.034 

.027 

4.06 

.032 

.025 

3-32 

U 

7.44 

.078 

.061 

5.77 

.000 

.047 

5.42 

.053 

.044 

H 

532 

9.30 

.121 

.095 

7.20 

.094 

.074 

6.75 

.088 

.OC9 

532 

3  16 

11.2 

.174 

.137 

8.66 

.135 

.100 

8.13 

.127 

.100 

8  16 

7-32 

13.0 

.237 

.187 

10.1 

.184 

.144 

9.50 

.173 

.130 

7-32 

X 

14.9 

.310 

.244 

11.5 

.240 

.189 

10.8 

.226 

.177 

M 

516 

18.6 

.485 

.381 

14.4 

.376 

.295 

13.5 

.353 

.277 

516 

H 

22.3 

.698 

.548 

17.3 

.541 

.425 

16.3 

.508 

.3D9 

% 

7-16 

26.0 

.950 

.746 

20.2 

.736 

.578 

19.0 

.691 

.543 

7-16 

H 

29.8 

1.24 

.974 

23.1 

.962 

.755 

21.7 

.903 

.709 

H 

9  16 

33.5 

1.57 

1.23 

26.0 

1.22 

.955 

24.3 

1.14 

.9CO 

9-16 

H 

37.2 

1.94 

1.52 

28.9 

1.50 

1.18 

27.1 

1  41 

1.11 

K 

11  16 

40.9 

2.34 

1.84 

31.7 

1.82 

1.43 

298 

1.70 

1.34 

11-16 

96 

44.6 

2.79 

2.19 

34.6 

216 

1.70 

32.5 

2.03 

1.60 

K 

13-16 

48.3 

3.27 

2.57 

37.5 

255 

1.99 

35  2 

2.38 

1.87 

13-  16 

X 

5-2.1 

3.80 

2.98 

40.4 

2.94 

2.31 

37.9 

2.76 

2.17 

% 

15-16 

5SO 

4.37 

3.42 

41.3 

3.38 

265 

40.6 

3.18 

2.49 

15-16 

1. 

59.5 

4.98 

390 

46.2 

3.85 

3.02 

4:5.3 

3.61 

2.81 

1. 

X 

66.9 

627 

4.92 

52.0 

4.87 

3.8-2 

48.7 

4.57 

3.60 

H 

% 

744 

7.75 

6.09 

57.7 

6.01 

4.72 

51.2 

5.64 

4.43 

8 

i 

81.8 

9.37 

7.37 

fi-5.5 

7.28 

5.7-' 

5!)  6 

682 

537 

M 

3 

83.3 

1.2 

8.77 

09.3 

8  65 

6.PO 

65.0 

8.12 

6.38 

H 

N 

9H.7 

3.1 

10.3 

75.1 

10.2 

7.98 

70.4 

9.53 

7.19 

K 

K 

iu4. 

5.2 

11.9 

80.8 

11.8 

9.25 

75.9 

11.1 

8.G8 

M 

K 

t. 

112. 
1  19 

75 

13.7 

86.6 

1H.5 

10.6 

81.3 
8fi.7 

12.7 
14.4 

9.97 
11.3 

% 

2. 

WEIGHT   OF   METALS. 


377 


ROOf  COpper  is  usually  in  sheets  of  2^  ft  X  5  ft:  or  12^  square  feet 
weighing  10  to  H  fts  per  sheet ;  and  is  laid  on  boards  *  No  solder  is  used  in 
the  horizontal  joints  as  it  is  in  tin  roofs ;  but  both  the  horizontal  aud  the  sloping 
joints  are  formed  by  ouly  overlapping  and  bending  the  sheets,  much  as  shown 
by  the  figs  "tinder  the  head  "  Tin  ;  "  except  that  the  horizontal  joints  are  bent  or 
locked  together,  as  in  this  figure  ;  and  then  flattened  down  close. 

Roof  lend  generally  weighs  4  to  6  tbs  persq  ft.  Its  great  contraction 
and  expansion,  however,  render  it  unfit  for  this  purpose,  as  it  is  liable  to  crack 
aud  leak.  It  may  be  used  for  flashings.  Price  in  1880  about  9  to  10  cts  per  Ib. 

WEIGHT  OF  BALLS. 


diameter 
in 
Inches. 

CAST 
LEAD. 

CAST 
COPPER. 

CAST 
BRASS. 

CAST 
IRON. 

Diameter 
•  in 

Inches. 

CAST 

LEAD. 

CAST 
COPPER. 

CAST 
BRASS. 

CAST 

IRON. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

H 

.026 

.021 

.019 

.017 

5M 

30.1 

24.1 

21.5 

19.8 

% 

.088 

.070 

.063 

.058 

J£ 

34.7 

27.7 

24.7 

22.7 

I. 

.209 

.167 

.148 

.136 

H 

39.6 

31.7 

28.3 

25.9 

y* 

.408 

.325 

.290 

.266 

6. 

45.0 

36.0 

32.0 

29.4 

.705 

.562 

.501 

.460 

/1j 

57.2 

45.8 

40.8 

37.4 

H 

1.12 

.893 

.795 

.731 

7. 

71.5 

57.2 

50.9 

46.8 

2. 

1.67 

1.33 

1.19 

1.07 

y% 

88.0 

70.3 

62.6 

57.5 

y± 

2.38 

1.90  ' 

1.69 

1.55 

8. 

106. 

85.3 

76.0 

69.8 

y* 

3.25 

2.60       i      2.32 

2.13 

y* 

127. 

102. 

91.2 

83.7 

H 

4.34 

3.47       i     3.09 

2.83 

9. 

151. 

121. 

108. 

99.4 

3. 

5.63 

4.50 

4.01 

3.68 

y* 

178. 

143. 

127. 

117. 

M 

7.15 

5.72 

5.10 

4.68 

10. 

208. 

167. 

148. 

136. 

H 

8.94 

7.14 

636 

5.85 

y* 

241. 

193. 

172. 

158. 

11.  0 

8.79 

7.83 

7.19 

11 

277. 

222. 

198. 

182. 

4.* 

13.4 

10.7 

9.50 

8.73 

y* 

317. 

253. 

226. 

207. 

y± 

16.0 

12.8 

11.4 

10.5 

12. 

360. 

288. 

257. 

236. 

18.9 

15.2 

13.5 

12.4 

,? 

22.7 
26.0 

17.9 

20.8 

15.9 
18.6 

14.6 

17.0 

The  weight!  of  balls  are  as  the  cubes  of  their 
diaius. 

LEAD  PIPES;  weight  of,  per  Foot  ran. 

Bore,  or 

THICKNESS  OF  METAL,  IN  INCHES. 

inner  diam. 

in  inches. 

rV 

i 

A 

? 

A 

t 

& 

i 

f 

t 

J 

1  inch. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs, 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

y 

.305 

.724 

1.28 

1.95 

2.74 

3.65 

4.53 

5.84 

8.  2 

1.7 

15.3 

19.5 

5-16 

.366 

.815 

1.47 

2.20 

3.05 

4.02 

4.96 

6.33 

9.  4 

2.4 

16.2 

20.5 

.427 

.967 

1.65 

2.44 

3.35 

4.38 

5.39 

6.82 

9.  6 

3.2 

17.0 

21.5 

7*16 

.488 

1.09 

1.83 

2.69 

3.66 

4.75 

5.82 

7.31 

0. 

3.9 

17.9 

22.4 

.548 

1.21 

2.01 

2.93 

3.96 

5.11 

6.24 

7.79 

1. 

4.6 

18.7 

23.4 

% 

.670 

1.16 

2.38 

3.42 

4.57 

5.85 

7.10 

8.77 

2. 

6.1 

20.4 

25.4 

% 

.791 

1.70 

2.74 

3.90 

5.18 

6.58 

7.96 

9.75 

3. 

76 

22.1 

27.3 

TX 

.911 

1  95 

3.11 

4.39 

5.79 

7.31 

8.82 

10.7 

4. 

9.1 

23.9 

29.3 

1. 

1.03 

2.19 

3.47 

4.88 

6.40 

8.04 

967 

11.7 

5. 

20.5 

25.6 

31.2 

.16 

2.44 

3.84 

5.37 

7.01 

8.77 

10.5 

127 

7! 

27.3 

33.2 

TX 

2.69 

4.21 

5.85 

7.62 

9.50 

11.4 

13.7 

8. 

23.4 

29.0 

35.1 

«/ 

AO 

2.9t 

4.58 

6.34 

8.23 

10.3 

12.3 

14.7 

9.5 

24.9 

30.7 

37.1 

y*> 

.52 

3.18 

4.94 

6.83 

8.84 

1.0 

13.1 

15.6 

20.7 

26.3 

32.4 

39.0 

fix 

.64 

3.43 

5.31 

7.32 

9.47 

1.7 

14.0 

16.6 

22.0 

27.8 

34.1 

41.0 

H 

.76 

3.67 

5.67 

7.81 

O.I 

2.4 

14.8 

17.6 

23.2 

29  3 

35.8 

42.9 

% 

.89 

3.92 

6.04 

8.30 

0.7 

3.2 

15.7 

18.6 

244 

30.8 

37.6 

44.9 

2. 

2.01 

4.16 

6.40 

8.78 

1.3 

3.9 

16.5 

19.5 

25.6 

32.2 

393 

46.8 

34 

2.25 

4.65 

7.13 

9.76 

2.5 

5.4 

18.2 

21.5 

28.1 

35.1 

42.7 

50.7 

y% 

2.49 

5  14 

7.86 

10.7 

3.7 

6.8 

20.0 

23.4 

30.5 

38.0 

46.1 

54.6 

% 

2.73 

5.83 

859 

11.7 

4  9 

8.3 

21.7 

25.4 

32.9 

41.0 

49.5 

58.5 

s. 

2.98 

612 

9.32 

12.7 

6.1 

9.7 

23.4 

27.3 

35.4 

43.9 

52.9 

62.4 

H 

3.22 

6.61 

10.1 

13.7 

7.4 

1.2 

25.1 

29.3 

87.8 

46.8 

56.4 

66.4 

Jrf£ 

3.46 

7.10 

10.8 

14.6 

8.6 

2.7 

268 

31.3 

40.3 

49.7 

59.8 

70.3 

% 

3.71 

7.59 

11.5 

156 

9.8 

4.1 

285 

33.2 

42.7 

52.7 

63.2 

74.2 

4. 

3.95 

8.08 

12.2 

16.6 

21.0 

5.6 

30.2 

35.2 

45.2 

55.6 

66.6 

78.1 

Lead  service  pipes  for  single  dwellings  in  Philadelphia  are  usually  of 
from  Jx£  inch  bore,  wt  1  to  2%  ft>s;  to  %  inch  bore,  wt  1%  to  3  ft>s  per  ft  run, 

nppnrninor  in  Viparl        T'ViPv  «•*!••**  Iv  Ikii  ••*!<•  frnm  snrlHpn  flnsincr  nf  sf  rmnnnlre  •  hiit 

. 
sometimes  do  so  from  the  freezing  of  the  contained  water.    See  pp  533,  573. 

Cost  of  lead  pipe  in  Philada,  1880,  about  8%  cts  per  ft>.    Tin-lined  15  cts. 
Made  on  a  large  scale  by  Messrs  Tathani  Brothers,  226  S  Fifth  St,  Phila. 

*  To  which  it  is  held  by  copper  cleats ;  as  at  Fig  y,  next  page. 
Price  of  roofing  copper  in  1880,  about  38  cts ;  and  copper  nails  45  cts  per  tt>. 


378 


TIN    AND    ZINC. 


BRAZED  COPPER  PIPES  j  weight  of,  per  Foot  run. 


Thickness 

INNER  DIAMETER  IN  INCHES. 

in 
Inches. 

1 

1H 

1H 

1« 

2 

2>4 

2X 

2J»' 

3 

W 

4 

ft* 

5 

6 



























_:  

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

1-16 

.800 

.989 

1.18 

1.37 

1.56 

1.75 

1.93 

2.12 

2.31 

269 

3.07 

3.45 

3.82 

4.58 

332 

1.24 

1.52 

1.80 

2.08 

2.36 

2.64 

2.92 

3.20 

3.48 

4.05 

4.61 

5.17 

5.73 

6.85 

M 

1.70 

2.08 

2.46 

2.84 

3.22 

2.61 

3.99 

4.37 

4.  75 

5.51 

6.27 

7.03 

7.80 

9.3'J 

3-16 

2.70 

3.27 

3.84 

4.41 

4.98 

5  55 

6.12 

6.69 

7.26 

8.40 

9.54 

10.7 

11.8 

14.1 

H 

3.79 

4.55 

5.IU 

6.07 

6.83 

?!  5!) 

8.35 

9.11 

9.87 

11.4 

12.9 

14.4 

16.0 

19.0 

516 

4.97 

5.92 

6.86 

7  81 

8  76 

9  70 

10  6 

11.6 

12  5 

14.4 

16.3 

18.2 

20.1 

r-t  a 

TIN  ANI>  ZINC. 

The  pure  metal  is  called  block  tin.  When  perfectly  pure,  (which  it 
rarely  is,  being  purposely  adulterated,  frequently  to  a  large  proportion,  with  the 
cheaper  metals  lead  or  zinc,)  its  sp  grav  is  7.29 ;  and  its  weight  per  cub  ft  is  455  K>s. 
It  is  sufficiently  malleable  to  be  beaten  into  tin  foil,  only  YoVo~  °^  an  ^nc^  ^"c^- 
Its  tensile  strength  is  but  about  4600  Ibs  per  sq  inch ;  or  about  7000  ft>s  when  made 
into  wire.  It  melts  at  the  moderate  temperature  of  442°  Fah.  Pure  block  tin  is 
not  used  for  common  building  purposes  ;  but  thin  plates  of  sheet  iron,  covered  with 
it  on  both  sides,  constitute  the  tinned  plates,  or,  as  they  are  called,  the  tin,  used  for 
covering  roofs,  rain  pipes,  and  many  domestic  utensils.  For  roofs  it  is  laid  on  boards. 

The  sheets 
of  tin  are  uni- 
ted as  shown  in 
this  fig.  First,  sev- 
eral sheets  are 
joined  together  in 
the  shop,  end  for 
end,  as  at  tt\  by 
being  first  bent 
over,  then  ham- 
mered flat, and  then 

^, soldered.  These  are 

then  formed  into  a 
J          roll   to   be  carried 
to  the  roof;  a  roll 

being  long  enough  to  reach  from  the  peak  to  the  eaves.  Different  rolls  being  spread 
up  and  down  the  roof,  are  then  united  along  their  sides  by  simply  being  bent  as  at  a 
and  s,  by  a  tool  for  that  purpose.  The  roofers  call  the  bending  at  s  a  double  groove, 
or  iloublf  loc .  ;  and  the  more  simple  ones  at  t,  a  single  groove,  or  lock. 

To  hold  the  tin  securely  to  the  sheeting  boards,  pieces  of  the  tin  3  or  4  ins  long, 
by  -I  ins  wide,  called  cleats,  are  nailed  to  the  boards  at  about  every  18  ins  along  the 
joints  of  the  rolls  that  are  to  be  united,  and  are  bent  over  with  the  double  groove  5. 
This  will  be  understood  from  y,  where  the  middle  piece  is  the  cleat,  before  being 
bent  over.  The  nails  should  be  4-penny  slating  nails,  which  have  broader  heads 
than  common  ones.  As  they  are  not  exposed  to  the  weather,  they  may  be  of  plain  iron. 
Much  use  is  made  of  what  is  called  leaded  tin,  or  tarns,  for  roofing.  It  is 
simply  sheet-iron  coated  with  lead,  instead  of  the  more  costly  metal  tin.  It  is  not 
as  durable  as  the  tinned  sheets,  but  is  somewhat  cheaper. 

The  best  plates,  both  for  tinning  and  for  tarns,  are  made  of  charcoal  iron :  which, 
being  tough,  bears  bending  better.  Coke  is  used  for  cheaper  plates,  but  inferior  as 
regards  bending.  In  giving  orders,  it  is  important  to  specify  whether  charcoal 
plates  or  coke  ones  are  required ;  *  also  whether  tinned  plates,  or  tarns. 

There  are  also  in  use  for  roofing,  certain  compound  metals  which  resist  tarnish  better  than  either 
lead,  tin,  or  zinc  ;  but  which  are  so  fusible  as  to  be  liable  to  be  melted  by  large  burning  cinders  fall- 
ing  on  the  roof  frrnn  a  neighboring  conflagration. 

A  roof  covered  with  tin  or  other  i  icUl  should,  if  possible,  slope  not  much  ZM«  than  five  decree*,  or 
about  an  inch,  to  a  foot;  and  at  the  eaves  there  should  be  a  sudden  fall  into  the  rain-gutter,  to  pre- 
vent rain  from  backing  up  so  as  to  overtop  the  double-groove  joint  *,  and  thus  causing  leaks.  Where 
eoal  is  used  for  fuel,  tiu  roofs  should  receive  two  coats  of  paint  when  first  put  up,  and  a  coat  at  every 

*  Price,  Phila.  1880.  Charcoal  iron,  tinned  plates,  $10  to  $13  per  box  ;  accord- 
ing to  weight  and  quality,  and  tarua  from  $8  to  $10.  In  1HS2  about  25  per  ct  lower. 


TIN    AND    ZINC. 


379 


2  or  3  rears  after.    Where  wood  only  is  ased,  this  is  not  necessary  ;  and  a  tin  roof,  with  l  good  pitch, 
will  lust  20  or  30  years.J 

Tinned  iron  plates  are  sold  by  the  box.  These  boxes,  unlike  glass,  have  not  equal 
areas  of.  contents.  They  may  be  designated  or  ordered  either  by  their  names  or 
sizes.  Many  rnaker^,  however,  have  their  private  brands  in  addition;  and  some  of 
these  have  a  much  higher  reputation  than  others. 

Contents  of  a  box  of  eitber  tinned  plates  or  tarns. 


Names. 

No.  in 
a  box. 

Sizes. 
lus. 

Wt.  of 

a  box. 
Lbs. 

Marks  on  the 

boxes. 

C1                N     1 

225 

13%  X  10 

iij 

CI 

13M  X     9% 

105 

CIl 

« 

12%  X     914 

98 

cm 

Cross  No  1    

n 

1314  x  10 

140 

XI 

Two  Cross.  No  1         

u. 

161 

XXI 

u 

ii 

182 

XXXI 

four  Cross  No  1               

<t 

u 

203 

XXXXI 

100 

16%  X  12^ 

98 

CD 

Cross  Doubles           

126 

XD 

II 

a 

147 

XXD 

Thrtf  Cross  Doubles         

t( 

u 

182 

XXXD 

u 

M 

203 

XXXXD 

Common  Small  Doubles     

200 

15  X  11 

163 

CSD 

Cross  tfmall  Doubles  
Two  Cross  Doubles        

188 
209 

XSD 
XXSD 

Three  Cross  Doubles 

it 

u 

230 

XXXSD 

four  Cross  Doubles    

u 

t( 

251 

XXXXSD 

Waiters  or  Wastes,  Common  No.  1 
Wastes  Cross  No.  1 

225 

13%  X  10 

112 
140 

>VCI 
\VXI 

112 
112 

14  X  20 
14  X  20 

112 
140* 

Sheets  of  larger  size  may  be  made  to  special  order ;  those  of  tinned  iron,  in  Enprland  ;  but  leaded 
tarns  are  made  in  Philada  also,  and  elsewhere.  Sheets  of  5)6  ins  by  33,  39.  45,  48,  52,  or  58,  on  hand. 
Larger  ones,  up  to  3  ftLy  7  ft,  and  up  to  No.  26  gauge,  (.013  inch.)  made  to  order. 

A  box  of  '225  sheets  of  13%  by  10.  contains  214.81  sq  ft;  but,  allowing  for  overlapping,  it  will  cover 
but  about  150  sq  ft  of  roof;  even  without  any  allowance  for  the  waste  which  occurs  in  cutting  away 
portions  in  order  to  fit  at  angles,  &c. 

To  find  the  area  of  roof  covered  by  any  sheet,  first  deduct  2  ins  from  its  width,  and  1  inch  from  Its 
length. 

ZillC,  in  sheets,  and  laid  in  the  same  manner  as  slates,  is  mnch  used  in  some  parts  of 
Knrope  for  roofing.  By  exposure  to  the  weather,  it  soon  becomes  covered  by  n  thin  film  of  white 
oxide,  which  protects  it  from  further  injury,  and  renders  the  roof  very  durable.t  Corrugated  sheet 
tine  is  also  used.  See  Galvanized  Sheet  Iron,  page  370. 

Ziuc  sheets  are  usually  about  3  ft  by  7  or  8  ft.  The  gauge  differs  from  that  of  iron ;  thus  No  13  is 
.032  of  an  inch  thick,  or  1.22  Ibs  per  sq  ft;  No  14,  =  .035  inch,  and  1.35  Ibs;  No  15,  =  .042  inch,  and 
1.49  Ibs  ;  No  16,  =  .049  inch,  and  1.62  Ibs  per  sq  ft.  Any  of  these  numbers  may  be  used  ou  root's,  for 
which  purpose  it  should  be  very  pure. 

WATER  KEPT  nr  ZINC  VESSELS  is  said  to  become  injurious  to  health  :  and  recently  an  ontcry  has  on 
that  account  arisen  again-a  jralvaniz-'d-iron  service-pipes  in  dwellings.  Yet  such  have  been  in  use 
for  many  years  in  New  England,  Philada.  and  elsewhere,  without  as  yet  any  deleterious  effects. 
This  is  possibly  owing  to  the  fact  that  service-pipes  being  short,  the  water  is  usually  all  drawn 
throuch  them  several  times  a  day  ;  and  hence  does  not  remain  in  contact  with  the  zinoor  lead  long 
enough,  to  acquire  a  poisonous  character.  In  taking  possession  of  a  house  in  which  the  water  has 
remained  stagnant  in  the  service-pipes  for  some  considerable  time,  such  water  should  all  be  run  to 
waste  ;  otherwise  sickness  may  ensue  from  its  use. 

•  Th»re  are  other  sizes,  as  10  X  20,  <fcc. 

f  The  price  Of  Sheet  Zinc  does  not  ordinarily  differ  much  from  that  of  sheet  lead, 
which  in  Philada  in  1880  was  about  8  to  9  cts  per  ft ;  or  in  pigs,  from  6  to  7  cts. 

The  price  of  block  tin,  made  into  either  pipes  or  sheets,  about  45  to  50  cts  per  tt.  Messrs 
Tatuaiu  a  Bros,  226  South  Fifth  St.  make  both,  in  bars  22  to  25  cts.  in  1880. 

+  The  cost  of  tin-roofing,  so-called,  but  actually  tarns,  in  Philadelphia,  1882.  is  about  7  or  8 

old  shingle  roofs.     Galvanized  iron  rain  water-pipes,  3  itis  diam,  about  20  eta  per  ft  run,  put  up. 


380 


WEIGHT    AND   STRENGTH    OF    WIRE    ROPES. 


Table  of  Wire  Rope,  manufactured  by  John  A.  Roebling's 
Sons,  Trenton,  IT.  J.     Prices  in  1880  about  as  per  table. 


Hope  of  133  Wires. 


Rope  of  49  Wires. 


1      i 

a 

s 

1 

| 

a 

a, 

a            o 

—  'on' 

2 

a         .2 

Wfi 
C 

~  «! 

1 

a 

a 

*i 

S-^s 

s 

ll 

a>°3  a 

a 

£8 

c  •£"* 

1 

I 

1 

CQ  ** 

So 

i  ^ 

5 

S 

B 

1 

1? 

&  5-5 

C   li     _ 

1 

a 

a 

i 

a 

ll 

III 

1 

F 

a 

3 

S 

£  a 
•£  o 

III 

£ 

.2 

p 

£  K    M 

I 

P 

£  M  ta 

3 

c 

S 

6 

£ 

B 

i 

6% 

1  20 

74  00 

j5^ 

11 

4% 

54 

36  00 

105i 

2 

6 

1  05 

65  00 

14^i 

12 

454 

47 

30  00 

10  * 

3 

5H 

91 

54  00 

13 

13 

3% 

41 

25  00 

4 

5 

78 

43  60 

12 

14 

3% 

35 

20  00 

8V 

5 

4% 

65 

35  00 

10^ 

15 

3 

29 

16  00 

7^ 

6 

4 

53 

27  20 

16 

23 

12  30 

7 

3^ 

41 

20  20 

8 

17 

2% 

18 

8  80 

534 

8 

31^ 

34 

16  00 

7 

18 

15 

7  60 

5 

9 

2% 

28 

11  40 

6 

19 

\yt 

13 

5  80 

10 

25 

8  64 

5 

20 

\y 

11 

4  09 

4  * 

10% 

2 

2* 

5  13 

43^ 

21     i     1« 

9 

2  83 

10.  ^ 

l£f 

23 

4  27 

4 

22     !     1M 

8 

2  13 

%H 

10H 

1>3 

22 

8  48 

3M 

23 
24 

1 

7 
6^ 

1  65 
1  38 

2Y 

Tiller  Rope,  %  in  diam,  26  cts. 

25 
26 

K 

P 

1  03 
0  81 

1« 

27 

N 

5  " 

0  56 

i>£ 

Ropes  from  No.  8  to  No.  10%  are  spe- 
cially adapted  for  hoisting-rope. 

27  J^ 

H 

4 

28 
2'J 

3 
2 

Large  Sash  Cord. 
Small     " 

All  kinds  of  shackles,  sockets,  swivel 
books,  and    fastenings,    put   on,    and 
splicer  made. 

Copper  Rope,  corresponding  to  the 
above  sizes,  made  to  order. 
Steel  ropes  are  much  stronger  and 
more  durable  than  irou. 

Notes  on  the  use  of  Wire  Rope,  by  Mr.  Roebling. 

"  Two  kinds  of  Wire  Rope  are  manufactured  ;  the  larger  sizes,  as  also  the*nost  pliable,  arecomposei 
of  133  wires,  and  are  generally  used  for  hoisting  or  running  rope.  Those  of  49  wires  are  stiffer.  and 
are  better  adapted  for  standing  rope,  guys,  and  rigging.  Orders  should  state  the  use  of  Rope,  and 
advice  will  be  given.  Ropes  up  to  3  inches  diameter  are  made  upon  special  application. 

For  safe  working  load,  allow  1-5  to  1-7  of  ultimate  strength,  according  to  speed  and  vibration.  When 
substituting  Wire  Rope  for  hemp  rope,  it  is  good  economy  to  allow  for  the  former  the  same  rate  per 
foot  run  which  experience  has  approved  of  for  the  latter. 

Wire  Rope  is  as  pliable  as  new  hemp  rope  of  the  same  strength ;  the  former  will  therefore  run  over 
the  same  sized  sheaves  and  pulleys  which  are  used  for  the  latter.  But  the  greater  the  diameter  of 
the  sheaves,  pulleys,  or  drums,  the  longer  Wire  Rope  will  last.  In  the  construction  of  machinery  for 
Wire  Rope  it  will 'be  found  good  economy  to  make  the  drums  and  sheaves  as  large  as  possible.  The 
size  of  drum  is  as  follows :  The  same  figure  which  expresses  the  circum  in  inches  in  the  second 
column  of  the  table  is  also  the  minimum  diam  of  drum  in  feet;  doubling  that  figure  will  give  the  max- 
imum. The  diameter  of  drum  should  be  no  less  th.rm  the  minimum,  nor  is  it  necessary  to  exceed  the 
maximum.  As  an  example,  take  a  No.  4 rope,  circumference  5  inches;  therefore  the  minimum  diam- 
eter of  drum  is  5  feet ;  and  the  maximum  10  feet.  Or  a  No.  10}^  rope,  circumference  2  inches ;  there- 
fore minimum  diam  is  2  feet;  and  maximum  4  feet.  A  smaller  diameter  of  drum  may  answer,  but 
the  short  bending  will  result  in  a  much  more  rapid  wear.  In  most  cases  the  Rope  will  wear  twice  as 
long  on  a  max  diam  as  on  a  minimum. 

Experience  h:»s  al«o  demonstrated  that  the  wear  increases  with  the  speed.  It  is  better  to  increase 
the  loHd  th:in  the  speed. 

Wire  Rope  is  manufactured  either  with  a  wire  or  hemp  centre.  The  latter  is  more  pliable  than  the 
former,  and  will  wear  hotter  where  there  is  short  bending.  Orders  should  specify. 

Wire  Rope  must  nut  be  coiled  or  uncoiled  like  hemp  rope.  When  mounted  on  a  reel  the  latter  should 
be  turned  on  a  spindle  to  pay  off  the  rope.  When  forwarded  in  a  coil  without  reel,  roll  it  over  the 
ground  like  a  wheel,  and  run  off  the  rope  in  that  way.  All  untwisting  must  be  avoided. 

To  preserve  Wire  Rope  apply  raw  linseed  oil  with  a  piece  of  sheepskin,  wool  inside;  or  mix  the  oil 
with  equal  parts  of  Spanish  brown  and  lampblack. 

To  preserve  Wire  Rope  under  water  or  under  ground,  take  mineral  or  vegetable  tar,  add  1  bushei 
of  fresh  slacked  lime  to  I  barrel  of  tar,  (which  will  neutralize  the  acid,)  and  boil  it  well,  then  saturate 
the  rope  with  the  boiling  tar. 

The  grooves  of  cast-iron  pulleys  and  sheaves  should  be  filled  with  well-seasoned  blocks  of  hard 
wood,  set  on  end.  to  be  reaewed  when  worn  out.  This  end  wood  will  save  the  rope  and  increase  ad- 
hesion. The  small  pulleys  or  rollers  which  support  the  ropes  on  inclined  planes  should  be  constructed 
on  the  same  plan.  When  large  sheaves  run  with  a  very  great  velocity,  the  grooves  must  be  lined  eithm- 


WEIGHT    AND   STRENGTH    OF    IRON    CHAINS. 


381 


with  leather  set  on  end ;  with  cork  ;  or  with  India  rubber.  This  is  done  in  the  case  of  all  sheaves  used 
in  the  transmission  of  power  between  distant  points  by  means  of  ropes;  which  frequently  run  at  the 
rate  of  4000  feet  per  minute.  Full  information  given,  if  desired,  on  the  size  of  rope,  aud  the  size  and 
speed  of  sheaves  to  be  used  for  transmitting  power.  Rope  %  ins  diam  will  transmit  100  horse  power 
to  a  great  dist. 

Notes  on  Ki^inj;. 

"  Wire  Rope  for  shrouds  and  stays  is  now  universally  superseding  hemp  rope,  for  the  following  rea- 
sons :  it  is  much  cheaper  than  hemp  rope ;  it  is  more  durable,  aud  it  will  not  stretch  permanently 
under  great  strains,  as  is  the  case  with  h«mp  rigging,  thus  saving  much  labor  in  setting  up;  and  it 
is  fully  as  elastic  as  hemp  rope  of  equivalent  size.  Tne  rope  with  49  wires  is  best  adapted  for  rigging. 

The  great  economy  in  using  wire  in  place  of  hemp  rigging  is  the  large  reduction  in  size  and  weight. 
The  bulk  of  wire  rigging  is  only  one-sixth  that  of  hemp,  while  the  weight  is  only  one-half.  The  ad- 
vantages of  lightness  are  apparent  at  a  glance  to  every  seaman  ;  the  removal  of  several  tons  of  weight 
from  the  height  occupied  by  the  standing  rigging  must  increase  both  the  steadiness  and  the  stability 
of  the  ship.  Again,  on  account  of  its  smaller  bulk,  less  resistance  is  offered  to  the  wind,  a  fact  fully 
appreciated  by  captains  of  steamers,  when  running  against  the  wind. 

For  protection  from  rust,  wire  rigging  is  galvanized;  when  not  galvanized,  apply  good  paint  with  a 
brush  or  a  piece  of  sheepskin,  once  or  twice  a  year,  and  a  good  set  of  wire  shrouds  aud  stays  will  out- 
last the  best  built  sailing  ship  or  steamer.  The  best  aud  cheapest  paint  is  linseed  oil  mixed  with 
equal  parts  of  lampblack  and  Spanish  brown  or  Venetian  red,  or  any  other  preparation  of  oxide  of  iron. 

All  vessels  in  the  U  S  Navy  are  now  rigged  with  Roebling's  Wire  Rope  exclusively,  it  having 
proved  the  best  in  the  test  made  by  the  Government  at  the  Washington  Navy  Yard." 

The  foregoing  is  a  copy  of  a  circular  by  Messrs  Roebling. 

On  planes  in  Schuylkill  Co.  <fec,  a  wire  rope  generally  lasts  long  enough  to  raise  one  mil- 
lion of  tons  of  coal  up  a  plane  half  a  mile  long,  and  rising  1  in  10.  The  ordinary  duration  on  inclined 
planes  throughout  the  country  is  from  1%  to  4  years,  according  to  the  amount  of  service;  and  also 
greatly  to  the  care  taken  of  them,  and  of  the  sheaves  and  rollers  upon  which  they  move. 

On  the  Mt  Pisgah  plane,  2500  ft  long,  rising  660  ft,  for  raising  empty  coal  oars,  aud  lowering  loaded 
ones,  thin  iron  bunds,  7  inches  wide,  and  abmo  %  inch  thick,  luive  been  used  instead  of  ropes.  They 
scarcely  exhibit  any  sign  of  wear  in  several  years.  They  should  be  riveted;  being  apt  to  break  if 
welded.  Steel  would  probably  be  the  best  material  in  many  cases. 

WEIGHT  AND  STRENGTH  OF  IRON  CHAINS. 

The  links  of  ordinary  iron  chains  are  usually  made  as  short  as  is 
consistent  with  easy  play,  in  order  that  they  may  not  become  bent  when  wound 
around  drums,  sheaves,  &c ;  and  that  they  may  he  more  easily  handled  in  slinging 
large  blocks  of  stone,  &e.  U.  S.  Govt.  expts,  1878,  prove  that  studs  weaken  the  links. 

When  so  made,  their  weight  per  foot  run  is  quite  approximately  3^  times  that  of  a  single  bar  of  the 
round  iron  of  which  they  are  composed.  Since  each  link  consists  of  two  thicknesses  of  bar  it  might 
be  supposed  that  a  chain  would  possess  about  double  the  strength  of  a  single  bar;  but  the  strength  of 
the  bar  becomes  reduced  about  y3^,  by  being  formed  into  links  ;  so  that  the  chain  really  has  but  about 
^  of  the  strength  of  two  bars.  As  a  thick  bar  of  iron  will  not  sustain  as  heavy  a  load  in'proportion  as  a 
thinner  one,  so  of  course,  stout  chains  are  proportionably  weaker  than  slighter  ones.  In  the  following 
table.  20  tons  per  sq  inch.,  is  assumed  as  the  average  breaking  strain  of  a  single  straight  bar  of  ordi- 
nary rolled  iron,  1  inch  in  diam;  or  1  inch  square;  19  tons,  from  1  to  2  ins;  and  18  tons,  from  2  to  3 
ins.  Deducting  -fa  from  each  of  these,  we  have  as  the  breaking  strain  of  the  two  bars  composing 
each  link,  as  follows:  14  tons  per  sq  inch,  up  to  1  inch  diam;  13.3  tons,  from  1  to  2  ins;  aud  12.6 
tons,  from  2  to  3  ins  diam;  and  upon  these  assumptions  the  table  is  based.* 

Table  of  strength  of  chains. 

Chains  of  superior  iron  will  require  ^  to  ^  more  to  break  them.    (Original.) 


Diam  of  rod 
of  which 
the  links 
are  made. 

Weight 
of  chain 
3er  ft  run. 

Breaking  strain 
of  the  chain. 

i 

Diam  of  rod 
of  which 
the  links 
are  made. 

Weight 
of  chain, 
per  ft  run. 

Breaking  strain 
of  the  chain. 

Ins. 

Pds. 

Pds. 

Tons. 

Ins. 

Pds. 

Pds. 

Tons. 

3-16 

£ 

.325 
.579 
.904 

1731 
3069 

4794 

.773 
1.37 
2.14 

i 

9.26 
11.7 
14.5 

49280 
59226 
73114 

2200 
26.44 
32.64 

/8 

1.30 

6922 

3.09 

m 

17.5 

88301 

3942 

I6 

134L6 

& 

1.78 
2.31 
2.93 
362 
4.38 
5.21 
6.11 
7  10 
8.14 

9408 
12320 
15590 
19219 
23274 
27687 
32301 
M7632 
43-277 

4.20 
5.50 
6.9fi 
8.58 
10.39 
1-2.36 
14.42 
16.80 
19.32 

ig 

214 

m 
p 

20.8 
24.4 

28.4 
32.6 
37.0 
40.9 
57.9 
70.0 
83.3 

105280 
123514 
143293 
164505 
187152 
224448 
277088 
335328 
398944 

47.00 
55.14 
6397 
73.44 
83.55 
100.2 
1  23.7 
149.7 
178.1 

Price  of  chain 

inch  ;  6«^  cts  for  1  inch, 


is,  Philada,  1880,  about  8%  cts  per  ft  for  %  inch  ;  iy2  cts  for 


382 


WEIGHT   OF    RAILROAD   SPIKES. 


Table  of  Manilla  rope. 


Diam. 
Ins. 

Oirc. 
Ins. 

Wt  per 
foot. 
Ibs. 

Breaking   load. 

Diam. 
Ins. 

Circ. 
Ins. 

Wt  pei- 
foot. 
Ibs. 

Breaking  load. 

Tons. 

Ibs. 

Tons. 

Ibs, 

.239 

% 

.019 

.25 

560 

1.91 

6 

1.19 

11.4 

25536 

.318 

1 

.033 

.35 

784 

2.07 

fA 

1.39 

13.0 

29120 

.477 

iM 

.074 

.70 

1568 

2.23 

7 

1.62 

14.6 

32704 

.636 

2         1      .132 

1.21 

2733 

2.39 

7K 

1.86 

16.2 

36288 

.795 

2^ 

.206 

1.91 

4278 

2.55 

8  2 

2.11 

17.8 

3987  2  ' 

.955 

3 

.297 

2.73 

6L15 

2.86 

9 

2.67 

21.0 

47040 

1.11 

3^ 

.404 

3.81 

8534 

3.18 

10 

3.30 

24.2 

54208 

1.27 

4 

.528 

6.16 

11558 

3.50 

11 

3.99 

27.4 

61376 

1.43 

VA 

.668 

6.60 

14784 

3.82 

12 

4.75 

30.6 

68544 

1.59 

5 

.825 

8.20 

18368 

4.14 

13 

5.58 

33.8 

75712 

1.75 

5^ 

.998 

9.80 

21952 

4.45 

14 

6.47 

37.0 

82880 

The  strength  of  Manilla  ropes,  like  that  of  bar  iron,  is  very  variable  ; 
and  so  with  hemp  ones.  The  above  table  supposes  an  average  quality.  Ropes  ot 
good  Italian  hemp  are  considerably  stronger  than  Manilla;  but  their  cost  excludes 
them  from  general  use.  The  tarring*  of  ropes  is  said  to  lessen  their  strength ; 
and,  when  exposed  to  the  weather,  their  durability  also.  We  believe  that  the  use  of 
it  in  standing  rigging  is  partly  to  diminish  contraction  and  expansion  by  alternate 
wet  and  dry  weather.  The  common  rules  for  finding  the  strength  of  rope 
by  multiplying  the  square  of  the  diam  or  circumf  by  a  given  coefficient  are  entirely 
erroneous.  Prices  in  Philada,  1880,  Manilla,  17  to  18  cts  per  ft) ;  Italian  hemp, 
25  cts;  American  hemp,  15  cts;  Sisel  hemp,  16  cts;  jute,  (E.  Indies,)  10  cts. 

The  strengths  of  pieces  from  the  same  coil  may  vary  25  per  ct. 

A  few  months  of  exposed  work  weakens  ropes  20  to  50  per  ct. 


t 


WEIGHT  OF  RAILROAD 

The  hook-headed  spikes  £,  commonly  used  for  confining  rails  to 
the  cross-ties,  vary  within  the  limits  of  the  following  table;  the  lightest  ones 
for  light  rails  on  short  local  branches ;  and  the  heaviest  ones  for  heavy  rails 
on  first-class  roads.  The  table  is  from  the  Phoenix  Iron  Co  of  Philadelphia. 
The  spikes  are  sold  in  kegs  usually  of  150  fibs.  For  the  weight  of  spikes  of 
larger  dimensions,  we  may  near  enough  take  that  of  a  square  bar  of  the 
same  length.  What  is  saved  at  the  point,  suffices  for  the  addition  at  the 
head. 


Size  in  ins. 
Length.    Side. 

No.  per  keg 
of  150  ft>s. 

No.  per  5) 

Size  in  ins. 
Length.    Side. 

No.  per  keg 
of  150  Ibs. 

No.  per  ft>. 

•  -i/  v    V 

350 

2.33 

526 

3.5 

4y    X    \& 

400 

2.66 

5i/  x   T7j 

289 

1.93 

6       X     ? 

705 

4.7 

**A    X    7» 

218 

1.46 

5       X   & 
5      X   *A 

48S 
390 

3.25 
2.6 

6       X   K 
6      X   A 

310 
262 

2.07 
1.75 

5        X    T97T 

295 

1.97 

6       X    ^ 

196 

1.C.O 

5       X   % 

257 

1,71 

A  size  in  very  common  use.  is  5^  X  tV ;  which  weiShs  about  H  ft  Per 
spike.  A  mile  of  single-track  road,  with  2112  cross-ties,  2%  feet  apart  from  center 
to  center:  and  with  rails  of  the  ordinary  length  of  24  feet,  or  10  ties  to  a  rail ;  thus 

*~Prlce  of  spike*  and  of  cut  nails,  in  Philada,  in  1880,  about  5  cts  per  ft.    Rivets  6  to 

ej? 

In  1882,  all  about  25  per  ct  less. 


SPECIFIC   GRAVITY. 


383 


having  440  rail-joints  per  mile;  with  4  spikes  to  each  tie,  except  at  the  rail-joints,  at 
each  of  which  there  will  be  4  spikes  ;  *  will  require  at  a  neat  calculation  9328  spikes. 
But  an  allowance  must  be  made  for  rail  guards  at  road-crossings,  which  we  mar  assume  to  be  '24  ft 
wide,  or  the  length  of  a  rail.  A  guard  will  usually  consist  of  4  extra  rails  for  protecting  the  track- 
rails,  and  spiked  to  the  11  ties  by  which  said  track  rails  are  sustained.  Consequently,  such  a  crossing 
requires  11.  X  8  —  88  spikes.  For  turnouts,  sidings,  loss,  &c.  we  may  roughly  average  584  t  spikes 
more  per  mile;  thus  making  in  all  if  we  assume  one  road-crossing  per  niilej  9328  -f-  b8  -f-  584  =  1UOOO 
spikes  per  mile;  or  5000  Ibs,  or  33#  kegs  of  150  tt>s. 

Adhesion  of  spikes.  Professor  W.  R.  Johnson  found  that  a  plain  spike 
.375,  or  %  inch  square,  driven  3%  ins  into  seasoned  Jersey  yellow  pine,  or  unseasoned 
chestnut,  required  about  2uOO  ft>s  force  to  extract  it;  from  seasoned  white  oak.  about 
4000;  and  from  well-seasoned  locust,  about  6000  ft>s.  Bevan  found  that  a  t^peuny 
nail,  driven  one  inch,  required  the  following  forces  to  extract  it:  Seasoned  oeech,  b'67 
Ibs;  oak,  507  ;  elm,  327  ;  pine,  187. 

Very  careful  experiments  in  Hanover,  Germany, 

BY  ENorNKER  PUNK,  give  from  2465  to  3'J40  Ibs,  (mean  of  many  experiments,  about  3000 
fts.)  as  the  force  necessary  to  extract  a  plain  ^  inch  square  iron  spike,  6  ins  long,  wedge- 
pointed  for  one  inch,  (twice  the  thickness  of  the  spike,)  and  driven  4)£  ins  into  white  or 
yellow  pine.  When  driven  5  ins,  the  force  reqd  was  about  -fa  part  greater.  Similar 
spikes,  -A-  inch  square,  7  ins  long,  driven  6  ins  deep,  reqd  from  3700  to  6745  Ibs  to  ex- 
tract them  from  pine  ;  the  mean  of  the  results  being  4873  Ibs.  In  all  cases,  about  twice  at 
much  force  wis  reqd  to  extract  them  from  oak.  The  spikes  were  all  driven  across  the 
rain  of  the  wood.  Experience  shows  that  wuen  driven  with  the  grain,  spikes  or  nails 
not  hold  with  much  more  than  half  as  much  force. 

Jagged  spikes,  or  twisted  ones,  (like  an  au?er,  t  or  those  which  were  either  swelled  or  diminished 
near  the  middle  of  their  length,  all  proved  inferior  to  plain  square  ones.   When  the  length  of  the  wedge 
point  was  increased  to  4  times  the  thickness  of  the  spike,  the  resistance  to  drawing  out  was  a  trine  less. 
When  the  length  of  the  spike  is  fixed,  there  is  prob-ibly  no  b  -tier  shape  than  the  plain  square  cross- 
section,  with  a  wedge  point  twice  as  long  as  the  width  of  the  spike,  as  per  this  tig. 

Boards  of  oak  or  pine,  nailed  together  by  from  Mo  i6tenpermy 

common  cut  nails,  and  then  pulled  apart  in  a  direction  lengthwise  of  the  hoards,  and  across  the  nails, 
tending  to  break  the  Utter  in  two  by  a  shearing  action,  averaged  about  300  to  400  Ibs  per  nail  to  sepa- 
rate them;  as  the  result  cf  many  trials. 

WEIGHT  OF 


gr 
do 


Name. 

Length. 
Inches. 

No.  per  flX 

Name. 

Length. 
Inches. 

No.  per  ft. 

3  penny 

\y 

557 

10  penny 

2% 

66 

4      " 

Wg 

336 

12      " 

3% 

50 

5       " 

1% 

210 

20      " 

3% 

32 

6      " 

2 

163 

30      " 

4^z 

19 

7      " 

214 

128 

40      " 

4% 

16 

8      " 

VA 

93 

50      u 

5% 

13 

The  sizes  and  weights  vary  considerably  with  different  makers      Ours  are  averages,     ^h^ 
above  are  machine-made,  or  cut  nails ;  in  distinction  to  the  wrought  nails  made  by  the  black sn •  5 1 ; . . 


SPECIFIC  GEAVITY. 

THE  sp  grav  of  a  body,  is  its  weight  as  compared  with  that  of  an  equal  bulk  of  some 
other  body,  which  is  adopted  as  a  standard  of  comparison.  For  other  substances  than 
air  and  gases  generally,  pure  water  is  the  usual  standard  ;  and  since  the  weight  of  a 
given  bulk  of  water  varies  somewhat  with  its  temperature ;  and  also  with  the  state 
of  the  air,  the  former  is  assumed  to  be  62°  Fah  ;  and  the  latter  at  30  ins,  at  sea-level. 
But  where  extreme  scientific  accuracy  is  not  aimed  at,  all  these  considerations  may 
be  neglected ;  and  any  clear  fresh  water,  at  any  ordinary  temperature,  say  from  (50° 
to  80°  may  be  used;  for  if  at  70°,  the  resulting  sp  gr  will  be  but  1  part  in  1176  too 
great;  at*75°,  1  in  670;  at  80°,  1  in  454;  at  85°,  1  in  336.  At  62°  pure  water  weighs 
62.355  fts  avoir  per  cub  ft. 

To  find  the  sp  srrav  of  a  body,  heavier  than  water.  Weigh  it 
first  in  the  air  ;  and  then  in  water  ;  and  find  the  diff.  The  diff  is  what  the  body  loses 
in  water:  and  is  the  weight  of  a  bulk  of  water  equal  to  the  bulk  of  the  body.  Then 
say  as  this,  Diff  :  wt  in  air  :  1  : :  sp  grav  of  body. 

*  This  suppose**  the  joint  and  chair  to  rest  upon  a  tie:  bat  when  long  chairs  are  u«ed  with  a  view 
of  placing  the  rail  joint  between  two  ties  laid  near  each  other,  there  will  be  8  spikes  to  a  joint :  or 
1780  per  mile  more  than  above;  equal  to  880  As;  making  in  all,  per  mile  single  track,  say  12000 
•pikes,  or  «000  Ibs,  or  40  kegs. 

t  This  allows  that  turnouts  and  sidings  amount  to  about  1  mile  of  extra  track  on  15  miles  of  road. 

4*  Price  in  Philada,  1880,  about  5  cts  per  Q>.  Roofing  nails  of  tinned  iron,  10  cents.  Copper 
nails,  45  cts. 


384 


SPECIFIC   GRAVITY. 


The  weight  of  a  given  bulk  of  a  substance  which  is  either  porous,  or  absorbent  of  water,  cannot  be 
Inferred  from  its  sp  gr.  Thus  pure  river  aaud,  is  pure  quartz  ;  and  of  course  has  the  same  sp  gr  ;  yet, 
a  solid  cub  ft  of  quartz,  weighs  nearly  twice  as  much  as  a  cub  ft  of  sand  ;  on  account  of  the  interstices 
of  the  latter.  A  brick,  some  sandstones,  &c.  absorb  water;  so  that  their  sp  gr  will  not  furnish  the 
weight  of  a  dry  mass  of  the  same.  In  such  cases,  the  engineer  will  generally  first  measure  the  con- 
tents of  a  piece  of  the  substance,  if  a  solid  ;  and  then  weigh  it;  thus  ascertaining  its  weight  per  cub 
ft,  &c.  If  it  is  in  grains,  or  dust,  he  will  measure,  and  then  weigh,  a  cub  ft  of  it. 

TO  find  the  Sp  grav  Of  a  liquid.  First  carefully  weigh  some  solid  body,  as  a 
piece  of  metal,  in  the  air.  Then  weigh  it  in  water,  and  note  the  loss,  sav  L.  Then  weigh  it  in  the 
other  liquid :  and  note  the  loss,  say  /.  Then  as  loss  L,  is  to  loss  I,  so  is  1.  or  the  sp  gr  of  water,  to  the 
sp  gr  of  the  liquid.  Or,  if  the  sp  gr,  and  weight  of  the  solid  body,  are  already  known,  merely  weigh 
it  in  the  liquid.  Then  as  its  weight  in  air,  is  to  its  loss  in  the  liquid,  so  is  its  sp  gr,  to  that  of  the  liquid. 

Timber,  when  first  purchased  from  lumber  yards,  even  under  shelter,  is  rarely,  if  ever,  perfectly 
dry ;  but  its  weight,  if  tolerably  seasoned,  will  be  about  %  part  greater  than  given  in  our  tables ,  or 
about  *4  to  %  part,  if  green. 

Table  of  specific  gravities,  and  weights. 

In  this  table,  the  sp  gr  of  air,  and  gases  also,  are  compared  with  that  of  water, 
instead  of  that  of  air ;  which  last  is  usual. 


Names  of  Substances. 


Average 
SpGr. 


Air,  atmospheric ;  at  60°  Fan,  and  under  the  pressure  of  one  atmosphere  or 

14.7  fts  per  sq  inch,  weighs  g^-j  part  as  much  as  water  at  60° .00123 

Alcohol,  pure .793 

of  commerce .834 

"        proof  spirit .916 

Ash,  perfectly  dry.    (See  footnote,  p  386.) average. .  .752 

1000  ft  beard  measure  weighs  1.748  tons. 

Ash,  American  white,  dry "      ..  .61 

1000  ft  "board  measure  weighs  1.414  tons. 

Alabaster,  falsely  so  called  ;  but  really  Marbles 2.7 

"          real;  a  compact  white  plaster  of  Paris average..         2.31 

Aluminium 2.6 

Antimony,  cast,  6.66  to  6.74 average  ..         6.70 

native "      ••         6.67 

Anthracite,  1.3  to  1.84.     Of  Penn  a,  1.3  to  1.7 usually  ..         1.5 

A  cubic  yard  solid,  averages  about  1.75  cub  yds, when  broken  to  any  mar- 

«        ketsize;  and  loose. 

Anthracite,  broken,  of  any  size.    Loose average.. 

•'  "        raodorately  shaken "    ... 

"  heaped  bushel,  loose,  77  to  as. 

A  ton,  loose,  averages  from  40  to  43  cub  ft " 

at  54  fts  per  cub  ft,  a  cub  yard  weighs  .651  ton. 

Asphaltum,  1  to  1.8 , "     ••         1.4 

Basalt.    See  Limestones,  quarried "     ••         2.9 

Bath  Stone,  Oolite "     ••         2.1 

Bismuth,  cast.     Also  native "     ••         9.74 

Bitumen,  solid.     See  Asphaltum. 

Brass,  (Copper  and  Zinc,)  cast,   7.8  to  8.4 "     ••         8.1 

••     rolled "     ••         8.4 

Bronze.    Copper  8  parts;  Tin  1.     (Gun  metal.)    8.4  to  8.6 '•     ••         8.5 

Brick,  best  pressed 

"       common  hard 

"       soft,  inferior 

Brickwork.     See  Masonry. 

Boxwood,  dry 

Calcite,  transparent.. '     ••          2.722 

Carbonic  Acid  Gas,  is  1^  times  as  heavy  as  air '     ••  .00187 

Charcoal,  of  pines  and  oaks '     

Chalk,  2.2  to  2.8.     See  Limestones,  quarried '     ••         2.5 

Clay,  potter's,  dry,  1.8  to  2.1 '     ••         1.9 

dry,  in  lump,  loose 

Coke,  loose,  of  pood  coal 

'      a  heaped  bushel,  loose,  35  to  42  Ibs 

"      a  ton  occupies  80  to  97  cub  ft 

In  coking,  coals  swell  from  25  to  50  per  cent. 
Equal  weights  of  coke  and  coal,  evaporate  about  equal  wts  of 
water :  and  each  abt  twice  as  much  as  the  same  wt  of  dry  wood. 

Corundum,  pure,  3.8  to  4 3.9 

Cherry,  perfectly  dry average  ..          .672 

1000  ft  board  measure  weighs  1.56'2  tons. 

Coal,  bituminous.  1.2  to  1.5 "     -.          1.35 

"  "  broken,  of  any  size;  loose " 

41  "  moderately  shaken " 

"  "  a  heaped  bushel,  loose.  70  to  78  Ibs. 

"  "  a  ton  occupies  43  to  48  cub  ft. 

A  cubic  yard  solid,  averages  about  1.75  yards  when  broken  to  any 
market  size,  and  )oo«e. 


SPECIFIC    GRAVITY. 


385 


Table  of  specific  gravities,  and  weights  — (Continued.) 


Names  of  Substances. 


Average 
Sp  Gr. 


Chestnut,  perfectly  dry .    (See  footnote,  p  386.) average . . 

1000  board  measure  weigas  I.o25  tons. 

Cement,  hydraulic.     American,  Rosendale;  ground,  ktose average  .. 

"    U.S.  Struck  bush,.  70  fts 

"  "  Louisville,   "          »«      62 

"  "  Copley,         "          "      67 

"  English  Portland,  U.S.  struck  bush,  by  Gillmore,  100tol28 
••        "  "        Various,  weighed  by  writer,  95  to  102.., 

"         "  "        a  barrel  400  to  430  Ibs. 

"  French  Boulogne  Portland,  struck  bush,  95  to  110 

Differences  of  4  or  5  pounds  either  more  or  less  than  we  here  give  per 
loose  struck  U.S.  bush,  often  occur  in  the  cement  from  the  same 
manufactory,  owing  not  only  to  the  difficulty  of  measuring  exactly, 
but  to  the  want  of  uniformity  in  the  composition  of  the  stone,  de- 
gree of  burning,  grinding,  dryness,  &c.  Moreover,  the  term  "loose" 
is  indefinite.  We  mean  by  it  the  average  looseness  which  it  has 
when  thrown  by  a  scoop  into  a  half  bushel  when  measuring  that 
quantity  for  sale.  By  shaking  it  may  easily  be  compacted  about  y 
part,  so  a.*  to  weigh  i  more  per  bush,  or  cub  ft.  And  by  ramming, 
about  %  part,  so  as  to  weigh  about  %  more.  So  with  lime,  plas- 
ter, &c. 

Copper,  cast, 8.6  to  8.8 8.7 

rolled 8.8  to  9.0 8.9 

Crystal,  pure  Quartz.     See  Quartz. 

Cork 25 

Diamond,  3.44  to  3.55  ;   usually  3.51  to  3.55 3.53 

Earth  ;  common  loam,  perfectly  dry.  loose 

"  •«  "  "  "     shaken 

"  "  "  "  "     moderately  rammed 

41  "    slightly  moist,  loose 

."  "  "    more  moist,         "    

•'  "  "  "  shaken 

moderately  packed 

11  "  "    as  a  soft  flowing  mud 

"  "  "    as  a  soft  mud,  well  pressed  into  a  box 

Ether . "6 

Elm,  perfectly  dry.    (See  footnote,  p  386.) average . .  .56 

1000  ft  board  measure  weighs  1.302  tons. 

Ebony,  dry.., '       ••         1-22 

Emerald,  2.63  to  2.76 l      ..         2.7 

Fat 

Flint "       ••         2.6 

Feldspar,   2.5  to  2.8 ..         2.65 

Garnet,  3.5  to  4.3;  Precious,  4.1  to  4.3 "      ..         4.2 

Glass,  2.5  to  3.45 

"     common   window "       ..          2.52 

"     Millville,  New  Jersey.     Thick  flooring  glass "       -.         2.53 

Granite,  2.56  to  2.88.     See  Limestone,  160  to  180 "       ••         2-72 

Gneiss,  common,  2.62  to  2.76 "       ..         2.69 

44      in  loose  piles 

"      Hornblendic 

"  "  quarried,  in  loose  piles 

Gypsum,  Plaster  of  Paris,    2.24  to  2.30 '       ..         2.27 

41        in  irregular  lumps 

"         ground,  loose,  per  struck  bushel,  70 

well  shaken.    "        "        80 

"  "       Calcined,  loose,  per  struck  bush,  65  to  75 , 

Greenstone,  trap,  2.8  to  3. 2 

41  '4      quarried,  in  loose  piles 

Gravel,  about  the  same  as  sand,  which  see. 

Gold,      cast,  pure,  or  24  carat "       ..       19.258 

"         native,  pure,  19.3  to  19.34 "       ..       19.32 

"  "         frequently  containing  silver,  15.6  to  19.3 

"        pure,  hammered,  19.4  to  19.6 "      ..       19.5 

Gutta  Percha 

Hornblende,  black,  3.1  to  3.4 

Hydrogen  Gas,  is  14J4  times  lighter  than  air;   and  16  times  lighter  than 

'      oxygen average.. 

Hemlock,  perfectly  dry.   (Footnote,  p  386.) "      ..  .4 

1000  feet  board  measure  weighs  .930  ton. 

Hickory,  perfectly  dry.    (See  footnote,  p  386.) 

1000  feet  board  measure  weighs  1.971  tons. 

Iron,  cast,  6.9  to  7.4 "      ..          7.15 

"        "         usually  assumed  at "       ..          7.21 

At  450  ft>s.  a  cub  inch  weighs  .2604  Tb  ;  8601.6  cub  inches  a  ton  ;  and 
a  ft  —  3.8400  cub  inches  ;  cast-iron  gun  metal 7.48 


386 


SPECIFIC    GRAVITY. 


Table  of  specific  gravities,  and  weights  —  (Continued.) 


Names  of  Substances. 


Average 
Sp  Gr. 


Iron,  wrought,  7.6  to  7.9;  the  purest  has  the  greatest  sp  gr average..  7.77 

"    large  rolled  bars ••       ..  7.6 

"    sheet .• «' 

At  480  lbn,  a  cub  inch  weighs  .2778  ft  ;  and  a  ft  =  3.6000  cub  ins. 
Light  iron  indicates  impurity. 

Ivory average..  1.82 

Ice,  .917  to  .922 "       ..  .92 

India  rubber " 

Lignum  vitae,  dry "       ..  1.33 

Lard " 

Lead,  of  commerce,  11.30  to  11.47;  either  rolled  or  cast "       ..         11.38 

Limestones  and  Marbles,  2.4  to  2.86,  150  to  178.8 

"        ordinarily  about 2.7 

"  "  "        quarried  in  irregular  fragments,  1  cub  yard  solid, 

makes  about  1.9  cub  yds  perfectly  loose;  or  about 
1%  yds  piled.  In  this  last  case,  .571  of  the  pile 
is  solid;  and  the  remaining  .429  part  of  it  is 

voids piled. . 

Lime,  quick,  of  ordinary  limestone  and  marbles  »2  to  98  fts  per  cub  ft 1.5 

'       either  in  small  irregular  lumps  ;  or  ground,  loose  50  to  58 

In  either  case  1  solid  measure  makes  about  1.8  meas  loose;  and  then 

.555  of  the  mass  is  solid,  and  .445  is  voids. 

To  measure  correctly,  none  of  the  lumps  should  exceed  about  %  or 
Y*(j  of  the  smallest  dimension  of  the  vessel  used  for  measuring. 

Lime,  quick,  ground,  loose,  per  struck  bushel  62  to  70  fts 

"        well  shaken,     "        '•     80      "    

•'          "  "        thoroughly  shaken,  "     93%  "    

Mahogany,  Spanish ,  dry  * t average . .  .85 

"  Honduras,  dry "       ..  .56 

Maple,  dry* "       ..  .79 

Marbles,  see  Limestones. 

Masonry,  of  granite  or  limestones,  well  dressed  throughout 

"         "        "      well-scabbled  mortar  rubble.     About  -^  of  the  mass 

will  be  mortar 

"         "        "      well-scabbled  dry  rubble 

"          "        "      roughly  scabbled  mortar  rubble.     About  %  to  X  part 

wiU  be  mortar 

"          "        "      roughly  scabbled  dry  rubble 

At  155  fts  per  cub  ft,  a  cub  yard  weighs  1.868  tons  ;  and  14.45  cub  ft, 

1  ton. 
Masonry  of  sandstone;  about  ^  part  less  than  the  foregoing. 

"  brickwork,  pressed  brick,  fine  joints  ........     average.. 

medium  quality " 

"        "          "  coarse ;  inferior  soft  bricks " 

At  125  fts  per  cub  ft,  a  cub  yard  weighs  1.507  tons;  and  17.92  cub 
ft,  1  ton. 

Mercury,  at  32°  Pah 13.62 

60°    " 13.58 

"  212°    " 

Mica,  2.75  to  3.1 '.  2.93 

Mortar,  hardened,  1.4  to  1.9 1.65 

Mud,  dry,  close 

"     wet,  moderately  pressed 

'«     wet,  fluid '. 

Naphtha .848 

Nitrogen  Gas  is  about  -^  part  lighter  than  air , 

Oak,  live,  perfectly  dry,  .88  to  1.02  * average. 

"     white,       "          "      .66  to    .88 

"    red,  black,  &c* " 

Oils,  whale;  olive "       ..I          .92 

"     of  turpentine "      . .  |          .87 

Oolites,  or  Roestones,  1.9  to  2.5 "      ..  2.2 

Oxygen  Gas,  a  little  more  than  ^  P»rt  heavier  than  air .00136 

Petroleum .878 

Peat,  dry,  unpressed t 

Pine,  white,  perfectly  dry,  .35  to  .45* .40 

1000  ft  board  measure  weighs  .930  ton.* 

"      yellow,  Northern,  .48  to  .62 (  .55 

1000  ft  board  measure  weighs  1.276  tons.* 

"          "        Southern, .64  to  .80 

1000  ft  board  measure  weighs  1.674  tons.* 

*  Oreen  timbers  usually  weigh  from  one-fifth  to  nearly  one-half  more  than 

dry  ;  and  ordinary  building  timbers  when  tolerably  seasoned  about  one-sixth  more  than  perfectly*  dry. 


SPECIFIC   GRAVITY. 


387 


Table  of  specific  gravities,  and  weights  — (Continued.) 


Names  of  Substances. 


Average 
SpGr. 


Pine,  heart  of  long-leafed  Southern  yellow,  unseas.    (Footnote,  p  386.) ...         1.04 
1000  ft  board  measure  weighs  2.418  tons. 

Pitch 7 

Plaster  of  Paris  ;  see  Gypsum. 

Powder,  slightly  shaken 1 

Porphyry,  2.66  to  2.8 2.73 

Platinum 21  to  22 21.5 

native,  in  grains 16  to  19 ,....        17.5 

Quartz,  common,  pure 2.64  to  2.67 2,65 

finely  pulverized,  loose 

"  "  "  "         well  shaken 

"  "  "  "         well  packed '. 

"    quarried,  loose.    One  measure  solid,  makes  full  1%   broken  and 

piled 

Ruby  and  Sapphire,  3.8  to  4.0 3.9 

Rosin 1.1 

Salt,  coarse,  per  struck  bushel ;  Syracuse,  N.  York 56  Ibs  . . 

"          "        "        "  "        Turk's  Island;  Cadiz;  Lisbon.  76  to  80  .. 

"        St.  Barts 84  to  90  .. 

"        "        "  "        some  well-dried  West  India. ...  90  to  96  .. 

"        "        "  "        Liverpool 50  to  55  . 

"    Liverpool  fine,  for  table  use 60  to  62 

Sand,  of  pure  quartz ,  perfectly  dried,  and  loose,  usually  112  to  133  Ibs  per 

struck  bushel 2.65 

At  the  average  of  98  Ibs  per  cub  ft,  a  struck  bushel  weighs  122}£  Ibs  j 
and  18.29  bushels,  1  ton  ;  a  cub  yd  =  1.181  tons  ;  22.86  cub  ft,  1  ton. 
Slight  shaking  compacts  it  about  2  to  3  per  ct ;  and  ramming  about 
12  per  ct  when  dry. 

"      perfectly  wet,  voids  full  of  water 

"  "          "      at  the  mean  of  124  Ibs,  a  cub  yard  weighs  1.495  tons  ; 

and  18.06  cub—  1  ton. 
"      sharp  angular  sand  of  pure  quartz  with  very  large  and  very  small 

grains  dry  may  weigh 

If  any  ordinary  pure  natural  sand  be  sifted  into  2  or  3  or  more  parcels 
of  differently  sized  grains,  a  measure  of  any  of  these  parcels  will 
weigh  considerably  less  than  an  equal  measure  of  the  original  sand. 
Thus,  a  sand  weighing  98  Ibs  per  cub  foot,  may  give  others  weighing 
not  more  than  70  to  80  Ibs.  At  98  Ibs  per  cub  ft.  1  bulk  of  pure  quartz, 
has  made  1.68  bulks  of  sand  ;  of  which  the  solid  occupies  .6  ;  and  the 
voids  .4.  But  if  this  same  sand  be  compacted  to  110  Ibs  per  cub  ft, 
then  1  measure  of  solid  quartz  makes  1  %  measures  of  sand ;  of  which 
%  are  solid,  and  %  voids.  Sand  is  very  retentive  of  moisture  ;  and 
when  in  large  bulks,  is  rarely  as  dry  as  that  above  in  this  table.  But 
•with  its  natural  moisture,  and  loose,  it  is  lighter  than  when  dry,  its 
average  weight  then  not  exceeding  about  85  to  90  Ibs  per  cub  ft ;  or 
106J4  to  112^  Ibs  per  struck  bushel.  See  Voids  in  Sand,  p  504. 

Sandstones,  fit  for  building,  dry,  2.1  to  2.73 131  to  171.  2.41 

"  quarried,  and  piled.  1  measure  solid,  makes  about  \%  piled 

Serpentines,  good 2.5  to  2.65 2.« 

Snow,  fresh  fallen 

"      moistened,  and  compacted  by  rain 

Sycamore,  perfectly  dry.     (See  footnote,  p  386.) .59 

1000  ft  board  measure  weighs  1.376  tons. 

Shales,  red  or  black 2.4  to  2.8 average..         2.6 

'"         quarried,  in  piles "      , 

Slate 2.7to2.9 "      ..         2.8 

Silver "      ..       10.5 

Soapstone,  or  Steatite 2.65  to  2.8 "      ..         2.73 

Steel,  7. 7  to  7.9.    The  heaviest  contains  least  carbon "      ..         7.85 

Steel  is  not  heavier  than  the  iron  from  which  it  is  made ;  unless  the 
iron  had  impurities  which  were  expelled  during  its  conversion  into 
steel. 

Bulphur average. .        2. 

Spruce,  perfectly  dry.       Footnote,  p  386 "  .4 

1000  ft  board  measure  weighs  .930  ton. 

flpelter,  or  Zinc 6.8  to  7.2 "      ..         7.00 

Sapphire ;  and  Ruby,  3.8  to  4 "      ..         3.9 

Tallow "      ..  .94 

Tar •«      ..         1. 

Trap,  compact,  2.8  to  3.2 "      ..         3. 

"     quarried;  in  piles "      

Topaz.  3.45  to  3.65 "      ..         8.55 


GRADES. 


Table  of  specific  gravities,  and  weights  — (Continued.) 


Names  of  Substances. 

Average 
Sp  Gr. 

Average 
Wt  of  a 
Cub  Ft. 
Lbs. 

7  35 

4/>9. 

20  to  30 

Water   pure  rain    or  distilled    at  32°  Fah    Barotn  30  ins  

62  417 

<.                  4,            ,<            ,<               .<                            g.jO        ««                   ««               «.       .4     . 

1 

62  H55 

<«          «       .1       t«         it              212°     "          "         "    "  . 

59  7 

At  60°,  a  cub  inch  weighs  .03607  Ib  ;  or  .57712  oz  avoir.      And  a  Ib  con- 
tains 27.724  cub  ins  ;  equal  to  a  cube  of  3.0263  inches  oil  each  edge. 
Water    sea   1  026  to  1  030                                                                     average 

1  028 

64  08 

Although  the  weight  of  fresh  water  is  in  practice  almost  invariably 
assumed  as  62^  Ibs  per  cub  ft,  yet  62^  would  be  nearer  the  truth,  at 
ordinary  temperatures  of  about  70°;  or  a  Ib  =  27.759  cub  ins  ;  and  a 
cub  in  zr  .5764  oz  avoir  ;  or  .4323  oz  troy  ;  or  252.175  grains.  The  grain 
is  the  same  in  troy,  avoir,  and  apoth. 
Wax,  bees  average  .  . 

97 

605 

Wines,  .993  to  1.04  

998 

62  3 

Walnut,  black   perfectly  drv      (See  footnote  p  386  )                           " 

61 

38 

1000  ft  board  measure  weighs  1.414  tons. 
Zinc,  or  Spelter,  6.8  to  7.2  " 

7  00 

437.5 

Zircon,  4.0  to  4.9  •' 

4.45 

Table  of  grades  per  mile;  or  per  1OO  feet  measured  hori- 
zontally.   See  p  629. 


Grade 
in  ft. 
per  mile. 

Grade 
in  ft. 
per  100  ft. 

Grade 
in  ft. 
per  mile. 

Grade 
in  ft. 
per  100  ft. 

Grade 
in  ft. 
per  mile. 

Grade 
iu  ft. 
per  100  ft. 

Grade 
in  ft. 
perniile 

Grade 
in  ft. 
per  100  ft. 

1 

.01894 

39 

.73S64 

77 

1.45833 

115 

2.17803 

2 

.03788 

40 

.75758 

78 

1.47727 

116 

2.19697 

3 

.05682 

41 

.77652 

79 

1.49621 

117 

2.21591 

4 

.07576 

42 

.79545 

80 

1.51515 

118 

2.23485 

5 

.09470 

43 

.81439 

81 

1.53409 

119 

2.25379 

6 

-11364 

44 

.83333 

82 

1.55303 

120 

2.27273 

7 

.13258 

45 

.85227 

83 

1.57197 

121 

2.29167 

8 

.15152 

46 

.87121 

84 

1.59091 

122 

2.31061 

9 

.17045 

47 

.89015 

85 

1.60985 

123 

2.32955 

10 

.18939 

48 

.90909 

86 

1.62879 

124 

2.34848 

11 

.20833 

49 

.92803 

87 

1.64773 

125 

2.36742 

12 

.22727 

50 

.94697 

88 

1.66666 

126 

2.38636 

13 

.24621 

51 

.96391 

89 

1.68561 

127 

2.40530 

14 

.26515 

52 

.98485 

90 

1.70455 

128 

2  42424 

15 

.28409 

53 

1.00379 

91 

1.72348 

129 

2.44318 

16 

.30303 

54 

1.02273 

92 

1.74242 

130 

2.46212 

17 

.32197 

55 

1.04167 

93 

1.76136 

131 

2.48106 

IS 

.34091 

56 

1.06061 

94 

1.78030 

132 

2.50000 

19 

.35985 

57 

1.07955 

95 

1.79924 

133 

2.51894 

20 

.37879 

58 

1.09848 

96 

1.81818 

134 

2.53788 

21 

.39773 

59 

1.11742 

97 

1.83712 

135 

2.55682 

22 

.41667 

60 

1.13636 

98 

1.85G06 

136 

2.57576 

23 

.43561 

61 

1.15530 

99 

1.87500 

137 

2.59470 

24 

.45455 

62 

1.17424 

100 

1.89394 

138 

2.61364 

25 

.47348 

63 

1.19318 

101 

1.91288 

139 

2.63258 

26 

.49242 

64 

1.21212 

102 

1.93182 

140 

2.65152 

27 

.51136 

65 

].  23106 

103 

1.95076 

141 

2.67045 

28 

.53030 

66 

1.25000 

104 

1.96969 

142 

2.6S939 

29 

.54924 

67 

1.26S94 

105 

1.98864 

143 

2.70833 

30 

.56818 

68 

1.28788 

105 

2.00758 

144 

2.72727 

31 

.58712 

69 

1.30682 

107 

2.02652 

145 

2.74621 

32 

.60606 

70 

1.32576 

108 

2.04545 

146 

2.76515 

33 

.62500 

71 

1.34470 

109 

2.06439 

147 

2.78409 

34 

.64394 

72 

1.36364 

110 

2.08333 

148 

2.80303 

35 

.66288 

73 

1.38258 

111 

2.10227 

149 

2.82197 

36 

.68182 

74 

1.40152 

112 

2.12121 

150 

2.84091 

37 

.70076 

75 

1.42045 

113 

2.14015 

151 

2.85985 

38 

.71970 

76 

1.43939 

114 

2.15909 

152 

2.87879 

GRADES. 


389 


Remark  on  preceding:  Table. 

If  the  grade  per  mile  should  consist  of  feet  and  tenths,  add  to  the  grade  per  100  feet  in  the  foregoing 
table,  that  corresponding  to  the  number  of  tenths  taken  from  the  table  below  ;  thus,  for  a  grade  of 
43.7  feet  per  mile,  we  have  .8H39  -4-  .01326=  .82765  feet  per  100  feet. 


Ft.  per  Mile. 

Per  100  Feet. 

Ft.  per  Mile. 

Per  100  Feet. 

Ft.  per  Mile. 

Per  100  Feet. 

.05 
.1 
.15 
.2 
.25 
.3 
.35 

.00094 
.00189 
.00283 
.00379 
.00473 
.00568 
.00662 

.4 
.45 
.5 
.55 
.6 
.65 

.00758 
.00852 
.00947 
.01041 
.01136 
.01230 

.7 
.75 
.8 
.85 
.9 
.95 

.01326 
.01420 
.01515 
.01609 
.01705 
.01799 

Table  of  grades  per  mile,  and  per  100  feet  measnred  hori- 
zontally, and  corresponding  to  different  angles  of  inclina- 
tion. For  another  table  per  100  ft  only,  see  p  629. 


II 

Feet  per 
mile. 

Feet  pei 

100  ft. 

H>   d 

Q    3 

Feet  per 
mile. 

Feet  pe 
100  ft. 

si>    a 
«    3 

Feet  per 
mile. 

Feet  pe 
100  ft. 

8?  .2 

c  S 

Feet  per 
mile. 

Feet  per 
100ft. 

0     1 

1.536 

.0291 

0  45 

69.11 

1.3090 

1  58 

181.3 

3.4341 

3  26 

316.8 

5.9994 

2 

3.072 

.0582 

46 

70.64 

1.3381 

2    0 

184.4 

3.4924 

28 

319.8 

6.0579 

3 

4.608 

.0873 

47 

72.18 

1.3672 

2 

187.5 

3.5506 

30 

322.9 

6.1163 

4 

6.144 

.1164 

48 

73.72 

1.3963 

4 

190.6 

3.6087 

82 

326.0 

6.1747 

5 

7.680 

.1455 

49 

75.26 

1.4254 

6 

193.6 

3.6669 

34 

329.1 

6.2330 

6 

9.216 

.1746 

50 

76.80 

1.4545 

8 

196.7 

8.7250 

36 

332.2 

6.2914 

7 

10.75 

.2037 

51 

78.33 

1.4837 

10 

199.8 

8.7833 

38 

335.3 

6.3498 

8 

12.29 

.2328 

52 

79.87 

1.5128 

12 

202.8 

3.8416 

40 

338.4 

6.4083 

9 

13.82 

.2619 

53 

81.40 

1.5419 

14 

205.9 

8.8999 

42 

341.4 

6.4664 

10 

15.36 

.2909 

54 

82.94 

1.5710 

16 

208.9 

3.9581 

44 

344.5 

6.5246 

11 

16.90 

.3200 

65 

84.47 

1.6000 

18 

212.0 

4.0163 

46 

347.6 

6.5832 

1-2 

18.43 

.3491 

56 

86.01 

1.6291 

20 

215.1 

4.0746 

48 

350.7 

6.6418 

13 

19.96 

.3782 

57 

87.54 

1.6583 

22 

218.1 

4.1329 

50 

353.8 

6.7004 

14 

21.50 

.4073 

58 

89.08 

1.6873 

24 

221.2 

4.1911 

52 

356.8 

6.7583 

15 

23.04 

.4364 

59 

90.62 

1.7164 

26 

224.3 

4.2494 

54 

359.9 

6.8163 

16 

24.58 

.4655 

1 

92.16 

1.7455 

28 

227.4 

4.3076 

56 

363.0 

6.8751 

17 

26.11 

.4946 

2 

95.23 

1.8038 

30 

230.5 

4.3659 

58 

366.1 

6.9339 

18 

27.64 

.5237 

4 

98.30 

1.8620 

32 

233.5 

4.4242 

4 

369.2 

6.9926 

19 

29.17 

.5528 

6 

101.4 

1.9202 

34 

236.6 

4.4826 

5 

376.9 

7.1384 

20 

30.72 

.5818 

8 

104.5 

1.9784 

36 

239.7 

4.5409 

10 

384.6 

7.2842 

21 

32.26 

.6109 

10 

107.5 

2.0366 

38 

242.8 

4.5993 

15 

392.3 

74300 

22 

33.80 

.6400 

12 

110.6 

2.0948 

40 

245.9 

4.6576 

20 

400.1 

7.5767 

23 

35.33 

.6691 

14 

113.6 

2.1530 

42 

248.9 

4.7159 

25 

407.8 

7.7234 

24 

36.86 

.6982 

16 

116.7 

2.2112 

44 

252.0 

4.7742 

30 

415.5 

7.8701 

25 

38.40 

.7273 

18 

119.8 

2.2094 

46 

255.1 

4.8325 

35 

423.2 

8.0163 

26 

39.94 

.7564 

20 

122.9 

2.3277 

48 

258.2 

4.8908 

40 

431.0 

8.1625 

27 

41.47 

.7855 

22 

126.0 

2.3859 

50 

261.3 

4.9492 

45 

438.7 

8.3087 

28 

43.01 

.8146 

24 

129.1 

2.4141 

52 

264.3 

5.0075 

50 

446.5 

8.4554 

29 

44.54 

.8436 

26 

132.1 

2.5023 

54 

267.4 

5.0658 

55 

454.2 

8.6021 

30 

46.08 

.8727 

28 

135.2 

2.5604 

56 

270.5 

5.1241 

5 

461.9 

8.7489 

31 

47.62 

.9018 

30 

138.3 

2.6186 

58 

273.6 

5.1824 

5 

469.6 

8.8951 

32 

49.16 

.9309 

32 

141.3 

2.6768 

3 

276.7 

5.2407 

10 

477.4 

9.0413 

33 

50.69 

.9600 

34 

144.4 

2.7350 

2 

279.7 

5.2990 

15 

485.1 

9.1875 

34 

52.23 

.9891 

36 

147.4 

2.7932 

4 

282.8 

5.3573 

20 

492.9 

9.3347 

35 

53  76 

1.0182 

38 

150.5 

2.8514 

6 

285.9 

5.4158 

25 

500.6 

9.4819 

36 

55.30 

1.0472 

40 

153.6 

2.9097 

8 

289.0 

5.4742 

30 

508.4 

9.6292 

37 

5683 

.0763 

42 

156.6 

2.9679 

0 

292.1 

5.5326 

35 

516.1 

9.7755 

38 

58.37 

.1054 

44 

159.7 

3.0262 

2 

295.1 

5.5909 

40 

523.9 

9.9218 

39 

59.90 

.1345 

46 

1(52.8 

3.0844 

4 

298.2 

5.6493 

45 

531.6 

10.068 

40 

61.44 

.1636 

48 

165.9 

3.1427 

6 

301.3 

5.7077 

50 

539.4 

10.215 

41 

62.97 

.1927 

50 

169.0 

3.2010 

8 

304.4 

i).7660 

55 

547.2 

10.362 

42 

61.51 

.2218 

52 

172.0 

3.2592 

20 

307.5 

5.8244 

6 

555. 

10.510 

43 

66.04 

1.2509 

54 

175.1 

3.3175 

22 

310.5 

5.8827 

44 

67.57 

1.2800 

56 

178.2 

3.3758 

24 

313.6 

5.9410 

On  a  turnpike  road  1°  38',  or  about  1  in  35,  or  151  ft  per  mile,  is  the 

greatest  slope  that  should  be  given  to  allow  horses  to  trot  down  rapidly  with  safety.  In  crossing 
mountains,  this  is  often  increased  to  3,  or  even  to  5°.  It  should  never  exceed  2%°,  except  when  abso- 
lutely necessary. 


390 


RAIL- JOINTS,  AND   CHAIRS. 


TABLE  OF  ACRES  REQUIRED  per  mile,  and  per  1OO  feet, 
for  different  widths. 


Width. 

Feet. 

Acres 
per 
Mile. 

Acres 
per 
100  Ft. 

Width. 
Feet. 

Acres 
per 
Mile. 

Acres 
100  Ft. 

Width. 
Feet. 

Acres 
per 
Mile. 

Acres 
per 
100  Ft. 

Width. 
Feet. 

Acres 
per 
Mile. 

Acres 
per 
100  Ft. 

1 

.121 

.002 

26 

3.15 

.06.) 

52 

6.30 

.119 

78 

9.45 

.179 

2 

.242 

.005 

27 

3.27 

.062 

53 

6.42 

.122 

79 

9.58 

.181 

3 

.364 

.007 

28 

3.39 

.064 

64 

6.55 

.124 

80 

9.70 

.184 

4 

.485 

.009 

29 

3.52 

.067 

55 

6.67 

.126 

81 

9.82 

.186 

5 

.606 

.011 

30 

3.64 

.069 

56 

6.79 

.129 

82 

9.94 

.188 

6 

.727 

.014 

31 

3.76 

.071 

57 

6.91 

.131 

Y* 

10. 

.189 

7 

.848 

.016 

32 

3.88 

.073 

*A 

7. 

.133 

83 

10.1 

.190 

8 

.970 

.018 

33 

4.00 

.076 

58 

7.03 

.133 

84 

10.2 

.193 

Y± 

1. 

.019 

34 

4.12 

.078 

59 

7.15* 

.135 

85 

10.3 

.195 

9 

1.09 

.021 

35 

4.24 

.080 

60 

7.27 

.138 

86 

10.4 

.197 

10 

1.21 

.023 

36 

4.36 

.083 

61 

7.39 

.140 

87 

10.5 

.200 

11 

1.33 

.025 

37 

4.48 

.085 

62 

7.52 

.142 

88 

10.7 

.202 

12 

1.46 

.028 

38 

4.61 

.087 

63 

7.64 

.145 

89 

10.8 

.204 

13 

1.58 

.030 

39 

4.73 

.090 

64 

7.76 

.147 

90 

10.9 

.207 

14 

1.70 

.032 

40 

4.85 

.092 

65 

7.88 

.149 

% 

11. 

.209 

15 

1.82 

.034 

41 

4.97 

.094 

66 

8. 

.151 

91 

11.0 

.209 

16 

1.94 

.037 

& 

5. 

.094 

67 

8.12 

.154 

92 

11.2 

.211 

X 

2. 

.038 

42 

5.09 

.096 

68 

8.24 

.156 

93 

11.3 

.213 

17 

2.06 

.039 

43 

5.21 

.099 

69 

8.36 

.158 

94 

11.4 

.216 

18 

2.18 

.041 

44 

5.33 

.101 

70 

8.48 

.161 

95 

11.5 

.218 

19 

2.30 

.044 

45 

5.45 

.103 

71 

8.61 

.163 

96 

11.6 

.220 

20 

2.42 

.04H 

46 

5.58 

.106 

72 

8.73 

.165 

97 

11.8 

.223 

21 

2.55 

.048 

47 

5.70 

.108 

73 

8.85 

.168 

98 

11.9 

.225 

22 

2.67 

.051 

48 

5.82 

.110 

74 

8.97 

.170 

99 

12. 

.227 

23 

2.79 

.053 

49 

5.94 

.112 

^ 

9. 

.170 

100 

12.1 

.230 

24 

2.91 

.055 

1A 

6. 

.114 

75 

9.09 

.172 

% 

3. 

.057 

50 

6.06 

.115 

76 

9.21 

.174 

25 

3.03 

.057 

51 

6.18 

.117 

77 

9.33 

.177 

RAIL-JOINTS  AXI>  CHAIRS. 

A  railroad  track  being  weakest  at  the  joints  between  the  rails,  where  they 
are  deprived  of  their  vertical  strength,  has  of  course  a  greater  tendency  to  bend  at 
those  points ;  and  this  bending  produces  an  irregularity  in  the  movement  of  the 
train,  which  is  detrimental  to  both  rolling-stock  and  track.  Moreover,  that  end  of  a 
rail  upon  which  a  loaded  wheel  is  moving,  bends  more  than  the  adjacent  unloaded 
end  of  the  next  rail ;  so  that  when  the  wheel  arrives  at  said  second  rail,  it  imparts  to 
its  end  a  severe  blow,  which  injures  it.  Thus,  the  ends  of  the  rails  are  exposed  to 
far  more  injury  than  its  other  portions.  Numerous  devices  have  been  resorted  to  for 
strengthening  the  joints  of  the  rails,  with  a  view  of  preventing  this  bending  entirely ; 
or,  at  least,  of  causing  the  two  adjacent  rail-ends  to  bend  equally,  and  together;  so 
as  to  avoid  the  blows  alluded  to.  None  of  these  joint-fastenings,  known  as  chairs, 
fish-plates,  wooden  blocks,  &c,  have  proved  entirely  satisfactory. 

Much  of  the  deficiency  ascribed  to  the  fastenings,  is,  however,  really  due  to  want  of  stability  in  th» 
'cross-ties  at  the  joints;  and  more  attention  must  be  directed  to  this  latter  consideration,  before  an 
efficient  fastening  can  be  obtained.  Observation  shows  that  when  the  joint-ties  are  very  firmly 
bedded,  almost  any  of  the  ordinary  fastenings  will,  (if  the  joint  is  placed  between  two  ties,  instead 
of  resting  upon  a  tie,)*  answer  very  well ;  whereas,  when  the  cross- ties  are  so  insecurely  bedded  as  to 
play  up  and  down  for  half  an  inch  or  more  under  the  driving-wheels  of  the  engines,  the  strongest  and 
most  effective  fastenings  soon  become  comparatively  inoperative.  All  the  parts  of  the  best  of  them 
will  in  that  case  become  gradually  loosened,  warped,  bent,  or  broken.  This  remark  applies  to  all  the 
fishing-splices,  chairs,  long  wooden  blocks,  bolts,  spikes.  &c,  in  present  use. 

Experience  has  established  the  superiority  of  suspended  joints,  over  supported 
ones.  On  a  portion  of  a  track  carrying  a  heavy  business,  with  joint-fastenings  closely 
resembling  that  in  Figs  10,  (with  long  wooden  blocks  B;  and  long  fish-pieces  c,) 
alternate  joints  were  suspended  between  two  ties ;  while  the  intermediate  one  rested 
upon  a  center  tie  ;  the  blocks,  however,  extending  over  three  ties.  The  ends  of  the 
rails  were  more  injured  by  crushing  and  brooming  in  the  latter  than  in  the  former. 
And  so  with  a  number  of  other  patterns  of  short  joint-fastenings  of  wrought  iron. 
Long  fastenings,  perhaps,  possess  but  little  superiority  over  short  ones,  where  the 
track  is  not  kept  in  good  repair;  for  the  great  bearing  of  the  former,  although  impart- 

•In  the  first  oaae  the  joint  i.s  called  a  SUSPENDED  one  ;  in  the  last  a  SUPPORTED  one. 


RAIL-JOINTS,  AND    CHAIRS- 


391 


ing  increased  firm  ness  on  a  good  track,  becomes  converted  into  a  powerful  leverage, 
by  which  it  accelerates  its  own  destruction,  in  a  bad  one.  An  element  in  the  injury  of 
joints,  is  the  omission  of  proper  fastenings  at  the  center  of  the  rails.  Each  rail 
should  be  so  firmly  attached  to  the  cross-ties  at  and  near  its  center,  as  to  compel  the 
contraction  and  expansion  to  take  place  equally  from  that  point,  toward  each  end. 

It  would  probably  be  somewhat  difficult  to  accomplish  this  perfectly.  The  at- 
tempts hitherto  made  have  failed. 

One  of  the  earliest  suggestions  for  a  joint  fastening,  was  the  limti-joint.  or 
fish-splice,  or  fish-plates,  Fig  1,  introduced  upon  the  Newcastle  and  French- 
town  R.R.  in  Delaware,  by  Kobt.  H.  Barr,  in  1843.* 

This  consists  of  two  strips  of  iron,  z  z.  called  fishes,  rolled  to  fit  the  sides  of  the  rail;  and  bolted 
together  and  to  the  rails,  by  either  keybolts,  or  screwbolts.  Usually  4  bolts  are  used;  sometimes  6. 
The  fishes  are  usually  about  %  to  %  inch  thick.  The  curved  shape  of  the  inner  top  and  bottom  edges 
of  these  splices  ;  and  of  the  top  and  bottom  of  the  stem  of  the  rai  1,  creates  a  tendency  to  yield  under  the 
bending  of  the  joint  under  heavy  loads,  as  shown  in  an  exaggerated  manner  at  Figs  2  and  3.  Either 
of  these  brings  a  great  strain  upon  the  bolt,  and  is  apt  to  pull  it  apart.  At  a  later  period,  Edward 
Miller,  C  E,  of  Philada,  in  order  to  remedy  these  defects,  introduced  the  rolling  of  the  rails  with  square 
•boulders,  as  in  Fig  4,  which  however  soon  fell  into  disuse. 


Fish-splices  are  frequently  made  with  a  shallow  groove  about  %  inch  deep,  on  their 
outer  side,  as  at  n  n,  and  F,  Figs  10.  This  groove  receives  either  the  square  head  of 
the  bolt,  which  is  then  inserted  first,  and  the  nut  afterwards  screwed  on ;  or  else  the 
nut  is  first  placed  in  the  groove,  and  the  bolt  afterwards  screwed  into  it.  It  was 
supposed  that  the  nut  and  bolt  would  thus  be  prevented  from  loosening  and  un- 
screwing under  the  jarring  of  the  trains.  The  result  however  has  not  proved  satis- 
factory ;  and  inasmuch  as  the  groove  also  weakens  the  plates,  it  is  falling  into 
disuse.  The  bolt  holes  through  either  the  plates  or  the  rail,  are  made  about  %  inch 
longer  than  high,  to  allow  the  rails  to  expand  and  contract.  Such  splices,  about  18 
to  24  ins  long,  and  ^  to  %  ins  thick,  with  4  bolts  about  4  ins  long,  by  %  tc,  %  inch 
diarn,  and  placed  between  two  joint  cross-ties  about  1  ft  apart  in  the  clear,  were  for 
many  years  the  most  approved  in  this  country;  but  on  some  roads  of  heavy  traffic, 
they  are  being  supplanted  by  the  angle-bar  splice,  or  fish  plate,  similar  to 
those  shown  in  Fig  11,  but  usually  without  the  enlargements  at  cand  a.  The  spikes 
which  confine  the  angle  bars  to  the  cross-ties,  serve  to  counteract  the  creeping 
alluded  to  3  or  4  lines  below,  and  which  the  common  splice  does  not  prevent. 

Under  the  extremes  of  temperature  in  the  United  States,  bar  iron  expands  or  contracts  about  1  part 
in  916 ;  or  I  inch  in  76}£  feet ;  consequently,  a  rail  30  ft  long  will  vary  J-~-  inch  ;  and  one  20  feet  long 
fullv  x  inch. 

Beside  this,  the  rails  are  very  liable  to  move  or  creep  bodily 
in  the  direction  of  the  heaviest  trade;  and  by  this  process  also  the 
joint-fastenings  are  exposed  to  additional  strain  and  derangement. 

The  patent  Stop-Chair,  Of  Mr  Jollll  A.  WilSOn,  €  E,  is  said  to  prevent  this. 
It  consists  simply  of  a  piece  of  plate-iron,  bent  to  fit  to  the  fish-plate,  and  to  the  outer  top  part  of  the 
base  of  the  rail.  It  is  about  3  ins  wide,  by  7  ins  long.  Its  lower  end  is  held  to  the  tie  by  two  common 
rail-spikes  ;  and  its  upper  end  is  bolted  to  the  outside  of  a  fish-plate  by  one  of  the  bolts  which  confine 
the  fish  to  the  rail.  Two  chairs  are  used  at  each  joint,  one  at  each  end  of  a  fish.  Made  by  the 
Wharton  Switch  Co,  Phila.  .\Vt.  I  ft>  e;ich- 

All  rails  appear  to  l>eoome  elongated  very  slightly  at  their  ends  by  use  ;  and  this 
renders  a  full  allowance  for  contraction  and  expansion  the  more  necessary. 

It  i»  a  remarRablo  fact,  not  satisfactorily  accounted  for,  that  when  lengths  of  from 
100  yards,  to  some  miles,  of  mils  hove  been  perfectly  welded,  or  riveted  together  tightly,  and  spiked 
to  the  ties  as  usual,  no  elongation  or  contraction  by  heat  or  cold  could  be  detected. 

What  is  called  the  compensating  fish-joint  is  in  use  on  many 
roads. 

Its  novelty  consists  in  a  peculiar  cup-washer,  enclosing  a  ring  of  India-rubber  2  inches  diam.  and 
^  inch  thick  ;  and  placed  under  the  nut  of  each  of  the  four  bolts.  It,  is  claimed  that  this  prevents  the 
nuts  from  unscrewing  themselves;  and  diminishes  the  jar  and  noise  of  passing  trains. 


-5f  Average  price  in  Philada,  in  1880,  of  wrought-iron  chairs,  about  4  to  5  cts  per  Ib.  Common 
fish-plates  the  same.  Bolts  and  nuts  6  to  7  cts,  rough  finish. 

t Price  of  common  fish-plates.  Phila,  1882.  about  2^  cts  per  Ib.  Angle-bar  ones,  about  3  cts. 
Bolts  aud  nuts  4  cts  per  Ib.  THE  I  HILA  IRON  &  STEEL  Co,  939  North  Delaware  Avenue,  make  both 
kinds. 


392 


KAIL-JOINTS,  AND    CHAIRS. 


Each  of  the  two  fish-plates  of  each  joint,  is  usually  from  16  to  24  ins  long,  fcy  %  to  %  ins  thick.  Th« 
weight  of  a  complete  joiut,  with  its  four  j^-iuch  bolts,  cup-washers,  &c,  from  Itt  to  24  Ibs.  They  an 
generally  used  without  chairs. 


A  new  form  of  fish-plates,  contrived  by  Mr.  S, 

E.  Pettier,  is  shown  at  Fig  4%. 

They  are  about  18  ins  long.  Near  the  bottom  they  are  connected  by  two 
bolts  c;  and  at  top  by  four  bolts  v.  These  last  have  the  billings  lock-nut 
•washer  a;  for  which  see  page  375.  This  joint  seems  to  be  much  more 
effective  than  the  common  tish,  against  both  vertical  and  lateral  force.  It  is 
of  course  a  suspended  joint ;  and  a  quadrant-shaped  piece  is  cut  out  from  the 
ends  of  the  vertical  flanges  c  c,  to  prevent  interference  with  the  cross-ties. 
It  has  been  tried  on  the  Reading  RR  with  good  results,  but  not  sufficient  to 
displace  the  common  fish-plates. 


Fig  5  was  an  early  form  of  supported  wrought-iror 
chair.  It  is  still  employed  on  some  roads  of  light 
traffic.  It  is  about  7  ins  square,  %  thick,  and  weight 
10K>8. 


The  rolled-iron  sleeve  -chair,  Fig  6, 

made  by  the  Phoenix  Iron  Co,  of  Philadel- 
phia, and  by  some  other  establishments,  is 
in  extensive  use. 

It  is  first  rolled  in  long  pieces  ;  and  then  sawed  into 
such  lengths  as  may  be  ordered.  When  supported,  il 
is  usually  either  9  or  10  inches  long;  requiring  4 
spikes  ;  and  when  suspended,  about  2  ft  long  ;  with  £ 
spikes.  It  is  chiefly  used  as  a  supported  chair;  and 
as  such,  is  a  favorite  on  many  roads  of  moderate  trade. 
Under  heavy  traffic,  it  is  deficient  in  vertical  strength, 
especially  when  suspended.  The  curved  lips  then  fre- 
quently break  off  at  the  ends;  and  occasionally  th« 
chair  breaks  entirely  across  at  the  spike-holes.  An 
objection  to  long  chairs  of  this  pattern,  is  the  diffi- 
culty of  sliding  them  upon  the  rails  ;  and  the  still 
greater  one  of  sliding  them  off,  when  either  a  chair, 
or  a  rail,  is  to  be  removed.  See  Fig  8. 

Ibs  per  inch  of  length;  so  that 
ft>s.    With  rails  20,  24,  or  30  feet 


The  sleeve-chair,  Fig1  6,  -weighs 

chairs  of  9,  10,  or  24  ins  long,  weigh  13%,  15,  or 

long,  there  would  in  one  mile  of  single-track  road,  be  528,  440,  or  352  chairs. 

The  9-inch  ones  weigh  7128.  5940,  or  4752  R>s  ;  the  10-inch,  weigh  7920,  6600,  or  5280  tts  ;  the  24- 
inch,  weigh  19008,  158W,  or  12672  fts.*  But  an  addition  should  be  made  to  these  weights  and  costs, 
of  say  5  per  cent,  or  -^  part,  or  %  mile-in  every  5  miles,  for  turnouts,  sidings,  &c.  Allowing  fom 
half-pound  spikes  to  a"  chair,  there  are  2112,  1760,  1408  chair-spikes  ;  or  1056,  880,  or  704  fts  per  mile. 

Figs  7  show  Fisher's  wrousrht-iroii  rail-Joint,  highly  esteemed  on  the 
Lehigh  Valley  road  ;  where  it  is  extensively  used  under  a  heavy  traffic  of  coal,  mer* 


*  Price  io  Philada,  1880,  5  cts  per  tt. 


RAIL- JOINTS,  AND   CHAIRS. 


393 


chandise,  and  passengers.  In  this,  the  disadvantages  before  alluded  to  of  the  long 
sleeve-chair,  are  entirely  got  rid  of.  At  A  is  shown  a  transverse  section  of  the  fas- 
tening, with  the  rail  in  its  place ;  while  the  other  portions  of  Figs  7  show  its  details 
separately.  It  consists  of  a  simple  rolled  chair  c,  6  ins  square,  and  %  inch  thick ; 
with  two  of  its  sides  turned  up  %  of  an  inch.  In  its  base  are  4  bolt-holes,  through 
which  are  passed  from  below,  two  curved  double  screw-bolts,  of  round  iron  1  inch 
dium ;  one  of  which  is  shown  at  "D.  Over  the  tops  of  these  bolts  are  let  down  the 
two  rolled  pieces  or  bars  1 1,  each  as  long  as  the  chair,  or  t>  ins;  and  shaped  to  fit  the 
top  of  the  base  of  the  rail.  The  nuts  n  n  bind  firmly  together  these  bars  1 1,  the  rails 
and  the  chair  c.  To  prevent  the  nuts  n  (see  Fig  B)  from  unscrewing  themselves,  a 
small  rod  s  g,  of  %  inch  square  iron,  notched  on  the  side  next  to  the  nut,  is  inserted 
between  the  nut  and  the  rail;  one  end  of  it,  s,  being  bent  to  clasp  the  nut.  In  Fig 
A,  the  ends  g  of  these  rods  are  shown  by  2  small  black  squares.  The  whole  is  sus- 
pended between  two  cross-ties,  7  inches  apart  in  the  clear;  on  each  of  which  the  rail 
is  simply  held  by  two  spikes,  as  on  the  other  ties  ;  without  any  chair.  The  chairs  c, 
when  suspended,  very  seldom,  if  ever,  break.  Those  of  them  first  used  on  this  road 
were  but  ^  inch  thick,  and  were  supported  upon  the  ties;  only  one  double  bolt 
being  used  instead  of  two.  They  frequently  broke  across  the  bolt-holes. 

This  excellent  fastening  was  patented  by  Mr  Mark  Fisher,  of  the  iron  firm  of  Fisher  &  Norris,  of 
Trenton,  N  Jersey.  The  same  firm  make  also  a  longer  chair,  with  3  double  bolts. 

Notwithstandingthe  fact  that  the  cross-ties  are  but  from  1%  to  8  ft  long,  (gauge  4  ft  8^,)  the  road  is 
a  very  pleasant  one  to  ride  upon:  there  is  but  little  jolting,  or  rattling  at  the  joints  ;  and  the  ends  of 
'  the  rails  appear  to  be  as  well  protected  as  by  any  joint  fastening  we  have  seen.  The  track  is  well 
watched,  and  carefully  kept  in  line  :  a  precaution  especially  necessary  in  a  road  doing  the  heavy  busi- 
ness of  the  Lehigh  Valley  R  R.  The  success  of  this  joint  confirms  our  opinion  that  a  great  length  of 
fastening,  or  of  cross-tie,  is  not  in  itself  essential  to  a  good  track. 

The  weight  of  this  fastening  complete  is  about  13*4  Ibs.  With  rails  20,  24,  or  30 
ft  long,  there  would  in  one  mile  of  single-track  road,  be  528,  440,  or  352  chairs, 
weighing  7128,  5940,  or  4752  Ibs;  or,  allowing  5  per  cent  for  turnouts  and  sidings, 
say  7484,  6237,  or  4990  ft>s. 

A  modification  of  the  sleeTe- 
chair.  Fig  6,  has  been  introduced  upon  the 
Hudson  River  R  R,  involving,  to  some  extent, 
the  same  principle  as  that  on  the  Lehigh  Val- 
ley. It  is  shown  at  Fig  8. 

As  but  one  bar  Is  here  used,  and  as  it  is  placed  on  the 
outside  of  the  rail,  its  nuts  are  not  in  danger  of  being 
struck  by  the  flanges  of  the  wheel  when  a  low  rail  is 
employed;  or  when  the  wheel  tires  become  worn,  thus 
bringing  the  flanges  lower  down.  It  is  probable  that  a 
suspended  chair  of  this  kind,  not  more  than  7  inches 
long,  and  with  its  lip  s  considerably  thicker  than  they 
are  usually  made,  would  equal  the  Lehigh  Valley  one  in 
efficiency.  k-j 

Figs  9  show  a  remarkable  snspended  fastening-,  called  the  King-- 
Joint: highly  approved  of  at  one  time  on  the  Camden  &  Amboy  road,  on  which 
it  was  employed  for  many  years,  under  a  heavy  traffic;  to  the  almost  entire  exclu- 
sion of  others,  except  for  experimental  comparison. 


K68 


It  was  invented  in  1851,  by  Edwin  A  Stevens,  President  of  the  road.  The  chief  engineer,  Col  Cook, 
informed  the  writer  that  after  several  years'  trial  of  a  great  variety  of  fastenings,  he  gave  a  decided 
preference  to  this.  It  has,  however,  one  very  serious  defect:  namely,  that  when,  in  consequence  of 
a  bad  foundation,  the  cross-ties  play  too  freely  up  and  down,  the  ends  of  the  rails  frequently  split 
off;  either  at  top,  as  shown  by  the  line  I ;  or  at  bottom,  as  shown  by  k.  We  have  never  learned,  how- 
ever,  that  this  was  productive  of  accident  to  trains.  When  the  break  occurs  above  the  slot  v  v  the 
jolting  of  course  gives  notice  of  the  fact,  and  a  new  rail  is  put  in  ;  but  when  below  the  slot  the 
ring  and  its  wedges  continue  to  uphold  the  broken  piece  ;  at  times  probably  for  days  before  the  frae- 
tare  is  observed.  Nearly  all  the  fractures  are  below  the  slet. 


394 


RAIL- JOINTS,  AND    CHAIRS. 


This  fastening  consists  of  a  simple  welded  triangular  ring  tea,  (in  the  end  view ;  or  m  in  the  aid* 
Tiew,)  %  an  inch  thick,  and  3>4  ins  wide.  This  ring  passes  through  a  slot  vv,  (see  middle  fig,)  4  tag 
long,  cut  into  the  adjacent  rail-ends.  Two  cast-iron  wedges  w  w,  6  to  8  ins  long,  of  a  shape  to  fit  the 
ring  and  the  rail,  are  inserted  between  them;  and  a  thinner  one  S8,  of  plate  iron,  below  the  rail. 
The  first  are  cast  around  a  cylindrical  rod  of  rolled  iron,  (w  of  the  end  view,  ii  of  the  side  view,) 
about  %  inch  diam;  aud  a  little  longer  than  the  wedges  ;  for  increasing  their  strength,  and  for  pre- 
venting them  from  falling  out  from  the  chair  in  case  they  should  break,  which  they  sometimes  do. 

The  joint  is  suspended  between  two  cross- ties,  1  ft  apart  in  the  clear. 

The  writer  examined  it  carefully  for  many  consecutive  years  ;  and  so  far  as  regards  protecting  the 
ends  of  the  rails  from  wear,  and  freedom  from  jolting,  it  certainly  appears  to  him  to  be  fully  as  ef- 
fective as  any  other.  He  has  seen  joints  laid  alternately  with  it;  with  heavy  cast-iron  supported 
chairs;  and  with  long  splices  somewhat  similar  to  the  one  next  to  be  described,  but  extending  over 
3  ties.  With  the  exception  of  the  breaking  alluded  to,  the  ring  appeared  to  be  decidedly  superior  to 
the  supported  chair  ;  and  by  no  means  inferior  to  the  long  splice. 

We  have  also  repeatedly  compared  it  with  the  heavy  rolled  joint-fastening  Pig  13,  which  is  5  feet 
long,  and  rests  on  three  cross-ties.  Under  similar  conditions  of  road-bed,  whether  good  or  bad,  we 
have  been  unable  to  perceive  that  the  ring-joint  (when  the  ties  are  but  a  foot  apart  in  the  clear) 
yielded,  under  heavv  trams,  any  more  than  it. 

The  weight  of  a  ring  itself  i?  5>£  Ibs  I  tne  three  wedges  about  1%  Ibs ;  total  13  fts :  or  nearly  the 
same  as  the  Lehigh  Valley  fastening,  Fig  7.  The  gauge  of  the  C  &  A  road  is  4  ft,  10  ins ;  and  th« 
cross- ties  average  about  8^  ft  long. 

Without  pretending  to  advocate  this  joint  in  its  present  shape,  we  have  thought  proper  to  describ* 
it,  as  furnishing  useful  hints  on  this  important  subject.  It  was  the  first  that  led  us  to  doubt  whether 
some  long  fastenings  did  not  furnish  the  elements  of  their  own  destruction,  by  the  long  leverage 
which  they  afforded  for  bending  and  breaking  themselves  to  pieces,  when  the  ties  are  badly  supported. 
Since  the  death  of  Col  Cook,  a  gradual  substitution  of  long  combined  wood  and  iron  fastenings,  in 
place  of  the  ring-joint, has  been  commenced ;  and  is  applied  wherever  the  latter  is  found  to  have  frac- 
tured a  rail. 

Figs  10  represent  the  combined  suspended  joint-fastening  introduced  upon  the 
Philada  &  Reading  railroad  by  J.  Duttoii  Steele,  C  E. 


The  very  heavy  tonnage  of  this  road,  exceeding  perhaps  that  of  any  other  in  existence,  demandt 
particular  attention  to  the  joints;  and  Mr  Steele's  combination  of  Trimble's  long  wooden  splice,  with 
a  long  wrought-iron  fish-splice,  and  a  longer  rolled-iron  chair  under  the  whole,  certainly  proved 
itself,  for  many  years,  superior  to  any  other  of  the  numerous  cast  and  wrought  iron  chairs  previously 
tried  on  that  road.  The  joint  is  suspended  between  2  cross- ties,  placed  1  ft  apart  in  the  clear.  B  is 
a  block  (known  as  Trimble's  splice)  of  oak.  3  by  3  ins.  and  3  ft  long,  dressed  on  one  side  to  fit  the  out 
side  of  the  rail :  and  c  is  a  rolled  fish,  17  ins  long,  (shown  more  in  detail  at  F.)  placed  on  the  insidt 
of  the  rail.  This  fish  and  the  oak  block  are  bolted  together,  through  the  rail,  by  two  %-inch  screw- 
bolts  a  a,  13  inches  apart.  Under  the  rail  is  a  rolled  chair  d.  2  ft,  8  ins  long.  5  ins  wide,  and  ^  inch 
thick ;  turned  up  ^  inch  along  each  of  its  2  sides,  and  fastened  to  the  2  wooden  cross-ties  by  4  hook- 
headed  spikes,  ^  inch  square,  by  5V$  ins  long.  The  heads  of  the  two  screw-bolts  are  made  somewhat 
oblong,  (about  %  inch  by  !>£,)  for  fitting  into  the  groove  nn.  seen  along  one  side  of  the  fish  ;  so  as  to 
prevent  the  tendency  of  the  bolts  to  revolve  under  the  action  of  the  trains,  and  thus  unscrew  the  nut 
at  the  other  end.  The  nuts,  however,  unscrew  themselves,  notwithstanding  this  precaution.  We 
consider  13  ins  to  be  too  far  apart  for  the  two  screw-bolts,  inasmuch  as  it  allows  the  fish  in  time  to 
become  bulged  out  from  the  rail  quite  perceptibly,  between  the  spikes,  under  the  downward  pressure 
on  the  ends  of  the  rails.  This  impairs  its  efficiency.  Four  bolts  would  be  better.  The  strain  on  the 
screw-bolts  is  great,  both  vert  and  nor;  and  it  becomes  greater  as  the  wooden  blocks  in  time  lose  (as 
they  do)  their  close  fit  to  the  sides  of  the  rails.  The  blocks  then  cease  to  act  in  perfect  unison  with 
the  other  parts  of  the  fastening,  in  sustaining  passing  loads;  and  when  the  track  is  not  kept  in  good 
order,  the  various  parts  mav  plainly  be  seen  to  yield  and  move  in  various  directions,  independently 
of  each  other.  The  bolt-nuts  then  loosen :  and  the  fish  pieces,  long  chairs,  and  long  bolts,  become 
bent:  and  sometimes  split,  or  break  entirely.  The  long  wooden  blocks  B  crush  and  decay  soonest 
near  where  their  tops  are  in  contact  with  the  rail.  They,  however,  have  an  average  life  of  6  to  8  years, 
upon  roads  kept  in  tolerable  order. 

These  remarks  are  more  or  less  applicable  to  all  joint-fastenines  with  long  splices,  when  the  ties  are 
allowed  to  become  unstable.  With  the  care  taken  to  prevent  this  on  the  Reading  mad.  Mr  Steele's 
fastening  haa  sustained  the  enormous  traffic  of  the  line  very  well  for  many  years.  It  will,  however, 
be  supplanted  by  better  ones  more  recently  introduced. 

The  iron  in  one  of  these  joints,  including  bolt*  and  spikes,  weiehs  nhnnt  32  ft><» :  and  with  rails  ot 
20,  24,  or  80  ft  long,  there  would  be  528,^40,  or  352  joints;  or  16896,  14080,  or  11264  D)8,  per  mile  ol 
single-track  road. 


RAIL- JOINTS,    AND   CHAIRS. 


395 


Fig  11  shows  the  Fritz  and  Sayre  Splice  Plate  now  (1883)  used  on  the 
Lehigh  Valley  R.  R.  of  very  heavy  traffic ;  together  with  a  cross  section  of  the  67 
Ibs  to  a  yard  steel  rail  of  that  road,  as  designed  by  Robert  H.  Sayre.  Esq.,  Chief  Eng 
and  Supt;  and  the  standard  cast-iron  car-wheel  tread  of  the  New  York  Central ;  all 
carefully  drawn  to  scale,  and  one-third  of  actual  size.  These  forms  of  rail  and 
splice  are  the  result  of  careful  study,  and  each  detail  has  been  modified  from  time 
to  time  as  experience  dictated,  until  now  they  are  probably  the  most  perfect  in  this 
country.  Mr.  Sayre  places  the  stems  of  the  two  plates  much  farther  apart  than  usual, 
thus  giving  the  joint  greater  lateral  strength ;  at  the  same  time  adding  to  its  vertical 
strength  by  the  support  given  to  the  entire  lower  side  of  the  rail-head  by  the  upper  en- 
largement c ;  while  the  lower  one  a  secures  a  full  bearing  on  the  foot  of  the  rail.  The 
oblong  head  6  of  the  bolts  prevents  them  from  unscrewing;  and  the  lock-nut  washer  v 
(Shaw's,  alias  the  Verona,  p  375)  is  intended  to  do  the  same  with  the  nut.  The 
weight  of  the  two  iron  splice-plates  (each  2  ft  long)  is  40  Ibs;  the  4  bolts  (5  ins 
long,  %  diam)  with  their  nuts  and  washers  v,  4  Ibs;  the  4  spikes  (5%  ins  long,  -r9g 
square,  two  to  each  plate,  their  heads  shown  at  o  o)  2  Ibs.  Total  46  Ibs  per  joint. 
The  drilled  bolt  holes  in  the  stem  of  the  rail  are  1  inch  diam,  to  allow  the  rails  to 
contract  and  expand. 

These  splices  are  made  by  the  Bethlehem  Iron  Co,  John  Fritz,  Supt, 
Bethlehem,  Lehigh  Co,  Penna. 


396 


RAIL-JOINTS,  AND   CHAIRS. 


Fig  13  is  Charles  E.  Smith,  Esqr's,  inverted  T  joint-splice,  or 

fastening,  of  rolled  iron,  for  the  U  rail  of  the  Camden  &  Atlantic  railroad. 

It  is  5  ft  long,  and  rests  on  3  cross- ties,  one  of  which  is  under  the 

P.    49iPi  rail-joint.    Its  base  is  4%  ins  wide,  by  ^  inch  thick.     The  vert  web  is 

ICTlJ       -       !  2  ins  high,  and  1  inch  thick  where  it  joins  the  base.     This  joint-piece  is 

riveted  loosely  to  the  rails  by  8  rivets  in  pairs ;  two  pairs  being  between 
each  two  ties.  Eight  hook  headed  spikes,  5%  inches  long,  by  %  inch 
square,  in  pairs,  (2  pairs  on  the  center  tie.)  confine  both  rails  and  s'plices 
to  the  cross-ties.  This  splice  keeps  the  rails  in  position  very  well ;  but 
it  would  be  easier  upon  the  ends  of  the  rails  if  it  rested  on  four  ties,  so 
as  to  suspend  the  rail-joint  between  two  of  them.  The  writer  has  been 
unable  to  perceive  that  the  deflection  under  trains  is  any  less  with  this 
splice,  than  with  such  as  Figs  7,  9,  &c,  when  the  ties  ara  not  firm. 

The  weight  of  iron  in  one  splice,  including  spikes,  is  about  70  Bs. 

With  rails  20,  24,  or  30  ft  long,  with  5  per  ct  allowance  for  turnouts,  &c,  there  would  be  36960,  30800, 
or  24640  8>s  per  mile  of  single  track;  consequently  it  is  an  expensive  fastening.  The  great  number 
of  spikes  driven  into  three  ties  prevents  the  rails  from  creeping. 

A  SIMILAR  JOINT,  but  shorter,  and  inverted,  and  suspended  between  ties,  has  been  suggested  for 
edge  rails;   see  Fig  17. 

Fig  14  is  a  joint- fastening  proposed  many  years  since  by  Alex. 
W.  Rae,  C  E,  of  Penna. 

It  certainly  possesses  merit.  With  a  length  of  about  16  ins,  and  with  the  addi- 
tion of  a  thin  wedge,  as  well  as  of  2  more  bolts,  all  below  the  rail,  in  the  positions 
shown  by  the  dotted  lines,  it  would  probably  constitute  as  effective  a  suspended 
joint-fastening  as  has  yet  been  tried.  There  would  be  very  little  strain  on  the  bolts. 
If  16  inches  long,  and  }4  inch  thick,  the  weight  per  joint  would  be  about  50  fts,  in- 
cluding 6  bolts,  and  a  wedge  beneath  the  base  of  the  rail.  As  in  all  other  joints, 
some  device  should  be  used  to  prevent  the  nuts  from  unscrewing  themselves  by  the 
jolting  of  passing  trains.  See  Star  Washer,  Fig  17  ;  also,  Billings  washer,  p  375. 
The  star  washer  being  of  very  thin  iron,  becomes  in  time  so  brittle  from  rust  as  to 
require  renewal.  The  Billings  washer  occasionally  breaks  under  great  strain.  A 
perfect  lock-nut-washer  of  simple  construction  is  still  u  desideratum. 

Pig  15  was  also  one  of  the  numerous  joint-fastenings  suggested  at  an  early  day ; 
bui.  like  the  foregoing,  it  never  came  into  use.  In  lengths  of  about  8  ins,  it  would 
probably  make  an  efficient  fastening;  especially  with  the  addition  of  a  broad  thin 
wedge  between  the  bottom  of  the  rail  and  the  foot  of  the  chair.  This  would  dimin- 
ish the  difficulty  of  sliding  the  chairs  on  or  off  of  the  rail :  and  would  thus  make  it 
easy  to  employ  larger  ones;  besides  insuring  a  firm  bearing  for  the  base  of  the  rail 
upon  the  fastening. 

A  fastening  of  tempered  steel,  like  Fig  15,  but  without 
any  bolt  and  key,  has  of  late  years  been  us^d  on  some  English  roads. 
Its  elasticity  permits  it  to  be  easily  slipped  upon  the  rails. 


Fig  16  is  the  Phcenix  Iron  Co's  suspended 

rail-joint,  devised  and  patented  by  Sain'l  J.  Reeves, 
Esqr,  'President  of  the  Co,  of  Philada. 

It  consists  of  three  pieces,  a,  b,  c,  of  rolled  iron,  each  14  ins  long, 
which  are  confined  to  the  rail  by  two  bolts  ;  one  of  which  is  shown  at 
nn.  The  weight,  including  bolts  and  nuts,  is  35  Ibs.  To  prevent  the 
six-sided  nuts  t  from  loosening  themselves,  a  small  strip  t t,  of  thin 
sheet  iron,  fits  around  them;  and  its  ends  are  bent  square  over  the  top 
and  bottom  of  the  piece  a:  or  a  star  washer  is  used.  .  This  fastening 
has  a  "stop,"  for  preventing  the  ends  of  the  rails  from  coming  toge- 
ther; but  it  is  not  only  ineffective,  but  decidedly  injurious  to  the  e» 
tire  fastening. 


Tig-15 


TURNOUTS. 


397 


TURNOUTS, 


the  switcti-raiis,  lonn  ine  luicrum  or  center  auoui  wnicn  tuey  move;  ana  are  called 
their  heels  ;  and  the  ends  o  and  s,  at  which  the  motion  is  greatest,  are  their  toes.f 
The  dist  x  o  reqd  for  this  motion  of  the  toe  of  each  switch- rail,  is  called  its  throw, 
see  Fig  5,  and  x  g  o  is  the  switch  angle.  The  throw  must  be  equal  at  least 
to  the  width  a  x  of  the  top  of  the  rail,  in  addition  to  a  width  o  «,  sufficient  to 
allow  the  flanges  of  the  wheels  on  the  track  A  B,  to  pass  along  readily  between  the 
rails  b  and  w.  The  tops  of  rails  are  generally  between  2  and  2%  ins  wide;  while 
about  1%  to  2  ins  will  usually  suffice  for  the  flanges.  The  throw,  x  o,  however,  is 
commonly  about  5  ins ;  that  is.  the  toes  o  and  s  are  moved  5  ins  from  their  original 
position  (shown  by  the  dotted  lines),  when  the  train  is  intended  to  leave  the  track 
A  B,  and  to  pass  along  the  turnout,  to  the  track  C  D.  The  motion  is  given  by  means 
of  a  switch-lever,  which  will  be  described  by-and-by.  The  entire  triangular 
part  P,  Fig  2,  is  the  tongue  of  the  frog;  and  its  sharp  end  is  the  point. 


Frog-making  has  become  one  of  the  specialties  of  the  day. 

In  ordering  them  it  is  not  necessary  to  furnish  more  than  the  frog-angle,  or  num- 
ber; and  an  exact  cross-section  of  the  rail  used  on  the  road.  The  same  frog  will 
answer  for  turning  out  either  to  the  right  hand  or  to  the  left,  except  the 
spring- rail  frog  described  farther  on. 

*  As  the  business  on  the  Pennsylvania  Central  increased,  it  was  found  expedient  to  increase  the 

length  of  the  sidings  to  one  mile. 

T  This  applies  to  the.  common  or  stub  switch,  Fips  1  and  H.     In  the  Wharton,  Lorenz  and  some 
others,  the  positions  of  heel  and  toe  are  'he  reverse  of  this.    See  rages  407,  408. 


398 


TURNOUTS. 


Fig  '2  is  a  plan ;  Figs  3  and  4  side  views,  and  Fig  Z  a  cross-section  of  a  frog  as  frequently 
made. 

Tae  entire  frog  was  formerly  made  of  cast  iron ;  hardened  by  chilling;  so  as  better  to  resist  the 
action  of  passing  wheels;  hut  even  with  this  precaution  they  wore  out  so  much  more  rapidly  than 
the  rails,  that  the  wings  and  tongue  are  now  capped  with  the  best  steel.  It  is  used  in  thicknesses  of 
from  ^  to  1  inch  or  more:  and  i.s  nrmly  bolted  down  to  the  cast-iron  portion;  besides  being  other 
wise  secured  to  it  in  the  best  makes.  It  is  not  necessary  to  steel  the  entire  length  of  the  wings ;  a 
very  trifling  economy  is  gained  by  leaving  their  ends  m  and  c  unprotected;  inasmuch  as  the  wheels 
do  not  run  upon  those  parts.  All  the  tongue  P,  however,  should  be  steeled. 

The  projections  t  t  are  merely  for  bolting  the  frog  down  to  ths  wooden  cross-ties.  The  wings  and 
tongue  must  evidently  be  raised  above  the  bottom  plate  of  the  frog  sufficiently  to  prevent  the  flanges 
of  wheels  from  touching  this  last.  About  \%  to  2  inches  will  suffice,  as  shown  at  Fig  Z  ;  which  is  a 
transverse  section  of  Fig  2,  taken  across  t  i  w  t. 

The  channel  is  called  the  MOUTH  of  the  frog  at  a,  Figs  2  and  8;  and  its  THROAT,  at  the  narrowest 
part  w  i.  Fig  2 ;  or  z  x,  Fig  8.  That  part  of  the  tongue  which  is  back  of  «,  Fig  8,  or  between  u  and  g, 
ia  called  its  HEEL. 

Although  the  frog  forms  a  part  of  the  turnout  curve,  still  its  shortness  warrants  us  in  making  it 
straight  from  j  to  i ;  and  from  s  to  t,  Fig.  8. 

Fig  'A  is  a  side  view  of  a  frog  of  uniform  depth.  This  depth  may  be  about  3J^  ins  ;  namely,  about 
2  ins  for  the  wings  and  tongue;  and  1&  for  the  base  plate. 

In  Fig  4,  which  is  also  a  side  view,  the  projections  1 1,  the  number  of  which  will  vary  with  the 
length  of  the  frog,  are  cast  in  one  piece  with  the  frog ;  but  are  entirely  below  it.  The  end  one  at  P. 
passes  also  under  the  end  of  the  adjoining  rails  ;  thus  forming  a  chair  for  them,  as  well  as  for  the 
frog.  At  the  end  R  is  shown  a  mode  of  proceeding  when  the  rails  are  high  ;  and  are  to  rest  upon  the 
ends  of  the  base  of  the  frog,  as  in  the  fig.  Whichever  mode  may  be  employed,  it  will  of  course  be 
used  at  both  ends  of  the  frog.  Such  details  are  pretty  much  a  matter  of  whim ;  and  vary  with  the 
notions  of  the  designers. 

Fig  6  is  a  frog,  as  often  made,  by  merely  bending  two  pieces,  /  g  and 
h  t,  of  ordinary  rails,  for  forming  the  wings  ;  and  by  cutting  and  bolting  together 
two  other  pieces,  c  and  c,  for  the  tongue  and  point.  Frogs  are  now  regularly  manu- 
factured on  the  principle  of  Fig  6,*  and  are  displacing  to  a  great  extent  the  old 
cast-iron  ones.  They  are  made  from  8  to  10  feet  long,  according  to  the  frog-number, 
(see  Art  2.)  At  their  ends  they  are  drilled  for  fish-plates,  by  which  they  are  joined 
to  the  track-rails.  They  thus  form,  as  it  were,  part  of  the  track,  and  are  therefore 
less  liable  than  the  ordinary  frogs  to  wear  loose  under  the  passage  of  trains.  The 
wing-rails  and  the  point  are  secured  at  the  proper  distance  apart  by  iron  pieces, 
which  fit  closely  to  the  rails, and  are  inserted  between  the  wing-rails  and  the  point. 
They  extend  beyond,  and  enclose,  the  point,  thus  bracing  and  staying  it.  They  are, 
of  course,  low  enough  to  clear  the  flanges  of  the  wheels.  The  parts  are  bolted  to- 
gether to  form  one  solid  frog. 


To  explain  the  nse  of  the  guide-rails,  c  C9  a  ft.  Fig  1. 

Suppose  wheels  to  be  rolling  from  A  toward  B,  Fig  1,  on  the  main  track  :  the  switch  rails  being  in 
the  dotted  positions.  On  arriving  opposite  the  frog,  some  irregularity  of  motion  might  cause  the 
flanges  of  the  wheels  running  along  the  rails  n  I,  to  press  hard  against  said  rails.  Consequently,  after 
passing  the  throat  w  i  (Fig  2),  they  would  press  against  the  wing  i  c. ;  and  passing  between  c  and  P, 
leave  the  track,  or  strike  the  sharp  end  of  P ;  breaking  it ;  and  endangering  the  train.  To  prevent  this, 
the  guide  rail  c  c  is  placed  so  near  the  rails  g  /t,  Fig  1,  (generally  about  \%  to  2  ins.)  that  the  flanges 
at  those  rails,  while  passing  between  them  and  the  guide-rail,  not  only  prevent  the  flanges  at  the 
opposite  rails  from  pressing  against  the  wing  t  c;  but  guide  them  safely  along  their  proper  channel 
from  i  to  m,  Fig  2,  without  striking  against  the  sharp  end  of  the  point." 


*  The  Pennsylvania  Steel  Co.,  office  208  S.  4th  St.,  Phila.,  manufacture 
patterns,  and  made  entirely  ef  steel  rails.     Prices  in  1880  from  $35  to  $50. 


such  frogs,  of  different 


TURNOUTS. 


399 


In  like  manner,  if  the  switch-rails  be  in  the  positions  g  o,  n  s.  (or  switched,)  and  if  wheels  be  rolling 
along  the  turnout,  Fig  1,  from  A  toward  D,  when  they  arrive  at  the  frog,  the  centrifugal  force  (being 

of  the  frog:  thus  rendering  liable  the  same  kind  of  accident  a*  in  the  preceding  case.  This  is  pre- 
vented iu  the  same  manner  as  before,  by  the  guide-rail  a  a ;  which  keeps  the  flanges  in  their  proper 
channel  from  w  to  c,  Fig  2. 

The  narrow  flange-way  between  the  guide-rail  c  c,  Fig  1,  and  the  rail  b  h,  need  not  extend  farther 
than  from  the  end  a,  Fig  2,  of  the  frog,  to  about  one  foot  toward  g,  from  the  sharp  end  of  the  point. 
In  a  distance  of  at  least  about  2  ft  more  at  each  of  its  ends,  the  guide-rail  should  flare  out  to  about  3 
ins  from  the  rail  b  h ;  so  as  to  guide  the  flanges  into  the  narrower  part  of  the  flange- way.  The  same 
with  a  a.  Guide  rails  should  be  very  firmly  confined  to  their  wooden  cross-ties  ;  inasmuch  as  they 
have  to  resist  a  strong  side  pressure.  This  is  usually  done  by  bolting  against  them  two  or  more  stout 
blocks  n  n,  Fig  5}<j,  of  wood,  or  cast  iron;  along  the  narrowest,  or  most  strained  portion  of  the 
flange- way. 

Art.  2.  Fig.  7.  A  substitute  for  a  frog1.  It  is  a  bar,  either  of  steel,  or 
of  cast  iron,  2  to  4  inches  deep,  with  a  steel  top  bolted  to  it.  It  is  merely  a  short  movable  switch, 
which  may  be  made  to  accommodate  either  line  of  rails,  by  means  of  a  switch  lever  attached  to  n.  It 
of  course  requires,  like  other  switches,  a  person  to  attend  to  it ;  which  the  frog  does  not. 

The  length  a,  ff,  Fig-  8,  of  a  frog,  usually  varies  from  4  to  8  feet :  and 
depends  upon  the  obliquity  at  which  the  rails  w  m,  Figs  1  and  8,  cross  the  rails  n  I. 


Frog-  Angle,  Frog  Number.  The  angle  at  which  the  turnout  curve 
crosses  the  rails  n  Z,  Fig  1,  determines  the  frog  angle  i  c  <,  Fig  8,  of  the  frog:  and 
how  many  times  the  length  g  cof  the  tongue,  or  rather,  how  many  times  the  length 
of  a  line  from  c  to  the  center  of  a  line  from  I  to  i,  exceeds  the  breadth  t  i.  The 
frog  is  called  a  No.  3,  4,  10,  &c,  according  as  this  is  3,  4,  or  10,  &c,  times  greater  than 
t  i ;  thus,  Fig  8  is  a  No.  3 ;  and  Fig  2,  No.  5.  They  are  usually  made  of  Nos.  4  to  12 ; 
sometimes  with  half  numbers.  Having  first  drawn  two  parallel  lines  22, 33,  for  the 
top  of  the  rails  w  m ;  and  4  4,  5  5,  for  the  top  of  the  rails  n  I ;  crossing  each  other  at 
the  reqd angle  or  No,  we  get  the  theoretical  point  c  of  the  frog;  and  the  dist  from  c 
toi  or  t,  need  be  no  greater  than  will  give  a  width,  t  i,  sufficient  to  accommodate 
the  ends  of  the  rails  /  and  tn,  and  the  heads  of  the  two  spikes  at  e,  which  confine 
them  to  their  cross  ties.  See  Art  3.  As  the  theoretical  point  c,  would  be  too 
narrow  and  weak  for  service,  it  is  in  practice  terminated  where  the  tongue  is  about 
%  iricn  wide,  as  at  r.  Draw  z  k  parallel  to  c  t ;  and  dist  from  it  about  1%  ins ;  or 
barely  sufficient  to  allow  the  ready  passage  of  the  flanges  of  the  wheels.  Also  in 
like  manner  draw  xp,  parallel  to  c  t.  Now,  the  length  from  x  to  .;' ;  or  from  z  to  s, 
need  be  no  greater  than  will  suffice  for  rounding  off  the  sharp  angles  at  x  and  z,  so 
as  to  prevent  their  being  struck  by  the  flanges  of  wheels.  About  6  to  8  ins  will  an- 
swer for  this ;  thus  giving  the  dotted  curve/  y,  about  one  foot  long.  The  same  will 
be  done  at  z.  At  first  these  angles  at  x  and  z  were  left  sharp  ;  but  experience  having 
shown  that  they  became  ground  away  by  the  flanges,  they  are  now  rounded  off;  and 
trains  traverse  more  pleasantly  in  consequence. 

For  the  length  of  the  wing  z  o,  draw  a  line  6  6,  parallel  to  x  p,  and  dist 
from  it  about  an  inch  more  than  the  entire  width  of  the  widest  tires  (probably  6% 
ins)  used  on  the  road.  Then  Jc  will  be  the  least  (list  from  z,  to  which  the  wing  should 
be  made  straight ;  from  k  to  o,  say  not  less  than  about  6  ins,  (more,  if  the  length  of  the 
fro<r  allows  it.)  the  inner  edge  of  the  wing  should  be  flared  outward,  (so  as  to  guide 
wheels  entering  that  end  of  the  frog,)  as  in  the  fig.  If  the  frog  is  of  cast  iron,  capped 
with  steel,  the  steel  need  not  extend  beyond  fc;  inasmuch  as  the  treads  of  the  wheels 
never  run  on  the  part  k  o.  The  hame  process  and  remarks  apply  to  the  wing  x  v. 

The  object  in  redneing  the  width  of  the  channels 

For  some  dist  each  way  from  the  point  r,  so  as  bnrelv  to  admit  the  flanges  freely,  is  to  nllow  the  treads 
of  the  wheels  to  have  as  much  bearing  as  possible  upon  the  wings,  and  upon  the  narrow  point  of  the 
tongue,  while  moving  over  the  broadest  part  of  the  channel  near  b,  y  and  r.  In  frogs  shorter  than 


400 


TURNOUTS. 


about  No.  4,  it  becomes  difficult  to  secure  a  sufficient  bearing  for  the  treads,  even  with  the  utmost 
allowable  contraction  of  the  channel,  when  the  width  of  the  tires  on  the  road  is,  a«  usual  about  6  ins 
In  the  frogs  firs  t  used  upon  our  railroads,  the  wheels  did  not  run  upon  their  treads,  while  pa.ssins 
along  the  portion  comprised  between  zxkp,  but  upon  their  flanges.  To  effect  this,  that  portion  of  the 
bottom  of  the  channels  was  gradually  raised,  so  that  its  depth  below  the  top  of  the  wings  was  less 
than  the  depth  of  the  flanges.  By  this  means,  even  the  shortest  possible  frogs,  or 'those  in  which  the 
channels  cross  each  other  at  right  angles,  afforded  a  satisfactory  bearing  for -wheels  passing  along 
them.  This  arrangement  is  still  employed  for  said  shortest  frogs' :  which  are  used  when  two  tracks 
of  railroad  cross  each  other  at  right  angles.  A  great  objection  to  it,  however,  and  one  which  excludes 
it  except  when  absolutely  necessary,  is,  that  the  flanges  very  soon  cut  gutters  in  the  bottoms  of  the 
channels,  so  as  to  cause  the  treads  to  run  upon  the  wings,  and  to  jolt  in  crossing  the  channels.  Said 
bottoms,  therefore,  require  very  frequent  renewal. 

Mr.  J.  Wood's  self-lie  ting  frog;.  Fig  8%,  was  contrived  with  a  view  to 
remedy  this  defect.  It  is  made  entirely  of  common  T  rails.  The  wings  a  b  c,  and 
TO  n  o,  are  confined  together  by  a  bolt  at .<?,  which  passes  loosely  through  the  tongue 
x  z,  as  shown  more  fully  in  the  cross-section  g.  Also  by  a  clamp  d;  shown  in  cross- 
section,  at  h.  The  ends  c  and  o,  are  fast  to  the  chair  e  e,  which  is  bolted  to  the  cross-tie  t. 


J. WOODS  SELF  ACTING    FROG 


The  wings  thus  united,  are  movable  :  the  ends  c  and  o  being  the  center  of  motion  ;  so  that  either  ft 
or  n  can  be  brought  into  contact  with  the  tongue  z ;  a  very  small  play  of  the  ends  c  and  o,  in  the  nixed 
chair  e,  e,  being  sufficient  to  admit  of  this.  The  clamp  d  moves  with  the  wings  ;  the  motions  of  which 
are  caused  by  the  flanges  of  the  wheels.  The  mode  of  action  is  this  :  Suppose  the  wings  to  be  in  i  he 
position  shown  by  the  fig.  Then  it  is  ready  for  a  train  moving  from  y  toward  k  ;  or  from  k  toward 
j/;  and  in  either  case  the  treads  of  the  wheels  have  a  full  bearing  throughout  the  entire  length  of  the 
frog  ;  and  there  is  no  channel  to  be  crossed.  Again,  if  a  wheel  on  w  x  be  moving  toward  t,  its  flange 
will  push  a  b  away  from  the  tongue;  and,  consequently,  will  draw  m  n  close  up  to  it;  so  that  said 
wheel  will  pass  along  the  frog  under  the  same  circumstances  as  the  preceding  one  did  ;  and  the  frog 
•will  remain  in  position  for  the  wheels  which  follow  ;  or  for  others  moving  from  i  toward  w.  An  iron 
plate  g  is  bolted  to  the  cross-tie,  for  the  wings  to  slide  on.  Again,  suppose  the  frog  to  be  as  in  the  fig ; 
and  an  engine  to  be  moving  from  t  toward  w.  Now,  in  order  to  open  the  wing  c  b  a,  and  to  close  the 
wing  o  n  w  up  to  the  tongue  z  x,  a  guide-rail  on  the  opposite  line  of  the  main  track,  and  not  shown  in 
the  fig,  draws  the  wheels  which  are  moving  from  i  toward  w,  over  against  the  wing  c  6,  forcing  it  open  ; 
and  of  course  closing  onm.  But  in  this  case,  it  will  be  observed  that  the  motion  of  the  wings  has  to 
take  place  while  the  weight  of  the  engine  itself  is  partly  resting  on  the  wing  n  o;  and,  consequently, 
their  sliding  transversely  of  the  track,  is  attended  with  great  friction.  This,  however,  does  not  in 
practice  seem  to  interfere  with  the  efficiency  of  the  frog.  It  was  in  extensive  use  for  many  years 
on  the  Camden  &  Amboy  R.  R,  where  it  was  daily  traversed  in  both  directions  by  numerous  trains  at 
high  speed.  When  the  frog  is  very  long,  one  or  two  additional  connecting  clamps  d,  and  slide-plates 
g  should  be  introduced.  A  little  attention  is  necessary  to  prevent  the  accumulation  of  dirt  and  stones 
between  the  wings  and  the  tongue.  The  whole  should  be  of  steel  if  possible.  Of  course,  two  guide- 
rails  are  necessary,  as  c  c,  a  a,  Fig  1. 

In  the  Spring- Rail  Frog1,*  which  has  supplanted  all  others  on  several  of 
our  principal  railroads,  the  turnout  rail  m  o,  Fig  *%,  and  the  point  x  z,  are  perma- 
nently fixed  at  such  a  dist  apart  as  to  allow  the  wheel-flanges  to  pass  between  them ; 
but  the  rail  a  b  c  is  left  unspiked  about  as  far  back  as  k.  At  about  c  o  it  is  connected 
with  the  fixed  rail  m  o  by  a  bolt  which  passes  through  both  of  them.  This  bolt  is 
furnished  with  a  spring  which  keeps  a  b  pressed  against  the  fixed  point  x  z  while  no 
trains  are  passing  to  or  from  the  turnout.  Wheels  passing  in  either  direction  on  the 
main  track  y  fc,  thus  have  a  full  bearing  throughout  the  length  of  the  frog.  Trains 
passing  to  or  from  the  turnout  operate  as  in  Mr.  Wood's  self-acting  frog  described 
above,  except  that  after  the  passage  of  each  wheel  the  spring  at  c  o  restores  the  wing- 
rail  a  6  to  its  position  close  to  w  z  as  in  the  Fig.  This  frog,  therefore,  does  not  pre- 
sent a  full  bearing  throughout  to  wheels  entering  or  leaving  the  turnout,  but  only 
to  those  passing  to  and  fro  on  the  main  track.  The  standard  length  of  these  frogs, 
for  any  angle,  is  15  feet.  In  ordering  them  it  is  necessary  to  specify  not  only  the 


•K-Made  by  the  Pennsylvania  Steel  Co.,  office  208  S.  4th  St.,  Phila. 


TURNOUTS. 


401 


frog-angle  or  number,  but  also  whether  right  hand  or  left;  which  is  not  re- 
quired in  ordering  the  stiff  frog. 

Art.  3.  With  rails  whose  bases  are,  as  usual,  not  more  than  4  ins  wide  the  lengths 
of  frogs  which  do  not  extend  under  the  ends  of  the  rails,  need  not  be  greater  than  the 
following,  in  ft  and  ins.  But  if  they  do  so  extend,  an  addition  of  from  8  to  12  ins 
must  be  made  to  the  bottom  of  the  frog  for  that  purpose. 


LENGTH. 

Ft.  Ins. 

2  9 

3  7 

4  4 


But  we  may  shorten  a  frog 

considerably  by  cutting  away  a  portion  of  the 
bases  of  the  ends  of  the  rails  I  and  m,  as  shown 
by  the  line  g  y,  Fig  9 ;  BO  that  the  tops  I  and  m,  of 
the  rails,  shall  come  into  contact  with  each  other 
at  g :  thus  diminishing  the  width  necessary  for 
t  i ;  and  of  course  the  distance  c  g.  When  this  is 
done,  the  length  will  not  exceed  as  follows,  even 
with  rails  2>j  ins  wide  at  the  head. 


LENGTH. 
Ft.  Ins. 
2  4 

2  11 

3  6 


No.  OF  FKOG. 


In  loner  frogs  especially,  this  reduction  is  advisable,  not  only  on  the  score  of  economy ;  but  of  ease 
of  handling ;  and  less  liability  to  be  broken  if  the  foundation  becomes  infirm. 

Art.  4.    The  laying-out  of  Turnouts. 

The  words  heel  and  toe  are  used  in  this  article  with  reference  to  the  common  or 
stub  sivitch,  Fig  1,  in  which  the  heels  are  at  g  and  n  ;  and  the  toes  at  o  and  s.  In  the 
Wharton.  Lorenz,  and  some  other  switches,  the  positions  of  heel  and  toe  will  be  seen 
to  be  the  reverse  of  this. 

The  formulas  given  in  our  former  editions  for  finding  frog  dist,  rad  of  turnout, 
etc,  were  based  upon  the  old  practice  of  regarding  the  straight  switch-rails  go,ns, 
Fig  1,  as  forming  a  tangent  to  the  turnout  curve,  which  last  was  considered  as  be- 
ginning at  the  toes,  o  and  s,  of  the  switch-rails.  The  modern  practice  is  to 
curve  the  switch-rails  so  as  to  form  a  part  of  the  turnout  curve;  the  latter  being 
supposed  to  begin  at  the  htels,  g  and  n,  of  the  switch.  This  view  of  the  case  admits 
of  simpler  formulas. 

In  each  of  the  Figs  10, 11,  and  12,  w  p  z  represents  the  main  track.  The  frog" 
distance,  j?/,  is  a  straight  line  drawn  from  the  theoretical  point  of  frog,  /,  to 
the  heel,  p,  of  that  switch-rail  which,  when  opened,  forms  the  inner  rail  of  the  turn- 
out. Formerly  when  the  turnout  curve  was  taken  as  starting  at  the  toe  of  the 
switch,  the  frog  dist  was  a  straight  line  from  the  theoretical  point  of  frog  to  the  toe, 
o,  Fig  1.  of  the  outer  switch-rail,  g  n,  when  opened. 

As  already  remarked  frogs  are  usually  made  of  Nos.  4  to  12;  sometimes  with 
half  numbers  ;  and  the  turnout  radii,  etc.,  are  made  to  conform  to  them. 

Scrupulous  accuracy  is  not  necessary  in  these  matters.  Thus,  a  deviation,  either 
way,  of  say  3  per  cent  in  the  length  of  the  turnout  radius  from  that  given  by  the 
table  or  the  formulas,  will  be  almost  inappreciable.  So  too,  if  a  frog  number  should 
be  used,  intermediate  of  those  in  the  first  column  of  the  table,  the  other  dimen- 
sions may  be  found,  approximately  enough,  by  using  quantities  similarly  interme- 
diate. A  rail  almost  always  has  to  be  cut  in  two  in.  order  to  fill  up  the  said  dist; 
and  the  exact  length  of  the  piece  can  be  found  by  actual  measurement  at  the  time 
of  cutting  it. 

When  the  turnout  is  to  be  traversed  by  passenger  trains  the  rad  should  not  be 
less  than  about  800  feet. 

Rein.  When  the  turnout  leaves  a  straight  track,  as  in  Fig  10,  the  frog  angle 
is  equal  to  the  central  angle  f  c  o.  When  the  main  track  is  curved,  and  the  turnout 
curves  in  the  opposite  direction  (Fig  11),  it  is  equal  to  the  sum  (v  f  n)  of  the 
central  angles/c  o,fn  o ;  and  when  the  two  curve  in  the  same  direction  (Fig 
32),  it  is  equal  to  the  diff  (n  f  c}  of  the  central  angles/c  o,  fn  o. 

Art.  5.  To  lay  out  a  turnout,  p  x.  Fig  1O,  from  a  straight 
track,  p  z.  From  the  column  of  radii  in  the  table  page  402,  select  one,  c  o,  suit- 


402 


TURNOUTS. 


able  for  the  turnout;  together  with  the  corresponding  frog  number,  frog  distp/and 
switch  length.  Place  the  frog  so  that  the  main-line  side  of  its  tongue  shall 
be  at  /  z, precisely  in  line  with  the  inner  edge  of  the  rail, 
wz:  and  its  theoretical  point,  J\ at  the  tabular  frog  dist 
pj,  from  the  starting-point  p.  Stretch  a  string  from 
q  (opposite  p)  to/;  and  from  it  lay  off  the  three  ordi- 
natrs  from  the  table;  thus  finding  three  points  (in 
addition  to  g  and  j)  in  the  outer  curve.  Do  not,  how- 
ever, drive  stakes  at  these  points  ;  but  as  each  of  them 
is  found,  measure  off  from  it,  inward,  half  the  gauge 
of  the  track  ;  and  there  drive  stakes.  Do  the  same 
from  q  and/.  The  five  stakes  will  all  then  be  in  the 
dotted  center  line  of  the  turnout,  Fig  30;  and  will 

q  serve  as  guides  to  the  work,  without  being  liable  to  be 
displaced. 

Remark.    The  dimensions  in  the  Table  below  are  found  by 
the  following  formulas;  the  main  track  being  straight. 

=  Gauge  -r-  Frog  dist. 


Tangent  of 
hall*  frog  angle 

Frog  No 

Or,  Frog  No 

Radius  c  o 

Or,  Radius  co 

Or,  Radius  co 

Frog  dist  p  f... 
Or,  Frog  dist  p  f... 
Or,  Frog  dist  p  f... 

Middle  ord 

Each  side  ord 

Switch  Length 

approx  enough 


{f  Radius  c  o  -f-  Twice  the  gauge, 
lalf  the  cotangent  of  half  the  frog  .angle. 
=  Twice  the  gauge  X  Square  of  frog  number. 
=  (Frog  distp  /  -f-  Sine  of  frog  angle)  —  half  the  gauge. 
=  (Gauge  H-  versed  sine  of  frog  angle)  —  half  the  gauge. 
=  Frog  number  X  Twice  the  gauge. 
=  Gauge  p  q  -~-  Tangent  of  half  the  frog  angle. 
=  (Rad  c  o  +  half  the  gauge)  X  Sine  of  frog  angle. 
-  %  gauge,  approx  enough. 
"£  mid  ord  =  -fa  (or  .188  of  the)  gauge,  approx  enough. 


V 


Throw  in  ft  X  10000 


Tangential  diat  for  chords  of  100  ft.  for  rad  c  o  of 
turnout  curve.     See  table  p  416. 

TABLE  OF  TURNOUTS  FROM  A  STRAIGHT  TRACK.    Fig  10. 

Gauge  4  ft  8^  ins.    Throw  of  switch  5  ins. 

For  any  other  gauge,  the  frog  angle  for  any  given  frog  number  remains 
the  same  as  in  the  table.  The  other  items  may  be  taken,  approx  enough,  to  vary  di- 
rectly as  the  gauge. 


Number 

£& 

Turnout 
Radius 

0  0 

DeflAngof 
Turnout 
Curve 

Frog 

Dist 
Pf 

Middle 

Ordlnato 

Side 
Ords 

Stub 
Switch 
Length 

o     /• 

Feet. 

o     r 

Feet. 

Feet. 

Feet. 

Feet. 

12 

4  46 

1356 

4  14 

113.0 

1.177 

.883 

34 

UH 

4  58 

1245 

4  36 

108.3 

1.177 

.883 

32 

11 

5  12 

1139 

5    2 

103.6 

1.177 

.883 

31 

H>H 

5  28 

1038 

5  31 

98.9 

1.177 

.883 

29 

10 

5  44 

942 

6    5 

94.2 

1.177 

.883 

28 

^A 

6    2 

850 

6  45 

89.5 

1.177 

.883 

27 

9 

6  22 

763 

7  31 

84.7 

1.177 

.883 

25 

&A 

6  44 

680 

8  26 

80.0 

1.177 

.883 

24 

8 

7  10 

603 

9  31 

75.3 

1.177 

.883 

22 

^A 

7  38 

530 

10  50 

70.6 

1.177 

.883 

21 

7 

8  10 

461 

12  27 

65.9 

1.177 

.883 

20 

&A 

8  48 

398 

14  26 

61.2 

1.177 

.883 

18 

6 

9  32 

339 

16  58 

56.5 

1.177 

.883 

17 

&A 

10  24 

285 

20  13 

51.8 

1.177 

.883 

15 

5 

11  26 

235 

24  32 

47.1 

1.177 

.883 

14 

V/2 

12  40 

191 

30  24 

42.4 

1.177 

.883 

13 

4 

14  14 

151 

38  46 

37.7 

1.177 

.883 

11 

Remark.  The  switch  lengths  in  the  Table  merely  denote  the 
shortest  length  of  Stub  switch  that  will  at  the  same  time  form  part  of  the 
turnout  curve,  and  give  5  ins  throw.  Pointed,  or  split- rail  switches,  like 
the  Lorenz,  Ac,  require  only  half  this  throw  ;  still,  to  suit  the  curve,  they  should 
be  as  long  as  the  Stub,  but  in  practice  all  kinds  seem  frequently  to  be  made  much 
shorter  than  the  table  requires,  thereby  sharpening  the  beginning  of  the  curve. 


TURNOUTS. 


403 


Art.  6.  To  lay  out  a  turnout  from  a  curved  main  track. 

There  are  two  Cases : 

Case  1,  Fig  11;  when 
the  two  curves  deflect  in  op- 
posite directions. 

Case  2,  Fig  12;  when 
the  two  curves  deflect  in  the 
same  direction. 

Exact  rules  for  the  di- 
mensions in  these  cases 
would  be  complicated  ;  and, 
moreover,  they  are  not  neces- 
sary, inasmuch  as  the  follow- 
ing method,  using  the  table 
in  Art  5,  is  sufficiently  close 
wherever  the  greater  of  the 
two  radii  is  not  less  than  say 
about  800  feet.  For  shorter 
radii  (and  indeed  for  all  cases)  the  method  by  means  of  a  drawing  to  a  large  scale 
(see  Art  7)  will  be  found  useful. 

Having  determined  approx  upon  a  radius  for  the  turnout  curve,  take  from 
the  table  p  416  its  corresponding  deflection  angle,  and  that  for  the  main  curve. 
In  Case  1,  find  the  sum  of  these  two  angles.  In  Case  2,  find  their  difference.  In  the 
table  p  402  find  the  deflection  angle  (not  the  frog  angle)  nearest  to  the  sum  or  diff 
just  found,  and  greater  or  less  than  it  according  as  it  is  preferred  to  have  the  turn- 
out radius  c  o  Jess  or  greater  than  the  approx  one  selected.  Thus,  suppose  it  is 
required  to  lay  out  a  turnout,  q  a;,  Fig  11,  Case  1,  rad  of  main  curve  to  be  2865  ft, 
and  that  selected  approx  for  the  turnout  716.8  ft.  The  def  angles  will  be,  respec- 
tively, 2°  and  8°,  and  their  sum  10°.  The  nearest  def  angles  in  table  p  402  are  10° 
50'  and  9°  3V.  Here  10°  50'  will  give  a  turnout  rad  c  o  a  trifle  shorter,  and  9°  31' 
one  somewhat  longer  than  716.8ft.  From  the  tabular  def  angle  thus  selected;  in 
Case  1,  subtract  the  def  angle  of  main  curve;  or  in  Case  2,  add  them  to- 
gether. The  resulting  diff  or  sum  is  the  def  angle  of  the  turnout  curve;  and 
table  p  416  will  give  its  rad  c  o.  Thus,  in  the  above  case,  tabular  angle  10°  50' — def 
angle  of  main  curve  2°  =  8°  50'  def  angle  of  turnout  curve ;  the  corresponding  rad 
(c  o)  of  which,  by  table  p.  416  is  649  feet ;  or,  if  9°  31<  instead  of  10°  50'  be  taken  as 
the  tabular  angle,  we  have  9°  31'  —  2°  =  7°  31',  rad  c  o  say  763  ft. 

With  the  same  data,  the  operation  in  Case  2  would  be.  Turnout  defl  angle  8°  — 
defl  angle  of  main  curve  2°=  6°.  Nearest  tabular  defl  angles  6°  5'  and  5°  31'.  Here 

either  6°  5'  -f  2°  =  8°  5',  rad  say  709  ft ;  or 
50  31/  +  2o  =  7°  31',  rad  say  763  ft. 

The  frog  number  and  switch 
length  in  the  table,  opposite  the  defl 
angle  thus  selected,  are  the  proper  ones  for 
the  turnout. 


The  frog  dist  (p  f )  is  found  thus : 

In  Case  1  add  to  the  tabul 


n 


ular  frog  dist  half  an 

inch  per  100  ft  for  each  degree  of  defl  angle 
of  the  easier  of  the  two  curves:  In  Case  2 
subtract  it.  Thus,  suppose  the  frog  number 
found  as  above,  to  be  10.  The  tabular  frog 
dist  is  94.2  ft.  Let  main  curve  be  one  of  2°. 
Then  in  Case  1  add  twice  %  inch,  (or  1  inch) 
per  100  ft,  to  94  2  feet ;  and  in  Case  2  sub- 
tract a  like  amount.  In  this  example  we 
have  (near  enough),  corrected  frog  dist  for 
Case  1,  94.3  ft ;  and  for  Case  2,  94.1  ft. 

TJ     49  I../         /  Place  the  frog  with  the  main-line  side  of 

riU»l£,  U//    j    I        its  tongue  at/*,  in  line  with  the  inner  edge 

of  the  frog-rail,  p  z,  of  the  main  line,  and  with 
its  theoretical  point/  at  the  distp  /(found  as  above)  from  the  heel,/),  of  the  inner 
switch  rail.  Stretch  a  string  from  /  to  the  heel  q,  of  the  outer  switch  rail.  Measure 
tho  dist,  qf,  divide  it  into  four  equal  parts,  and  lay  off  three  ordinates,  found  thus: 
Middle  ord  =  (Square  of  half  qf)  -f-  twice  the  rad  of  turnout  curve.  Each  side 
ord  =  three-fourths  of  middle  ord. 

These  three  ords,  and  the  points  q  and  /,  give  us  5  points  of  the  outer  rail  of 
the  turnout  curve ;  and  from  these  we  measure,  inward,  half  the  gauge,  and  drive 
6  center  guide  stakes,  as  in  Art  5. 


404 


TURNOUTS. 


Art.  7.  To  find  frog:  <li*t*,  etc.,  by  means  of  a  drawing-  to 
scale.  The  frog  dist  can  generally  be  fouud  near  enough  for  practice,  from  a 
drawing  on  a  scale  of  about  ^  or  %  inch  to  a  foot.  And  so  in  the  m:-my  cases 
where  turnouts  cross  tracks  in  various  directions,  in  and  about  stations,  depots,  &c. 

Figs  13  and  13*4  are  intended  merely  to  furnish  a  few  general  hints  in  regard  to  such  drawings. 
For  instance,  the  curves  of  a  main  track,  as  well  as  those  of  a  turnout,  generally  have  radii  too  large 
to  admit  of  being  drawn  on  a  scale  of  %  inch  to  a  foot,  by  a  pair  of  dividers  or  compasses.  But  they 
may  be  managed  thus  :  Draw  any  straight  line  a  b,  Fig  13,  to  represent  by  scale  a  100  ft  chord  of  the 
curve,  divide  it  into  twenty  5  ft  parts,  a  1,  1  2,  2  3,  Ac.,  and  lay  off  by  scale  the  19  corresponding 
ordinates,  1  1,  2  2,  3  3,  &c,  taken  from  the  table  on  page  633.  By  joining  the  ends  of  these,  we 


obtain  the  reqd  curve,  a  c  b,  of  the  main  track ;  and  of  course  can  draw  the  inner  line  y  t,  distant 
from  it  by  scale  the  width  of  track,  say  4  ft  8%  ins.  Now  let  a  c  b  and  y  t,  Fig  13^,  be  a  curved 
main  track  so  drawn ;  and  let  any  point  m  be  taken  as  the  starting-point  of  the  turnout  m  v,  &c.  On 
each  side  of  m  measure  off  any  two  equidistant  points  n  and  n,  in  the  same  curve;  and  through  m 
draw  «  g,  parallel  to  n  n.  Then  is  m  g  a  tang  to  the  curve  y  m  t ,  at  m.  Having  determined  on  the 
rad  of  the  turnout  curve  m  v  e,  draw  that  curve  by  the  same  process  as  before  ;  first  laying  off  the 
angle  g  m  i,  equal  to  the  tangential  angle  of  the  curve,  taken  from  the  table  p  416.  Then,  beginning 
at  m,  lay  off  5  feet  dists  along  m  i;  and  from  them,  as  in  Fig  13,  draw  the  ords  corresponding  to  the 
turnout  curve.  Through  the  ends  of  these  ords,  draw  the  curve  m  v  e  itself.  Then  the  frog  dist  will 
be  the  straight  dist  from  c  to  v,  and  can  be  measured  by  the  scale,  within  a  few  inches  ;  or  near  enough 
for  practice.  The  middle  ord  of  the  arc  m  v  cannot  be  found  correctly  by  so  small  a  scale  as  34  iacn 


to  a  foot,  but  should  be  calculated  thus :  From  the  square  of  the  rad,  take  the  square  of  half  the 
chord  m  v.  Take  the  sq  rt  of  the  rem.  Subtract  this  sq  rt  from  the  rad.  If  two  other  ords  should 
be  desired,  half-way  between  m  and  v,  and  the  center  cue,  they  may  each  be  taken  as  %  of  the 
center  one. 

The  frog  angle  at  t;  will  be  equal  to  the  angle  r  v  d,  formed  between  the  tang  v  r,  to  the  curve  a  c  6  ; 
and  the  tang  v  d,  to  the  curve  mv  e.  These  tangs  are  found  in  the  same  way  as  m  g ;  namely,  for  the 
tang  v  r,  lay  off  from  v  two  equidistant  points  A  and  h,  on  the  curve  a  c  b;  and  through  v  draw  v  r 
parallel  to  h  h.  Also,  for  v  d,  lay  off  from  v  any  equidistant  points  u  and  u,  on  the  curve  mv  e,  and 
through  v  draw  v  d  parallel  to  them.  This  angle  may  be  measured  by  a  protractor.  Or  if  on  the  two 
tangs  we  make  v  4  and  v  4,  equal  to  each  other,  and  draw  the  dotted  line  4  4 ;  and  from -its  center  at  6, 
draw  6  v ;  then  6t>  divided  by  4  4  will  give  the  No  of  the  frog.  With  care,  and  a  little  ingenuity,  the 


%13/a 

X 


young  student  will  be  able,  by  similar  processes,  to  solve  graphically  any  turnout  case  that  may  pre- 
sent itself.  The  method  by  a  drawing  has  great  advantages  over  the  tedious  and  complicated  calcu- 
lations which  otherwise  become  necessary  in  cases  where  curved  and  straight  tracks  intersect  each 
other  in  various  directions.  The  drawing  serves  as  a  check  against  serious  errors,  which  would  be 
detected  at  once  by  eye.  None  of  the  graphical  measurements  will  be  strictly  accurate;  but  with 
care,  none  of  the  errors  need  be  of  practical  importance.  The  ordinates  for  bending  rails  so  as  to 
suit  turnout  curves  can  be  found  from  the  table,  p  418. 

Art.  8.  An  experienced  track-layer,  with  a  <rood  eye,  can  place  his  own  guide- 
stakes  by  trial  on  the  ground ;  and  by  them  lay  his  turnouts  with  an  accuracy  as 
practically  useful  as  the  most  scrupulous  calculations  of  the  engineer  can  secure. 


TURNOUTS. 


405 


The  following  example,  Fig  13^,  of  a  turnout  from  a  straight  track  Y  Z,  exhibits  a  common  case, 
in  which  all  the  work  may  be  performed  on  the  ground,  without  previous  calculation.  Let  i  v  o  be 
the  tongue  of  a  frog,  with  which  the  assistant  has  been  directed  to  make  a  turnout  from  Y  Z ;  and 
that  he  has  received  no  instructions  more  than  that  the  turnout  must  start  at  d,  and  terminate  in  a 
track  W,  to  be  laid  parallel  to  Y  Z,  and  distant  from  it  r  x  or  r  x,  equal  to  6  ft. 

Place  the  tongue  of  the  frog  by  guess  near  where  it  must  come,  having  its  edge  v  i  precisely  in  line 
with  the  inner  or  flange  edge,  of  the  rail  6  r.  Then  stretch  another  piece  of  twine  along  the  edge  o  v 
of  the  frog,  and  extending  to  d  g.  Try  by  measure  whether  v  e  is  then  equal  to  e  d;  and  if  it  is  not, 
move  the  frog  along  the  line  6  r,  until  those  two  dists  become  equal.  Then  is  v  the  proper  place  for 
the  point  of  the  frog ;  b  v  is  the  frog  dist ;  one-half  of  c  e  is  the  length  of  the  middle  ord  of  the  turn- 
out curve  d  v ;  and  if  two  intermediate  ords  are  needed  at «  and  »,  each  of  them  will  be  %  of  said 
middle  one. 

The  frog  being  now  placed,  proceed  thus:  Place  two  stakes  and  tacks,  x  and  x,  at  the  reqd  inter- 
track  dist,  rx  and  rx,  of  6  ft  from  the  rails  b  r.  Then  range  by  pieces  of  twine  xx  and  v/,  to  find 
the  point  n  of  intersection.  Then  measure  n  v,  and  make  n  m  equal  to  it.  Then  is  m  the  end  of  the 
reverse  curve  v  m  of  the  turnout.  The  ords  of  this  curve  may  be  found  as  before ;  one-half  of  n  A 
being  the  middle  one,  &c. 

RKM.  It  may  frequently  be  of  use  to  remember  that  in  any  arc,  as  v  m,  of  a.  circle  ;  v  n  and  m  n 
being  tangs  from  the  ends  of  the  arc ;  one-half  of  the  dist  knis  the  middle  ord  ft  z  of  the  curve ;  near 
enough  for  most  practical  purposes,  whenever  the  length  of  the  chord  v  m  of  the  arc  is  not  greater 
than  one-half  the  rod  of  the  circle  of  which  the  arc  is  a  part.  Or  within  the  same  limit,  vice  versa, 


if  we  make  k  n  equal  to  twice  k  z,  then  will  n  be  very  approximately  the  point  at  which  two  tangs 
from  the  ends  of  the  arc  will  meet.  Also,  the  middle  ord  of  the  half  arc  v  z,  or  z  m,  may  be  taken  as 
Yi  of  the  middle  ord  k  z  of  the  whole  arc. 

Art.  9.    The  common  blunt-ended,  or  stub  switch,  Fig  14,  is 

usually  from  20  to  26  feet  long  from  heel  to  toe,  on  main  lines.  The  switch-rails  are  generally  spiked 
unyieldingly  to  the  ties  for  4  to  8  feet  back  from  their  heels,  a  a,  so  that  when  the  switches  are  opened 
they  bend  so  as  to  form  a  part  of  the  turnout  curve.  Their  toes,  o  o,  rest  and  slide  upon  cast-iron 
chairs,  c  c,  shown  in  detail  at  Figs  X  Y  Z. 

Safety  castings  are  sometimes  used  with  these  switches.  Each  switch- rail  is  then  furnished 
at  its  toe  with  two  safety  castings  :  one  like  H,  Fig  20,  on  its  inner  side,  and  one  like  Y  on  its  outer 
side.  These  castings  are  bolted  to  the  switch-rails,  and  in  the  stub  switch  move  with  them.  In  case 
a  train,  either  on  main  track  or  on  turnout,  approaches  the  switch  from  the  right  in  Fig  14,  while 
the  switch  is  misplaced,  these  castings  receive  the  wheels  and  guide  them  on  to  the  switch-rails,  a  a. 

The  switch-rails  are  connected  together  by  from  3  to  5  transverse  wrought-iron  clamp-rods,  »,  t, 
full  1J4  ins  diam;  shown  more  plainly  at  Fig  M  ;  and  at  i  in  Figs,  15,  16,  17.  One  of  these  clamp- 
bars,  t  t  n,  is  near  the  toe  o ;  and  is  made  lonjrer  than  the  others,  inasmuch  as  it  is  used  for  opening 
and  shutting  the  switches;  which  operation  requires  it  also  to  be  jointed,  as  shown  at  c.  Figs  15,  16, 
17.  It  is  myved  by  a  lever,  such  as  x  s  w,  Fig  15  ;  attached  to  a  cast-iron  switch-stand,  I,  d.  h,  y ; 
or  such  as  e,  m.  Fig  16,  about  2  ft  long,  by  aid  of  the  crank  B.  This  last  arrangement  is  called  a 
tumbling-switch.  J  Zis  one  of  the  pi u miner  blocks  for  the  journals  of  the  crank  to  revolve  in.  These 
blocks,  as  also  the  switch-stand  of  Fig  15,  are  firmly  bolted  down  to  the  long  cross-tie  s  s,  on  which  the 
toes  of  the  switches  rest;  or  should  the  tie  not  be  wide  enough, to  a  piece  of  timber  securely  laid  for  the  ex- 


406 


TURNOUTS. 


press  purpose ;  or  the  snitch-rod,  as  t  n,  Fig  14,  may  be  bent  at  an  angle,  or  at  a  re- 
verse curve,  so  as  to  bring  its  outer  end  n  over  the  cross-tie. 


03  ti  a  12  in 


The  switch-stand  may  be  used  wherever  it  will  not  be  in  the  way  of  passing  trains.  Its  lever  w  x 
is  generally  used  also  as  a  vane-rod,  lor  supporting  a  vane  w,  about  a  foot  or  two  high,  painted  white, 
and  reaching  to  about  7  ft  above  the  ground.  lu  the  upright  position  of  the  lever,  as  in  the  fig,  tl.e 
vaue  indicates  to  an  approaching  engine  that  the  main  track  is  clear.  If  the  rod  leans  to  the  right 
then  the  turnout  to  the  left  is  ready  for  travel ;  and  if  there  be  two  turnouts,  one  to  the  right,  and 
the  other  to  the  left  of  the  main  track,  (in  which  case  the  switch  is  called  a  three-throw  one,)  then  the 
opposite  inclination  of  the  rod  shows  the  right-hand  one  to  be  ready  for  the  train. 


TURNOUTS. 


407 


The  tumbling*  switch  may  be  iiseri  in  the  space  between  two 
tracfcft  ;  tor  its  lever  «  m  is  always  lying  flat  upon  the  cross-tie,  except  during  the 
act  of  reversing  it  for  moving  the  switch  ;  consequently  it  is  not  in  the  way  of  the  trains. 

The  oblonc  hole  near  the  end  of  the  lever  is  for  allowing  it  to  fit  over  a  staple  bolted  to  the  tie.  The 
hasp  of  a  padlock  is  then  attached  to  the  staple,  for  preventing  mischievous  persons  from  misplacing 
the  switch.  A  second  staple  receives  the  lever  when  it  is  reversed  or  turned  back.  The  dist  n  n, 
Fig  16,  must  plainly  be  equal  to  one-half  of  the  throw  of  the  switch -rail. 

The  vane-rod,  or  lever,  w  s  x,  Fig  15,  is  held  in  position  at  b  6,  I Z,  or  I  /,  by 

means  of  a  hinged  collar  w,  Figs  W,  fitting  over  the  staple  £;  to  which  it  is  held  by 
the  hasp  r,  of  a  padlock  p.  When  the  lever  is  to  be  shifted  from  6  6,  p  is  unlocked 
and  removed;  the  collar  n  is  raised  up  to  clear  the  staple  t-  and  the  lever  is  first 
drawn  back,  or  toward  the  reader,  about  an  inch,  in  order  to  clear  the  projecting 
bead  g.  The  elasticity  of  the  long  lever  permits  this  last  to  be  done  readily.  The 
lever  is  then  shifted  to  1 1,  where  its  elasticity  again  brings  it  flat  against  the  face  a, 
of  the  switch-stand,  where  the  bead  g  retains  it.  The  collar  w  then  falls  into  its 
place,  on  each  side  of  the  staple  t,  and  the  padlock  is  replaced.  Figs  W  will  make 
the  details  plain.  These  admit,  of  course,  of  much  variety ;  what  we  have  giveii  are 
perhaps  as  good  as  any. 

Usually,  pieces  of  flat  iron  about  3  ins  wide  by  H  inch  thick,  are  spiked  to  the  tops  of  all  the  cross 
ties  which  sustain  the  switches  ;  in  order  to  facilitate  the  sliding  of  the  latter.  These  may  be  very 
short,  toward  the  heel  ends  a ;  and  increase  to  about  15  inches  in  length  at  the  toe  ends.  Sonieticuea 
these  are  omitted  entirely. 

Fig  18  is  a  side-view  of  another  arrangement  which  may  be  -. 

used  between  tracks.    It  is  called  the  Monkey-Switch.     A  stand  K 

tt  is  cast  in  one  with  the  hollow  cone  nn.   The  latter  is  smoothly  ^>__<^^ 

bored  to  admit  the  cylindrical  pin  a  o,  about  2  ins  diam  ;  with  a 
square  head  s,  for  the  handle  or  lever  h  h,  about  3  feet  long.  A 
wrought  collar  e  e  is  fastened  to  9  o  by  a  pin  through  both,  and 
turns  with  s  o.  The  nor  piece  o  t  is  fast  to  s  o  :  and  near  its  end 
t  is  fastened  the  short  vert  round  pin  i  v,  about  1  inch  diam, 
which  passes  loosely  through  an  opening  in  the  end  c  of  the  long 
rod  ttn,  Fig  14,  which  opens  and  shuts  the  switch.  A  small  pin 
passes  through  iujust  below  c,  to  prevent  the  latter  from  falling. 
For  a  throw  of  5  ins,  the  nor  dist  between  the  centers  of  s  o  and 
i  v  must  plainly  be  2!^  ins,  or  half  the  throw.  In  the  fig,  o  i,  Ac, 
is  in  the  position  of  only  half  open.  When  the  switch  is  a  three- 
throw  one,  this  dist  between  centers  must  be  five  ins:  and  the 
length  of  the  piece  o  i,  and  the  length  1 1  of  the  stand,  must  be 
increased  to  suit.  In  a  3  throw,  the  position  of  o  i  c  as  seen  in 
the  fig,  is  after  o  i  has  described  a  quadrant,  thus  opening  one 
turnout  and  closing  another.  In  this  position  it  must  be  held 
for  the  time  by  a  bolt  passing  through  the  top  of  the  standard, 
and  through  o  t  near  the  end  i ;  otherwise  a  passing  train  would 
be  apt  to  move  it,  thus  partially  closing  the  turnout,  and  throwing  the  train  off  the  track.  By  turn- 
ing o  i  around  through  another  quadrant,  the  remaining  third  track  becomes  passable.  In  a  2  throw, 
o  i  is  never  left  in  the  position  shown  in  Fig  18,  butdescribes  an  entire  semicircle  each  time  the  turn- 
out is  opened  or  closed,  (t  therefore  requires  no  bolt  for  confining  it  temporarily.  The  breadth  of 
the  stand  t  t  does  not  show  in  the  fig  :  it  is  usually  6  ins  at  every  part. 

Art.  1O.  Whurton'B  patent  safety  switch.  Fig  19  shows  the  switches  set  for  the  main 
track,  and  Fig  2'J  for  the  turnout.  A  and  B  are  the  main-track  rails.  They  are  continuous,  and 


Fig-18 


mcn.es 
MONKEY  SWITCH 


WHARTON. 


*                      mcam 

2P  >y 

a                              -"^PZ      1 

E 

9 

c 

z.  ^n 
%20^ 

Ki? 

V     , 

Itt 

D 

10 

n 

•piked  to  the  ties  throughout,  and  are  not  moved  or  broken  in  the  working  <>f  the  switch.     K  and  D 
»r«  the  switch-rails,  with  thuir  heti*  at  n  (the  place  of  (he  toes  in  the  stub  twitch,  Art  9.)    The 


408 


TURNOUTS. 


switch-rajls,  like  others,  are  connected  together  bj  clamp-bars  g,  and  slide  on  the  ties.  The  inntr 
switch-rail  D  (called  the  elevating  rail)  is  blunt  ended.  At  its  toe,  its  top  is  level  with  that  of 
A  ;  but  from  that  point  it  rises,  until  at  ra,  about  4  feet  from  the  toe,  its  top  is  about  2}£  inches  (more 
recently  1%  inches)  higher  than  that  of  A.  This  enables  the  flange  of  the  wheel  K,  in  passing  to  or 
from  the  turnout,  to  pass  clear  over  the  rail  A.  D  retains  this  elevation  from  m  to  n.  Bevond  n  it 
falls  gently,  until  its  top  is  again  at  the  track  level.  The  unbroken  main  rails,  and  this  carrying  of 
the  flanges  over  the  main  rail,  are  special  features  of  the  Whartou.  The  recent  reduction  from  '2% 
to  1%  inches  in  the  rise  was  made  in  order  to  lessen  the  jar  which  it  gave  to  trains  using  the  turnout 
at  more  than  quite  moderate  speed.  The  tread  of  the  outer  wheel  L,  in  passing  to  or  from  the  turn- 
out, runs  upon  the  outer  flange  (the  upper  one  in  the  Figs)  of  the  outer  switch  rail,  or  grooved 
rail  E.  This  flange  is  pointed,  so  that  when  E  is  placed  as  in  Fig  '20,  its  point  x  fits  close  up  against 
the  stem,  and  under  the  head,  of  the  main  rail  B,  so  that  the  wheel  L,  moving  in  either  direction,  can 

Sass  it  without  jar.  The  fixed  guard-rail  «  tends  to  draw  L  away  from  B,  and  thus  to  avoid  any 
anger  of  its  striking  the  point  x.  Rail  E  is  elevated  like  D,  to  prevent  the  rocking  of  cars,  etc., 
which  would  result  if  D  alone  was  elevated.  Its  raistd  inner  edge  z  z  prevents  the  flange  of  wheel  K 
from  striking  rail  A  in  going  up  or  down  the  incline  at  the  toe  of  D  in  Fig  20. 

The  switch  is  operated  by  cranks  v  w,  and  a  tumbling  lever  i.  This  lever,  when  the  switch  is  set 
either  way,  lies  so  as  to  bring  the  orank  v  a  little  below  its  center,  so  that  any  lateral  pressure  on  the 
switch-rails,  instead  of  misplacing  the  switch,  only  presses  the  lever  closer  to  the  ground.  The  weight 
on  the  end  of  the  lever  tends  to  bring  it  clear  down  to  its  proper  position. 

Provision  against  accident  In  case  of  misplacement  of  the  switch.    Case  1.   A 

train  moving  in  a  directiou  opposite  to  the  arrow,  must  ot  course  go  as  the  switch  is  set,  whether 
right  or  wrong.  On  this  account  it  is  usual  (when  possible)  on  double-track  roads  to  so  place  switches 
that  trains  not  intended  for  the  turnout  will  approach  them  only  in  the  directiou  of  the  arrow.  For 
single-track  roads  the  Wharton  is  so  arranged  that  it  will  not  remain  set  for  the  turnout  unless 
actually  held  so  by  the  switchman.  Case  2.  If  a  train  comes  from  the  turnout  while  the  switch  has 
been  wrongly  left  set  for  the  main  line.  Fig  19,  the  wheels  run  upon  the  fixed  "safety-castings" 
H  Y,  and  the  flange  o  o,  on  H,  guides  them  safely  on  to  the  main-track  rails  A  B.  Case  8.  A  train 
moving  along  the  main  track  in  the  direction  of  the  arrow  while  the  switch  is  wronglv  set  for  the 
turnout,  Fig  20.  The  cranks  v  w  are  so  arranged  (see  Figs)  that  when  the  switches  are  pushed  up 
against  the  main-track  rails,  Fig  20,  the  end  c  of  the  movable  guard-rail  is  moved  in  the  opposite 
direction,  and  brought  close  up  to  the  main  rail  A.  In  this  case  the  flange  of  the  first  wheel  K  pushes 
aside  c,  which,  pulling  the  rod  attached  to  it,  throws  the  crank  w  (and  through  it  the  entire  switch) 
into  the  proper  position,  Fig  19,  for  the  main  track. 

A 


-*-    g       ^i 

n       D 

P 

9 

F«,.  21 

->,„      S           rv 

n       E 

I    LORENZ. 

-,           ,,    B 
A,. 

s    """TC 

n       D 

P 

9 

Eg.  22 

n        E 

Art.  11.  The  Lorenz  Safety  Switch*  of  Wm.  Lorenz,  Esq.,  Chief  Engineer  Reading  R.  R.,  in 
use  ou  many  ot  our  main  roads,  is  a  "  split-rail  "or  "  point"  one.  The  movable  parts  of  the  point- 
rails,  on  main  lines,  are  usually  from  15  to  18  feet  long.  Fig  21  shows  this  switch  set  for  the  main 
line,  Fig  22  for  the  turnout.  The  rails  A  A,  B  B,  are  continuous,  and  spiked  to  the  ties.  The  switches 
or  point-rails,  D  E,  are  spiked  to  the  ties  for  from  4  to  8  feet  back  from  their  heels,  n,  which,  as  in 
the  Wharton,  are  in  the  places  occupied  by  the  toes  in  the  common  blunt-ended  or  stub  switch.  The 
point-rail  B  is  so  arranged  as  to  form  a  part  of  the  turnout  curve  in  Fig  22.  The  switches  are  oper- 
ated by  any  ordinary  lever,  see  p.  406.  Short  guard-rails,  a  a,  are  sometimes  added  to  prevent  wheels 
from  running  against  the  points  of  the  switches. 

Provision  against  accident  In  case  of  misplacement.  The  remarks  on  the  Wharton 
switch  uu'der  Case  1,  Art.  10,  apply  also  to  the  Lorenz  on  double-track  roads.  If  a  train  moves  in 
the  direction  of  the  arrow,  in  either  Fig.  when  the  switches  are  misplaced,  the  wheel  flanges  push 
aside  the  point-rails,  D  and  E,  and  thus  force  a  passage  through  the  switch.  A  spring,  of  rubber  or 
coiled  steel,  placed  on  the  switch-rod,  sometimes  at  p,  sometimes  outside  of  the  track,  permits  this 
motion  of  the  points  ;  and  after  a  wheel  has  thus  passed  through,  the  spring  returns  the  rails  to  their 
former  places. 

•X- Manufactured  by  the  Pennsylvania  Steol  Co.,  office  208  S.  4th  St.,  Phila.  Price  in  1882  of  a 
switch  with  point  rails  18  feet  long  $125,  including  rails  A  and  B  and  guard-rails  s  s,  but  exclusive 
of  switch-stand  and  putting  in  place.  In  ordering,  give  gauge  of  track,  a  full-sized  section  of  rail 
used,  directions  for  drilling  fishing-holes,  and  number  of  frog.  Also  state  whether  the  turnout  ia 
right-hand  or  left,  and  whether  the  main  line  is  straight  or  curved.  If  curved,  state  radius,  and 
whether  in  same  direction  as  turnout  (Fig  12),  or  opposite  (Fig.  11). 


RAILROADS. 


409 


Platform  weigh-scales.      Since  the  making  of  these  is  a  specialty,  we 

shall  not  describe  their  construction.  Large  ones  from  30  to  150  ft  long,  for  weighing  from  30  to  150 
tons  of  loaded  cars  at  once,  cost  at  the  shop  from  $25  to  $30  per  ft  of  length.  The  preparation  of  the 
pit  under  the  track,  with  its  lining,  foundations,  &c,  and  putting  the  scales  into  place,  will  cost  about 
35  to  40  per  ct  in  addition.  Superior  ones  are  made  by  Messrs  Riehle,  9th  St  above  Master,  Philada, 
and  by  Fairbanks  &  Co,  St  Johnsbury,  Vermont ;  office,  715  Chestnut  St,  Philada. 

Testing1  machines  for  either  tension  or  compression,  up  to  some  hundreds 
of  tons,  are  also  specialties  of  Messrs  Riehle. 


KAILKOADS. 


The  total  annual  expenses,  on  U  States  railroads,  usually 
range  from  4a  to  75  per  cent  of  the  receipts;  the  mean  being  60  per  cent.  Few  fall 
below  the  lower  limit ;  while  few  exceed  the  higher  one.  The  average  of  all  the 
railroads  in  England  and  Wales,  previous  to  1865,  was  about  48  per  ct ;  in  1866,  they 
were  48.8;  and  in  1867,  60.6  per  ct. 

Table  of  average  per  centages  of  the  various  general  items 
which  constitute  the  annual  expenses  of  railroads. 


ITEMS. 

Approximate 
Average  of 
U.  S.  Roads. 

Average 
English  Re 

•of 
ads. 

Mctintencmcp  of  way  cind  worJcs*  

Per  ct. 
25 

Per  ct. 
19 

30f 

28 

j?£t>ctiVs  and,  rcnevxils  of  COLTS  %  

10 

9 

30 

28 

(rcneral  expenses  II         

5 

6/<2 

Damages  to  persons  and  goods   

^A 

8 

AfUttJ,  .fjwr  i           , 

100 

100 

In  1869,  the  proportions  which  the  several  items  of  expense  on  the  Penna  Central,  between  Phila 
and  Pittsburg,  bore  to  the  total  expense,  were  very  nearly  as  follows:  Conducting  transportation, 
28.5  per  ct;  motive  power,  30.4;  maintaining  roadway,  27.3;  maintaining  cars,  12;  general  ex- 
penses, 1.8;  total,  100. 

Each  of  these  items  is,  however,  subject  to  great  variation,  not  only  on  diff  roads,  but  on  the  same 
road,  from  year  to  year.  A  road  with  many  bridges,  deep  cuts,  high  embkts,  Ac.  to  keep  in  repair, 
will  have  heavier  maintenance  of  way  than  one  which  has  but  few  ;  and  this  item  may  be  but  small 
one  year,  and  twice  as  great  the  next.  Fuel  may  be  cheap  on  one  road,  and  dear  on  another:  thus 
materially  affecting  the  item  of  motive  power.  And  so  with  the  other  items.  Sometimes  mainte- 
nance of  way  exceeds  motive  power  and  cars  together;  at  others,  conducting  transportation  is  fully 
half  the  total  expense. 

Tiie  gross  receipts  per  mile  of  road  in  1868,  according  to  Poor's  Rail- 
road Manual,  were,  in  Massachusetts,  $15400;  New  York,  $i;>142;  Pennsylvania, 
$13900;  average  of  the  whole  United  States,  about  $10000.  And  the  earnings  from 
freight  averaged  about  2%  times  that  from  passengers. 

The  tonnage  on  the  N  York  roads  was  3625  tons  per  mile ;  Massachusetts, 
5438  tons;  Pennsylvania,  8000  tons;  average  of  U  S,  about  2500  tons. 

The  total  annual  expenses  on  railroads  in  the  United  States 
usually  range  between  65  and  130  cents  per  train  mile:  that  is,  per  mile  actually 
run  by  trains.  Also,  between  1  and  2  cents  per  ton  of  freight,  and  per  passenger 
carried  one  mile.  When  a  road  does  a  very  large  business,  and  of  such  a  character 
that  the  trains  may  be  heavy,  and  the  cars'full,  (as  in  coal-carrying  roads,)  the  ex- 
pense per  train  mile  becomes  large;  but  that  per  ton  or  passenger,  small ;  and  vice 
versa,  although  on  coal  roads  half  the  train  miles  are  with  empty  cars. 

*  Including  bridges,  depots,  stations,  and  other  buildings,  roadway,  superstructure,  fences,  &c. 

t  Of  this  30  per  ct  of  total  expenses,  about  1 1  (7  to  14)  is  for  fuel ;  about  7  (5  to  10)  for  repairs  of 
engines;  and  12  for  pay  of  engine  drivers,  firemen,  cleaners,  oil,  waste,  &c ;  as  averages  of  a  great 
number  of  roads  of  every  character  as  to  length,  traffic,  &c.  Fuel  range  from  5  to  12  cts  per  train- 
mile  ;  usually  6  to  9  cts.  The  repairs  of  engines  from  6  to  14  cts  :  usually  about  8  to  12. 

+  The  annual  repairs  of  a  large  8-whee!  passenger  car,  ranges  between  about  $300  and  $000:  mail 
and  express  cars,  from  $150  to  $300;  freight  cars,  from  $75  to  $150;  coal  cars,  from  $20  to  $30,  (4 
wheels.) 

§  Chiefly  wages  of  clerks,  ticket-sellers,  conductors,  brakemen,  bridge  and  switch-tenders,  tele- 
graphers, porters,  weighers,  &c.  Ac;  lighting  and  warming  of  ears,  depots,  stations,  Ac. 

II  Salaries  of  the  higher  officers  of  the  Company ;  law,  stationery,  advertising,  office  rents,  &c,  &c. 
Generally  varies  from  2^  to  7  per  ct. 


410 


RAILROADS. 


Table  of  Annual  Expenses  of  some  U  S  Railroads.* 


Names  of  Companies. 


Lehigh  Valley,  1860 

1862 

"     "    1868  and  1869 about. 

"     "    1872   

Baltimore  &  Ohio,  main  stem,  1859 4434 

"  "         "      I860 4254 

"      1865 

"  "          '      «        "      1866 

"   1872 ..; 

East  Tennessee  A  Georgia,  1872 

Memphis  &  Charleston,  1860 2617 

Georgia  Central,  1872 4180 

Penna  Central,  main  line  from  Phila  to  Pittsburg,  358  miles, 

1859,  exclusive  of  State  tonnage  tax 7848 

"  "        I860,        "  "  "  "     

"  "        1861,        "  "  "  "     

"  "        1868,  tonnage  tax  repealed ... 

"  «        1869,        "         "          •*         about...    32000 

"        1872,        «'         «          «          

Phila  &  Reading,  1859 

1860 

"          1868,  365  miles  of  main  road  and  branches...   17200 

"  "          1869  

1872  

North  Pennsylvania,  1860,  54  miles  long 3213 

"  "  1862 3240 

"  "  1867 9534 

"  "  1868 ... 

"  "  1872. 

Connecticut;  average  of  all  the  railroads,  1861 3781 

Massachusetts;     "          "        "          "  1861 3785 

{2700 
to 
4300 

1867    average.. 

Galena  &  Chicago,  1859 

I860     3102 

Phila,  Wilmington  &  Baltimore,  main  stem,  1859 4586 

"  "  "  "        "       1860 7100 

"  "  "  "        "       1861 7785 

"  "        "       1867 17380 

New  York;  all  the  U  R  in  the  State,  average,!  1859 4964 

"         "        "          "  "  1861 5100 

"  "          "         "  "  "  1867 13856 

New  Jersey  R  K  and  Transportation,  1861.. 12213 

Louisville  &  Nashville,  1861.     

Phila  &  West  Chester,  1861,  27  miles 2274 

1862 2282 

"  1872 7030 

Phila,  Germantown  &  Norristown,  1861,  20  miles 6K)5 

"  1862 6405 

"  "  "  1867 18208 

New  York  &  Erie,  1861  6461 

18  7,  with  its  branches,  784  miles  in  all 14545 

New  York  Central,  1861    .    8360 

'•        1867,  with  its  branches,  696  miles  in  all....    15620 

English  R  R,  averages  for  1856-7-8     

Scotch      '•  '•          "       "     "" 

Irish         "  "          "       "    "  " 

*  Annu  -I  t-fpot-t-i  ofren  omit  the  lengths  of  the  roads  and  branches:  and  as  these  frequently  vary 
from  year  to  \  ear,  it  is  possible  that  tne  table  may  ooutain  some  errors  in  the  first  column. 

f  2528  miles  in  operation.  Total  exps  equalled  1.56  cts  per  passenger  or  ton  carried  1  mile.  Dead 
weight  of  cars,  equal  to  1.19  tons  per  passenger ;  and  to  1.74  tons  per  ton  of  freight. 


Per  Per 
Mile  of  I  Train 
Road. 

$ 


RAILROADS.  411 

Fuel  for  locomotives.  As  nearly  as  the  writer  can  judge  from  a  mass  of 
Yery  conflicting  testimony,  a  ton  of  good  anthracite,  or  of  bituminous  coal,  is  equal 
to  1%  cords  of  good  dry  hard  mixed  woods,  (chiefly  white  oak  ;)  or  to  from  t%  to  '2.% 
cords  *  of  such  soft  ones  as  hemlock,  white,  and  common  yellow  pine.  Much  ol  the 
inferior  bituminous  coal  of  Illinois  is  hardly  equal  to  a  cord  of  average  wood.  On 
the  Illinois  1'eiitral  in  1660,  the  engines  ran  about  200,000  miles,  with  average 
trains  of  10%  loaded  8-vvheel  cars.  The  wood-burners  averaged  39  miles  per  cord  ; 
or  '£%  cords  per  100  uiiles.  The  coal-burners,  with  this  inferior  coal,  36}^  miles  per 
ton  ;  say  2%  tons  per  100  miles  ;  making  1  cord  of  wood  equal  to  ly1^-  tons  of  coal. 
On  the  Chicago,  Burlington  «V  Quiiicy  K  K.  a  very  full  trial  (about 
11000  rniles  with  each  kind)  was  made  with  this  coal,  and  with  another  of  good 

Duality  from  another  locality.  The  trains  averaged  2  %  eight-wheel  cars,  with  their 
aily  loads  ;  and  the  result  was  3.5  tons  of  the  bad  coal"  and  2.2  of  the  good,  per  100 
miles  ;  making  1  ton  of  the  good  coal  equal  to  1.45  cords  mixed  woods.  On  the 
Great  Western  R  R  ol"  Mass,  the  same  trains  required  2%  tons  of  anthra- 
cite per  100  miles;  or  4  coids  of  hemlock,  (an  inferior  fuel;)  or  i  ton  anth  =  1.6 
cords  hemlock. 

On  the  Philada  A  Reading,  passenger  trains  of  5  large  cars,  and  weighing  54 
tons,  exclusive  of  engine  and  tender,  (or  about  90  tons  with  them.)  used  2.7  cord>  of  good  mixed 
woods,  or  1.5  tons  of  anth  per  100  miles.  Here  1  ton  anth~  1.8  cords  of  wood.  In  1867.  passenger  trains 
of  6fe  tons,  (about  10*2  with  engine  and  tender.)  are  reported  to  use  2.22  tons  per  100  miles.  This  is 
much  greater  in  proportion  to  the  increased  weight  of  train  than  in  the  preceding  case  ;  and  there 
would  seem  to  be  some  discrepancy.  The  freight  trains  use  3.45  tons,  and  the  very  heavv  coal  trains 
4.75  tons  per  100  miles.  These  last  consist  of  about  96  four-wheel  coal-cars,  weighing  2}'  tons  each, 
or  241  tons  in  all  ;  96  loads  of  coal,  of  nearly  4%  tons  each,  or  443  tons  in  all  :  and  engine  and  tender 
about  50  tons;  making  the  entire  weight  of  the  train  734  tons.  The  returning  empty  trains  up  the 
road  weigh  291  tons  in  all.  and  consume  about  the  same  quantity  of  fuel  as  the  descending  loaded 
ones.  The  grade  of  the  95  miles  of  road  in  the  direction  of  the  loaded  trains  is  descending  :  most  of 
it  al  less  than  6  feet  per  mile;  and  none  exceeding  14  ft,  except  3  continuous  miles  of  22.4  ft.  and  a 
stretch  of  less  than  2  miles  of  35  ft  per  mile  ascending;  on  which  last  auxiliary  power  is  used.  When 
the  coal  trains  burned  wood  they  used  about  Y^  cords  to  the  present  ton  of  coal. 

The  mean  weight  of  the  loaded  coal  trains  down,  and  of  the  empty  ones  up,  is  512  tons  ;  and  that 
of  the  passenger  trains  each  way  is  102  tons,  or  only  -JL-  as  much  ;  yet  the  quantity  of  fuel  consumed 
b<  the  first  is  but  little  more  than  twice  as  great  as  by  the  last.  The  first  run  about  10  miles  per 
hour  ;  the  last,  25  miles. 

On  the  Penn'a  Central,  experiments  made  1  ton  of  Pittsburg  bituminous  coal  equal 
to  1.49  cords  of  wood  (chiefly  white  oak)  weighing  5215  Ibs,  or  2.328  tons  ;  or  3500  Ibs  per  cord.  Trains 
of  4  eight-wheel  freight  cars,  weighing  30.8  tons,  and  loaded  with  33  tons;  engine  and  tender  41.  56 
tons,  making  105.36  tons  in  all  ;  in  ascending  12  miles  of  continuous  grade  of  95  ft  per  mile,  at  a  speed 
of  20^  miles  per  hour,  consumed  either  1073  fts  (.479  ton  of  coal  ;  or  2483  Ibs  (1.108  ton.  or  .7094  cord) 
of  wood.  This  in  100  rniles  gives  3.992  tons  of  coal,  or  5.91  cords  of  wood  ;  or  say  4  tons,  or  6  cords  ; 
or  1  ton  coal  =  \%  cords  wood.  And  this  accords  closely  with  the  comparative  consumption  of  an- 
thracite and  wood,  in  hauling  heavy  coal  trains,  on  the  Reading  R  R.  The  passenger  engine  used  in 
the  foregoing  experiments  weighed  "28%  tons,  of  which  18.16  tons  rested  on  4  drivers,  5%  ft  diam.  It 
was  arranged  to  be  equally  well  adapted  to  coal  and  to  wood.  The  trials  show  the  effect  of  steep  up- 
grades in  increasing  the  consumption  of  fuel.  Thus,  on  the  Reading  road,  an  engine  with  20  tons  on 

with  engine  and  tender  about  625  tons  ;  consuming  at  the  rate  of  \%  tons  of  anthracite,  or  7.1  cords 
of  wood  per  100  miles;  which  in  12  miles  would  be  .57  ton,  or  .84  cord  ;  against  the  .48  ton,  or  .71 
cord,  required  on  the  steep  grade,  with  a  train  weighing  but  %  as  much. 

The  run  of  freight  and  passenger  trains  throughout  the  U  States  is  20  to  40  miles  per  cord,  and  30 
to  60  miles  per  ton.  Much  depends  upon  the  adaptation  of  the  engine  to  the  kind  of  fuel  used.  A  good 
coal-hnrner  may  be  bad  for  wood,  and  vice  versa;  so  that  trials  with  the  same  engine  may  give  very 
erroneous  results  as  to  the  comparative  merits  of  the  two  kinds  of  fuel.  When  wood  is  used,  about 
^  cord  :  or  when  coal,  about  %  cord  of  wood,  must  be  used  for  kindling,  and  getting  up  steam  rendy  for 
running;  and  this  item  is  the  same  for  a  long  run  as  for  a  short  one  :  so  that  long  roads  have  in  this 
respect  an  advantage  over  short  ones,  in  economy  of  fuel.  Wood  has  the  disadvantage  of  emitting 
•parks;  and  is,  moreover,  nearly  twice  as  heavy  as  coal,  for  the  performance  of  equal  duty  ;  and  is, 
therefore,  more  expensive  to  handle.  It  also  occupies  4  or  5  times  as  much  space  as  coal. 


traffic  woul 
$18250  per  mle. 

To  find  the  tractive  power  of  a  locomotive,  mult  together  the 

*  A  oord  is  4  X  4  X  8  ft,  or  128  cub  ft.  A  cord  of  good  dry  white  oak.  (next  to  hickory,  the  best 
wood  for  fuel,)  weighs  3500  Ibs  or  1.563  tons.  Dry  hemlock,  white,  or  common  yellow  pine,  (all 
of  them  inferior  for  fuel.)  about  .9  ton.  Perfectly  green  woods  generally  weigh  about  •£  to  ^  more 
than  when  pnrtinlly  dried  for  locomotive  use:  in  other  words,  a  cord  of  wood,  in  its  partial  drving, 
loses  from  X  to  %  ton  of  water,  and  still  contains  a  large  quantity  of  it.  Since  this  water  causes  a 
great  waste  of  heat,  green  wood  should  never  be  used  as  fuel.  The  values  of  woods  as  fuel  are  in 
nearly  the  same  proportion  as  their  weights  per  cord  when  perfectly  dry. 


412 


RAILROADS. 


square  of  the  diam  of  one  piston  in  ins;  the  single  length  of  stroke  in  ins;  and  the 
cylinder  pressure  of  the  steam  in  Ibs  per  sq  inch.  Divide  the  prod  by  the  diam  of  a 
driver  in  ins.  The  quot  will  be  the  power  in  tbs.  Whether  the  engine  can  really 
employ  that  power  or  not,  depends  upon  the  weight  on  its  driving-wheels.  And  the 
velocity  depends  upon  tne  steam-generating  capability  of  the  boiler.* 

Or  ordinarily  file  tractive  power  =  adhesion  =  at  least,  one-fifth  of 
the  wt  on  the  drivers.  With  the  rails  clean  and  in  good  order  it  may  rise  even  to 
full  one-third  of  said  weight  at  low  speeds.  It  becomes  less  at  high  speeds. 

Table  of  appro x  greatest  wt  of  train  which  a  good  locomotive 
weighing  27  tons,  or  60480  ft>s,  all  of  it  on  the  drivers,  can  (in  addition  to  the  wt 
of  itself  and  tender,  45  tons,)  take  up  straight  grades,  at  about  8  or  10  miles  an 
hour,  when  all  is  in  good  order,  taking  the  resistance  at  8  Ibs  per  ton  on  a  level ; 
and  the  traction  at  one-fifth  the  wt  on  drivers.  For  other  engines  the  loads 
will  be  about  as  the  wts  on  drivers.  With  everything  in  perfect  order,  friction  ut 
times  falls  to  5  Ibs,  or  even  to  4  Ibs,  per  ton  wt  of  train. 


Grade  in 
Ft.  per 

Mile. 

Tons. 

Grade  in 
Ft.  per 

Mile. 

Tons. 

Grade  in 
Ft.  per 

Mile. 

Tons. 

Grade  in 
Ft.  per 

Mile. 

Tons. 

Level. 

1458 

16 

769 

60 

315 

160 

113 

1 

1383 

18 

725 

65 

294 

not 

105 

2 

1315 

20 

686 

70 

275 

180 

98 

3 

1253 

22 

650 

75 

258 

190 

91 

4 

1197 

24 

617 

80 

242 

200 

84 

5 

1144 

26 

587 

85 

228 

225 

71 

6 

1096 

28 

561 

90 

216 

250 

60 

7 

1052 

30 

536 

95 

204 

275 

61 

8 

1012 

32 

513 

100 

194 

300 

44 

9 

974 

35 

482 

110 

175 

350 

32 

10 

939 

40 

437 

120 

159 

400 

22 

11 

905 

45 

400 

130 

146 

450 

15 

12 

875 

50 

367 

140 

134 

500 

9 

14 

819 

55 

340 

150 

123 

528 

7 

In  practice,  trains  must  not,  as  a  general  rule,  weigh  more  than  from  ^  to  %  of  those  in  the  table. 
This  reduction  is  necessary  in  order  to  have  the  trains  under  more  complete  control;  to  admit  of 
greater  speed  in  case  of  detention ;  to  allow  for  curves,  slippery  rails,  head  winds,  &c,  &c. 

The  animal  expense  of  running  a  locomotive  and  tender, 

averaging  75  miles  a  day,  for  267  days  in  the  year,  or  20000  miles  annually,  which 
is  common,  (have  run  60000)  may  be  found  in  the  following  manner : 

Fuel,  say  2\£  cords  per  75  miles,  at  $3.50  per  cord ;  $7.87%p?r  day $2100 

Repairs,  at  Sets  per  mile  run 1800 

Engineer,  or  driver,  12  months,  at  $90 1080 

Fireman,  12  months,  at  $50 600 

Oil  and  waste,  at  1  ct  per  mile  run 200 

Sawing,  and  loading-up  wood;  1%  cts  per  mile  run < 

Supplying  water let  "        "  200 

Putting  away,  cleaning,  and  getting  out,  say 120 

Locomotive,  superintendence "   • 100 

Total $6500 

Equal  to,  say  34  cts  per  train  mile ;  or  $24.35  per  running  day;  or  $17. 81,  for  every 
day  in  the  year. 

This  is  all  'that  is  usually  stated  in  annual  reports  of  expenditures;  but  inasmuch  as  an  engine  in 
active  service,  even  under  a  judicious  system  of  repairs,  generally  becomes  worthless,  (except  as  old 
iron.)  in  say  16  years  on  an  average,  an  additional  allowance  of  nbout  6  per  ct  on  the  first  cost,  or 
about  $500  to  $800,  should  be  made  annually  for  DEPRECIATION  OF  EACH  ENGINE. 

For  Evaporation  by  Locomotives,  see  p  434. 

*  The  cylinder  pres  is  always  less  than  the  boiler  pres ;  and  the  disproportion  increases  with  the 
speed.  Thus,  at  8  or  10  miles  an  hour,  the  boiler  pres  may  be  about  110  fts  per  sqiuch;  and  the 
cylinder  pres  from  90  to  100  fts  ;  while  at  a  speed  of  30  or  +0  miles,  the  proportion  may  be  as  110  to  60 

°YBaldwin's  8-driver  engines  of  35  tons,  all  on  the  drivers,  draw  40  empty  coal  cars,  weighing  100 
tons,  up  a  continuous  grade  of  175  ft  per  mile  for  3^  miles,  with  many  curves  and  reversed  curves  of 
from  450  to  600  ft  rad.  at  8  miles  an  hour,  as  an  every -day  operation. 

7  On  the  Penn'a  Central,  engines  of  29  ton*,  all  of  it  on  8  drivers,  take  200  tons  of  cars  and  freight  up 
a  continuous  grade  of  95  ft  per  mile  for  »%  miles.  The  passenger  engines  take  up  3  loaded  8-wheel 


RAILROADS.  413 

The  length  of  Locos  out  to  out  is  usually  about  a  ft  per  ton  wt.  But  some  very  heavy 
freighters  weigh  1.5  tons  per  ft.  Extreme  width  9  ft  4  ina  for  4  ft  8}$  gauge.* 

The  height  Of  the  pipe,  or  smoke  stack,  is  usually  13  to  15  ft  above  the  rails. 
Good  StOUt  Steel  driving-tires,!  bearing  about  3  to  3^  tons,  will  run  aboutSOOOO 

or  J4  iuch  in  diaiu  ;  and  will  bear  3  or  i  turnings-down  before  wearing  too  thin  for  safety.  Driving 
•xies  and  journals  are  about  5><j  to  6  ius  diam.  Flanges  are  about  \y±  ins  deep. 

Tenders  Of  passenger  engines,  usually  weigh  5  to  6  tons  empty;  and  12  to 
15,  with  water  and  fuel ;  those  of  freight  engines,  6  to  8  tons  empty  ;  and  15  to  20,  with  water  and  fuel. 
Their  length  from  out  to  out,  is  usually  about  %  that  of  the  engine.  Extreme  width,  9  ft. 

Passenger  engines  usually  carry  fuel  and  water  sufficient  for  20 

or  30  miles;  some  50  to  70.  Freight  trains,  enough  for  10  to  20  miles.  Roads,  or  divisions  with 
steep  grades,  require  the  fuel  and  water  stations  to  be  nearer  together  than  where  the  grades  are  easy. 
The  tanks  of  passenger  engines  usually  contain  from  1200  to  1800  gallons ;  and  those  of  freight  engines, 
from  1600  to  2200.  Twelve  hundred  galls  —  160  cub  ft  =  4.47  tons  ;  and  2200  galls  =  294.cub  ft-  8.15 
tons.  Stations  for  fuel  and  water,  as  well  as  those  for  passengers,  &c,  should,  as  far  as  possible,  be 
placed  at  the  summit  of  two  up-grades,  for  convenience  of  both  stopping  and  starting. 

Passenger  ears.  Eight  wheels ;  for  56  passengers  ;  with  water-closet,  and 
space  tor  a  stove,  are  about  46  to  48  ft  extreme  length  from  out  to  out  of  bumpers.  On 
the  4  ft  8%  inch  gauge,  the  extreme  width,  out  to  out  of  the  slightly  projecting  roof- 
cornice,  is  usually  8^  to  10  feet ;  the  last  especially  on  sleeping-cars.  On  the  6  ft  gauge 
of  the  N.  York  &  Erie,  11  ft.  Penna.  R.  R.  standard,  9  ft  7%;  gauge  4ft  S% 
Weight  empty  about  16  tons ;  or  when  filled  with  passengers,  about  20  tons. 

Therefore,  even  when  the  cars  are  full,  there  is  a  dead  load  of  about  4  tons,  to  1  paying  ton  of 
passengers.  But  as  a  general  rule,  passenger  trains  are  not  more  than  one-half  filled;  thus  making 
the  proportion  at  least  8  to  1.  The  annual  reports  of  the  State  of  N.  York,  show  there  about  18  to  1 ! 
or,  say  l£.  ton  of  dead  load  for  each  passenger !  In  England,  with  their  heavier  passenger-cars,  the 
dead  load  on  some  lines  is  from  20  to  30  times  the  paying  one  I  A  ton  of  paying  freight,  with  the  cars 
only  partially  loaded  as  usual,  requires  on  an  average  about  1*4  of  dead  car- weight ;  and  since  freight 
is  hauled  at  a  less  expensive  speed  than  passengers,  It  is  seen  that  the  carrying  of  a  single  passenger 
may  readily  cost  as  much  as  that  of  a  ton  of  freight;  especially  when,  as  is  too  often  the  case,  a  con- 
siderable portion  of  the  passengers  are  dead-heads  ;  a  class  which  travel  very  often. 

The  platforms  of  passenger  cars  are  usually  3%  to  3%  ft  above  the  rails. 
Weight  of  cars.    Passenger  cars  for  about  50  to  60  persons,  14  to  18 

tons.  Baggage  and  mail  cars,  8  wheels ;  length  from  out  to  out  of  bump- 
ers, about  40  to  45  ft ;  width  8^  to  9%  ft  out  and  out  of  all,  10  to  14  tons.  Freight 
cars,  (house  or  box-cars,)  length  30  to  36  ft  out  to  out;  width  7^ to  9  ft;  8  wheels, 
7  to  9  tons.  Platform,  and  gondola  cars,  for  stone,  lumber,  iron,  Ac,  8 
wheels,  25  to  30  ft  long  out  to  out ;  width  7  to  9  ft,  6  to  8  tons.  Coal  cars,  single, 
on  4  wheels;  2%  to  2%  tons;  and  carry  about  5  tons  of  coal.  Size  about  12  feet 
long,  by  6^  ft  wide  out  to  out  of  everything.  Double  ones,  on  8  wheels,  are  most  in 
use.  They  weigh  from  5  to  6  tons;  and  carry  about  10  tons  of  coal.  Ordinary 
sleeping  cars  are  about  the  same  size  as  day-cars ;  and  weigh  from  20  to  25 
tons ;  but  some  recent  ones,  to  accommodate  56  persons,  are  nearly  70  ft  long ;  by 
11  ft  wide  out  to  out ;  with  16  Wheels ;  and  weigh,  empty,  about  33  tous.J 

The  wheels  for  passenger  and  freight  cars,  are  usually  about 
30  to  33  ins  diam ;  and  those  of  coal  cars  26  to  28.  The  cast-iron  wheels  made  by 
Messrs.  A.  Whitney  &  Sons,  of  Philada,  weigh  as  follows,  per  single  wheel.g  Combined 
width  of  tread  and  flange,  5%  ias.  Cone,  1  in  21. 

D.iam-     36.        33.        30.        28.        26.        24. 
ins. 


W  fi»ht>  585       50°       44°       4°°        35°       335 

The  weights  by  other  makers  do  not  differ  from  these  materially. 

The  diam  of  car  or  engine  wheels  does  not  include  the  flanges ;  but  is  the  least 
diam  from  tread  to  tread.  Good  chilled  cast  wheels  will  run  about  100,000  miles ; 
usually  however  they  hardly  average  50000k 

*  Cost  of  Locomoti  ves,  Philada,  1880,  $250  to  $300  per  ton  of  their  weight ; 

the  tender  being  supposed  to  be  thrown  in. 

t  Price,  in  Philada,  1880,  11  to  13  cts  per  ft. 

I  APPROXIMATE  AVERAGE  PRICKS  OF  CARS  in  the  Eastern  States  in  1880.  Passenger-cars  for  about 
50  to  60  persons,  $4500  to  $5500.  Dining-room  cars,  from  $6000  to  $8000.  Sleeping-cars,  $6000  to 
$10000.  Some  very  extravagant  ones,  for  60  persons,  have  cost  from  $20000  to  $25000.  Mail  and 
baggage  cars,  $2800.  Freight  cars,  (house,  or  box-cars)  $750.  Gondola,  and  platform  cars,  $400  to 
$500.  All  the  foregoing  ou  3  wheels.  Coal  cars,  single,  4  wheels,  wooden  bodies,  $200;  with  iron 
bodies,  $300. 

^  Cost  in  Philadelphia,  in  1880,  about  3£$  cts  per  fl>.  Wrought-iron  axles,  hammered,  about  5  ct» 
per  ft ;  and  rolled,  ift  eta. 

27 


414 


RAILROADS. 


T°?  ???i£OIMlSI-,0f  Peill«yl  v 

rage,  about  $58000  per  mile  :  aud  have  about 


with  their  equipments,  have  cost  on  an  ar*. 


1      Locomotive  to  every  .........................................  4  miie8  of  road. 

Engine-house  to  "     ......................  ...........  .....  20      "  " 

1      Passenger  car  to  "     .........................................  8      "  *< 

1      Baggage,  mail,  or  express  car  to  every..  ..............  18      "  " 

2%  Freight  cars,  and  trucks,  to  .........     "      ................  1      «  « 

6      Coal  cars  to  .............................     "     ...............  1      «  « 

1      Depot,  or  station..    ...................     "    ...........  .....  7      "  « 

1      Fuel  and  water  station  to  .........     "     ................  9      "  " 

The  average  cost  per  mile  of  the  railroads  of  New  England  has  been 

about  $40500;  of  the  Middle  States,  $55000;  of  the  Southern  States,  $30000. 

Miles  of  railroad  in  the  world,  at  the  close  of  1879,  about  203000;  or 

I  here  ''"IT  ^  circumf  °f  -the  .earth-     In  Nor'h  America,  90000  ;  Europe,  and  entire  Eastern  hemi- 

Standurd  size  of  axles  for  passenger  or  freight  cars,  on  a  4  ft  8%  track.  JLength,  total  6  ft 
11^  ins  ;  between  hubs  4  ft  0}^  in  ;  each  wheel-seat  7  ins;  each  journal  7  ins.  Diam,  at  middle 
Z%  ins  ;  at  hubs  4  %  ins  ;  at  journals  3%  ins.  Weight,  347  fts  per  axle. 

Table  of  cubic  yards  of  ballast  per  mile  of  road. 

Side-slope  of  the  ballast  1  to  1.  \Vidth  in  clear  between  2  tracks  6  ft.  The  ties 
and  rails  may  be  laid  first,  for  carrying  the  ballast  along  the  line;  then  raised  a 
few  ft  at  a  time,  and  the  ballast  placed  under  them. 


Depth 

TOP  WIDTH, 
SINGLE  TRACK. 

TOP  WIDTH, 

DOUBLE  TRACK. 

10  Ft. 

11  Ft. 

12  Ft. 

21  Ft. 

22  Ft. 

23  Ft. 

Cub.  Y. 

Cub.  Y. 

Cub.  Y. 

Cub.  Y. 

Cub.  Y. 

Cub.  Y. 

12 

2152 

2347 

2543 

4303 

4499 

4695 

18 

8374 

3667 

3960 

6600 

6894 

7188 

24 

46f)4 

5085 

5474 

8996 

9388 

9780 

30 

6111 

6600 

7087 

11490 

11S80 

12470 

A  mail  Can  break  3  to  4  cub  yds  per  day,  of  hard  quarried  .stone,  to  a  size  suitable 
for  ballast ;  say  averaging  cubes  of  3  ins  on  au  edge.  Where  other  ballast  cannot  be  had,  hard-burnt 
clay  is  a  good  substitute.  The  slag  from  iron  furnaces  is  excellent.  The  ties  decay  more  rapidly 
when  gravel  or  sand  is  used  instead  of  broken  stone,  because  these  do  not  drain  off  the  rain,  but  keep 
the  ties  damp  longer. 

CrOSS-tieS  of  8^  ft,  by  9  ins,  "by  7  ins,  contain  3.719  cub  ft  each;  and  if  placed  2%  ft  apart 
from  center  to  center,  there  will  be  2112  of  them  per  mile,  amounting  to  291  cub  yds.  Therefore,  if 
they  are  completely  embedded  in  the  ballast,  they  will  diminish  its  quantity  by  that  amount.  At  2  ft 
apart  there  will  be  2640  of  them,  occupying  364  cub  yds  ;  and  at  3  ft  apart,  1760  of  them ;  243  cub  yds. 

Cubic  feet  contained  in  cross-ties  of  different  sizes. 


Dimensions. 

Dimensions. 

Dimensions. 

Ft.  Ins.  Ins. 

Cub.  Ft. 

Ft.      Ins.   Ins. 

Cub.  Ft. 

Ft.    Ins.   Ins. 

Cub.  Ft. 

8  by  8  by  6 
896 

2.667 
3.000 

&H  by  8  by  6 
Sh       9        6 

2.833 

3.188 

9  by  8  by  6 
996 

3.oOO 
3.375 

897 

3.500 

8^       9        7 

3.719 

997 

3.938 

8     10        6 

3.333 

8^      10        6 

3.542 

9      10        6 

3.750 

8     10        7 

3.889 

8^      10        7 

4.132 

9      10        7 

4.375 

8     10        8 

4.444 

8^      10        8 

4.722 

9      10        8                     5.000 

8     12        8 

5.333 

8^      12        8 

5.667 

9      12        8                     6.000 

The  average  life  of  ties  is  6  to  8  years;  and  the  expense  for  renewing 

them  is  serious  ,  amounting  probably  to  not  less  than  ten  millions  of  dollars  anuually  in  the  United 
States  alone.  The  process  of  Seely  of  preserving  timber  by  the  injection  of  creosote  (see  p  358)  is 
altogether  the  best  that  has  been  devised,  and  there  cau  be  no  doubt  that  its  employment  will  in  many 
cases  be  true  economy.  This  remark  applies  as  well  to  timber  for  bridges,  and  for  engineering  and 
architectural  purposes  generally.  The  process  is  said  to  preserve  timber  from  the  Limnoria,  and 
Teredo  Navalit,  sea-worms;  but  in  our  opinion,  there  is,  as  yet,  no  good  foundation  for  this  state- 
ment. No  remedy  is,  we  believe,  yet  known  for  this  evil,  although  saturation  with  creosote  certainly 
delays  their  attacks  for  many  years  if  thoroughly  done.  The  limnoria  works  from  near  H.  W.  mark, 
down  to  a  little  below  the  surface  of  mud  bottom  ;  the  teredo  within  somewhat  less  limits. 

The  writer  believes  that  most  of  the  fault  usually  ascribed  to  cross-ties,  as  well  as  to  rail-joints,  is 
in  reality  due  to  imperfect  drainage  of  the  roadbed.  Hence,  he  does  not  agree  with  those  who  advo- 
cate VERY  LONG  TIES  ;  but  considers  that  with  good  ballast,  on  a  well-drained  roadbed,  8^  ft  is  as  good 
as  more ;  and  that  «^  ft,  by  9  ins.  by  7  ins :  aud  2%  ft  apart  from  center  to  center,  is  sufficient  for  the 
heaviest  traffic.  On  many  important  roads  they  are  but  8  ft;  and  on  some  only  1%  ft  long;  track 
4  ft8>£.  The  actual  COST  OF  CUTTING  DOWN  THE  TREES,  topping  off  the  branches,  and  hewing  the  ties 
ready  for  hauling  away  to  be  laid,  is  about  6  to  9  cts  per  tie,  at  $1.75  per  day  per  hewer. 


RAILROADS.  415 

The  narrow  bases  of  rails  resting  immediately  on  the  cross-ties,  without  chairs,  frequently  produce 
in  time  such  an  amount  of  crushing  in  the  ties  as  to  injure  them  materially  even  before  decay  begins. 
Mr  P.  H.  Whitman,  of  Portland,  Maine,  has  patented  a  mode  of  obviating  this  by  letting  into  the  tie  a 
thin  block  of  hard  wood  under  each  rail.  He  sells  machines  for  cutting  the  grooves  for  receiving 
these  blocks.  Burnetis'id  ties  rust  the  spikes  away  rapidly.  Creosoted  ones  preserve  them. 

Post-and-rail  fences,  in  panels  8%  ft  long;  5  rails;  usually  cost  between 
40  to  100  cents  per  panel,  including  the  putting  up;  or  from  $512  to  $1280  per  mile 
of  road  fenced  on  both  sides,  with  1280  panels. 

Worm  fences  seven  rails  high,  with  two  rails  on  end  at  each  angle,  cost  about 
%th  less.  Labor  $1.75  per  day.  The  scarcity  or  abundance  of  timber  chiefly  in- 
fluences the  price  ;  as  is  also  the  case  with  ties. 

The  Glidden  Barbed  Steel  Wire  fence,  made  by  .1.  L.  Ellwood  &  Co., 
at  De  Kalb,  Illinois,  has  (1882)  come  into  extensive  use;  many  thousands  of  miles 
of  it  having  already  been  put  up.  Its  cost  per  mile  of  single  row  of  fence,  put  up, 
including  the  wooden  posts  and  all  labor,  will  usually  range  from  $150  to  $250,  de- 
pending on  the  height  of  fence,  the  varying  market  price  of  wire,  labor,  &c. 


Every  sq  inch  of  sectional  area  of  rail,  corresponds  to  10  Ibs  per  yard  of  a  single 
rail  ;  or  to  15.7143  tons  per  mile  of  single-track  road.    Consequently, 


15.7143 


Thus,  a  rail  of  100  tons  per  mile  of  single  track,  will  have  a  section  of  6.364  sq  ins  ;  and  will  weigh 
63.64  Ibs  per  yd  of  single  rail.  Add  for  turnouts,  sidings,  road-crossings,  and  a  trifle  for  waste  in 
cutting.  Steel  must  inevitably  take  the  place  of  iron  for  rails,  or  at  least  for  their  tops.*  When  the  ties 
are  in  place,  and  the  rails  distributed  in  piles  at  short  intervals,  a  gang  of  6  men  can  lay  %  a  mile  of 
rails  per  day,  of  single  track  ;  or  after  the  ballast  is  in  place,  a  gang  of  15  men  will  lay  about  one 
mile  of  complete  single-track  superstructure  per  week. 

Approximate  average  estimate  for  a  mile  of  single-track 

railway.     Labor  $1.75  per  day. 
Grubbing  and  clearing,  (average  of  entire  road,).  3  acres  at  $50  ........................  $  150 

Grading;  20000  cub  yds  of  earth  excavation,  at  35  cts  ..................................    7000 

"  2000  cub  yds  of  rock  excavation,  at  $1.00  .....................................     2000 

Masonry  of  culverts,  drains,  abutments  of  small  bridges,  retaining-walls,  &c; 

400  cub  yds,  at  $8,  average  ...................................................................    3200 

Ballast;   3000  cub  yds  broken  stone,  at  $1.00  .................................................    3000 

Cross-ties;  2112,  at  60  cts,  delivered  ...............................................................     1267 

Rails;  (60  Ibs  to  a  yard;)  96  tons,  at  $100,  delivered  ..............  .  .....................     9600 

Spikes  .................  ...................................................................................       275 

Mail-joints,  or  chairs  .................................................................................      525 

Sub-delivery  of  materials  along  the  line  ...................................  .....................      300 

Laying  track.  ..................................  ....................................................      600 

Fencing  ;  (average  of  entire  road,)  supposing  only  %  of  its  length  to  be  fenced..      450 
Small  wooden  bridges,  trestles,  sidings,  road-crossings,  cattle  guards,  <£c,  <$c  ......     1000 

Land  damages  .............  ................................................................     1000 

Engineering,  superintendence,  officers  of  Co,   stationery,  instruments,  rents, 

printing,  law  expenses,  and  other  incidentals  ............................................      2033 

Total  .............  .......  .........................  $32400 

Add  for  depots,  Shops,  Engine-houses,  Passenger  and  Freight  Stations,  Platforms, 
Wood  Sheds,  Water  Stations  with  their  tanks  and  pumps,  Telegraph,  Engines,  Cars,  Weigh  Scales, 
Tools,  &c,  &c.  Also  for  large  bridges,  tunnels,  Ac.t  Engines  and  cars  for  starting  business  will  cost 
$2000  to  $10000  per  mile  ;  increasing  in  time  with  the  traffic. 

Under  peculiarly  favorable  circumstances,  such  as  a  level  region  requiring  very  light  grading  ;  and 
which  furnishes  plenty  of  sand  or  gravel  convenient  for  ballasting  ;  and  by  the  use  of  a  lighter  rail, 
&c,  a  good  road  may  be  built  for  $25000,  or  even  considerably  less,  per  mile.  And,  on  the  other  hand, 
circumstances  must  be  somewhat  unfavorable  that  raise  the  cost  to  more  than  $40000  per  mile,  exclu* 
sive  of  buildings  and  outfit,  at  prices  in  1873. 

*  Price  of  steel  rails,  Phil,  1882,  about  $50  per  ton  ;  iron  ones  the  same. 
f  The  cost  of  a  complete  set  of  shops  of  brick,  for  the  thorough  re- 

pair of  about  '20  locomotives,  and  of  the  corresponding  number  of  passenger  and  other  cars  :  together 
with  suitable  smith  shop,  foundry,  car  shop,  boiler  shop,  copper  and  brass  shop,  paint  shop,  store 
rooms,  lumber  shed,  offices,  &c;  completely  furnished  with  steam  power,  lathes,  planing  machines, 
scales,  and  all  other  necessary  tools  and  appliances,  will  be  about  from  $75000  to  $100000  exclusive  of 
ground.  A  large  yard,  of  at  least  an  acre,  should  adjoin  the  biyldings.  A  moderate  establishment, 
for  the  repairs  of  a  few  engines  only,  may  be  built  and  furnished  for  $'25000. 

£n?iiie  houses  of  brick  will  cost  about  $1500  to  $2000  per  engine  stall. 

A  way-station  house,  30  X  60  ft,  surrounded  by  a  platform  12  ft  wide, 

protected  by  projecting  roof;  for  passengers  and  freight;  will  cost  from  $6000  to  $10000,  according  to 
finish  and  completeness,  at  eastern  city  prices. 


416 


RAILROADS. 


Table  of  Radii,  Middle  Ordiiiates,  Ac,  of  Carves.    Chord  100  feet. 
Contains  no  error  as  great  as  1  in  the  last  figure. 


Ang.  of 
Defl. 

Bad. 
in  ft. 

Defl. 
Dist. 
in  ft. 

Tang. 
Dlst. 
in  ft. 

Mid. 
Ord. 

Angi  of 

Defl. 

Bad. 
In  ft. 

Defl. 

Dist. 

in  ft. 

Tang. 
Dist. 
in  ft. 

Mid. 
Ord. 

0 

1 

343761 

.029 

.014 

.004 

1    36 

8681 

2.793 

1.396 

.349 

2 

171880 

.058 

.029 

.007 

38 

3508 

2.851 

1.425 

.856 

3 

114587 

.087 

.043 

.011 

40 

34H8 

2.909 

1.454 

.864 

4 

85940 

.116 

.058 

.014 

42 

3370 

2.967 

1.483 

.871 

6 

6875-2 

.145 

.072 

.018 

44 

3306 

3.025 

1.512 

.378 

6 

57293 

.175 

.087 

.022 

46 

3243 

3.084 

1.542 

.385 

T 

49109 

.204 

.102 

.025 

48 

3183 

3.142 

1.571 

.393 

8 

42970 

.233 

.116 

.029 

50 

3125 

8.200 

1.600 

.400 

9 

88197 

.262 

.131 

.033 

52 

8070 

8.257 

1.629 

.407 

10 

34378 

.291 

.145 

.036 

54 

3016 

8.316 

1.658 

.414 

11 

31252 

.320 

.160 

.040 

56 

2964 

8.374 

1.687 

.422 

12 

28648 

.349 

.174 

.043 

58 

2913 

8.433 

1.716 

.429 

13 

26444 

.378 

.189 

.047 

2 

2865 

3.490 

1.745 

.436 

14 

24555 

.407 

.203 

.051 

2 

2818 

8.549 

1.774 

.443 

15 

22918 

.436 

.218 

.054 

4 

2778 

3.606 

1.803 

.451 

16 

21486 

.465 

.232 

.058 

6 

2729 

3.664 

1.832 

.468 

17 

20222 

.494 

.247 

.062 

8 

2686 

8.723 

1.861 

.465 

18 

19098 

.524 

.262 

.065 

10 

2645 

3.781 

1.890 

.473 

19 

18094 

.553 

.276 

.069 

12 

2605 

3.839 

1.919 

.480 

20 

17189 

.582 

.291 

.073 

14 

2566 

3.897 

1.948 

.487 

21 

16370 

.611 

.305 

.076 

16 

2528 

8.956 

1.978 

.495 

22 

15626 

.640 

.320 

.080 

18 

2491 

4.014 

2.007 

.602 

23 

14947 

.669 

.334 

.083 

20 

2456 

4.072 

2.086 

.609 

24 

14324 

.698 

.349 

.087 

22 

2421 

4.131 

2.065 

.516 

25 

13751 

.727 

.363 

.091 

24 

2387 

4.189 

2.094 

.623 

26 

13222 

.756 

.378 

.095 

26 

2355 

4.246 

2.123 

.631 

27 

12732 

.785 

.392 

.098 

28 

2323 

4.305 

2.152 

.638 

28 

12278 

.814 

.407 

.102 

80 

2292 

4.363 

2.182 

.545 

29 

11854 

.844 

.422 

.106 

32 

2262 

4.421 

2.210 

.552 

SO 

11459 

.873 

.436 

.109 

34 

2232 

4.480 

2.240 

.660 

31 

11090 

.902 

.451 

.113 

36 

2204 

4.537 

2.268 

.667 

32 

10743 

.931 

.465 

.116 

38 

2176 

4.596 

2.298 

.574 

33 

10417 

.960 

.480 

.120 

40 

2149 

4.654 

2.327 

.682 

34 

10111 

.989 

.494 

.123 

42 

2122 

4.713 

2.356 

.689 

35 

9822 

1.018 

.509 

.127 

44 

2096 

4.771 

2.385 

.596 

36 

9549 

1.047 

.523 

.131 

46 

2071 

4.829 

2.414 

.603 

37 

9291 

1.076 

.538 

.134 

48 

2046 

4.888 

2.444 

.611 

38 

9047 

1.105 

.552 

.138 

50 

2022 

4.946 

2.473 

.618 

39 

8815 

1.134 

.567 

.142 

62 

1999 

6.003 

2.501 

.625 

40 

8594 

1.164 

.582 

.145 

54 

1976 

6.061 

2.530 

.632 

41 

8385 

1.193 

.596 

.149 

56 

1953 

5.120 

2.560 

.640 

42 

8185 

1.222 

.611 

.153 

58 

1932 

6.176 

2.588 

.647 

43 

7995 

1.251 

.625 

.156 

3 

1910 

5.235 

2.618 

.664 

44 

7813 

1.280 

.640 

.160 

2 

1889 

6.294 

2.647 

.662 

45 

7639 

1.309 

.654 

.164 

4 

1869 

5.350 

2.675 

.669 

46 

7473 

1.338 

.669 

.167 

6 

1848 

5.411 

2.705 

.676 

47 

7314 

1.367 

.683 

.171 

8 

1829 

6.467 

2.734 

.683 

48 

7162 

1.396 

.698 

.174 

10 

1810 

5.526 

2.763 

.691 

49 

7016 

1.425 

.712 

.178 

12 

1791 

5.588 

2.792 

.698 

50 

6876 

1.454 

.727 

.182 

14 

1772 

5.643 

2.821 

.705 

51 

6741 

1.483 

.741 

.185 

16 

1754 

5.707 

2.850 

.713 

52 

6611 

1.513 

.757 

.189 

18 

1736 

5760 

2.880 

.720 

53 

6486 

1.542 

.771 

.193 

20 

1719 

5.817 

2.908 

.727 

54 

6366 

1.571 

.786 

.197 

22 

1702 

5.875 

2.937 

.734 

55 

6251 

1.600 

.800 

.200 

24 

1685 

5.935 

2.967 

.742 

56 

6139 

1.629 

.815 

.204 

26 

1669 

5.992 

2.996 

.749 

57 

6031 

1.658 

.829 

.207 

28 

1663 

6.050 

3.025 

.756 

58 

5927 

1.687 

.844 

.211 

30 

1637 

6.108 

3.054 

.764 

59 

5827 

1.716 

.858 

.214 

32 

1622 

6.166 

3.083 

.771 

1 

5730 

1.745 

.872 

.218 

34 

1607 

6.223 

3.112 

.778 

*     2 

5545 

1.803 

.902 

.225 

36 

1592 

6.281 

3.140 

.785 

4 

5372 

1.862 

.931 

.233 

38 

1577 

6.341 

3.170 

.793 

6 

5209 

1.920 

.960 

.240 

40 

1563 

6.398 

3.199 

.800 

8 

5056 

1.978 

.989 

.247 

42 

1549 

6.456 

8.228 

.807 

10 

4911 

2.036 

1.018 

.255 

44 

1535 

6.515 

3.257 

.814 

12 

4775 

2.094 

1.047 

.262 

46 

1521 

6.575 

3.287 

.822 

14 

4646 

2.152 

1.076 

.269 

48 

1508 

6.631 

3.316 

.829 

16 

4523 

2.211 

1.105 

.276 

50 

1495 

6.689 

3.345 

.836 

18 

4407 

2.269 

1.134 

.284 

52 

1482 

6.748 

3.374 

.843 

20 

4297 

2.327 

1.163 

.291 

54 

1469 

6.807 

3.403 

.851 

22 

4192 

2.385 

1.192 

.298 

56 

1457 

6.863 

3.432 

.858 

24 

4093 

2.443 

1.221 

.305 

58 

1445 

6.920 

3.460 

.865 

26 

3997 

2.502 

1.251 

.313 

4 

1433 

6.980 

3.490 

.873 

28 

3907 

2.560 

1.280 

.520 

5 

1403 

7.125 

3.562 

.891 

30 

3820 

2.618 

1.309 

.327 

10 

1375 

7.271 

3.635 

.909 

32 

3737 

2.676 

1.338 

.334 

15 

1348 

7.416 

8.708 

.927 

34 

3657           2.734 

1  .367 

.342 

20 

1323 

7.561 

3.781 

.946 

RAILROADS. 


417 


Table  of  Radii,  Middle  Ordinates,  «fcc,  of  Curves.   Chord  100  feet. 

(Continued.) 
The  Tangential  Angle  is  always  one-half  of  the  Angle  of  Deflection. 


Ang.  of 
Defl. 

Bad. 
in  ft. 

Defl. 
Dist. 
in  ft. 

Tang. 
Dist. 
in  ft. 

Mid. 
Ord. 

Ang.  of 
Din. 

Bad. 
in  ft. 

Defl. 
Dist. 
in  ft. 

Tang. 
Dist. 
in  ft. 

Mid.* 
Ord. 

O      ' 

4  25 

1298 

7.707 

3.854 

.963 

o    • 
10  15 

559.7 

17.87 

8.942 

2.238 

SO 

1274 

7.852 

3.927 

.982 

30 

546.4 

18.30 

9.160 

2.292 

85 

1250 

7.997 

3.999 

1.000 

45 

533.8 

1873 

9.378 

2.347 

40 

12-28 

8.143 

4.072 

1.018 

11 

521.7 

19.17 

9.596 

2.402 

45 

1207 

8.288 

4.145 

1.036 

15 

510.1 

19.60 

9.814 

2.456 

50 

1186 

8.433 

4.218 

1.054 

30 

499.1 

20.04 

10.03 

2.511 

55 

1166 

8.579 

4.290 

1.072 

45 

488.5 

20.47 

10.25 

2.566 

5 

1146 

8.724 

4.363 

1.091 

12 

478.3 

20.91 

10.47 

2.620 

5 

1128 

8.869 

4.436 

1.109 

16 

468.6 

21.34 

10.69 

2.675 

10 

1109 

9.014 

4.508 

1.127 

30 

459.3 

21.77 

10.90 

2.730 

15 

1092 

9.160 

4.581 

1.145 

45 

450.3 

22.21 

11.12 

2.785 

20 

1075 

9.305 

4.654 

1.164 

13 

441.7 

22.64 

11.34 

2839 

25 

1058 

9.450 

4.727 

1.182 

15 

433.4 

23.07 

11.56 

2.894 

30 

1042 

9.596 

4.799 

1.200 

30 

425.4 

23.51 

11.77 

2.949 

85 

1027 

9.741 

4.872 

1.218 

45 

417.7 

23.94 

11.99 

3.003 

40 

1012 

9886 

4.945 

1  236 

14 

410.3 

24.37 

12.21 

3.058 

45 

996.9 

10.03 

5.017 

1.255 

15 

403.1 

24.81 

12.43 

3.113 

50 

982.6 

10.18 

6.090 

1.273 

30 

396.2 

25.24 

12.65 

3  168 

55 

968.8 

10.32 

5.163 

1.291 

45 

389.5 

25.67 

12.86 

3.223 

6 

955.4 

10.47 

5.235 

1.309 

15 

383.1 

26.11 

13  08 

3.277 

5 

942.3 

10.61 

5.308 

1.327 

15 

376.8 

26.54 

13.30 

3.332 

10 

929.6 

10.76 

5.381 

1.346 

30 

370.8 

26.97 

13.52 

3.387 

15 

917.2 

10.90 

5.453 

1364 

45 

3649 

27.40 

13.73 

3.442 

20 

905.1 

11.05 

5.526 

1.382 

16 

359.3 

27.84 

13.95 

3.496 

25 

893.4 

11.19 

5.599 

1.400 

30 

348.5 

28.70 

14.39 

3.606 

80 

882.0 

11.34 

5.672 

1.418 

17 

338.3 

29.56 

14.82 

3.716 

35 

870.8 

11.48 

5.744 

1.437 

30 

328.7 

30.43 

15.26 

3.825 

40 

859.9 

11.63 

5.817 

1.455 

18 

319.6 

31  29 

15.69 

3.935 

45 

849.3 

11.77 

5.890 

1.473 

30 

311.1 

32.15 

16.13 

4.045 

50 

839.0 

11.92 

5.962 

1.491 

19 

302.9 

3301 

16.56 

4.155 

55 

828.9 

12.07 

6.035 

1.510 

30 

295.3 

33.87 

17.00 

4.265 

7 

819.0 

12.21 

6.108 

1528 

20 

287.9 

34.73 

17.43 

4.375 

5 

809.4 

12.36 

6.180 

1.546 

21 

274.4 

36.44 

18.30 

4.594 

10 

800.0 

12.50 

6  253 

1.564 

22 

262.0 

38.17 

19.17 

4.815 

15 

790.8 

12.65 

6.326 

1.582 

23 

250.8 

39.87 

20.04 

5.035 

20 

781.8 

12.79 

6.398 

1.600 

24 

240.5 

41.58 

20.91 

5.255 

25 

773.1 

12.94 

6.471 

1.619 

25 

231.0 

43.29 

21.77 

5.476 

30 

764.5 

13.08 

6.544 

1.637 

26 

222.3 

44.98 

22.64 

5.696 

35 

756.1 

13.23 

6.616 

1.655 

27 

214.2 

46.69 

2351 

5.917 

40 

747.9 

13.37 

6.68J 

1.673 

28 

206.7 

48.38 

24.37 

6.139 

45 

739.9 

13.52 

6.762 

1.691 

29 

199.7 

50.08 

25.24 

6.361 

50 

732.0 

1366 

6.835 

1.710 

30 

193  2 

51.76 

2611 

6.582 

55 

724.3 

13.81 

6.907 

1.728 

31 

187.1 

53.45 

26.97 

6.805 

8 

716.8 

13.95 

6.980 

1.746 

32 

181.4 

55.13 

27.88 

7.027 

15 

695.1 

14.39 

7.198 

1.801 

33 

176.0 

56.S2 

28.70 

7.252 

30 

674.7 

14.82 

7.416 

1.855 

34 

171.0 

58.48 

29.56 

7.473 

45 

655.5 

15.26 

7.634 

1.910 

35 

166.3 

60.13 

30.42 

7.695 

9 

637.3 

15.69 

7.852 

1.965 

36 

161.8 

61.80 

31  29 

7.H19 

15 

620.1 

16.13 

8070 

2.019 

37 

157.6 

63.45 

3215 

8.142 

30 

603.8 

16.56 

8.288 

2.074 

38 

153.6 

65.10 

33.01 

8.366 

45 

588.4 

17.00 

8.506 

2.1?8 

39 

149.8 

66.76 

33.87 

8.591 

10 

573.7 

17.43 

8.724 

2.183 

40 

146.2 

68  40 

34.73 

8.816 

For  ordinates  5  ft  apart,  for  Chords  of  100  ft,  see  p  633. 

To  find  tangential  and  deflection  angle*  for  any  given  rad  and 
chord.  Divide  half  the  chord  by  the  rad.  The  qnot  will  be  nat  sine  of  the  tangl 
ang.  Find  this  tangl  ang  in  the  table  of  nat  sines  ;  and  mult  it  by  2  for  the  def  ang. 

To  find  the  def  dist  for  chords  1OO  ft  long.  Div  10000  by  the 
rad  in  feet. 

To  find  the  def  dist  for  equal  chords  of  any  given  length. 
Div  chord  by  rad.  Mult  quot  by  chord.  Or  div  sq  of  chord  by  rad. 

To  find  the  tangl  dist  for  equal  <*lior«Iwol" any  given  length. 
First  find  the  tangl  ung  as  above.  Divide  it  by  2  Find  in  the  table  of  nat  sines 
the  nat  sine  of  the  quot.  Mult  this  nat  sine  by  the  given  chord.  Mult  prod  by  2. 

To  find  I  he  rad  to  any  given  def  ang.  for  qual  chords  of  any 
length.  Divide  the  def  ang  by  2.  Find  iiat  sine  of  the  quotient.  Divide  half 
the  chord  by  this  nat  sine. 

*  The  middle  ordinal e  for  a  rad  of  600  ft  or  more,  (chord  100  ft,   im;y  in 

practice  be  taken  at  one-fourth  of  the  taug  dist.     Even  iu  400  it  rad  it  will  be  too  shmt  only  b  iu  th» 
third  decimal.    See  foot  of  p  20. 


418 


RAILROADS. 


Table  of  Middle  Ordinates,  to  be  used  for  the  bending  of  rails  of  different 
lengths,  so  as  to  form  portions  of  curves  of  different  radii.  Ordinates  for  lengths 
or  radii  intermediate  of  those  in  the  table,  may  be  found  by  simple  proportion. 


LENGTHS  OF  RAILS. 

Def. 

Aug. 

Radius. 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

Deg. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet.  Feet. 

Feet. 

Feet. 

Feet.  Feet. 

.5 

11460. 

.010 

.008 

.006 

.005 

.004 

.004 

.003 

.002  |  .002 

.001 

.001 

.000 

.000 

1. 
1.5 

5730. 
3820. 

.020 
.029 

.016 
.026 

.013 
.021 

.011 

.018 

.009 
.016 

.008 
.013 

.006 
.010 

.005     .004     .003 
.008     .006  1  .004 

.002 

.003 

.001 
.002 

.001 
.001 

2. 

2865. 

.038 

.034 

.029 

.025 

.021 

.017 

.014 

.011 

.008 

.006 

.004 

.003 

.001 

2.5 

2292. 

.049 

.043 

.037 

.031 

.027 

.022 

.018 

.014 

.010 

.007 

.005 

.003 

.002 

3. 

1910. 

.058 

.051 

.044 

.037 

.031 

.026 

.022 

.017 

.012 

.009 

.006 

.004 

.002 

3.5 

1637. 

.070 

.061 

.052 

.043 

.037 

.031 

.025 

.020 

.015 

.011 

.008 

.005 

.003 

4. 

1433. 

.019 

.069 

.060 

.050 

.042 

.035 

.029 

.023 

.018 

.013 

.009 

.006 

.003 

4.5 

1274. 

.088 

.077 

.067 

.056 

.047 

.039 

.032 

026 

.020 

.015 

.010 

.007 

.004 

5. 

1146. 

.099 

.086 

.074 

.063 

.053 

.044 

.035 

.029 

.022 

.016 

.011 

.007 

.004 

5.5 

1042. 

.108 

.094 

.082 

.070 

.059 

.048 

.039 

.032 

.024 

.018 

.012 

.008 

.004 

6. 

955.4 

.117 

.102 

.088 

.076 

.064 

.052 

.042 

.034 

.026 

.019 

.013 

.008 

.005 

6.5 

882 

128 

112 

.097 

.082 

.069 

.057 

.046 

.037 

.02S 

.021 

014 

.009 

.005 

7. 

819. 

.137 

.120 

.104 

.088 

.074 

.061 

.049 

.039 

.030 

.022 

.015 

.010 

.005 

7.5 

764.5 

.146 

.127 

.111 

.094 

.079 

.065 

.053 

.042 

.032 

.024 

.016 

.010 

.006 

8. 

716.8 

.158 

.137 

.119 

.100 

.085 

.070 

.056 

.045 

.034 

.025 

.017 

.011 

.006 

8.5 

674.6 

.166 

.145 

.126 

.106 

.090 

.074 

.060 

.048 

.036 

.027 

.018 

.012 

.007 

9. 

637.3 

.175 

.153 

.133 

.112 

.095 

.078 

.063 

.050 

.038 

.029 

.019 

.012 

.007 

9.5 

603.8 

.187 

.163 

.141 

.119 

.101 

.083 

.067 

.054 

.042 

.031 

.021 

.013 

.008 

10 

573.7 

.196 

.171 

.148 

.125 

.106 

.087 

.071 

.057 

.045 

032 

.022 

.014 

.008 

1 

521  7 

216 

188 

163 

.139 

.117 

.096 

.078 

.063 

.049 

.036 

.024 

.016 

009 

2 

47H.3 

.236 

.206 

.179 

.151 

.128 

.105 

.085 

.069 

.053 

.039 

.026 

.017 

.010 

3 

441  7 

254 

222 

.192 

.163 

.138 

.113 

.092 

.075 

.057 

.012 

.028 

019 

.010 

4 

410.3 

.275 

.239 

.207 

.175 

.148 

.122 

.099 

.080 

.061 

.045 

.030 

.020 

.011 

5 

383.1 

.295 

.257 

.223 

.188 

.159 

.131 

.106 

.085 

.065 

.049 

.033 

.021 

.012 

6 

359.3 

.313 

.273 

.236 

.200 

.170 

.139 

.113 

.091 

.070 

.052 

.035 

.023 

.013 

7 

338.3 

.333 

.290 

.252 

.213 

.180 

.148 

.120 

.096 

.074 

.055 

.037 

.024 

.014 

8 

319.6 

.351 

.306 

.265 

.225 

.190 

.156 

.127 

.102 

.078 

.058 

.039 

.025 

.014 

9 

302.9 

.371 

.324 

.280 

.238 

.201 

.165 

.134 

.108 

.082 

.061 

.041 

.027 

.015 

20 

287.9 

.392 

.341 

.296 

.250 

.212 

.174 

.141 

.114 

.087 

.066 

.044 

.028 

.016 

21 

274.4 

.410 

.357 

.309 

.262 

.222 

.182 

.148 

.120 

.091 

.069 

.046 

.030 

.017 

22 

262. 

.430 

.375 

.325 

.275 

.233 

.191 

.155 

.126 

.096 

.072 

.048 

.031 

.018 

23 

250.8 

.450 

.390 

.338 

.287 

.243 

.199 

.162 

.131 

.100 

.075 

.050 

.033 

.019 

24 

240.5 

.469 

.408 

.354 

.299 

.253 

.208 

.169 

.137 

.104 

.078 

.052 

.034 

.019 

25 

231. 

.486 

.424 

.367 

.311 

.263 

.216 

.176 

.142 

.108 

.081 

.054 

.035 

.020 

26 

222.3 

.506 

.441 

.382 

.323 

.274 

.225 

.183  |  .148 

.112 

.084 

.056 

.037 

.021 

27 

214.? 

.524 

.457 

.396 

.335 

.284 

.233 

.190  I  .153 

.116 

.087 

.058 

.038 

.022 

28 

206.  f 

.545 

.475 

.411 

.348 

.294 

.242 

.197     .158 

.120 

.090  |  .060 

.039 

.022 

29 

199.7 

564 

491 

424 

.361 

.303 

.250 

.203  !  -163 

.124 

.093 

.062 

.041 

023 

To  prepare  a  Table,  T,  (next  page,)  of  Level  Cuttings,  for 
every  -^o  °*  a  foot  °*  height,  or  depth. 

Let  the  fig  represent  the  cutting ;  or,  if  inverted, 
-/-   the  filling ;  in  which  the  horizontal  Hues  are  sup- 
posed to  be  y1^  foot  apart.    First  calculate  the 
area  in  square  feet,  of  the  layer  ab  co,  adjoining 
the  roadway  a  b.     Then  find  how  many  cubic 

?ards  that  area  gives  in  a  distance  of  100  feet, 
hese  cubic  yards  we  will  call  Y ;  they  form  the 
first  amount  to  be  put  into  the  Table  T. 
Next  calculate  the  area  in  square  feet  of  the  triangle  a  no.     Multiply  this  area  by  4.     Find  how 
many  cubic  yards  this  increased  area  gives  in  a  distance  of  100  feet.     Or  they  will  be  found  ready 
calculated  on  the  next  page.     We  will  call  them  y.     This  is  all  the  preparation  that  is  needed  before 
commencing  the  table. 

"Exam.-Let  the  roadbed  a  b  be  18  feet,  and  the  side-slopes  1%  to  1.  Then  for  the  area  of  a  b  c  o  : 
since  the  side-slopes  are  1%  to  1 ;  and  8  t  is  .1  foot;  c  o  must  be  18.3  feet;  and  the  mean  length  of 
a  b  co  must  be  18.15  feet.  Consequently,  the  area  is  18.15  X  .1=  1.815  square  feet;  which,  in  a 

distance  of  100  feet,  gives  181.5  cubic  feet ;  which  is  equal  to  ^^  rr  6.7222  cubic  yards  ;  or  Y. 

Next,  as  to  the  triangle  a  no:  its  height  a  n  being  .1  foot,  and  its  base  »  o  .15  feet ;  its  area 
—  'l  X  r=  '°i?— .0075  square  ft.  This  multiplied  by  4,  gives  .03  square  feet ;  which,  in  a  distance  of 


Having  thus  found  Y  and  y,  proceed  to  make  out  the  table  in  the  manner  following,  which  is  BO 
plain  as  to  require  no  explanation.    The  work  should  be  tested  about  every  5  feet,  by  calculating  the 


RAILROADS. 


419 


area  of  the  full  depth  arrived  at ;  multiply  it  by  100,  and  divide  the  product  by  27  for  the  cubic  yards 
The  cubic  yards  thus  found  should  agree  with  the  table. 


Y 6.7222  . 

y 1111 


6.8333 
.1111 


6.9444 
.1111 


7.0555 
.1111 


7.1666 
.1111 


7.2777 


....  Y.  6.722     .1 


6.8333 


13.5555 
6.9444 


205000 
7.0555 


27.5555 
7.1666 


34.7222 

7.2777 


42.0000    .« 


TABI 

E   T. 

Height. 
Feet. 

Cub.  Yds. 

.1  

6.72  Y. 

.2 

13  6 

.3  

20.5 

.4  

27.6 

.5  

34.7 

.6  

42.0 

&( 

The  following  table  contains  y.  ready  calculated  for  different  side-slopes.  It  plainly 
remains  the  same  for  all  widths  of  roadbed. 


Side-slope. 

y 

Side-slope. 

y 

14  to  1  

0185 

1%  to  1  

1296 

0370 

2      to  1  

1482 

%  to  1 

0556 

214  to  1   

1667 

1       to  1 

0741 

2V£  to  1  ... 

1852 

li^  to  1 

0926 

3      to  1  

2222 

ji/  to  1 

1111 

4      to  1 

2963 

Table  of  Long  Chords. 

Lengths  of  Chord  in  ft,  required  to  subtend  from  1  to  4  stations  of  100  ft  each, 


Ang. 
of 
Defl. 

ISta. 

2  Sta. 

3Sta. 

4Sta. 

Ang. 
of 
Defl. 

ISta. 

2Sta. 

3Sta. 

4Sta. 

1° 

100 

200.0 

300.0 

400.0 

% 

100 

199.7 

298.9 

397.5 

\/ 

100 

200.0 

300.0 

399.9 

6° 

100 

199.7 

298.8 

397.3 

iz 

100 

2000 

300.0 

399.9 

1^ 

100 

199.7 

298.7 

397.0 

S/ 

100 

200.0 

300.0 

399.8 

i^ 

100 

199.7 

298.6 

396.7 

2° 

100 

200.0 

299.9 

399.7 

3Z 

100 

199.6 

298.5 

396.5 

y± 

100 

200.0 

299.9 

399.6 

7° 

100 

199.6 

298.4 

396.2 

g 

100 

200.0 

299.8 

399.5 

& 

100 

199.6 

298.3 

396.0 

X 

100 

200.0 

299.8 

399.4 

100 

199.6 

298.2 

'    395.7 

3° 

100 

200.0 

299.7 

399.3 

74 

100 

199.6 

298.1 

395.4 

X 

100 

200.0 

299.7 

399.2 

8° 

100 

199.6 

298.0 

395.1 

<A 

100 

200.0 

299.6 

399.1 

i/£ 

100 

199.5 

297.9 

394.8 

% 

100 

200.0 

299.6 

399.0 

M 

100 

199.5 

297.8 

394.5 

4° 

100 

199.9 

299.6 

398.9 

3Z 

100 

199.4 

297.7 

394.3 

\/ 

100 

199.9 

299.5 

398.7 

9° 

100 

199.4 

2975 

394.1 

\/ 

100 

199.9 

299.4 

398.5 

y± 

100 

199.4 

297.4 

3937 

ax 

100 

1999 

299.3 

398.3 

u 

100 

199.3 

297.3 

393.2 

5° 

100 

199.9 

299.2 

398.0 

% 

100 

199.2 

297.2 

392.8 

K 

100 

1998 

29H.1 

397.8 

10° 

100 

199.2 

297.0 

392.4 

| 

100 

199.8 

299.0 

397.6 

Elevation  of  outer  rail  in  curves  theoretically  is  equal  in  ins  to  (square 
of  vel  in  ft  per  sec  X  gauge  in  ins)  -f-  (Rad  of  curve  in  ft  X  32.2).  Experience 
has  shown  that  half  an  inch  for  each  degree  of  def  angle  (100  ft  chords)  does  very 
well  for  4  it  8.5  ins  gauge  up  to  40  miles  per  hour.  At  60  miles  use  1  inch  per  deg. 
In  dangerous  places  this  may  be  increased  for  safety  against  high  winds.  Ap- 
proaching the  curve  raise  the  outer  rail  at  the  rate  of  1  inch  in  about  60  or  80  ft. 


420 


RAILROADS. 


Table  1.     Level  Cuttings.* 

Roadway  14  feet  wide,  side-slopes  1%  to  1. 
For  single-track  embankment. 


Height 
in  Ft. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

0 

5.24 

10.6 

16.1 

21.6 

27.3 

33.1 

390 

45.0 

51.2 

1 

57.4 

63.8 

70.2 

76.8 

83.5 

90.3 

97.2 

104.2 

111.3 

118.6 

2 

1259 

133.4 

141.0 

148.6 

156.4 

164.4 

172.4 

180.5 

188.7 

197.1 

3 

205.6 

214.1 

222.8 

231.6 

240.5 

249.5 

258.7 

267.9 

277.3 

286,7 

4 

296.3 

306.0 

315.8 

325.7 

335.7 

345.8 

356.1 

366.4 

376.9 

387.5 

5 

398.1 

408.9 

419.9 

430.9 

442.0 

453.2 

464.6 

476.1 

487.6 

4993 

6 

511.1 

523.0 

535.0 

547.2 

559.4 

571.8 

584.2 

596.8 

609.5 

622.3 

7 

6352 

618.2 

661.3 

674.6 

687.9 

701.4 

714.9 

728.6 

742.4 

756.3 

8 

770.3 

784.5 

798.7 

813.1 

827.5 

842.1 

856.8 

871.6 

886.5 

901.5 

9 

916.7 

931.9 

947.3 

962.7 

978.3 

994.0 

1010 

1026 

1042 

1058 

10 

1074 

1090 

1107 

1123 

1140 

1157 

1174 

1191 

1208 

1225 

11 

1243 

1260 

1278 

1295 

1313 

1331 

1349 

1367 

1385 

1404 

12 

1422 

1441 

1459 

1478 

1497 

1516 

1535 

1554 

1574 

1593 

13 

1613 

1633 

1652 

1672 

1692 

1712 

1733 

1753 

1773 

1794 

14 

1815 

1835 

1856 

1877 

1898 

1920 

1941 

1962 

1984 

2006 

15 

2028 

2050 

2072 

2094 

2116 

2138 

2161 

2183 

2206 

2229 

16 

2252 

2275 

2298 

2321 

2344 

2368 

2391 

2415 

2439 

2463 

17 

2487 

2511 

2535 

2559 

2584 

2608 

2633 

2658 

2683 

2708 

18 

27:33 

2759 

2784 

2809 

2835 

2861 

2886 

2912 

2938 

2964 

19 

2991 

3017 

3044 

3070 

3097 

3124 

3151 

3178 

3205 

3232 

20 

3259 

3287 

3314 

3342 

3370 

3398 

3426 

3454 

3482 

3510 

21 

3539 

3567 

3596 

3625 

3654 

3683 

3712 

3741 

3771 

3800 

22 

3830 

3859 

3889 

3919 

3949 

3979 

4009 

4040 

4070 

4101 

23 

4132 

4162 

4193 

4224 

4255 

4287 

4318 

4349 

4381 

4413 

24 

4444 

4476 

4508 

4541 

4573 

4605 

4638 

4670 

4703 

4736 

25 

4769 

4802 

4835 

4868 

4901 

4935 

4968 

5002 

5036 

5070 

26 

5104 

5138 

5172 

5206 

5241 

5275 

5310 

5345 

5380 

5415 

27 

5450 

5485 

5521 

5556 

5592 

5627 

5663 

5699 

5735 

5771 

2S 

5807 

5844 

5880 

5917 

5953 

5990 

6027 

6064 

6101 

6139 

29 

6176 

6213 

6251 

6289 

6326 

6364 

6402 

6440 

6479 

6517 

30 

6556 

6594 

6633 

6672 

6711 

6750 

6789 

6828 

6867 

6907 

31 

69  16 

6386 

7026 

7066 

7106 

7146 

7186 

7226 

7267 

7307 

32 

7318 

7389 

7430 

7471 

7512 

7553 

7595 

7636 

7678 

7719 

33 

7761 

7803 

7845 

7887 

7929 

7972 

8014 

8057 

8099 

8142 

34 

8185 

8228 

8271 

8315 

8358 

8401 

8445 

8489 

8532 

8576 

35 

8620 

8664 

8709 

8753 

8798 

8842 

8887 

8932 

8976 

9022 

36 

9067 

9112 

9157 

9203 

9248 

9294 

9340 

9386 

9432 

9478 

37 

9524 

9570 

9617 

9663 

9710 

9757 

9804 

9851 

9898 

9945 

3S 

9993 

10040 

10088 

10135 

10183 

10231 

10279 

10327 

10375 

10424 

39 

10472 

10521 

10569 

10618 

10667 

10716  110765 

10815 

10864 

10913 

40 

10963 

11013 

11062 

11112 

11162 

11212 

11263 

11313 

11364 

11414 

41 

11465 

11516 

11567 

11618 

11669 

11720 

11771 

11823 

11874 

11926 

42 

11978 

12029 

120S1 

12134 

12186 

12238 

12291 

12343 

12396 

12449 

43 

1-2502 

12555 

12608  !  12661 

12715 

12768 

12822 

12875 

12929 

12983 

44 

13037 

13091 

13145 

13200 

13254 

13309 

133H3 

13418 

13473 

13528 

45 

13583 

13639 

13694 

13749 

13805 

13861 

13916 

13972 

14028  (14084 

46 

U141 

in  97 

14254 

14310 

14367  14424  114480 

14537 

14r>95  14652 

47  114709 

14767 

14S24  14882 

14940  14998  15056 

15114 

15172 

152:30 

48 

15289 

15347 

15406  i  15465 

15524  15583  15642  15701 

15761  15826 

49 

L>880 

15939 

15999 

16059 

16119 

16179 

16239  16300 

16360  16421 

50 

16181 

16542 

16603 

16664 

16725 

16787 

16848  16909 

16971  !l7033 

51 

170J4 

17156 

17218 

17280 

17343 

17405 

17467  117530 

17593  117656 

52 

17719 

17782 

17845 

17908 

17971 

18035 

18098 

18162 

18226 

18-29C 

53 

18354 

18418 

18482 

18546 

18611 

18675 

18740 

18805 

18870 

18935 

n4 

19000 

19065 

19131 

19196 

19262 

19327 

19393 

19459 

19525 

19591 

55 

196YT 

19724 

19790 

19857 

19923 

19990 

20057 

20124 

20191 

20259 

56 

20326 

20393 

20461  120529 

20596 

20664 

20732 

20800 

2()«69 

20937 

57 

21005 

21074 

21143  '21212 

21-280 

21349 

21419 

21488 

21557 

21627 

58 

21696 

21766 

21836  J21906 

2197  ri 

22046 

22116 

22186 

22257 

22327 

59 

22398 

22469 

22540 

22611 

22682 

22753 

22825 

22896 

22968 

23039 

60 

23111 

23183 

23255 

23327 

23399 

23472 

23544 

23617 

23689 

23762 

-y-  From  the  Author'*  "  Measurement  of  Excavation  and  Embankment." 


RAILROADS. 


421 


Table  2.      Level  Cuttings. 

Roadway  24  feet  wide,  side-slopes  1%  to  1. 
For  double-track  embankment, 


Height 
in  Ft. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

CuTYds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

0 

8.94 

18.0 

27.2 

36.4 

45.8 

55.3 

64.9 

74.7 

84.5 

1 

94.4 

104.5 

114.7 

124.9 

135.3 

145.8 

156.4 

167.2 

178.0 

188.9 

2 

200.0 

211.2 

222.4 

233.8 

245.3 

256.9 

268.6 

280.5 

292.4 

304.4 

3 

316.6 

328.9 

341.2 

353.7 

366.3 

379.0 

391.9 

4048 

417.8 

431.0 

4 

444.4 

457-8 

471.3 

484.9 

498.6 

512.4 

526.4 

540.4 

554.6 

568.8 

5 

583.3 

597.8 

612.4 

627-1 

642.0 

656.9 

671.9 

687.1 

702.3 

717.7 

6 

733.3 

748.9 

764.7 

780.5 

796.4 

812.5 

828.7 

844.9 

861.3 

877.8 

7 

894.4 

911.2 

928.0 

944.9 

962.0 

979.2 

996.4 

1014 

1031 

1049 

8 

1067 

1085 

1102 

1121 

1139 

1157 

1175 

1194 

1212 

1231 

9 

1250 

1269 

1288 

1307 

1326 

1346 

1365 

1385 

1405 

1425 

10 

1444 

1465 

1485 

1505 

1525 

1546 

1566 

1587 

1608 

1629 

11 

1650 

1671 

1692 

1714 

1735 

1757 

1779 

1800 

1822 

1845 

12 

1867 

1889 

1911 

1934 

1956 

1979 

2002 

2025 

2048 

2071 

13 

2094 

2118 

2141 

2165 

2189 

2213 

2236 

2261 

2285 

2309 

14 

2333 

2358 

2382 

2407 

2432 

2457 

2482 

2507 

2532 

2558 

15 

2583 

2e;o9 

2635 

2661 

2686 

2713 

2739 

2765 

2791 

2818 

16 

2844 

2871 

2898 

2925 

2952 

2979 

3006 

3034 

3061 

3089 

17 

3117 

3145 

3172 

3201 

3229 

3257 

3285 

3314 

3342 

3371 

18 

3400 

3429 

3458 

3487 

3516 

3546 

3575 

3605 

3635 

3665 

19 

3694 

3725 

3755 

3785 

3815 

3846 

3876 

3907 

3938 

3969 

20 

4000 

4081 

4062 

4094 

4125 

4157 

4189 

4221 

4252 

4285 

21 

4317 

4349 

4381 

4414 

4446 

4479 

4512 

4545 

4578 

4611 

22 

4644 

4678 

4711 

4745 

4779 

4813 

4846 

4881 

4915 

4949 

23 

4983 

5018 

5052 

5087 

5122 

5157 

5192 

5227 

5262 

5298 

24 

5333 

5369 

5405 

5441 

5476 

5513 

5549 

5585 

5621 

5658 

25 

5694 

5731 

5768 

5805 

5842 

5879 

5916 

5954 

5991 

6029 

26 

6067 

6105 

6142 

6181 

6219 

6257 

6295 

6334 

6372 

6411 

27 

6450 

6489 

6528 

6567 

6606 

6646 

6P85 

6725 

6765 

6805 

28 

6844 

6885  . 

6925 

6965 

7005 

7046 

7086 

7127 

7168 

7209 

29 

7250 

7291 

7332 

7374 

7415 

7457 

7499 

7541 

7582 

7625 

30 

7667 

7709 

7751 

7794 

7836 

7879 

7922 

7965 

8008 

8051 

31 

8094 

8138 

8181 

8225 

8269 

8313 

8356 

8401 

8445 

8489 

32 

8533 

8578 

8622 

8667 

8712 

8757 

8802 

8847 

8892 

8938 

33 

8983 

9029 

9075 

9121 

9166 

9212 

9259 

9305 

9351 

9398 

34 

9444 

9491 

9538 

9585 

9632 

9679 

9726 

9774 

9821 

9869 

35 

9917 

9965 

10012 

10061 

10109 

10157 

10205 

10254 

10302 

10351 

36 

1.0400 

10449 

10498 

10547 

10596 

10646 

10695 

10745 

10795 

10845 

37 

10894 

10945 

10995 

11045 

11095 

11146 

11196 

11247 

11298 

11349 

38 

11400 

11451 

11502 

11554 

11605 

11657 

11709 

11761 

11812 

11865 

39 

11917 

11969 

12021 

12074 

12126 

12179 

12232 

12285 

12338 

12391 

40 

12444 

12498 

12551 

12605 

12659 

12713 

12766 

12821 

12875 

12929 

41 

12983 

13038 

13092 

13147 

13202 

13257 

13312 

13367 

13422 

13478 

42 

13533 

13589 

13645 

13701 

13756 

13813 

13869 

13925 

13981 

14038 

43 

14094 

14151 

14208 

14265 

14322 

14379 

14436 

14494 

14551 

14609 

44 

14667 

14725 

14782 

14840 

14899 

14957 

15015 

15074 

15132 

15191 

45 

15250 

15309 

15368 

15427 

15486 

15546 

15605 

15665 

15725 

15785 

46 

15844 

15905 

15965 

16025 

16085 

16146 

16206 

16267 

16328 

16389 

47 

16450 

16511 

16572 

16634 

16695 

16757 

16819 

16881 

16942 

17005 

48 

17067 

17129 

17191 

17254 

17316 

17379 

17442 

17505 

17568 

17631 

49 

176P4 

17758 

17821 

17885 

17949 

18013 

18076 

18141 

18205 

18269 

50 

18333 

18398 

18462 

18527 

18592 

18657 

18722 

18787 

18852 

18918 

51 

18983 

19049 

19115 

19181 

19246 

19313 

19379 

19445 

19511 

19578 

52 

19644 

19711 

19778 

19845 

19912 

19979 

20046 

20114 

20181 

20249 

53 

20317 

20385 

20452 

20521 

20589 

20657 

20725 

20794 

20862 

20931 

54 

21000 

21069 

21138 

21207 

21276 

21346 

21415 

21485 

21555 

21625 

55 

21694 

21765 

21835 

21905 

21975 

22046 

22116 

22187 

22258 

22329 

56 

22400 

22471 

22542 

22614 

22685 

22757 

22829 

22901 

22972 

23045 

57 

23117 

•23189 

23261 

23334 

2:5406 

23479 

23552 

23625 

23698 

23771 

58 

23844 

23918 

23991 

24065 

24139 

24213 

24286 

24361 

24435 

24509 

59 

24583 

24658 

24732 

24S07 

24882 

24957 

25032 

25107 

25182 

25258 

60 

25333 

25409 

25485 

25561 

25636 

25713 

25789 

25865 

25941 

•26018 

For  continuation  to  100  feet,  see  TABLE  7. 


422 


RAILROADS. 


Table  3.      Level  Cuttings. 

Roadway  18  feet  wide,  side-slopes  1  to  1. 
For  single-track  excavation, 


Depth 

in  Ft. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

0 

6.70 

13.5 

203 

27.3 

34.3 

41.3 

48.5 

55.7 

63.0 

1 

70.4 

77.8 

85.3 

929 

100.6 

1083 

116.1 

124.0 

132.0 

140.0 

2 

148.1 

156.3 

164.6 

172.9 

181.3 

189.8 

198.4 

207.0 

215.7 

224.5 

3 

233.3 

242.3 

251.3 

260.3 

269.5 

278.7 

288.0 

297.4 

306.8 

316.3 

4 

325.9 

335.6 

345.3 

355.1 

365.0 

375.0 

385.0 

395.1 

405.3 

415.6 

5 

425.9 

436.3 

446.8 

457.4 

468.0 

478.7 

489.5 

500.3 

511.3 

522.3 

6 

533.3 

544.5 

555.7 

567.0 

578.4 

589.8 

601.3 

612.9 

624.6 

636.3 

7 

648.1 

660.0 

672.0 

684.0 

696.1 

708.3 

720.6 

732.9 

745.3 

757.8 

8 

770.4 

7830 

795.7 

808.5 

821.3 

834.3 

847.3 

860.3 

873.5 

886.7 

9 

900.0 

913.4 

926.8 

940.3 

953.9 

967.6 

981.3 

995.1 

1009 

1023 

10 

1037 

1051 

1065 

1080 

1094 

1108 

1123 

1137 

1152 

1167 

It 

1181 

1196 

1211 

1226 

1241 

1256 

1272 

1287 

1302 

1318 

12 

1333 

1349 

1365 

1380 

1396 

1412 

1428 

1444 

1460 

1476 

13 

1493 

1509 

1525 

1542 

1558 

1575 

1592 

1608 

1625 

1642 

U 

1659 

1676 

1693 

1711 

1728 

1745 

1763 

1780 

1798 

1816 

15 

1833 

1851 

1869 

1887 

1905 

1923 

1941 

1960 

1978 

1996 

16 

2015 

2033 

2052 

2071 

2089 

2108 

2127 

2146 

2165 

2184 

17 

2204 

2223 

2242 

2262 

2281 

2301 

2321 

2340 

2360 

2380 

18 

2100 

2420 

2440 

2460 

2481 

2501 

2521 

2542 

2562 

2583 

19 

2604 

2624 

2645 

2666 

2687 

2708 

2729 

2751 

2772 

2793 

20 

2815 

2836 

2858 

2880 

2901 

2923 

2945 

2967 

2989 

3011 

21 

3053 

3056 

3078 

3100 

3123 

3145 

3168 

3191 

3213 

3236 

22 

3259 

3282 

3305 

3328 

3352 

3375 

3398 

3422 

3445 

3469 

23 

3493 

3516 

3540 

3564 

3588 

3612 

3636 

3660 

3685 

3709 

24 

3733 

3758 

3782 

3807 

3832 

3856 

3881 

3906 

3931 

3956 

25 

3981 

4007 

4032 

4057 

4083 

4108 

4134 

4160 

4185 

4211 

'26 

4237 

4263 

4289 

4315 

4341 

4368 

4394 

4420 

4447 

4473 

27 

4500 

4527 

4553 

4580 

4607 

4634 

4661 

4688 

4716 

4743 

28 

4770 

4798 

4825 

4853 

4881 

4908 

4936 

4964 

4992 

5020 

29 

5048 

5076 

5105 

5133 

5161 

5190 

5218 

5247 

5276 

5304 

30 

5333 

5362 

5391 

5420 

5449 

5479 

5508 

5537 

5567 

5596 

31 

5626 

5656 

5685 

5715 

5745 

5775 

5805 

5835 

5865 

5896 

32 

5926 

5956 

5987 

6017 

6048 

6079 

6109 

6140 

6171 

6202 

33 

6233 

6264 

6296 

6327 

6358 

6390 

6421 

6453 

6485 

6516 

34 

6548 

6580 

6612 

6644 

6676 

6708 

6741 

6773 

6S05 

6838 

35 

6870 

6903 

6936 

6968 

7001 

7034 

7067 

7100 

7133 

7167 

3tf 

7200 

7233 

7267 

7300 

7334 

7368 

7401 

7435 

7469 

7503 

37 

7537 

7571 

7605 

7640 

7674 

7708 

7743 

7777 

7812 

7847 

38 

7881 

7916 

7951 

7986 

8021 

8056 

8092 

8127 

8162 

8198 

39 

8233 

S269 

8305 

8340 

8376 

8412 

8448 

8484 

8520 

8556 

10 

8593 

8629 

8665 

8702 

8738 

8775 

8812 

8848 

8885 

8922 

41 

8959 

8996 

9033 

9071 

9108 

9145 

9183 

9220 

9258 

9296 

42 

9333 

9371 

9409 

9447 

9485 

9523 

9561 

9600 

9638 

9676 

43 

9715 

9753 

9792 

9831 

9869 

9908 

9947 

9986 

10025 

10064 

44 

10104 

10143 

10182 

10222 

10261 

10301 

10341 

10380 

10420 

10460 

45 

10500 

10540 

10580 

10620 

10661 

10701 

10741 

10782 

10822 

10863 

46 

10304 

10944 

10985 

11026 

11067 

11108 

11149 

11191 

11-232 

11273 

47 

11315 

11356 

11398 

11440 

11481 

11523 

11565 

11607 

11649 

11691 

48 

11733 

11776 

11818 

11860 

11903 

11945 

11988 

12031 

12073 

12116 

49 

1-2159 

12202 

12245 

12288 

12332 

12375 

12418 

12462 

12505 

1-2549 

50 

12593 

12636 

12680 

12724 

12768 

12812 

12856 

12900 

12945 

12989 

51 

13033 

13078 

13122 

13167 

13212 

13256 

13301 

13346 

13391 

13436 

52  '13481 

13527   i  13572 

13617 

13663 

13708 

13754 

13*00 

13845 

13891 

53 

13937 

13983 

14029 

14075 

14121  114168 

14214 

14-260 

14307 

14353 

54 

14400 

14447 

14493 

14540 

14587 

14634 

14681 

1.47-28 

14776 

14823 

55 

14870 

14918 

14965 

15013 

15061 

15108 

15156 

15204 

15252 

15300 

56 

15348  15396 

15445 

15493 

15541, 

15590 

15638 

15687 

15736 

15784 

57 

15833  15882 

15931 

15980 

16029 

16079 

16128 

17177 

16227 

16276 

58 

16^26 

16376 

16425 

16475 

16525 

16575 

166-25 

16675 

16725 

16776 

59 

16826 

1*876 

16927 

16977 

17028 

17079 

17129 

17180 

17231 

17282 

60  (17333 

17384 

17436 

17487 

17538 

17590 

17641 

17693 

17745 

17796 

For  continuation  to  100  feet  deep,  see  Table  7. 


]££fLKO 


,KOADS. 

T«*6ie  4.     Level  Cuttings. 

dway  18  feet,  side-slopes  1^  to  1. 
For  single-track  excavation, 


423 


Depth 
in  Ft. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds.  Cu.Yds. 

0 

6.72 

13.6 

20.5 

27.6 

34.7 

42.0 

49.4 

56.9   64.5 

1 

72.2 

80.1 

88.0 

96.1 

104.2 

112.5 

120.9 

129.4 

138.0   146.7 

2 

155.5 

164.5 

173.5 

182.7 

191.9 

201.3 

210.8 

220.4 

230.1   240.0 

3 

249.9 

260.0 

270.1 

280.4 

290.8 

301.3 

311.9 

322.6 

333.4   344.5 

4 

355.5 

366.7 

378.0 

389.4 

400.9 

412.5 

424.2 

4360 

448.0!  460.0 

5 

472.2 

484.5 

496.9 

509.4 

522.0 

534.7 

547.6 

560.5 

573.6   586.7 

6 

600.0 

613.4 

626.9 

640.5 

654.2 

668.1 

682.0 

696.1 

710.21  724.5 

7 

738.9 

753.4 

768.0 

782.7 

797.6 

812.5 

827.6 

842.7 

858.0 

873.4 

8 

888.9 

904.5 

920.2 

936.1 

952.0 

968.1 

984.2 

1001 

1017 

1033 

9 

1050 

1067 

1084 

1101 

1118 

1135 

1152 

1169 

1187 

1205 

10 

1222 

1240 

1258 

1276 

1294 

1313 

1331 

1349 

1368 

1387 

11 

1406 

1425 

1444 

1463 

1482 

1501 

1521 

1541 

1560 

1580 

12 

1600 

1620 

1640 

1661 

1681 

1701 

1722 

1743 

1764 

1785 

13 

1806 

1^27 

1848 

1869 

1891 

1913 

1934 

1956 

1978 

2000 

14 

20-22 

2045 

2067 

2089 

2112 

2135 

2158 

2181 

2204 

2227 

15 

2250 

2273 

2-297 

2321 

2344 

2368 

2392 

2416 

2440 

2465 

16 

2489 

2513 

2538 

2563 

2588 

2613 

2638 

2663 

2688 

2713 

17 

2739 

2765 

2790 

2816 

2842 

2868 

2894 

2921 

2947 

2973 

18 

3000 

3027 

3054 

3081 

3108 

3135 

3162 

3189 

3217 

3245 

19 

3272 

3300 

3328 

3356 

3384 

3413 

3441 

3469 

3498 

3527 

20 

3556 

3585 

3614 

3643 

3672 

3701 

3731 

3761 

3790 

3820 

21 

3850 

3880 

3910 

3941 

3971 

4001 

4032 

4063 

4094 

4125 

22 

4156 

4187 

4218 

4249 

4281 

4313 

4344 

4376 

4408 

4440 

23 

4472 

4505 

4537 

4569 

4602 

4635 

4668 

4701 

4734 

4767 

24 

4800 

4833 

4867 

4901 

4934 

4968 

5002 

5036 

5070 

5105 

25 

5139 

5173 

5208 

5243 

5278 

5'il3 

5348 

5383 

5418 

5453 

26 

5489 

5525 

5560 

5596 

5632 

5668 

5704 

5741 

5777 

5813 

27 

5850 

5887 

5924 

5961 

5998 

6035 

6072 

6109 

6147 

6185 

28 

6222 

6260 

6298 

6336 

6374 

6413 

6451 

6489 

6528 

6567 

29 

6606 

6645 

6684 

6723 

6762 

6801 

6841 

6881 

6920 

6960 

30 

7000 

7040 

7080 

7121 

7161 

7201 

7242 

7283 

7324 

7365 

31 

7406 

7447 

7488 

7529 

7571 

7613 

7654 

7696 

7738 

7780 

32 

7822 

7865 

7907 

7949 

7992 

8035 

8078 

8121 

8164 

8207 

33 

8250 

8293 

8337 

8381 

8424 

8468 

8512 

8556 

8600 

8645 

34 

8689 

8733 

8778 

8823 

8868 

8913 

8958 

9003 

9048 

9093 

35 

9139 

9185 

9230 

9276 

9322 

9368 

9414 

9461 

9507 

9553 

36 

9600 

9647 

9694 

9741 

9788 

9835 

9882 

9929 

9977 

10025 

37 

10072 

10120 

10168 

10216 

10264 

10313 

10361 

10409 

10458 

10507 

38 

10556 

10605 

10654 

10703 

10752 

10801 

10851 

10901 

10950 

11000 

39 

11050 

11100 

11150 

11200 

11251 

11301 

11352 

11403 

11454 

11505 

40 

11556 

11607 

11658 

11709 

11761 

11813 

11864 

11916 

11968 

12020 

41 

12072 

12126 

12177 

122-29 

12282 

12335 

12388 

12441 

12494 

12547 

42 

12600 

12653 

12707 

12761 

12814 

12868 

12922 

12976 

13030 

13085 

43 

13139 

13193 

13248 

13303 

13358 

13413 

13468 

13523 

13578 

13fi33 

44 

13689 

13745 

13800 

13856 

13912 

13968 

14024 

14081 

14137  114193 

45 

14250 

14307 

14364 

14421 

14478 

14535 

14592 

14C49 

14707 

14765 

46 

14822 

14880 

14938 

149P6 

15054 

15113 

15171 

15229 

152*8 

15347 

47 

15406 

15465 

15524 

15583 

15642 

15701 

15761 

15821 

15880 

15940 

48 

16000 

16060 

16120 

16181 

16241 

16301 

16362 

16423 

16484 

16545 

49 

16606 

16667 

16728 

16789 

16851 

16913 

16974 

17036 

17098 

17160 

50 

17222 

17285 

17347 

17409 

17472 

17535 

17598 

17661 

17724  I177S7 

51 

17850 

17913 

17977 

18041 

18104 

18168 

18232 

18296 

18360  18425 

52 

18489 

18553 

18618 

18683 

18748 

18813 

18878 

18943 

19008  119073 

53 

19139 

19205 

19270 

19336 

19402 

19468 

1U534 

19601 

19667  119733 

54 

19800 

19867 

19934 

20000 

200(38 

20135 

20202 

20269 

20337 

20405 

55 

20472 

20540 

20608 

20676 

20744 

20813 

20881 

20949 

21018 

21087 

56 

21156 

21225 

21  '294 

21363 

21432 

21501 

21571 

21641 

21710 

21780 

57 

21850 

21920 

21990 

22061 

22131 

22201 

22272 

22343 

22414 

22485 

58 

22556 

22627 

22698 

22769 

22841 

22913 

22984 

23056 

23128 

23200 

59 

23272 

23345 

23417 

23489 

23562 

23635 

23708 

23781 

23854 

23927 

60 

24000 

24073 

24147 

24221 

24294 

24368 

24442 

24516 

24590 

24665 

For  continuation  to  TOO  feet  deep,  see  Table  7. 


424 


RAILROADS. 


Table  5.     Level  Cuttings. 

Roadway  28  feet  wide,  side-slopes  1  to  1. 
For  double -track  excavation, 


Depth 
<n  Ft. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Cu.Yd8. 

Cu.Yds 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu  Yds 

Cu.Yds. 

0 

10.4 

20.9 

31.4 

42.1 

52.8 

63.6 

74.4 

85.3 

96.3 

1 

107.4 

118.6 

129.8 

141.1 

152.4 

363.9 

175.4 

187.0 

198.7 

210.4 

2 

222.2 

234.1 

246.1 

258.1 

270.2 

282.4 

294.7 

307.0 

319.4 

331.9 

3 

344.4 

357.1 

369.8 

382.6 

395.4 

408.3 

421.3 

434.4 

447.6 

460.8 

4 

474.1 

487.4 

5009 

514.4 

528.0 

541.7 

555.4 

569.2 

583.1 

597.1 

5 

611.1 

625.2 

639.4 

653.7 

668.0 

682.4 

696.9 

711.4 

726.1 

740.8 

6 

755.6 

770.4 

785.4 

800.4 

815.5 

830.6 

845.8 

861.1 

876.5 

891.9 

7 

907.5 

923.0 

938.7 

954.5 

970.3 

986.2 

1002 

1018 

1034 

1050 

8 

1067 

1083 

1099 

1116 

1132 

1149 

1166 

1182 

1199 

1216 

9 

1233 

1250 

1267 

1285 

1302 

1319 

1337 

1354 

1372 

1390 

10 

1407 

1425 

1443 

1461 

1479 

1497 

1515 

1534 

1552 

1570 

11 

1589 

1607 

1626 

1645 

1664 

1682 

1701 

1720 

1739 

1759 

12 

1778 

1797 

1816 

1836 

1855 

1875 

1895 

1914 

1934 

1954 

13 

1974 

1994 

2014 

2034 

2055 

2075 

2095 

2116 

2136 

2157 

14 

2178 

2199 

2219 

2240 

2261 

228  1 

2304 

232* 

2346 

2367 

15 

2389 

2410 

2432 

2454 

2475 

2497 

2519 

2541 

2563 

2585 

16 

2607 

2630 

2652 

2674 

2697 

2719 

2742 

2765 

2788 

2810 

17 

2833 

2856 

2879 

291)3 

2926 

2949 

2972 

2996 

3019 

3043 

18 

3067 

3090 

3114 

3138 

3162 

3186 

3210 

3234 

3259 

3283 

19 

3307 

3:332 

3356 

3381 

3406 

3431 

3455 

3480 

3505 

3530 

20 

3556 

3581 

3606 

3631 

3657 

3682 

3708 

3734 

3759 

3785 

21 

3811 

3837 

3863 

3889 

3915 

3942 

3968 

3994 

4021 

4047 

22 

4074 

4101 

4123 

4154 

4181 

4208 

4235 

4263 

4290 

4317 

23 

4344 

4372 

4399 

4427 

4455 

4482 

4510 

4538 

4566 

4594 

24 

4622 

4650 

4679 

4707 

4735 

4764 

4792 

4821 

4850 

4879 

25 

4907 

4936 

4965 

49  J  4 

5024 

5053 

5082 

5111 

5141 

5170 

26 

5200 

5230 

5259 

5289 

5319 

5349 

5379 

5409 

5439 

5470 

27 

5500 

5530 

5561 

5591 

5622 

5653 

5634 

5714 

5745 

5776 

28 

5807 

5839 

5870 

5901 

5932 

5964 

5995 

6027 

6059 

6090 

29 

6122 

6154 

6186 

6218 

6250 

6282 

6315 

6347 

6379 

6412 

30 

6444 

6477 

6510 

6543 

6575 

6>08 

6641 

6674 

6708 

6741 

31 

6774 

6807 

6341 

6874 

6308 

6942 

6975 

7009 

7043 

7077 

32 

7111 

7145 

7179 

7214 

7248 

7282 

7317 

7351 

7386 

7421 

33 

7456 

7490 

7525 

7560 

7595 

7631 

7666 

7701 

7736 

7772 

34 

7807 

7  8  A3 

7879 

7914 

7950 

7986 

8022 

8058 

8094 

8130 

35 

8t67 

8203 

8239 

8276 

8312 

8349 

8386 

8423 

8459 

8496 

36 

8533 

8570 

8608 

8645 

8682 

8719 

8757 

8794 

8S32 

8870 

37 

8907 

8945 

8983 

9021 

90  >9 

9097 

9135 

9174 

9212 

9250 

38 

9289 

9327 

9366 

9405 

9444 

9482 

9521 

9560 

9599 

9639 

39 

9678 

9717 

9756 

9796 

9835 

9875 

9915 

9954 

9994 

10034 

40 

10074 

10114 

10154 

10194 

10235 

10275 

10315 

10356 

10396 

10437 

41 

10178 

10519 

10559 

10600 

10641 

10682 

10724 

107^5 

10806 

10847 

42 

108S9 

10330 

10972 

11014 

11055 

11097 

11139 

11181 

11223 

11265 

43 

U307 

11350 

U3J2 

11434 

11477 

11519 

11562 

11605 

11648 

11690 

44 

11733 

11776 

11819 

11863 

11906 

11949 

11992 

12036 

12079 

12123 

45 

12167 

12210 

12254 

12298 

12342 

123^6 

12430 

12474 

12519 

12563 

46 

12607 

12652 

12696 

12741 

12786 

12831 

12875 

12920 

12965 

13010 

47 

13056 

13101 

13146 

13191 

13237 

13282 

13328 

13374 

13419 

13465 

48 

13511 

13587 

13603 

13649 

13695 

13742 

13788 

13834 

13S81 

13927 

49 

13974 

14021 

14068 

14114 

14161 

14208 

14255 

14303 

14350 

14397 

50 

14444 

14492 

14539 

14587 

14635 

14682 

14730 

14778 

14826 

14874 

61 

14922 

14970 

15019 

15067 

15115 

15164 

15212 

15261 

15310 

15359 

52 

15407 

15456 

15505 

15554 

15804 

15653 

15702 

15751 

15801 

1  5850 

53 

15900 

15950 

15999 

16049 

16099 

16149 

16199 

16249 

1T299 

16350 

54 

16400 

16150 

16501 

16551 

16602 

1»!653 

I1H704 

16754 

16805 

[16856 

55 

16907 

16959 

17010 

17061 

17112 

17164 

17215 

17267 

17319 

i  17370 

56 

17  422 

17474 

175'>6 

1"578 

17630 

17682 

17735 

17787 

17839 

117892 

57 

17944 

17997 

18050 

18103 

18155 

18208 

18261 

18314 

18368 

118421 

58 

18474 

18527 

18581 

186'?4 

1^6S8 

18742 

18795 

18S49 

18903 

18957 

59 

[19011 

19065 

19119 

19174 

19228 

19282 

19337 

19391 

19446 

19501 

60 

[19556 

19610 

19665 

197?0 

19775 

19831 

19886 

19941 

19996 

(20052 

For  continuation  to  100  feet,  »ee  Table  7. 


425 


TaCble  6.      Level  Cuttings. 

^Koadway  28  ft  wide,  side-slopes  1^  to  1. 
^//  For  double-track  excavation, 


Depth 

™ 

1 

in  Ft. 

.0 

.1 

.2 

.3   !   .4 

.5 

.6 

.7 

.8 

.9 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

0 

10.4 

21.0 

31.6 

42.4 

53.2 

64.2 

753 

86.5 

97.9 

1 

109.3 

120.8 

132.5 

144.3 

156.1 

168.1 

180.2 

192.4 

204.8 

217.2 

2 

229.6 

242.3 

255.0 

267.9 

280.9 

294.0 

307.2 

320.5 

334.0 

347.5 

3 

361.2 

374.9 

388.8 

402.8 

416.9   431.1 

445.4 

459.9 

474.4 

489.1 

4 

503.7 

618.6 

533.6 

548.6 

5639   579.3 

594.7 

610.2 

625.8 

641.6 

5 

657.5 

673.4 

689.5 

705.7 

722.1 

738.5   7550 

771.7 

788.4 

805.3 

6 

822.2 

8o9.3 

856.5 

873.8   891.2 

908.8   926.4 

944.2 

962.0 

980.0 

7 

998.1 

1016 

1035 

1053 

1072 

1090 

1109 

1128 

1147 

1166 

8 

1186 

1204 

1224 

1243 

1263 

1283 

1303 

1322 

1343 

1863 

9 

1383 

1403 

1424 

1445 

1465 

1486 

1507 

1528 

1549 

1571 

10 

1692 

1614 

1635 

1657 

1679 

1701 

1723 

1745 

1767 

1790 

11 

1812 

1836 

1858 

1881 

1904 

1927 

1950 

1973 

1997 

2020 

12 

2044 

2068 

2092 

2116 

2140 

2164 

2189 

2213 

2238 

2262 

13 

2287 

2312 

2337 

2842 

2387 

2413 

2438 

2464 

2489 

2515 

14 

2541 

2567 

2593 

2619 

2645 

•2672 

2698 

2725 

2752 

2779 

15 

2806 

2833 

2860 

2887 

2915 

2942 

2970 

2997 

3025 

3053 

16 

3081 

3109 

3138 

3166 

3195 

3223 

3252 

8281 

3310 

3339 

17 

3368 

3397 

3427 

3456 

3486 

3516 

3546 

3576 

3606 

£636 

18 

3667 

3697 

3728   3758 

3789 

3820 

3851 

3882 

3913 

3944 

19 

3976 

4007 

4039   4070 

4102 

4134 

4166 

4198 

4231 

4'263 

20 

4296 

4328 

4361 

4394 

4427 

4460 

4493 

4527 

4560 

4594 

21 

4627 

4661 

4695 

4729 

4763 

4797 

4832 

4866 

4900 

4935 

22 

4970 

6005 

5040 

5075 

5111 

5146 

5181 

5217 

5253 

6288 

23 

5324 

5360 

5396 

6432 

5469 

5505 

5542 

5578 

5615 

6652 

24 

5689 

5726 

5763 

5800 

6838 

5875 

5913 

5951 

5989 

6027 

25 

6065 

6103 

6141 

6179 

6218 

6267 

6295 

6334 

6373 

6412 

26 

6461 

6491 

6530 

6570 

6609 

6649 

6689 

6729 

6769 

6809 

27 

6850 

6890 

6931 

6971 

7012 

7053 

7094 

7135 

7176 

7217 

28 

7259 

7300 

7342 

7384 

7426 

7468 

7510 

7552 

7594 

7637 

29 

7680 

7722 

7765 

7808 

7851 

7894 

7937 

7981 

8024 

8067 

.30 

8111 

8155 

8199 

8243 

8287 

8331 

8375 

8420 

8464 

8509 

31 

8554 

8698 

8643 

8688 

8734 

8779 

8824 

8870 

8915 

8961 

32 

9007 

9053 

9099 

9145 

9191 

9238 

9284 

9331 

9378 

9425 

33 

9472 

9519 

9566 

9613 

9661 

9708 

9756 

9804 

9851 

9900 

34 

9948 

9997 

10045 

10093 

10142 

10190 

10239 

10288 

10337 

10386 

35 

10435 

10484 

10534 

10583 

10633 

10683 

10732 

10782 

10832 

10882 

36 

10933 

10983 

11034 

11084 

11135 

11186 

11237 

11288 

11339 

11391 

37 

11443 

11494 

11546 

11598 

11649 

11701 

11753 

11806 

11858 

11910 

38 

11963 

12016 

12068 

12121 

12174 

12227 

12281 

12384 

12387 

12441 

39 

12494 

12548 

12602 

12656 

12710 

12764 

12819 

12873 

12928 

12982 

40 

13037 

13092 

13147 

13202 

13257 

12312 

13368 

13423 

13479 

135S5 

41 

13591 

13647 

13703 

13759 

13816 

13872 

13928 

13985 

14042 

14099 

42 

14156 

14213 

14270 

14327 

14385 

14442 

14600 

14558 

14615 

14673 

43 

14731 

14790 

14848 

14906 

14965 

15024 

15082 

15141 

15200 

16259 

44 

15318 

15378  15437 

15497 

15556 

15616 

15676 

15736 

15796 

15856 

45 

15917 

15977 

16038 

16098 

16159  16220 

16281 

16342 

16403 

16465 

46 

16526 

16587 

16649 

16711 

16773 

16835 

16897 

16969 

170-21 

17084 

47 

17146 

17209 

17272 

17335 

17o98 

17461 

17524 

17587 

17651 

17714 

48 

17778 

17842 

17905 

17969 

18033  (18098 

18162 

182*6 

18-291 

18356 

49 

18420 

18485 

18550 

18615 

18680 

18746 

18811 

18877 

18942 

19008 

50 

19074 

19140 

19206 

19272 

19339 

19405 

19472 

19538 

16605 

19672 

51 

19739 

19806 

19873 

19940 

20008 

20075 

20143 

20211 

20279 

20347 

52 

20415 

20483 

20551 

20620 

20688 

20757 

20826 

20894 

20963 

21032 

53 

21102 

21171 

21241 

21310 

213x0 

21450 

21519 

21589 

21659 

21730 

54 

21800 

21870 

21941 

22012 

22082 

22153 

22224 

22'295 

•22366 

22438 

55 

22509 

22581 

22652 

22724 

22796 

22868 

22940 

23012 

23085 

23157 

56 

23230 

23302 

23375 

23448 

23521 

23594 

23667 

23741 

23814 

23888 

57 

23961 

24035 

24109 

24183 

24257 

24331 

24405 

24480 

24554 

24629 

58 

24704 

24779 

24854 

24929 

25004 

25079 

25155 

25230 

25306 

25381 

59 

25457 

25533 

25609 

25686 

25762 

25838 

25915 

25992 

26068 

26145 

60 

26222 

26299 

26376 

26454 

26531 

26609 

26686 

26764 

26842 

26920 

For  continuation  to  100  fret,  see  Table  7- 


426 


RAILROADS. 


Table  7.      Level  Cuttings. 

Continuation  of  the  six  foregoing  Tables  of  Cubic  Contents,  to  100  feet  of  height  or  depth. 


Height 
or  Depth 
in  Feet. 

Table 

1 

Table 
2 

Table 
3 

Table 
4 

Table 
5 

Table 
6 

Cu.  Yds. 

Cu.  Yds. 

Cu.  Yds. 

Cu.  Yds. 

Cu.  Yds. 

Cu.  Yds. 

61 

23835 

26094 

17848 

24739 

20107 

26998 

.5 

24201 

26479 

18108 

25113 

20386 

27390 

62 

24570 

26S67 

18370 

25489 

20667 

27785 

.5 

24942 

27257 

18634 

25868 

20949 

28183 

63 

25317 

27650 

1S900 

26250 

21233 

28583 

.5 

25694 

28046 

19168 

26635 

21519 

28986 

64 

26074 

28444 

19437 

27022 

21807 

29393 

.5 

26457 

28846 

19708 

27413 

22097 

29801 

65 

26843 

29250 

19981 

27806 

22389 

30213 

.5 

27231 

29657 

20256 

28201 

22682 

30627 

66 

27622 

30007 

20533 

2SGOO 

22978 

31044 

.5 

28016 

30479 

20812 

29001 

23275 

31464 

67 

28413 

30894 

21093 

29406 

23574 

31887 

.5 

28812 

31313 

21375 

29813 

23875 

32312 

68 

29215 

31733 

21659 

30222 

24178 

32741 

.5 

29G20 

32157 

21945 

30635 

24482 

33172 

69 

30028 

32583 

22233 

31050 

24789 

33605 

.5 

30438 

33013 

22523 

31468 

25097 

34042 

70 

30852 

33444 

22814 

31889 

25107 

34481 

.5 

31268 

33879 

23108 

33313 

25719 

34924 

71 

31687 

34317 

23404 

32739 

26033 

35369 

.5 

32108 

34757 

23701 

33168 

26349 

35816 

72 

32533 

35200 

24000 

33600 

26667 

36267 

.5 

32960 

35646 

24301 

34035 

26986 

36720 

73 

33390 

36094 

24604 

34472 

27307 

37176 

.5 

33823 

36546 

24907 

34913 

27631 

37635 

74 

34259 

37000 

25214 

35356 

27956 

38096 

.5 

34697 

37457 

25522 

35801 

28282 

38561 

75 

35139 

37917 

25832 

36250 

28611 

39028 

.5 

35582 

38379 

26144 

36701 

28942 

39498 

76 

36029 

38844 

26458 

37156 

29174 

39970 

.5 

36479 

39313 

26774 

37613 

29608 

40446 

77 

36931 

39783 

27092 

38072 

29944 

40924 

.5 

37386 

40257 

27411 

38535 

30282 

41405 

78 

37844 

40733 

27733 

39000 

30622 

41889 

.5 

38305 

41213 

28056 

39468 

30964 

42375 

79 

38768 

41694 

28381 

39939 

31307 

42865 

.5 

39235 

42179 

28708 

40413 

31653 

43357 

80 

39704 

42667 

29037 

40889 

32000 

43852 

81 

40650 

43650 

29700 

41850 

32700 

44850 

82 

41607 

44644 

30370 

42822 

33407 

45859 

83 

42576 

45650 

31048 

43806 

34122 

46880 

84 

43555 

46667 

31733 

44800 

34844 

47911 

85 

44546 

47694 

32426 

45806 

35574 

48954 

86 

45548 

48733 

33126 

46822 

S6311 

60008 

87 

46561 

49783 

33833 

47850 

37056 

51072 

88 

47585 

50844 

34548 

48889 

37807 

52148 

89 

48620 

51917 

35270 

49939 

38567 

53235 

90 

49667 

53000 

36000 

51000 

39333 

54333 

91 

50724 

54094 

36737 

52072 

40107 

55443 

92 

51793 

55200 

37481 

53156 

40889 

56563 

93 

52872 

56317 

38233 

64250 

41678 

57694 

94 

53963 

57444 

38993 

55356 

42474 

58837 

95 

55065 

58583 

39759 

56472 

43278 

59990 

96 

56178 

59733 

40533 

57600 

44089 

61155 

97 

57302 

60894 

41315 

58739 

44907 

62331 

98 

58437 

62067 

42104 

59889 

45733 

63518 

99 

59583 

63250 

42900 

61050 

46567 

64716 

100 

60741 

64444 

43704 

62222 

47407 

65926 

RAILROADS. 


427 


Table  8, 

Of  Cubic  Yards  in  a  100-foot  station  of  level  cutting  or  filling,  to  be  added  to,  or  sub- 
tracted from,  the  quantities  in  the  preceding  seven  tables,  in  case  the  excava- 
tioua  or  embankments  should  be  increased  or  diminished  2  feet  in  width. 

Cubic  Yards  iu  a  length  of  100  feet;  breadth  2  feet;  and  of  different  depths. 


Height  or 
Depth 
in  Feet. 

Cubic 
Yards. 

Height  or 
Depth 
in  Feet. 

Cubic 
Yards. 

Height  or 
Depth 

in  Feet. 

Cubic 
Yards. 

Height  or 
Depth 
in  Feet. 

Cubic 
Yards. 

Height  or 
Depth 

in  Feet. 

Cubic 
Yards. 

.5 

3.70 

.5 

152 

.5 

300 

.5 

448 

.5 

696 

1 

7.41 

21 

156 

41 

304 

61 

452 

81 

600 

.5 

11.1 

.5 

159 

.5 

307 

.5 

456 

.5 

604' 

2 

14.8 

22 

163 

42 

311 

62 

459 

82 

607 

.5 

18.5 

.5 

167 

.5 

315 

.5 

463 

.5 

611 

3 

22.2 

23 

170 

43 

319 

63 

467 

83 

615 

.5 

25.9 

.5 

174 

.5 

322 

.5 

470 

.5 

619 

4 

29.6 

24 

178 

44 

326 

64 

474 

84 

622 

.5 

33.3 

.5 

181 

.5 

330 

.5 

478 

.5 

626 

5 

37.0 

25 

185 

45 

333 

65 

481 

85 

630 

.5 

40.7 

.5 

189 

.5 

337 

.5 

485 

.5 

633 

6 

44.4 

26 

193 

46 

341 

66 

489 

86 

637 

.5 

48.1 

.5 

196 

.5 

344 

.5 

493 

.5 

641 

7 

51.9 

27 

200 

47 

348 

67 

496 

87 

644 

.5 

55.6 

.5 

204 

.5 

352 

.5 

500 

.5 

648 

8 

5'J.3 

28 

207 

48 

356 

68 

504 

88 

652 

.5 

63.0 

.5 

211 

.5 

359 

.5 

507 

.5 

656 

9 

66.7 

29 

215 

49 

363 

69 

511 

89 

659 

.5 

70.4 

.5 

219 

.5 

367 

.5 

515 

.5 

663 

10 

741 

30 

222 

50 

370 

70 

519 

90 

667 

.5 

77.8 

.5 

226 

.5 

374 

.5 

522 

.5 

670 

11 

81.5 

31 

230 

51 

378 

71 

526 

91 

674 

.5 

852 

.5 

233 

.5 

381 

.5 

530 

.5 

678 

12 

88.9 

32 

237 

62 

385 

72 

533 

92 

681 

.5 

92.6 

.5 

241 

.5 

389 

.5 

537 

.5 

685 

13 

96.3 

33 

244 

53 

393 

73 

541 

93 

689 

.5 

100 

.5 

248 

.5 

396 

.5 

544 

.5 

693 

14 

104 

34 

252 

,54 

400 

74 

648 

94 

696 

.5 

107 

.5 

256 

.5 

404 

.5 

652 

.5 

700 

15  _ 

111 

35 

259 

55 

407 

75 

556 

95 

704 

.5 

115 

.5 

263 

.5 

411 

.5 

659 

.5 

707 

16 

119 

36 

267 

56 

415 

76 

563 

96 

711 

.5 

122 

.5 

270 

.5 

419 

.5 

567 

.5 

715 

17 

126 

37 

274 

57 

422 

77 

670 

97 

719 

.5 

130 

.5 

278 

.5 

426 

.5 

674 

.5 

722 

18 

133 

38 

281 

58 

430 

78 

578 

98 

726 

.5 

137 

.5 

285 

.5 

433 

.5 

581 

.5 

730 

19 

141 

39 

289 

59 

437 

79 

585 

99 

733 

.5 

144 

.5 

293 

.5 

441 

.5 

589 

.5 

737 

20 

148 

40 

296 

60 

444 

80 

593 

100 

741 

REMARK.  The  foreg-oinsr  tables  of  level  cntting-s  may  also  be 
used*  for  widths  of  roadway  greater  than  those  at  the  heacta 
Of  the  tables.  Thus,  suppose  we  wish  to  use  Table  1,  for  a  roadbed  m  n,  16  ft 
wide,  instead  of  c  6,  which  is  only  14  ft,  and  for  which  the  table  was  calculated.  It 
is  only  necessary  first  to  find  the  vert  dist  s  a,  between  these  two  roadbeds ;  and  to 
add  it  mentally  to  each  height  t  s,  of  the  given  embkt,  when  taking  out  from  the 

c      a      \ 


428 


RAILROADS. 


table  the  numbers  of  cub  yds  corresponding  to  the  heights.  By  this  means  we  obtaim 
the  contents  of  the  embkt  c  b  a  p,  for  any  required  dist.  Next,  from  these  contents 
subtract  that  corresponding  to  the  height  *  a,  for  the  same  dist.  The  remainder  will 
plainly  be  the  embkt  m  n  op. 

In  practice  it  will  be  sufficiently  correct,  to  take  8  a  to  the  nearest  tenth  of  a  foot,  which  will  save 
trouble  in  adding  it  mentally  to  the  heights  in  the  tables. 

If  the  roadbed  is  narrower  than  the  table,  as.  for  instance,  if  mnbe 

the  width  in  the  table,  but  wo  wish  to  flnd  the  contents  for  the  width  c  b,  then  first  find  s  a,  and  cal- 
culate the  cub  yds  in  100  ft  length  of  c  6  TO  n.  Then,  in  taking  out  the  cub  yds  from  the  table,  first 
subtract  «  a  mentally  from  each  height;  and  to  the  cub  yds  taken  out  for  each  100  ft,  opposite  this 
reduced  height,  add  the  cub  yds  in  100  ft  of  c  6  m  n. 

To  avoid  trouble  with  contractors  about  the  measurement  of  rock  cuts,  stipulate  in  the 

of  say  about  2  ft  of  width  of  cut  will  be  made,  to  cover  the  unavoidable  irregularities  of  the  sides. 


TURNTABLES. 


429 


TURNTABLES, 


A  turntable  is  a  platform,  usually  from  40  to  60  ft  long,  and  about  6 

to  10  ft  wide,  (see  Fig  1,)  upon  which  a  locomotive  and  its  tender  may  be  run,  and  then  be  turned 
around  horizontally  through  any  portion  of  a  circle;  and  thus  be  transferred  from  one  track  to 
another  forming  any  angle  with  it.  The  table  is  supported  by  a  pivot  under  its  center;  and  by 
wheels  or  rollers  under  its  f.vo  ends.  Frequently  other  rollers  are  added  between  the  center  and  ends. 
Beneath  the  platform  is  excavated  a  circular  pit  about  4  or  5  ft  deep,  having  its  circumf  lined  with 
a  wall  of  masonry  or  brick  about  2  ft  thick,  capped  with  either  cut  stone  or  wood.  The  diam  of  the 
pit  in  clear  of  this  liuing  is  about  2  ins  greater  than  the  length  of  the  turntable.  The  lining  is  gen- 
erally built  with  a  step,  as  seen  in  Pig  1,  for  supporting  the  circular  rail  on  which  the  end  rollers  travel ; 
or,  instead  of  this  step,  a  detached  support  may  be  used  for  this  circular  rail,  as  at  u,  in  Figs  7.  At  the 
center  of  the  pit  is  a  solid  well-founded  mass  of  masonry  or  timber,  for  the  pivot  to  rest  on,  as  seen 
in  Fig  1.  This,  as  well  as  the  step  for  the  end  rollers,  should  be  very  firm,  and  perfectly  level ;  other- 
wise the  platform  will  be  hard  to  work.  The  platform  is  frequently  floored  across  for  a  width  of  6  to 
10  ft,  to  furnish  a  pathway  across  the  pit,  without  stepping  down  into  it;  especially  when  under 
cover  of  a  building.  At  first  they  were  floored  over  so  as  to  cover  the  entire  circular  pit;  but  this  in- 
creased not  only  their  cost,  but  their  wt,  so  as  to  make  them  difficult  to  turn ;  besides  causing  much 
expense  for  repairs ;  with  greater  trouble  in  making  them.  It  is  therefore  rarely  done  at  present, 
except  where  want  of  space  sometimes  renders  it  necessary  in  indoor  turntables. 

A  turntable  should  be  several  feet  longer  than  is  necessary  for  merely  allowing  the  engine  and 
tender  to  stand  on  it ;  for  the  increased  length  enables  the  engine-men  to  move  them  a  little  backward 
or  forward,  so  as  to  balance  them  chiefly  upon  the  central  support;  and  thus  relieve  the  end  rollers. 
By  this  means  the  friction  while  turning  is  confined  as  much  as  possible  to  the  center  of  motion  ;  and 
is  therefore  more  readily  overcome  than  if  it  were  allowed  to  act  at  the  circumf.  The  eugine-mea 
soon  learn,  by  feeling,  the  proper  spot  for  stopping  the  engine  so  as  thus  to  balance  the  platform. 

The  Sellers  cast-iron  turntable* of  the  well-known  firm  of  Wm.  Sel- 
lers &  Co,  machinists,  Philada,  shown  in  Figs  1  to  6.  is  probably  the  most  perfect  one  that  has  been 
devised.  It  is  expensive  in  first  cost,  but  economical  in  the  long  run.  One  man  can  readily  turn  it 
without  the  aid  of  machinery,  when  loaded  with  a  heavy  engine  and  tender.  It  is  extensively  used 
ail  over  the  U.  S. ;  and  the  principle  i's  also  largely  applied  to  great  turning  bridges.  It  consists  of 
two  heavy  cast-iron  girders,  one  of  which  is  seen  in  Fig  1  ;  and  parts  of  one  in  Figs  3  and  4.  The 

curved.    ~- 


tops  of  these  girders  are  straight,  and  the  bottoms  cu; 


They  are  perforated  by  circular  openings 


Scale  of  Feet  forFigs  2.  3.  4 


•X-  Wrought  Iron  turntables  are  now  coming  into  coiumou  use. 

28 


430 


TURNTABLES. 


to  save  metal.  A  transverse  section  of  one  is  shown  at  g,  Fig  3.  The  metal  averages  abont  \%  Ins 
thick.  Each  of  these  girders  is  in  two  separate  pieces,  which  are  fastened  to  a  kind  of  central  hol- 
low cast-iron  boxing,  a  transverse  section  of  which  is  shown  by  A  B  J  ,1  D  L,  Pig  2 ;  and  a  top  view 
by  A  B  C,  Fig  4.  This  boxing  is  all  cast  in  one  piece.  Its  four  vertical  sides,  about  \%  ins 'thick, 
are  solid ;  the  opening  shown  in  Fig  1  does  not  exist,  but  was  inserted  in  the  cut  in  order  to  show  the 
central  pivot.  The  top  and  bottom  have  openings,  as  in  Figs  2  and  4.  The  girders  are  fastened  to 
the  central  boxing,  by  means  of  heavy  bars  3%  ins  square,  of  rolled  iron,  o,  o,  Figs  2  and  4,  or  x,  S, 
Fig  3,  tilting  iuto  sunk  recesses  on  top  of  the  boxing,  and  tightened  in  place  by  wedges  j,  i,  screw- 
bolted  beneath,  as  shown  below  S,  Fig  3;  and  by  two  2%-iuch  key-bolts/.  Fig  3,  which  pass  through 
the  bolt  holes  h,  fi,  Fig  2;  and  are  confined  to  the  girders  by  horizontal  keys  just  below  R,  Fig  3, 
where  the  keyhole  is  seen. 

The  central  portion  D  L,  Fig  2,  of  the  boxing,  is  a  hollow  cone,  open  at  top  and  bottom,  and  loosely 
surrounding  the  hollow  conical  pivot-post  P.  To  save  material,  two  openings  II  are  left  in  its  cir- 
cumf.  The  post  P  is  about  \%  ins  thick ;  and  is  firmly  bolted  down  to  the  large  block  of  cut  stone 
M.  which  caps  the  supporting  masonry  of  the  post.  On  top  of  the  post  rests  loosely  a  rough  cast  iron 

it,  without  at  all  ^training  the  friction  rollers.  This  cap  supports  the  steel  box.  seen  above  it  in 
Fig  2,  or  at  tt,  Fig  5,  or  U  V,  Fig  6,  which  contains  the  steel  frictioil  rollers,  r,  d.  &c, 

Figs  5  and  «.  There  are  about  15  of  these  ;  each  about  2%  ins,  both  in  length,  and  in  greatest  diam. 
They  have  no  axles ;  but  merely  lie  loosely  in  the  lower  part  of  the  box ;  filling  its  circumf  with  the 
exception  of  about  ^  an  inch  left  for  play.  In  the  direction  of  their  axes  they  have  but  ^  inch  play 
in  the  box.  The  lid  U,  Fig  6,  of  the  roller-box,  rests  on  top  of  the  rollers  themselves ;  and  does  not 
come  down  to  the  lower  part  V  of  the  box  by  about  5*  inch.  Both  the  rollers  themselves,  and  the 
insides  of  the  box  in  contact  with  them,  are  finished  with  mathematical  accuracy,  so  as  to  insure  a 
perfect  bearing  between  them.  The  rollers  are  kept  constantly  well  oiled,  as  much  of  the  ease  of 
turning  the  platform  depends  upon  it.  To  oil  them,  the  screws  of  the  cap  C  are  loosened  ;  and  a  mix- 
ture of  oil  and  tallow  sufficient  for  several  months  is  put  into  the  roller-box.  On  top  of  the  roller- 
box  is  the  cast  cap  C  C,  which  is  bolted  down  by  8  screw-bolts  of  1^  diam.  These  bolts  sustain  all 
the  weight  of  the  entire  platform  and  its  load,  except  what  little  may  rest  on  the  two  rollers  at  each 
of  its  ends  in  case  the  engine  is  not  perfectly  balanced  upon  the  rollers  alone.  By  partly  unscrew- 
ing them,  all  the  platform  is  lowered  around  the  post;  only  n,  the  rollers  above  it,  and  the  cap  C  C, 
remaining  unmoved  on  top  of  the  post.  This  furnishes  the  means  of  adjusting  the  height  of  the 
platform  above  the  bottom  of  the  pit,  so  as  to  bring  the  end  rollers  to  their  due  bearing  upon  the  cir- 
cular rail  upon  which  they  travel.  These  end  rollers  should  barely  touch  lightly  on  their  rail;  be- 
cause the  object  in  the  Sellers  table,  and  one  of  its  important  features,  is  to  throw  all  the  weight 
upon  the  friction  rollers.  The  friction  being  thus  kept  near  the  center  of  motion,  has  but  little 
leverage  with  which  to  resist  the  turning  of  the  table.  The  diam  of  the  roller-box  being  as  great  as 
15  inches,  it  is  not  difficult  to  balance  the  engine  and  tender  upon  it  alone;  but  this  cannot  be  done 
upon  a  central  pivot  only  about  6  ins  diam;  and  more  especially  because  the  foot  of  such  a  pivot 
should  be  convex,  as  at  "f.  Figs  7,  so  as  to  adjust  itself  to  the  slight  tilting  of  the  platform  when  the 
load  enters  or  leaves  it.  Under  C  C,  Fig  2,  are  blocks,  ww,  of  hard  wood,  which  are  driven  between 
the  eight  screws,  so  as  to  hold  all  in  place  after  this  adjustment  is  completed.  * 

All  turntables  should  have  the  means  of  making  such  adjustment.  In  Figs  7  (of  another  design 
of  platform)  it  is  attained  by  placing  the  central  pivot,/,  at  the  foot  of  a  large  screw,  by  turning 
which  the  whole  platform  can  be  raised  or  lowered  at  pleasure.  After  thus  making  the  adjustment, 
the  screw  is  keyed  fast  to  the  platform,  so  as  to  revolve  with  it  without  screwing  or  unscrewing  itself. 

The  ends  of  the  two  girders  of  the  Sellers  are  firmly  connected  transversely  by  heavy  cast-iron 
beams ;  the  ends  of  which  project  sideways  beyond  the  girders  ;  and  carry  the  cast  iron  end  rollers,  20 
ins  diam;  two  at  each  end  of  the  platform.  Intermediate  transverse  connection  is  secured  by  the 
wooden  cross-ties  notched  upon  the  girders  to  support  the  rails,  and  frequently  10  or  12  feet  long,  for 
giving  a  wide  footway  across  the  pit.  A  lever  8  or  10  ft  long,  fitting  iutp  a  staple,  is  used  for  turning 
these  platforms,  not  on  account  of  friction,  but  to  afford  a  handhold  to  the  man  who  turns  them. 

Wooden  turntables,  with  none  but  two  common  wheel  rollers  at  each  end 
of  the  platform  are  sometimes  resorted  to  from  motives  of  original  cost.  They  are,  however,  much 
harder  to  turn,  generally  requiring  two  men,  aided  by  wheelwork  :  and  are  more  liable  to  get  out  of 
order ;  and  more  expensive  to  repair.  They  are  made  of  a  great  variety  of  patterns,  both  as  regards 
the  girders  and  the  central  pivots,  end  rollers,  &c.  Frequently  an  addition  is  made  of  8  to  12  small 
rollers  travelling  on  a  circular  rail  of  6  to  12  ft  diam,  around  the  pivot  as  a  center.  These  are  in- 
tended to  sustain  the  whole  weight ;  the  end  rollers  being  so  adjusted  as  to  touch  their  rail  only  when 
the  platform  rocks  or  tilts  as  the  eneine  enters  or  leaves  it.  Therefore,  there  is  less  resistance  from 
friction  than  when,  as  in  Figs  7,  there  are  only  the  end  rollers  r.  In  this  last  case,  the  engine  and 


tender  cannot  be  balanced  so  precisely  upon  the  slender  central  pivot,  as  to  prevent  a  great  part  of 
the  wt  from  being  thrown  upon  the  end  rollers  ;  thus  materially  increasing  the  frictional  resistance. 
Tn  plan,  these  wooden  platforms  are  generally  in  shape  of  a  cross ;  that  is,  in  i      ition  to  the  main 


ve,^e  platform  is  intended  to  carry  the  wheelwork  R  x  a;,  for  turning  the  platform  ;  and  the  other  arm 
aerves  merely  as  a  balance  to  it;  therefore,  neither  of  them  requires  to  be  very  strong.     It 
tant  to  connect  the  four  ends  of  the  two  platforms  by  four  beams,  as  the  whole  structure  is  thereby  ma- 
terially stiffened.     In  the  figs  the  wheelwork  Exxis  for  convenience  improperly  shown  as  if  it  stood 
upon  the  main  platform. 


*  The  price  of  the  Sellers  turntable,  at  the  shop  in  Philada  in  1880, 

is  about  as  follows  :  30  ft  long,  $800:  40  ft,  $1200;  45  ft,  $1300;  50  ft,  $1400;  o4  ft,  $1800;  60  ft,  $1900. 
A  45-ft  one  weighs  about  23000  B>s ;  a  50-ft,  24700  K»s  ;  a  54-ft,  33000  ;  thus  making  the  average  in  the 
shop  say  5>£  cts  per  ft.  These  prices  do  not  include  cross-ties  and  other  woodwork  ;  nor  the  circular 
railway  for  the  end  rollers.  Machinery  for  turning,  being  considered  unnecessary,  is  not  attached, 
unless  specially  ordered.  Its  cost  is  extra. 

The  entire  cost  of  excavating  and  lining  the  pit;  foundation  for  pivot;  circular  rail  for  end  rollers, 
&c,  complete  for  a  5«-ft  turntable  will  vary  from  $1200  to  $2500  in  addition,  depending  on  the  class 
of  materials  and  workmanship;  and  whether  the  bottom  of  the  pit  is  paved  or  not. 


TURNTABLES. 


431 


The  Figs  7  need  butflttle  explanation.  They  represent  an  actual  45  ft  platform,  which  has  been  in 
use  for  some  years./ The  convex  foot  /  of  the  ceutral  pivot,  about  6  ins  diaru,  should  be  faced  witr- 
steel ;  and  should  pest  ou  a  steel  step  n  s.  This  should  be  kept  well  oiled  ;  and  protected  from  dust  by 
a  leather  collar ylrouud  p,  and  resting  on  «/  </.  Its  upper  part,  about  4  ius  diain,  is  cut  into  a  screw 
with  square  threads  about  >±  inch  thick,  tor  a  distance  of  about  15  ius.  It  works  in  a  female  sorew  ia 


TR.SEC.AT  CENTER 


I  '  '  M  '  I  '  '  I  "  I 

or    IFOOT 


the  strong  cast-Iron  nut  y  y.  and  serves  for  raising  the  whole  platform  when  necessary.  When  not 
in  use  for  this  purpose,  it  is  keyed  tight  to  the  platform,  (by  a  key  at  its  head  «.)  so  as  to  revolve  with 
It.  Strong  screw  bolts  t  j  connect  the  several  timbers  at  the  center  of  the  platform. 

R  is  a  light  cast-iron  stand  supporting  two  bevel  wheels  about  1  foot  diam.  which  give  motion  by 
means  of  an  axle  d.  I  %  ins  diam  to  two  similar  ones  below,  shown  more  plainly  at  W  and  Y.  These  last 
give  motion  by  the  axle  x  to  the  pinion  e. (6  ins  diam,  and  '2%  ins  face,)  which  turns  the  platform  by  work- 
ing into  a  circular  rack  t,  (teeth  horizontal,  1  inch  ntch  ;  3^  inches  face.)  which  surrounds  the  entire 
pit.  This  rack  is  spiked  to  the  under  side  of  a  continuous  wooden  curb  H  which  is  upheld  bv  pieces  F, 
a  few  feet  apart,  which  are  let  into  the  wall  J  J,  which  lines  the  pit.  The  short  beam  M  N,  (about  6 
ft,)  which  carries  the  lower  wheelwork,  is  suspended  stronglv  from  the  beams  of  the  transverse  plat- 
form  above  it.  Instead  of  the  t-vo  lower  bevel  wheels  W  Y,  and  the  horizontal  axle  x.  a  more  simple 
arrangement  is  to  place  the  pinion  «  at  the  lower  end  of  the  vertical  axle  d;  and  let  it  work  into  a 
rack  with  vertical  teeth  at  u,  on  the  inner  face  of  the  stone  foundation  of  the  circular  rail.  For  this 
purpose  the  stand  R  should  be  directly  over  u.  There  are  two  cast-iron  rollers  r.  2  ft  diam,  3  inch 
face,  under  each  end  of  the  main  platform;  and  one  under  each  end  of  the  secondary  one. 

Although  this  kind  of  platform  necessarily  has  much  friction,  yet  one  man  can  generally  tnrn  a  45- 
ft  or.e  by  means  of  the  wheelwork,  when  loaded  with  a  heavy  engine  and  tender.  Indoed^  he  may  do 
it  with  some  difficulty  by  hand  only,  while  all  is  new  and  in  perfect  order :  but  when  old.  and  the'cir- 
cular  railway  uneven  and  dirty,  it  requires  two  men  at  the  winches  to  do  it  with  entire  ease. 

As  before  remarked,  the  resistance  to  turning  is  di- 
minished by  employing  a  set  of  from  8  to  12  rollers  or 
wheels  r.  Figs  8,  about  a  foot  to  15  ins  in  diam,  so  ar- 
ranged as  to  form  a  circle  8  to  12  ft  diam  around  the 
pivot.  When  this  is  done,  the  main  girders  of  the  plat- 
form are  placed  8  to  12  ft  apart :  and  long  cross-ties 
are  used  for  supporting  the  railway  track.  Also,  the 
main  girders  are  sometimes  trussed  by  iron  rods,  as  in 
the  swing  bridge  on  page  271 :  but  instead  of  one  post 
a  c.  it  is  best  to  have  two,  6  or  8  ft  apart  at  foot,  and 
meeting  at  top.  The  width  of  platform  must  then  be 
sufficient  to  allow  the  engine  to  pass  the  posts  ou  either 
side  of  it.  Ten  feet  will  suffice. 

Fig  8  shows  the  arrangement  of  these  rollers  r, 
which  revolve  upon  a  circular  trxck  « ;  while  the  plat- 
form rests  on  their  tops  hy  the  track  u.  The  rollers  r  are  held  between  two  wrought-iron  rings  o,  o, 


about  3  ins  deep,  by  ^  inch  thick,  which  also  are  carried  by  the  rollers.  From  each  rolle 
tie-rod  t.  1  inch  diam,  extends  to  a  ring  n  n.  which  surrounds  the  pivot  p.  closelv,  hut  not  t 
as  to  revolve  independently  of  it.  These  tie-rods  keep  the  rings  oo  at  their  proper  dist  from 
so  that  the  rollers  cannot  leave  the  rails  «  and  u.  Between  each  two  rollers,  the  rings  o  o 
strengthened  by  some  arrangement  like  a,  to  prevent  change  of  shape.  The  pivot  p  may  be . 
lal  two  rollers  under  each  end  of  the  platform,  for  susta 


,  he  the 


radial 
ghtly.  so 
he  pivot, 
hould  b« 
inFig, 


ning  the 


There  must,  of  coun 

the  central  rollers.  Such  a  platform  of  50  ft  length  can."if"carefulVy  made7be"ta'rned.''tnget>ner  wi'th 
an  engine  and  tender,  by  one  man.  by  means  of  a  wooden  lever  12  to  15  ft  long,  inserted  in  a  staple 
for  that  purpose :  and  therefore  may  dispense  with  the  transverse  platform  for  sustaining  wheelwork. 
;ver,  is  usually  added. 


This, 


Such  rollers  as  have  just  been  described,  in  connection  with  the  friction  rollers  Fig  5  is  perhaps 

the  best  arrangement  for  a  large  turning  bridge.   At  least  one  end 


432 


WATER   STATIONS. 


of  a  platform  must  be  provided  with  a  Catch  or  stop  for  arresting  its  motion  at  the  moment 
it  has  reached  the  proper  spot.  A  common  mode  is 
shown  at  Figs  9.  It  consists  of  a  wrought-iron  bar  m  n,  4 
ft  long,  3  ins  wide,  and  %  thick ;  hinged  at  its  end  m,  which 
is  confined  to  the  top  of  the  platform.  Its  outer  end  71  is 
formed  into  a  ring  v  for  lifting  it.  A  strong  casting  e  e,  (or 
in  longitudinal  section  at  t  t,)  about  15  ins  long.  3  ins  wide, 
and  1  inch  thicK,  is  also  firmly  bolted  to  the  top  of  the  plat- 
form ;  and  the  stop-bar  m  n  rests  in  its  recess  r,  while  the 
platform  is  being  turned.  A  similar  casting  a  a  is  well 
bolted  to  the  wooden  or  stone  coping  c  c.  which  surrounds 
the  top  of  the  lining  wall  of  the  pit.  When  the  stop-bar 
reaches  this  last  casting,  as  the  platform  revolves,  it  rises 
up  one  of  its  little  inclined  planes  t  t,  and  falls  into  the  re- 
cess of  a  a,  bringing  the  platform  to  a  stand.  When  the 
platform  is  to  be  started  again,  tne  bar  is  lifted  out  of  its 
recess  by  the  ring  v.  until  it  passes  the  casting;  when  it  is 
again  laid  upon  the  coping  c  c,  and  moves  with  the  plat- 
form;  or,  if  required,  the  hinge  at  m  allows  it  to  be  turned 
entirely  over  on  its  back.  When  there  is  a  transverse  platform,  the  proper  place  for  the  stop  is  at  thnt 
end  which  carries  the  turning  gear;  as  it  is  there  handy  to  the  men  who  do  the  turning.  If  there  is 
only  a  main  platform,  the  stop  may  be  placed  midway  of  the  rails.  Sometimes  a  Spring  Catch 
is  placed  at  each  end  of  the  platform  ;  and  each  catch  is  loosened  from  its  hold  at  the  same  instant 
by  a  long  double-acting  lever.  All  the  details  of  a  platform  admit  of  much  variety.  Sometimes 
the  girders  are  made  of  plate  iron,  like  Figs  22  to  24,  p  214. 
Instead  of  the  friction  rollers.  Fig  5,  friction  balls  5  or  6  ins 
diam.  of  polished  steel,  are  sometimes  used.  The  pivots  also 
are  made  in  many  shapes. 

Plat  forms  like  on,  Fig  1O,  revolving 
around  one  end  o  as  a  center  of  motion,  are 

sometimes  useful.  The  shaded  space  is  the  pit.  If  an  engine 
approaching  along  the  track  W,  is  intended  to  pass  on  to  any 
one  of  the  tracks  1,  2,  3,  4.  the  platform  is  first  put  into  the 
required  position,  and  the  engine  passes  at  once  without  de- 
tention. If  the  platform  is  long,  it  will  be  necessary  to  have 

roller- wheels  not  only  under  the  moving  end  a,  but  at  one  or  two  other  points,  as- indicated  by  the 

roller  rails  cc. 


10 


WATEE  STATIONS, 


Water  stations  are  points  along  a  railroad,  at  which  the  engines  stop  to  take 

In  water.  Their  distance  apart  varies  (like  that  of  the  fuel  stations,  which  accompany  them.)  from 
about  6  miles,  on  roads  doing  a  very  large  business  ;  to  15  or  20  miles  on  those  which  run  but  few  trains. 
Much  depends,  however,  upon  where  water  can  be  had.  It  has  at  times  to  be  conducted  in  pipes  for 
2  or  3  miles  or  more.  The  object  in  having  them  near  together  is  to  prevent  delay  from  many  engines 
being  obliged  to  use  the  same  station.  To  prevent  interruption  to  travel,  they  are  frequently  placed 
upon  a  side  track.  A  supply  of  water  is  kept  on  hand  at  the  station  usually  in  large  wooden  tubs 
or  tanks,  enclosed  in  frame  tank-houses.  The  tank  house  stands  near  the  track,  leaving  only  about 
2  to  4  ft  clearance  for  the  cars.  It  is  two  stories  high  ;  the  tank  being  in  the  upper  one  ;  and  having 
Its  bottom  about  10  or  1 2  feet  above  the  rails.  In  the  lower  story  is  usually  the  pump  for  pumping  up  the 
water  into  the  tank;  and  a  stove  for  preventing  the  water  from  freezing  in  winter.* 

The  tanks  are  usually  circular;  and  a  few  inches  greater  in  diam  at  the  bottom  than  at  the  top,  so 
that  the  iron  hoops  may  drive  tight.  Their  capacity  generally  varies  from  6000  to  40000  gallons, 
(rarely  80000  or  more.)  depending  on  the  number  of  engines  to  be  supplied.  A  tender-tank  holds  from 
1200  to  2400  gallons  ;  and  an  onerine  evaporates  from  20  to  150  gallons  per  mile,  depending  on  the  class 
of  engine;  weight  of  train  ;  steepness  of  grade,  Ac.  Perhaps  40  gallons  will  be  a  tolerably  full  aver- 
age for  passenger,  and  80  for  freight  engines.  The  following  are  the  contents  of  tanks  of  different 
inner  diams,  and  depths  of  water.  U.  S.  galls  of  231  cub  ins  ;  or  7.4805  galls  to  a  cub  ft.  See  p.  434. 


Diam. 

Depth. 

Gallons. 

Cub.  Ft. 

Diam. 

Depth. 

Gallons. 

Cub.  Ft. 

Ft. 

Ft. 

Ft. 

Ft. 

12 

8 

6767 

905 

24 

12 

40607 

5429 

14 

9 

10363 

1385 

26 

13 

51628 

6902 

16 

9 

13535 

1810 

28 

14 

64481 

8621 

18 

10 

19034 

2545 

30 

15 

79310 

10603 

20 

10 

23499 

3142 

32 

16 

96253 

12868 

22 

11 

31277 

4181 

34 

17 

115451 

15435 

Cypress  or  any  of  the  pines  answer  very  well  for  tanks.  The  staves  may  be  about  2%  ins  thick  for 
the  smaller  ones  ;  to  4  or  5  ins  for  the  largest.  The  bottoms  may  be  the  same.  The  staves  should  be 
planed  by  machinery  to  suit  the  curve  precisely.  Nothing  is  then  needed  between  the  staves  to  pro 
duce  tightness.  A  single  wooden  dowel  is  inserted  between  each  two  near  the  top,  merely  to  hold  them 
in  place  while  being  put  together.  The  bottom  isdowelled  together  ;  and  simply  inserted  into  a  groove 
very  accurately  cut,  about  an  inch  deep,  around  the  inner  circumf  of  the  tub,  at  a  few  inches  above 
the  bottoms  of  the  staves.  (For  iron  ones  see  p  434.) 


*  A  frame  tank-house.  18  ft  square,  with  stone  foundations  for  both  bouse  and  tank,  will  by  itsel* 
cost  $400  to  $600.     A  brick  or  stone  one  somewhat  more. 


ATER  STATIONS. 


433 


One  of  2O  fi  cliaiii,  and  12  ft  deep,  may  have  9  hoops  of  good  iron; 
placed  several  incjrtfs  nearer  together  at  the  bottom  of  the  tank  than  at  the  top.  Their  width  3  ins; 
the  thickness  of/ftie  lower  two,  y±  iuch ;  thence  gradually  diminishing  until  the  top  one  is  but  half 
as  thick.  The^iower  two  are  driven  close  together.  These  dimensions  will  allow  for  the  rivet-holes 

into  place.*  Tnree  rivets  of  %  iuch  diam,  and  3  ins  apart,  in  line,  are  sufficient  for  a  joint  of  a  lower 
hoop.  One  of  34  ft  diam,  17  deep,  may  have  12  hoops  ;  the  lower  ones  4  ins  by  ^  ;  with  three  %  inch 
rivets  to  a  lower  hoop-joint. 

The  bottom  planks  of  the  tank  must  bear  firmly  upon  their  supporting  joists,  or  bearers. 

A  tank  must  have  an  inlet-pipe  by  which  the  water  may  enter  it;  a  waste-pipe  for  preventing 
overflow;  and  a  discharge  or  feed-pipe  7  or  8  ias  diam.  in  or  near  the  bottom;  through  which  the 
water  flows  out  to  the  tender.  The  inner  end  of  the  discharge-pipe  is  covered  by  a  valve,  to  be  opened 
at  will  by  the  engine  man,  by  means  of  an  outside  cord  and  lever.  To  its  outer  end  is  generally  at- 
tached a  flexible  canvas  and  gum-elastic  hose  about  7  o:  8  ins  diam,  and  8  or  10  ft  long,  through  which 
the  water  enters  the  tender-tank.  Or,  instead  of  a  hose,  the  feed-pipe  may  be  prolonged  by  a  metallic 
pipe,  or  nozzle,  sufficiently  long  to  reach  the  tender;  and  so  jointed  as,  when  not  in  use,  to  swing  to 
one  side,  or  to  be  raised  to  a  vertical  position,  (in  the  last  case  it  is  called  a  drop,)  so  as  not  to  be  in 
the  way  of  passing  trains. 

The  same  tank  may  supply  two  engines  on  different  tracks,  at  once.    The  tanks  are  very  durable. 

The  patent  frost-proof  tank  of  John  Bnriiham,  Batavia,  Illinois, 

is  simply  an  ordinary  tank,  in  which  the  water  is  prevented  from  freezing  by  means,  1st,  of  a  circular 
roof  which  protects  a  ceiling  of  joists,  between  which  is  a  layer  of  sawdust;  2d,  by  an  air-space  ob- 
tained by  a  ceiling  of  boards  beneath  the  timbers  on  which  the  tank  rests.  Although  the  sides  are 
entirely  unprotected,  no  house  is  necessary :  but  merely  strong  posts  and  beams  on  a  stone  founda- 
tion, for  the  support  of  the  tauk.t 

Tanks  are  frequently  made  rectangular,  with  vertical  sides  of 

posts  lined  with  plank,  and  braced  across  in  both  directions  by  iron  rods.  They  are  more  apt  to  leak 
than  circular  ones.  They  have  been  made  of  iron ;  but  wood  seems  to  be  preferred. 

The  water  for  supplying  the  tanks,   may  be   pumped    by  hand, 

steam,  horse,  wind,  hydraulic  ram,  or  otherwise,  from  a  running  stream ;  from  a  pond  made  by 
damming  the  stream  if  very  small  or  irregular:  from  a  cistern  below  the  tank  ;  or  from  a  common 
well.  Many  roads  doing  a  business  of  10  or  12  engines  daily  in  each  direction,  depend  entirely  upon 
wells  ;  and  pump  by  hand ;  generally  two  men  to  a  pump.  Those  doing  a  very  large  business,  when 
the  supply  cannot  be  obtained  by  gravity,  mostly  use  steam.  The  windmill  is  the  most  economical 
power;  and  when  well  made,  is  very  little  liable  to  get  out  of  order.  Of  course  it  will  not  work  during 
a  calta  ;  but  this  objection  may  be  obviated  in  most  cases  by  having  the  tanks  large  enough  to  hold 
a  supply  for  several  days.i  Steam,  however,  is  most  reliable. 

The  following*  table  will  give  some  idea  of  the  power  reqd  in  a  steam  en- 
gine for  the  pumping.  In  ordering  an  engine,  specify  not  its  number  of  horse-powers,  but  the  num- 
ber of  gallons  it  must  raise  in  a  given  number  of  hours,  to  a  given  height;  with  a  given  steam  pres- 
sure, (say  about  60  to  80  fts  per  sq  inch.)  The  pump  should  be  sufficiently  powerful  not  to  have  to 
work  at  night;  and  should  be  capable  of  performing  at  least  25  per  cent  more  than  its  reqd  duty. 

A  fair  average  horse  should  pump  in  8  hours  the  quantities  con- 
tained in  the  first  3  cols  ;  to  the  height  in  the  4th  col ;  or  sufficient  to  supply  the  number  of  locomo- 
tives in  the  5th  col,  with  about  2000  gals  each.  Two  men  should  do  about  ^  as  much.** 


Cub.  Ft. 

Lbs. 

Gals. 

Ht.  Ft. 

No.  of 
Locos. 

Cub.  Ft. 

Lbs. 

Gals. 

Ht.  Ft. 

No.  of 
Locos. 

1600 
2000 
2667 
3200 
3555 
4000 

100000 
125000 
166666 
200000 
222222 
250000 

11968 
14960 
19946 
23936 
26596 
29920 

100 
80 
60 
50 
45 
40 

6 
7J* 
10 
12 

lly< 

4571 
5333 
64^0 
8000 
10607 
16000 

285714 
333333 
400000 
500000 
666667 
1000000 

34194 
39893 

47872 
59840 
79787 
119680 

35 
30 
25 
20 
15 
10 

17 
20 
24 
30 
40 
60 

A  reservoir,  with  a  stand-pipe,  or  water  column,  is  preferable  to 

the  ordinary  tank,  when  the  locality  admits  of  it ;  being  less  liable  than  the  pump  to  get  out  of  order ; 
and  being  cheaper  in  the  end.     The  reservoir  is  supposed  to  be  filled  by  water  flowing  into  it  by 


*  Such  a  tank,  put  up  in  its  place,  will  cost  about  $400.  Greo.  J.  Burkhardt  &  Co,  1341  Buttonwood 
Street,  Philada,  make  tanks  their  specialty ;  and  are  provided  with  machinery  which  secures  perfect 
accuracy  of  joint  in  every  part.  Their  work  is  sought  from  great  distances. 

i  THE  "  RAILWAY  FROST-PROOF  TANK  Co,'1  OF  BATAVIA,  111,  make  a  specialty  of  the  construction  and 
erection  of  these  tanks,  complete  in  every  detail,  ready  for  use.  They  also  make  windmills. 

J  ANDREW  J.  CORCORAN,  No  76  John  St.  N.  York,  furnishes  excellent  machines.  He  also,  when  de- 
sired, provides  pumps,  &c,  complete.  The  cost  of  windmill  alone,  for  railway  stations,  varies  from 
about  $iOO  for  18  ft  diam  ;  to  $1400  for  36  ft  diara,  at  the  factory. 

*  *  The  <?ost  of  a  direct  acting  steam  pump,  with  its  boilers,  &c.  fixed 

in  place,  ready  for  work,  :ind  capable  of  the  duty  of  the  above  table,  may  be  roughly  set  dowu  at  about 
$150;  twice  t,he  duty,  $600:  4  times,  $750;  6  times.  $900:  10  times,  $1300:  20  times,  $2000.  Add 
cost  of  engine  house.  Made  by  Henry  R.  Worthiugton.  61  Be«-kman  St,  N.  York;  the  Philadelphia 
Hydraulic  Works,  Eveline  St,  Fhilada  ;  and  by  many  establishments  in  most  of  our  large  cities. 


4S4 


WATER  STATIONS. 


at  a  considerable  distance  from'  them,  according  to  circumstances.*  Prom  its  bottom,  an  iron  pipe 
from  8  to  12  ins  diam,  is  carried  (generally  underground,  )  to  within  a  few  ft  of  the  track.  At  that 
point  it  turns  vertically  upward  to  about  8  or  10  ft  above  the  track,  forming  a 


ciple  of  those  for  street  pipes,  (page  572,)  is  best. 


Fast  trains  sometimes  scoop  up  water  while  in  motion, 
from  a  long:  trough  laid  between  the  rails.  Length  of  trough  about 
i<£  mile  ;  width  18  ins  ;  depth  4  ins. 

Evaporation  from  Locomotives.    In  addition  to  what  is  said  on  pace 

432,  in  the  passage  preceding  the  table,  we  may  state  that  the  evaporation  is  usually  from  6  to  7  fts 
of  water  to  1  ft  of  fair  coal.  Hence  if  we  take  the  average  at  6^  E>*.  or  say  .8  of  a  gallon  of  water  to 
1  B>  of  coal,  and  assume,  as  on  page  432,  that  a  passenger  engine  evaporntes  an  average  of  40  trallons 
per  mile,  and  a  freight  engine  80  galls,  we  shall  have  very  nearlv  2%  tons  of  coal  consumed  per  100 
miles  by  the  former;  and  4^  tons  by  the  latter.  The  evaporation  from  a  heavily  tasked  powerful 
engine  may  amount  to  150  galls  or  more  per  mile;  but  such  ia  an  exceptional  case. 


Thickness  near  bottom  of  sheet-iron   water  tanks,  single 

iveted;  safety*;  ultima-  -x iW • 

punching  the  rivet  holes. 


riveted  ;  safety  4;  ultimate  strength  of  the  iron  40,000  tts  per  sq  inch,  but  reduced  say  one-half  by 
"--ugh  safe  from  the  water,  some  are  plainly  far  too  thin  for  handling. 


1NNEB  DIAMETER  IN  FEET. 


Depth 
in 
Feet. 

5 

10 

15 

20 

25 

3O 

35 

40 

THICKNESS  IN  INCHES.                          Original. 

1 
5 

10 
15 
20 
25 
30 

.0026 
.0130 
-0260 
.0391 
.0521 
.0651 
.0781 

.0052 
.0260 
0521 
.0781 
.1042 
.1302 
.1562 

.0078 
.0391 
.0781 
1172 
.1562 
.1953 
.2344 

.0104 
.0520 
.1042 
.1562 
2084 
.2604 
.3124 

.0130 
.0651 
.1302 
.1953 
.2604 
3255 
.3906 

.0156 
.0781 
.1562 
.2344 
.3125 
.3906 
.4687 

.0182 
.0911 
.1823 
.2734 
.3645 
.4557 
.5470 

.0208 
.1042 
.2083 
.3125 
.4166 
.5208 
.6250 

Within  the  limit*  of  the  table  we  may  assume  both  the  pres  and  the  thickness  to  vary  as 
either  diam  or  depth.  See  Table,  p  532. 

CIRCULAR  ARCS  IN  FREQUENT  USE. 

The  fifth  column  is  of  use  for  finding  points  for  drawing  arcs  too  large  for  the  beam-compass,  on 
the  principle  given  near  foot  of  p  17.  In  even  the  largest  office  drawings  it  will  not  be  necessary  to 
use  more  than  the  first  three  decimals  of  the  fifth  column  :  and  after  the  arc  is  subdivided  into  parts 
smaller  than  about  35°  each,  the  first  two  decimals  .25  will  generally  suffice.  Original. 


Rise 

For 

For  rise 

Rise 

For 

For 

in 

Degin 

For  rad 

length  of 

of  half 

in 

Degin 

For  rad 

length  of 

rise  of 

of 

arc. 

by 

chord 

mult  rise 

of 

arc. 

mu^rise 

chord 

mult 

chord. 

by 

by 

chord. 

by 

rise  by 

1-60 

o     • 
9   9.75 

313. 

1  .00107 

.2501 

X 

0       ' 

56   8.70 

8.5 

1.04116 

.2538 

1-45 

10  10.75 

253.625 

1.00132 

.2501 

1-7 

6346.90 

6.625 

1.05356 

.2549 

1-40 

11  26.98 

200.5 

1.00167 

.2502 

.155 

68  53.63 

5.70291 

1.06288 

.2557 

1-35 

13    4.92 

153.625 

1.00219 

.2502 

1-6 

73  44.39 

5. 

1.07250 

.2566 

1-30 

15  15.38 

113. 

1.00296 

.2503 

.18 

7911.73 

4.35803 

1.08428 

.2576 

1-25 

1817.74 

78.625 

1.00426 

.2504 

1-5 

87  12.34 

3.625 

1.10347 

.2593 

1-20 

22  50.54 

505 

1.00665 

.2506 

.207107 

90 

3.41422 

1.11072 

.2599 

1-19 

24    2.16 

45.625 

1.0073T 

.2507 

.225 

96  54.67 

2.96913 

1.12997 

.2615 

1-18 

25  21.65 

41. 

1.00821 

.2508 

X 

106  15.61 

2.5 

1.15912 

.2639 

117 

26  50.36 

36625 

1.00920 

.2509 

.275 

11514.59 

2.15289 

1.19082 

.2665 

1-16 

28  30.00 

32.5 

1.01038 

.2510 

.3 

12351.30 

1.88889 

1.22495 

.2692 

1-15 

30  22.71 

28.625 

1.01181 

.2511 

K 

134  45.62 

1.625 

1.27401 

.2729 

1-14 

32  31.22 

25. 

1.01355 

.2513 

.365 

144  30  98 

1.43827 

1.32413 

.2766 

1-13 

34  59.08 

21.625 

1.01571 

.2515 

.4 

154  38.35 

1.28125 

1.38322 

.2808 

1-12 

37  50.96 

18.5 

1.01812 

.2517 

.425 

161  27.52 

1.19204 

1.42764 

.2838 

1-11 

41  13.16 

15.625 

1.02189 

.2520 

.45 

167  5633 

1.11728 

1.47377 

.2868 

1-10 

45  14.38 

13. 

1.02646 

.2525 

.475 

174    7.49 

1.05402 

1.52152 

.2699 

1-9 

50    6.91 

10.625 

1.03260 

.2530 

.5 

180 

1. 

1.57080 

.2929 

*  AN  UNCOVERED  RESERVOIR  50  ft  diam  by  12  ft  deep,  lined  with,  brick  or  masonry,  will  usually  cost 
from  $2500  to  $3500,  according  to  circumstances. 

t  THE  PRICE  OP  A  CAST-IRON  WATER-COLUMN,  of  6-inch  bore;  with  bed-plate;  holding-down  bolts, 
and  washers  ;  connecting-pipes  ;  swing-joint  with  copper  arm  9  ft  long  ;  valve ;  hand- wheel ;  &c  ;  com- 
plete, ready  to  set  up,  (by  the  Pascal  Iron  Works,  Phtlada,  in  1871,)  is  $475  at  the  shop. 


COS'I    OF    EARTHWORK.  435 

COST  OF  EARTHWORK. 


THE  following  is  takeu  from  the  last  edition  of  the  writer's  volume  on  the  Measurement  of  Excava- 
tions and  Embankments. 

Al*t.  1.  It  is  advisable  to  pay  for  this  kind  of  work  by  the  cubic  yard  of  excavation  only  ;  in- 
utead  of  allowing  separate  prices  for  excavation  and  embankment.  By  this  means  we  get  rid  of  th« 
difficulty  of  measurements,  as  well  as  the  controversies  and  lawsuits  which  often  attend  the  deter- 
mination of  the  allowance  to  be  made  for  the  settlement  or  subsidence  of  the  embankments. 

It  is,  moreover,  our  opinion  that  justice  to  the  contractor  should  lead  to  the  Ellg'lisll  prac- 
tice of  paying1  the  laborers  by  the  cubic  yard,  instead  of  by  the  day. 

Kxperience  fully  proves  that  when  laborers  are  scarce  and  wages  high,  men  can  scarcely  be  depended 
upon  to  do  three-fourths  of  the  work  which  they  readily  accomplish  when  wages  are  low,  and  when 
fresh  hands  are  waiting  to  be  hired  in  case  any  are  discharged.  The  contractor  is  thus  placed  at  the 
mercy  of  his  men.  The  writer  has  known  the  most  satisfactory  results  to  attend  a  system  of  task- 
work, accompanied  by  liberal  premiums  for  all  overwork.  By  this  means  the  interests  of  the  laborers 
are  identified  with  that  of  the  contractor ;  and  every  man  takes  care  that  the  others  shall  do  their 
fair  share  of  the  task. 

Ellwood  Morris.  C  E,  of  Philadelphia.  Was,  we  believe,  the  first  person  who  properly  investigated 
the  elements  of  cost  of  earthwork,  and  reduced  them  to  such  a  form  as  to  enable  us  to  calculate  the 
total  with  a  considerable  degree  of  accuracy.  He  published  his  results  in  the  Journal  of  the  Franklin 
Institute  in  1841.  His  paper  forms  the  basis  on  which,  with  some  variations,  we  shall  consider  the 
matter ;  and  on  which  we  shall  extend  it  to  wheelbarrows,  as  well  as  to  carts.  Throughout  this  paper 
we  speak  of  a  cubic  yard  considered  only  as  solid  in  its  place,  or  before  it  is  loosened  for  removal.  It 
Is  scarcely  necessary  to  add  that  the  various  items  can  of  course  only  be  regarded  as  tolerably  close 
approximations,  or  averages.  As  before  stated,  the  men  do  less  work" when  wages  are  high  ;  and  more 
when  they  are  low.  A  great  deal  besides  depends  on  the  skill,  observation,  and  energy  of  the  con- 
tractor and  his  superintendents.  It  is  no  unusual  thing  to  see  two  contrp^tors  working  at  the  same 
prices,  in  precisely  similar  material,  where  one  is  making  money,  and  the  other  losing  it,  from  a  want 
of  tact  in  the  proper  distribution  of  his  forces,  keeping  his  road's  in  order,  having  his  carts  and  bar- 
rows well  filled,  &c.  &c.  Uncommonly  long  spells  of  wet  weather  may  seriously  affect  the  cost  of  exe- 
cuting earthwork,  by  making  it  more  diflicult  to  loosen,  load,  or  empty  ;  beside's  keeping  the  roads  in 
bad  order  for  hauling. 
The  aggregate  cost  of  excavating  and  removing  earth  is  made  up  by  the  following  items,  namely  : 

1st.    Loosening  the  earth  ready  for  the  nhoreUers. 

2d.     Loading  it  by  shovels  into  the  carts  or  barrowi. 

3d.     Hauling,  or  wheeling  it  away,  including  emptying  and  returning. 

4th.   Spreading  it  out  into  successive  layers  on  the  embankment. 

5th.  Keeping  the  hauling -road  for  carts,  or  the  plank  gangways  for  barrows,  in  good  order. 

6th.  Wear,  sharpening,  depreciation,  and  interest  on  cost  of  tools. 

7th.   Superintendence,  and  water  carriers. 

8th.   Profit  to  the  contractor. 

We  will  consider  these  items  a  little  in  detail,  baaing  our  calculations  on  the  assumption  that  com- 
mon labor  costs  $1  per  day.  of  10  working  hours.  The  results  in  our  tables  must  therefore  be  in- 
creased or  diminished  in  about  the  same  proportion  as  common  labor  costs  more  or  less  than  this. 

Art.  2.     loosen  i  n:r  the  earth  ready  for  the  shovellers.    This  is 

generally  done  either  by  ploughs  or  l>y  picks  ;  more  cheaply  by  the  first.  A  plough  with  two  horses, 
and  two'men  to  manage  them,  at  $1  'per  day  for  labor,  75  cents  per  day  for  each  horse  and  87  cents 
per  day  for  plough,  including  harness,  wear,  repairs.  &c.  or  a  total  of  $3  87,  will  loosen,  of  strong 
heavy  soils,  from  200  to  300  oubic  yards  a  day,  at  from  1.93  to  1/29  cents  per  yard ;  or  of  ordinary 
loam,  from  400  to  600  cubic  yards  a  day,  at  from  .97  to  .64  of  a  cent  per  yard.  Therefore,  as  an  ordi- 
nary average,  we  may  assume  the  actual  cost  to  the  contractor  for  loosening  by  the  plough,  as  fol- 
lows: strong  heavy  soils,  1.6  cents  ;  common  loam,  .8  cei.t :  light  sandy  soils,  .4  cent.  Very  stiff  pure 
olav,  or  obstinate  cemented  gravel,  may  be  set  down  at  2.5  cents  :  they  require  three  or  four  horses. 
By  the  pick,  a  fair  day's  work  is  about  14  yards  of  stiff  pure  clay,  or  of  cemented  gravel ;  25  yards 
of  strong  heavy  soils;  40  yards  of  common" loam  ;  60  yards  of  light  sandy  soils  — all  measured  in 
place;  which,  at  $1  per  day  for  labor,  gives,  for  stiff  clny.  7  cents;  heavy  soils.  4  cents;  loam,  2.5 
cents;  light  sandy  soil,  1.668  cents.  Pure  sand  requires  but  very  little  labor  for  loosening;  .5  of  a 
cent  will  cover  it. 

Art.  3.    Shovelling  the  loosened  earth  into  carts.   The  amount 

shovelled  per  day  depends  partly  upon  the  weight  of  the  material,  but  more  upon  so  proportioning 
the  number  of  pickers  and  of  carts  to  that  of  shovellers,  as  not  to  keep  the  latter  waiting  for  either 
material  or  carts.  Tn  fairlv  regulated  gangs,  the  shovellers  into  carts  are  not  actually  engaged  in 
shovelling  for  more  than  six-tenths  of  their  time,  thus  being  unoccupied  but  four- tenths  of  it:  while, 
under  bad  management,  they  lose  considerably  more  than  one-half  of  it.  A  shoveller  can  readily 
load  into  a  cart  one-third  of  a  cubic  yard  measured  in  place  (and  which  is  an  average  working  cart- 
load), of  sandy  soil,  in  five  minutes  :  "of  loam,  in  six  minutes:  and  of  any  of  the  heavy  soils,  in  seven 
minutes.  This  would  give,  for  a  day  of  10  working  hours,  ]'20  loads,  or  40  cubic  yards  of  light  sandy 
soil ;  100  loads,  or  33^  cubic  yards  of  loam  :  or  86  loads,  or  28.7  yards  of  the  heavy  soils.  But  from 
these  amounts  we  must  deduct  four  tenths  for  time  necessarily  lost:  thus  reducing  the  actual  work- 
ing quantities  to  24  yards  of  light  sandy  soil,  20  yards  of  loam,  17.2  yards  of  the  heavy  soils.  When 
the  shovellers  do  less  than  this,  there  is  some  mismanagement. 

Assuming  these  as  fair  quantities,  then,  at  $1  per  day  for  labor,  the  actual  cost  to  the  contractor 
for  shovelling  per  cubic  yard  measured  in  place,  will  be,  for  sandy  soils,  4.167  cents;  loam,  5  cents; 
heavy  soils,  clays.  Ac,  5.81  cents. 

In  practice,  the  carts  are  not  usually  loaded  to  any  leas  extent  with  the  heavier  soils  than  with  the 
lighter  ones.  Nor.  indeed,  is  there  any  necessity  for  so  doing,  inasmuch  as  the  difference  of  weight 
of  a  cart  and  one  third  of  a  cubic  yard  of  the  various  soils  is  too  slight  to  need  any  attention  ;  espe- 
cially when  the  cart  road  is  kept  in  good  order,  as  it  will  be  by  any  contractor  who  understands  hi* 


436  COST   OF    EARTHWORK. 

own  interest.  Neither  is  it  necessary  to  modify  the  load  on  account  of  any  slight  inclinations  which 
may  occur  iu  the  grading  of  roads.  An  earth-cart  weighs  by  itself  about  %  a  ton. 

Art.  4.    Hauling  away  the  earth;  dumping-,  or  emptying; 

and  returning  to  reload.  The  average  speed  of  horses  iu  hauling  is  about  2^  miles 
per  hour,  or  200  feet  per  minute ;  which  is  equal  to  100  feet  of  trip  each  way  ;  or  to  100  feet  of  lead, 
as  the  distance  to  which  the  earth  is  hauled  is  technically  called.*  Besides  this,  there  is  a. loss  of 
about  four  minutes  in  every  trip,  whether  long  or  short,  in  waiting  to  load,  dumping,  turning,  &c. 
Hence,  every  trip  will  occupy  as  rnauy  minutes  as  there  are  lengths  of  100  feet  each  in  the  lead ;  aud 
four  minutes  besides.  Therefore,  to  find  the  number  of  trips  per  day  over  any  given  average  lead,  we 
divide  the  number  of  minutes  in  a  working  day  by  the  sum  of  4  added  to  the  number  of  100  feet 
lengths  contained  in  the  distance  to  which  the  earth  has  to  be  removed ;  that  is, 

The  number  (600)  of  minutes  in  a  working  day  _  the  number  of  trips,  or  loads 
4  -f-  the  number  of  100- feet  lengths  in  the  lead   ~~  removed  per  day,  per  cart. 

And  since  %  of  a  cubic  yard  measured  before  being  loosened,  makes  an  average  cart-load,  the  num- 
ber of  loads,  divided  by  3,  will  give  the  number  of  cubic  yards  removed  per  day  by  each  cart;  aud 
the  cubic  yards  divided  into  the  total  expense  of  a  cart  per  day,  will  give  the  cost  per  cubic  yard  !V>r 
hauling. 

In  leads  of  ordinary  length  one  driver  can  attend  to  4  carts  ;  which,  at  $1  per  day,  is  25  cents  per 
cart.  When  labor  is"$l  per  dav.  the  expense  of  a  horse  is  usually  about  75  cents;  and  that  of  tiie 
cart,  including  harness,  tar,  repairs,  &c,  25  cents,  making  the  total  daily  cost  per  care  $1.25.  The 
expense  of  the  horse  is  the  same  on  Sundays  and  on  rainy  days,  as  when  at  work  :  and  this  consid- 
eration is  included  iu  the  75  cents.  Some  contractors  employ  a  greater  number  of  drivers,  who  also 
help  to  load  the  carts,  so  that  the  expense  is  about  the  same  in  either  case. 

EXAMPLE.  How  many  cubic  yards  of  loam,  measured  in  the  cut.  can  be  hauled  by  a  horse  and  cart 
in  a  day  of  10  working  hours,  (600  minutes,)  the  lead,  or  length  of  haul  of  earth  being  1000  feet,  (or 
10  lengths  of  100  feet,)  and  what  will  be  the  expense  to  the  contractor  for  hauling,  per  cubic  yard, 
assuming  the  total  cost  of  cart,  horse,  and  driver,  at  $1.25? 

600  minutes  600  ,  43  loads 

Here,  — ; —  =  —  =  43  loads.        And =:  14.3  cubic  yards. 

4+ W  lengths  of  100  feet,          14  3 

And    —, f-     - =  8.74  cents  per  cubic  yard. 

14.3  cub  yds 

In  this  manner  the  2d  and  3d  columns  of  the  following  tables  have  been  calculated. 

Art.  5.    Spreading,  or  levelling  off*  the  earth  into  regular 

thin  layers  Oil  the  embankment.  A  bankraan  will  spread  from  50  to  100 cubic 
yards  of  either  common  loam,  or  any  of  the  heavier  soils,  clays,  &c.  depending  on  their  dryness. 
This,  at  $1  per  day,  is  1  to  2  cents  per  cubic  yard;  and  we  may  assume  13^  cents  as  a  fair  average 
for  such  soils :  while  I  cent  will  suffice  for  light  sandy  soils. 

This  expense  for  spreading  is  saved  when  the  ear*h  is  either  dumped  over  the  end  of  the  embank- 
ment, or  is  wasted;  still,  about  y±  cent  per  yard  should  be  allowed  iu  either  case  for  keeping  the 
dumping-places  clear  and  in  order. 

REMARK.  When  removing  loose  rock,  which  requires  more  time  for  loading,  say, 
No.  of  minutes  (600)  >>i  a  working  day   __  No.  of  loads  removed, 
6  +  No.  of  100-/eet  lengths  of  load      ~    per  day,  per  cart. 

Art.  6.  Keeping  the  cart-road  in  good  order  for  hauling* 

No  ruts  or  puddles  should  be  allowed  to  remain  unfilled:  rain  should  at  once  be  led  off  by  shallow 
ditches  ;  and  the  road  be  carefully  kept  in  good  order;  otherwise  the  labor  of  the  horses,  and  the  wear 
of  carts,  will  be  very  greatly  increased.  It  is  usual  to  allow  so  much  per  cubic  yard  for  road  repairs ; 
but  we  suggest  so  much  per  cubic  yard,  per  100  feet  of  load  ;  say  -^  of  a  cent. 

Art.  7.   Wear,  sharpening,  and  depreciation  of  picks  and 

Shovels.    Experience  shows  that  about  %  of  a  cent  per  cubic  yard  will  cover  this  item. 

Superintendence  and  water-carriers.    These  expenses  win  vary  with 

lorsal  circumstances  ;  but  we  agree  with  Mr.  Morris,  that  1  %  cents  per  cubic  varrt  will,  underordinary 
circumstances,  cover  both  of  them.     An  allowance  of  about  y±  cent  mav  in  justice  be  added  for  extra 
trouble  in  digging  the  side-ditches  ;  levelling  off  the  bottom  of  the  cut  to  {trade ;  and  general  trimming 
up.     In  very  light  cuttings  this  may  be  increased  to  J^  cent  per  every  yard. 
At  y±  cent,  all  the  items  in  this  article  amount  to  2  cents  per  cubic  yard  of  cut. 

Art.  8.  Profit  tO  the  Contractor.  This  may  generally  be  set  down  at  from  6 to 
15  per  cent,  according  to  the  magnitude  of  the  work,  the  risks  incurred,  and  various  incidental  cir- 
cumstances. Out  of  this  item  the  contractor  generally  has  to  pay  clerks,  storekeepers,  and  other 
agents,  as  well  as  the  expenses  of  shanties.  &c  ;  although  these  are  in  most  cases  repnid  by  the  profits 
of  the  stores;  and  by  the  rates  of  boarding  and  lodging  paid  to  the  contractors  by  the  laborers. 

Art.  9.  A  knowledge  of  the  foregoing  items  enables  its  to 
calculate  with  tolerable  accuracy  the  cost  of  removing  earth. 

For  example,  let  it  be  required  to  ascertain  the  cost  per  cubic  yard  of  excavating  common  loam,  meas- 
ured in  place;  and  of  removing  it  into  embankment,  with  »n  aroraee  haul  or  l^ad  of  1000  feet:  the 
wages  of  laborers  being  $1  per  day  of  10  working  hours  ;  a  horse  75  cts  a  day  ;  and  a  cart  25  cts.  One 
driver  to  four  carts. 


•Jf  When  an  entire  cut  is  made  into  an  embankment,  the  mean  haul  is  the  dist  between  centers 
of  gravity  of  the  cut  aud  embkt. 


EARTHWORK. 


437 


Cents. 

Here  we  hayrtiost  of  loosening,  say  by  pick,  Art  2,  per  cubic  yard,  say,  2.50 

Loadinajfito  carts,  Art.  3,  "  "  5.00 

Hauling  1000  feet,  as  calculated  previously  in  example,  Art.  4,  "  8-74 

Spr/ading  into  layers.  Art.  5,  "  1.50 

Keeping  cart-road  in  repair,  Art.  6,  10  lengths  of  100  ft,  1.00 

Various  items  in  Art.  7,  2.00 

Total  cost  to  contractor,  20.74 

Add  contractor's  profit,  say  10  per  cent,        2.074 

Total  cost  per  cubic  yard  to  the  company,    22.814 
It  is  easy  to  construct  a  table  like  the  following,  of  costs  per  cubic  yard,  for  different  lengths  of  lead. 
Columns  2  and  3  are  first  obtained  by  the  Rule  in  Article  4 ;  then  to  each  amount  in  column  3  is  added 


it  particular  circumstances,    in  tnis  manner  the  tables  have  been  prepared. 

By  Carts.     Labor  .SI  per  day,  of  1O  working1  hours. 


1 

t 

s 

Common  Loam, 

Strong  Heavy  Soils, 

II 

\\ 

If 

K 

-si 

a.2 

TOTAL     COST     PER     CUBIC 

TOTAL     COST     PER     CUBIC 

32~ 

>,£ 

"2  ft 

YARD,      EXCLUSIVE     OF 

YARD,     EXCLUSIVE     OF 

£  js 

it 

Is 

PROFIT  TO  CONTRACTOR. 

PROFIT  TO  CONTRACTOR. 

^5  a> 

|5 

a 

rl 

csl 

M 

i-1 

3  i 

!lj 

1*1 

iji 

•o 

fi! 

1     •« 

| 

3 

i 

»« 

* 

£      OJ 

b     £ 

Feet. 

Cu.Yds. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

25 

470 

2.66 

13.69 

1244 

11.99 

10.74 

6.00 

14.75 

13.50 

12.25 

50 

44.4 

2.81 

13.86 

12.61 

12.16 

10.91 

6.17 

14.92 

13.67 

12.42 

75 

42.1 

2.97 

14.05 

12.80 

12.35 

11.10 

6.36 

15.11 

13.86 

12.61 

100 

40.0 

3.12 

14.22 

12.97 

12.52 

11.27 

6.53 

15.28 

14.03 

12.78 

150 

36.4 

343 

14.58 

13.33 

12.88 

11.63 

6.89 

1564 

14.39 

13.14 

200 

33.3 

3.75 

14.95 

13.70 

13.25 

12.00 

7.26 

16.01 

14.76 

13.51 

300 

28.6 

4.37 

15.67 

14.42 

13.97 

12.72 

7.98 

16.73 

15.48 

14.23 

400 

25  0 

5.00 

16.40 

15.15 

14.70 

13.45 

18.71 

17.46 

16.21 

14.96 

500 

22.2 

5.63 

17.13 

1588 

.  15.43 

14.18 

19.44 

18.19 

16.94 

15.69 

600 
700 

20.0 
18.2 

6.25 

17.85 

16.60 

16.15 

14.90 

20.16 

18.91 

17.66 

16.41 

800 

16^7 

7.48 

19.28 

18.03 

17.58 

16.33 

20.88 
21.59 

19.63 
20.34 

19.09 

17.13 

17.84 

900 

15.4 

8  12 

19.92 

18.67 

18.22 

16.97 

22.23 

20.98 

19.73 

18.48 

1000 

14.3 

8.74 

20.74 

19.49 

19.04 

17.79 

23.05 

21  80 

20.55 

19.30 

1100 

13.3 

9.40 

21.50 

20.25 

1980 

18.55 

23.81 

22.56 

21.31 

20.06 

1200 

12.5 

10.0 

22.20 

20.95 

20.50 

19  25 

24.51 

23.26 

22.01 

20.76 

1300 

11.8 

10.6 

22.90 

21.65 

21.20 

19.95 

25.21 

23.96 

22.71 

21.46 

1400 

11.1 

11.2 

23.60 

22.35 

21.90 

20.65 

25.91 

24.66 

23.41 

22.16 

1500 

10.5 

11.9 

24.40 

23.15 

23.70 

21.45 

26.71 

25.46 

24.21 

22.96 

1600 

10.0 

12.5 

25.10 

23.85 

23.40 

22.15 

27.41 

26.16 

24.91 

23.66 

1700 

.52 

13.1 

25.80 

24.55 

24.10 

22.85 

28.11 

26.86 

2561 

24.36 

1800 

.09 

13.7 

26.50 

25.25 

24.80 

23.55 

28.81 

27.56 

26.31 

25.06 

1900 

.70 

14.4 

27.30 

2605 

25.60 

24.35 

29.61 

28.36 

27.11 

25.86 

2000 

.33 

15.0 

28.00 

26.75 

26.30 

25.05 

30.31  • 

29.06 

27.81 

26.56 

2250 

54 

16.6 

29.85 

28.60 

28.15 

26.90 

32.16 

30.91 

29.66 

28.41 

2500 

90 

18.1 

31.60 

30.35 

29.90 

28.65 

33.91 

32.66 

31.41 

30.16 

M  mile 

.58 

19.0 

32.64 

31.39 

30.94 

29.69 

34.95 

33.70 

32.45 

31.20 

3000 

.88 

21.2 

35.20 

33.95 

33.50 

32.25 

37.51 

36.26 

35.01 

33.76 

3250 

.48 

22.8 

37.05 

35.80 

35.35 

34.10 

39.36 

38.11 

36.86 

35.61 

3500 

.13 

24.3 

38.80 

37.55 

37.10 

35.85 

41.11 

39.86 

38.61 

37.36 

3750 

.82 

259 

40.65 

39.40 

38.95 

37.70 

42.96 

41.71 

40.46 

39.21 

4000 

.54 

27.5 

42.50 

41.25 

40.80 

39.55 

44.81 

43.56 

42.31 

41.06 

4250 

.30 

29.1 

44.35 

43.10 

42.65 

41.40 

46.66 

45.41 

44.16 

42.91 

4500 

.08 

30.6 

46.10 

44.85 

44.40 

43.15 

48.41 

47.16        45.91 

44.66 

4750 

3.88 

32.2 

47.95 

46.70 

46.25 

45.00 

50.26 

49.01     !    47.76 

46:51 

5000 

3.70 

33.8 

49.80 

48.55 

48.10 

46.85 

52.11 

5086         49.61 

48.36 

1     mile 

352 

35.5 

51.78 

50.53 

50.08 

48.83 

54.09 

52.84        51.59 

50.34 

134m. 

2.86 

43.8 

61.40         60.15 

59.70 

58.45 

63.71 

62.46        61.21 

59.% 

\Yi  m. 

2.40 

52.1 

71.02        69.77 

69.32 

68.07 

73.33 

72.08         70.83 

69.58 

1%  DV. 

2.07 

60.4 

80.64    1    79.39 

78.94 

77.69 

82.95 

81.70        80.45 

79.20 

2     m. 

1.82 

68.7 

90.26         89.01 

88.56 

87.31 

92.57 

91.32    !    90.07 

88.82 

438  COST    OF    EAKTHWORK. 

By  Carts.     Labor  $1  per  clay,  of  1O  working  hours. 


«! 


II 


47.0 
44.4 

42.1 

40.0 

36.4 

33.3 

28.6 

25.0 

22.2 

20.0 

18.2 

16.7 

15.4 

14.3 

13.3 

12.5 

11.8 

11.1 

10.5 

10.0 
9.52 
9.09 
8.70 
8.33 
7.54 
6.90 
6.58 
5.88 
5.48 
5.13 
4.82 
454 
4.30 
4.08 
3.88 
3.70 
3.52 
2.86 
2.40T 
2.07 
1.82 


2 

.11 

e-M 
_a.S 

if 


2.66 
2.81 
2.97 
3.12 
3.43 
375 
4.37 
5.00' 
5.63 

6'.'87 
7.48 
8.12 
874 
9.40 

10.0 

10.6 

11.2 

11.9 

12.5 

13.1 

13  7 

14.4 

15.0 

16.6 

18.1 

19.0 

21.2 

22.8 

24.3 

25.9 

27.5 

29.1 

30.6 

32.2 

33.8 

35.5 

43.8 

52.1 


Pure  stiff  Clay,  or  cemented 
Gravel, 

Light  Sandy  Soils, 

TOTAL      COST     PER     CUBIC 

TOTAL     COST      PER     CUBIC 

YARD,     EXCLUSIVE     OF 

YARD,     EXCLUSIVE     OF 

PROFIT  TO  CONTRACTOR. 

PROFIT  TO  CONTRACTOR. 

lit 

H 

1  ' 

*u 

111 

It! 

|,| 

L| 

1*1 

fr<     oo 

s"£ 

J2 

i** 

£*£ 

s-g 

£** 

£-* 

Cts. 
1900 

Cts. 
17.75 

Cts. 
14.50 

Cts. 
13.25 

Cw. 

11.52 

Cts. 
10.77 

Cts. 
10.25 

Cts. 
9.50 

19.17 

17.92 

14.67 

13.42 

11.69 

10.94 

10.42 

9.67 

19.36 

18.11 

14.86 

13.61 

11.88 

11.13 

10.61 

9.86 

19.53 

18.28 

15.03 

13.78 

12.05 

11.30 

10.78 

10.03 

20'26 

19^01 

15'.76 

14.51 

12.78 

12.03 

11.51 

10.76 

20^98 

19.73 

15.48 

15.23 

13.50 

12.75 

12.23 

11.48 

21.71 

20.46 

17.21 

15.96 

14.23 

13.48 

12.46 

12.21 

22.44 

21.19 

17.94 

16.69 

14.96 

14.21 

13.69 

12.94 

23.16 

21.91 

18.66 

17.41 

15.68 

14.93 

14.41 

13.66 

23.88 

22.63 

19.38 

18.13 

16.40 

15.65 

15.13 

14.38 

24.59 

23.34 

20.09 

18.84 

17.11 

16.36 

15.84 

1509 

25.23 

23.98 

20.73 

19.48 

17.75 

17.00 

16.48 

15.73 

.26.05 

24.80 

21.55 

20.30 

18.57 

17.82 

17.30 

16.55 

26.81 

25.56 

22.31 

21.06 

19.33 

18.58 

18.06 

17.31 

27.51 

26.26 

23.01 

21.76 

20.03 

19.28 

18.76 

18.01 

28  21 

26.96 

23.71 

22.46 

20.73 

19.98 

19.46 

18.71 

28.91 

27.66 

24.41 

23.16 

21.43 

-  20.68 

20.16 

19.41 

29.71 

28.46 

25.21 

23.96 

22.23 

21.48 

20.96 

20.21 

30.41 

29.16 

25.91 

24.66 

22.93 

22.18 

21.66    !    20.91 

31.11 

29.86 

26.61 

25.36 

23.63 

22.88 

22.36 

21.61 

31.81 

30.56 

2731 

26.06 

24.33 

23.58 

23.06 

22.31 

32.61 

31.36 

28.11 

26.86 

25.13 

24.38 

23.86 

23.11 

33.31 

32.06 

28.81 

27.56 

25-83 

25.08 

24.56 

23.81 

35.16 

33.91 

30.66 

29.41 

27.68 

26.93 

2641 

25.66 

36.91 

35.66 

32.41 

31.16 

29.43 

28.68 

28.16 

27.41 

37.95 

36.70 

33.45 

32.20 

30.47 

29.72 

29.20 

28.45 

40.51 

39.26 

36.01 

34.76 

33.03 

32.28 

31.76 

31.01 

42.36 

41.11 

37.86 

36.61 

34.88 

34.13 

33.61 

32.86 

44.11 

42.86 

39.61 

38.36 

36.63 

35.88 

35.36 

34.61 

45.96 

44.71 

41.46 

40.21 

38.48 

37.73 

37.21 

36.46 

47.81 

46.56 

43.31 

42.06 

40.33 

39.58 

39.06 

38.31 

49.66 

48.41 

45.16 

43.91 

42.18 

41.45 

40.93 

4018 

51.41 

50.16 

46.91 

45.66 

43.93 

43.18 

42.66 

41.91 

53.26 

52.01 

48.76 

47.51 

45.78 

45.03 

4451 

4376 

55.U 

53.86 

50.61 

49.36 

47.63 

46.88 

46.36 

45.61 

57.09 

55  84 

52.59 

51.34 

4961 

48.86 

4834 

47.59 

66.91 

65.46 

62.21 

60.96 

59.23 

58.48 

57.96 

57.21 

7633 

75.08 

71.83 

70.58 

68.85 

68.10 

67.58 

66.83 

85.95 

84.70 

81.45 

80.20 

7847 

77.72 

77.20 

76.45 

95.57 

94.32 

91.07 

89.82 

88.09 

87.34 

86.82 

86.07 

Art.  1O.  By  WtieelbarrOWS.  The  cost  by  barrows  may  be  estimated  In  the  same 
manner  as  by  carts.  See  Articles  1,  Ac.  Men  in  wheeling  move  at  about  the  same  average  rate  as 
horses  do  in  hauling,  that  is,  2#  miles  an  hour,  or  200  feet  per  minute,  or  1  minute  per  every  100-feet 
length  of  lead.  The  time  occupied  in  loading,  emptying,  &c  (when,  as  is  usual,  the  wheeler  loads  his 
own  barrow,)  is  about  1.25  minutes,  without  regard" to  length  of  lead;  besides  which,  the  time  lost  in 
occasional  short  rests,  in  adjusting  the  wheeling  plank,  and  in  other  incidental  causes,  amounts  to 
about  1  part  of  his  whole  time;  so  that  we  must  in  practice  consider  him  as  actually  working  but 
9  hours  out  of  his  10  working  o^es.  at  the  rate  of  2.25  minutes  per  100  feet  of  lead.  To  find.  then, 
the  number  of  Barrow-loads  which  he  can  remove  in  a  day,  multiply  the  number  of  minutes  (600)  in 
a  working  day  by  .9:  and  divide  the  product  by  the  sum  of  1.25,  added  to  the  number  of  100-feet 
lengths  in  the  lead ;  that  is, 

The  number  of  minutes  in  a  working  day  X  .9   _  the  number  of  trips  or  of  loads 
1 .25  +  the  number  of  100- feet  lengths  of  lea d     ~    removed  per  day  per  barrow. 

Se*  Remark,  next  page. 

The  number  of  loads  divided  by  14  will  rrive  the  number  of  cub  yards,  since  a  cub  yard,  measured 
in  place,  averages  about  14  loads.  And  the  cost  of  a  tvheelrr  and  barrow  per  day.  (say  $1  per  man 
and  5  cents  per  barrow.)  divided  by  the  number  of  oub  yards,  will  give  the  cost  pt>r  yard  fjr  leading, 
wheeling,  and  emptying. 


COST   OF    EARTHWORK. 


439 


_    nmanir  cubic  yards  of  common  loam,  measured  in  place,  will  one  man  load,  wheel, 

and  empty,  per  day  of  10  working  hours,  (or  600  minutes  :)  the  lead,  or  distance  to  which  the  earth  is 
removed  being  1000  feet,  (or  10  lengths  of  100  feet ;)  and  what  will  be  the  expense  per  yard,  supposing 
the  laborer  and  barrow  to  cost  SI. 05  per  day  ? 

u<\ 

r  loads  per  day. 


And         =  3.43  cub  yd*  per  day.     And 


-  30.6  cent, 


per  cub  yard  for  loading,  wheeling  away,  emptying,  and  returning.     This  would  be  increased  almost 
inappreciably  by  the  cost  of  the  shovel,  which,  in  the  following  tables,  however,  is  included  in  the 
cose  of  tools. 
Item.  For  rock,  which  requires  more  time  for  loading,  say 

No  of  minutes  in  a  working  day  X  .9    _  Xo  of  loads  removed 
1.6  -4-  JVo  o/  100-/ee«  lengths  of  lead     ~  per  day,  per  barrow. 

Al*t.  11.  The  following  tables  are  calculated  as  in  the  case  of  carts,  by  first  finding  columns  2 
and  3  by  means  of  the  Rule  iu  Art  4,  and  then  adding  to  each  sum  in  column  3,  the  variable  quantity 
of  .1  of  a  cent  per  cubic  yard  per  100  feet  of  lead  for  keeping  the  wheeling-planks  in  order;  and  the 
prices  of  loosening,  spreading,  superintendence,  water-carrying,  &c,  per  cubic  yard,  as  given  in  th» 
preceding  Articles  2  to  7. 

By  Wheelbarrows.      Labor  $1  per  day,  of  1O  working  hours, 


3 

!l 

ft 

S* 

Common  Loam, 

Strong,  Heavy  Soils, 

«  a 

sl 

If 

li 

"Sw 

TOTAL     COST     PER     CUBIC 

TOTAL     COST     PER     CUBIC 

fcj 

"E  * 

YARD,     EXCLUSIVE     OF 

YARD,    EXCLUSIVE     OF 

c| 

f* 

>! 

PROFIT  TO  CONTRACTOR. 

PROFIT  TO  CONTRACTOR. 

•sf 

§1* 

3s 
*s$ 

tl 

*;] 

1^1 

ii 
if 

Ill 

|,1 

£33 

111 

y 

Ill 

ill 

u 

e  *  c. 

1     1 
111 

B 

j 

pi 

11 

u 

&<       OQ 

*    EC 

E    « 

£** 

CO 

* 

E     M 

E-* 

^Feet. 

Cu.Yds. 

~~Ct" 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

25 

25.7 

4.09 

10.12 

8.87 

8.42 

7.17 

11.62 

10.37 

9.12 

7.87 

50 

22.1 

4.75 

10.80 

9.55 

9.10 

7.85 

12.30 

11.05 

9.80 

8.55 

75 

19.3 

5.44 

1.52 

10.27 

9.82 

8.57 

13.02 

11  77 

10.52 

9.27 

100 

17.1 

6.14 

2.24 

10.99 

10.54 

9.29 

13.74 

12.49 

11.24 

9.99 

150 

14.0 

7.50 

3.65 

12.40 

11.95 

10.70 

15.15 

13.90 

12.65 

11.40 

200 

11.9 

8.82 

5.02 

13.77 

13.32 

12.07 

16.52 

15.27 

14.02 

12.77 

250 

10.3 

10.2 

6.45 

15.20 

14.75 

13.50 

17.95 

16.70 

1545 

14.20 

300 

9.07 

11.6 

7.90 

16.65 

16.20 

14.95 

19.40 

18.15 

16.90 

156S 

350 

8.14 

12.9 

19.25 

18.00 

17.55 

16.30 

20.75 

19.50 

1825 

17.00 

400 

7.36 

14.3 

20.70 

19.45 

19.00 

17.75 

22.20 

20.95 

19.70 

18.45 

450 

6.71 

15.6 

22.05 

20.80 

20.35 

19.10 

23.55 

22.30 

21.05 

19.80 

500 

6.17 

17.0 

23.50 

22.25 

21.80 

20.55 

25.00 

23.75 

22.50 

21.25 

600 

5.32 

19.7 

26.30 

25.05 

24.60 

23.35 

27.80 

26.55 

25.30 

24.05 

700 

4.67 

22.5 

29.20 

27.95 

27.50 

26.25 

30.70 

29.45 

28.20 

26.95 

800 

4.17 

25.2 

32.00 

30.75 

30.30 

29.05 

33.50 

32  25 

31.00 

29.75 

900 

3.76 

27.9 

34.80 

33.55 

33.10 

31.85 

36.30 

35.05 

33.80 

32.55 

1000 

3.43 

30.6 

37.60 

36.35 

35.90 

34.65 

39.10 

87.85 

36.60 

35.35 

1200 

2.91 

36.1 

43.30 

42.05 

41.60 

40.35 

44.80 

43.55 

42.30 

41.05 

1400 

2.53 

41.5 

48.90 

47.65 

47.20 

45.95 

50.40 

49.15 

47.90 

46.65 

1600 

2.24 

46.9 

54.50 

53.45 

52.80 

51.55 

56.00 

54.75 

53.50 

52.25 

1800 

2.00 

52.5 

60.30 

59.05 

58.60 

57.35 

61.80 

60.55 

59.30 

58.05 

2000 

1.81 

58.0 

6600 

64.75 

64.30 

63.05 

67.50 

66.25 

65.00 

63.75 

2200 

1.66 

63.3 

71.50 

70.25 

69.80 

68.55 

73.00 

71.75 

70.50 

69.25 

2400 

1.53 

686 

77.00 

75.75 

75.30 

74.05 

78.50 

77.25 

76.00 

74.75 

K  mile. 

1.39 

75.5 

84.14 

82.89 

82.44 

81.19 

85.64 

84.39 

83.14 

81.89 

440 


COST   OF    EARTHWORK. 


By  Wheelbarrows.      Labor  $1  per  day,  of  1O  working  boars. 


3 

I'S 

P 

If 

Pure  Stiff  Clay,  or  Ce- 
mented Gravel, 

Light  Sandy  Soils, 

If 

•2  I 

!•! 

•5  ^ 

^  if 

•2  a 

TOTAL     COST     PER    CUBIC 

TOTAL    COST     PER     CUBIC 

o'2 

«•£; 

•o  * 

YARD,     EXCLUSIVE      OF 

YARD,     EXCLUSIVE     OF 

•o^  " 

.2  * 

Nf 

PROFIT  TO  CONTRACTOR. 

PROFIT  TO  CONTRACTOR. 

2  J 

.3 

ii 

to*    . 

Ifj 

§* 
if 

Ill 

2-gl 

u 

O  88  D. 

III 

Ill 

III 

U 

M 

5 

S5~ 

I5 

75 

£   fc 

S       « 

£;* 

S   £ 

*     * 

£*g 

BTP 

Feet. 

Cu.Yds. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

25 

25.7 

4.09 

14.62 

13.37 

10.12 

8.87 

8.79 

8.04 

7.52 

6.77 

50 

22.1 

4.75 

15.30 

14.05 

10.80 

9.55 

9.47, 

8.72 

8.20 

7.45 

75 

19.3 

544 

16.02 

14.77 

11.52 

10.27 

10.19 

9.44 

8.92 

8.17 

100 

17.1 

6.14 

16.74 

15.49 

12.24 

10.99 

10.91 

10.16 

9.64 

8.89 

150 

14.0 

7.50 

18.15 

16.90 

13.65 

12.40 

12.32 

11.57 

11.05 

10.30 

200 

11.9 

8.82 

19.52 

18.27 

15.02 

13.77 

13.69 

12.94 

12.42 

11.67 

250 

10.3 

10.2 

20.95 

19.70 

16.45 

15.20 

15.12 

14.37 

13.85 

13.10 

300 

9.07 

11.6 

22.40 

21.15 

17.90 

16.65 

16.57 

15.82 

15.30 

14.55 

350 

8.14 

12.9 

23.75 

22.50 

19.25 

18.00 

17.92 

17.17 

16.65 

15.90 

400 

7.36 

14.3 

25.20 

23.95 

?070 

19.45 

19.37 

18.62 

18.10 

17.35 

450 

6.71 

15.6 

26.55 

25.30 

22.05 

20.80 

20.72 

19.97 

19.45 

18.70 

500 

6.17 

17.0 

28.00 

26.75 

23.50 

22.25 

22.17 

21.42 

20.90 

20.15 

600 

5.32 

19.7 

30.80 

29.55 

26.30 

25.05 

24.97 

24.22 

23.70 

22.95 

700 

4.67 

22.5 

33.70 

32.45 

29.20 

27.95 

27.87 

27.12 

26.60 

25.85 

800 

4.17 

25.2 

36.50 

35.25 

32.00 

30.75 

30.67 

29.92 

29.40 

28.65 

900 

3.76 

27.9 

39.30 

38.05 

34.80 

33.55 

33.47 

32.72 

32.20 

31.45 

1000 

3.43 

30.6 

42.10 

40.85 

37.60 

36.35 

36.27 

35.52 

35.00 

34.25 

1200 

2.91 

36.1 

47.80 

46.55 

43.30 

4205 

41.97 

41.22 

40.70 

39.90 

1400 

2.53 

41.5 

53.40 

52,  15 

48.90 

47.65 

4757 

46.82 

46.30 

45.55 

1600 

2.24 

46.9 

59.00 

57.75 

54.50 

53.25 

53.17 

52.42 

51.90 

51.15 

1800 

2.00 

52.5 

6480 

63.55 

60.30 

59.05 

58.97 

58.22 

57.70 

56.95 

2000 

1.81 

58.0 

70.50 

69.25 

66.00 

64.75 

64.67 

63.92 

63.40 

62.65 

2200 

1.66 

63.3 

76.00 

74.75 

71.50 

70.25 

70  17 

69.42 

68.90 

68.15 

2400 

1.53 

68.6 

81.50 

80.25 

77.00 

75.75 

75-67 

74.92 

74.40 

73.65 

%  mile. 

1.39 

75.5 

88.64 

87.39 

84.14 

82.89 

82.81 

82.06 

81.54 

80.79 

Art.  14.     Removing1    rock    excavation    by    wheelbarrows. 

A  cubic  yard  of  hard  rock,  in  place,  or  before  being  blasted,  will  weigh  about  1.8  tons,  if  sandstone 
or  conglomerate,  (150  Ibs  per  cubic  foot;)  or  2  tons  if  good  compact  granite,  gneiss,  limestone,  or 
marble,  (168  ft>s  per  cubic  foot.)  So  that,  near  enough  for  practice  in  the  case  before  us,  we  may  as- 
sume the  weight  of  any  of  them  to  be  about  1.9  tons,  or  4256  Ibs  per  cubic  yard,  in  place ;  or  158  Ibs 
per  cubic  foot. 

Now,  a  solid  cubic  yard,  when  broken  up  by  blasting  for  removal  by  wheelbarrows 
or  carts,  will  occupy  a  space  of  about  1.8,  or  l4  cubic  yards  ;  whereas  average  earth,  when  loosened, 
swells  to  but  about  1.2.  or  li  of  its  original  bulk  in  place;  although,  after  being  made  into  embank- 
ment, it  eventually  shrinks  into  less  than  its  original  bulk.  In  estimating  for  enrth.  it  is  Resumed 
that  yV  cubic  yard,  in  place,  is  a  fair  load  for  a  wheelbarrow.  Such  a  cubic  yard  will  weigh  on  an 

average  2430  B>s,  or  1.09  tons  ;  therefore,  — —  —  174  Ibs,  is  the  weight  of  a  barrow-load,  of  2.31  cubic 
feet  of  loose  earth.  Assuming  that  a  barrow  of  loose  rock  should  weigh  about  the  same  as  one  of 
earth,  we  may  take  it  at  -^  of  a  cubic  yard;  which  gives  —^-  =  177  Ibs  per  load  of  loose  rock, 

occupying  2  cubic  feet  of  space. 

In  the  following  table,  columns  2  and  3  are  prepared  on  the  same  principle  as  for  earth,  as  directed 
in  Article  4.  Column  4  is  made  up  by  adding  to  each  amount  in  column  3,  .2  of  a  cent  for  each  100 
feet  length  of  lead,  for  keeping  the  wheeling-planks  in  order;  and  45  cents  per  cubic  yard,  in  place, 
as  the  actual  cost  for  loosening,  including  tools,  drilling,  powder.  Ac:  as  well  as  moderate  drainage, 
and  everv  ordinary  contingency  not  embraced  in  column  3.  Contractor's  profits,  of  course,  are  not 
here  included. 

Ample  experience  shows  that  when  labor  is  at  $1  per  day.  the  foregoing  45  cents  per  cubic  yard,  in 
place,  is  a  sufficiently  liberal  allowance  for  loosening  hard  rock  under  all  ordinary  circumstances. 
In  practice  it  will  generally  range  between  30  and  60  cents  ;  depending  on  the  position  of  the  strata, 
hardness,  toughness,  water,  and  other  considerations.  Soft  shales,  and  other  allied  rocks,  may  fre- 
quently be  loosened  by  pick  and  plough,  as  low  as  15  to  20  cents ;  while,  on  the  other  hand,  shallow 
cuttings  of  very  tough  rock,  with  an  unfavorable  position  of  strata,  especially  in  the  bottoms  of  ex- 
cavations, may  cost  $1,  or  even  considerably  more.  These,  however,  are  exceptional  cases,  of  com- 
paratively rare  occurrence.  The  quarrying  of  average  hard  rock  requires  about  ^  to  %  Ib  of  powder 
per  cubic  yard,  in  place;  but  the  nature  of  the  rook,  the  position  of  the  strata.  &c.  may  increase  it 


>F    EARTHWORK. 


441 


to  H  B>.  or  more.  Soft  rockCr£quently  requires  more  powder  than  hard.  A  good  churn-driller  will 
drill  8  to  10  feet  in  depth,  of  holes  about  2*4  feet  deep,  and  2  inches  diameter,  per  day,  in  average 
hard  rook,  at  from  12  to  le  cents  per  toot.  Drillers  receive  higher  wages  than  common  laborers. 

Hard  Rock,  by  Wheelbarrows. 

Labor  $1  per  day,  of  10  working  hours. 


Length  of 
Lead,  or  dis- 
tance to 
which  the 
rock  is 
wheeled. 

Number  of 
cubic  yards, 
in  place, 
wheeled  per 
day  by  each 
barrow. 

Cost  per 
cubic  yard, 
in  place, 
for  loading, 
wheeling, 
and 
emptying. 

Total  cost 
per  cubic 
yard,  iu 
place,  ex- 
clusive of 
profit  to 
contractor. 

Length  of 
Lead,  or  dis- 
tance to 
which  the 
rock  is 
wheeled. 

Number  of 
cubic  yards, 
in  place, 
wheeled  per 
day  by  each 
barrow. 

Cost  per 
cubic  yard, 
in  pface, 
for  loading, 
wheeling, 

emptying. 

Total  cost 
per  cubic 
yard,  in 
place,  eX' 
elusive  of 
profit  to 
contractor. 

Feet. 

Cubic  Yds. 

Cents. 

Cents. 

Feet. 

Cubic  Yds. 

Cents. 

Cents. 

25 

12.2 

8.64 

53.7 

600 

2.96 

35.5 

81.7 

50 

10.7 

9.81 

54.9 

700 

2.62 

40.1 

86.5 

75 

9.58 

11.0 

56.2 

800 

2.34 

44.8 

91.4 

100 

8.66 

12.1 

57.3    , 

900 

2.12 

49.5 

96.3 

150 

7.26 

14.5 

59.8 

1000 

1.94 

54.1 

101.  1 

200 

6.25 

16.8 

62.2 

1200 

1.65 

63.6 

115.0 

250 

5.49 

19.1 

64.6 

1400 

1.44 

72.9 

120.7 

300 

4.89 

21  5 

67.1 

1600 

1.28 

82.2 

130.4 

350 

4.41 

23.8 

69.5 

1800 

1.15 

91.5 

140.1 

400 

4.02 

26.1 

71.9 

2000 

1.04 

100.8 

149.8 

450 

3.69 

28.5 

71  4 

2200 

.953 

110.2 

159.6 

500 

3.41 

308 

76.8 

2400 

.879 

119.5 

169.3 

Art.  15.    Removing  rock  excavation   by  carts.     A  cart-load  of 

I  rock  may  be  taken  at«i  of  a  cubic  yard,  in  place.     This  will  weigh,  on  an  average,  851  S>s ;  or  but  41 

i  Tbs  more  than  a  cart-load  of  average  soil.     Since  the  cart  itself  will  weigh  about  %  a  ton,  the  total 

;  loads  are  very  nearly  equal  in  both  cases.    Columns  2  and  3  of  the  following  table  are  prepared  on  the 

i  same  principle  as  for  earth,  as  directed  in  Art.  4.     Column  4  is  made  up  by  adding  to  each  amount  in 

|  column  3,  the  following  items  :   For  blasting,  (and  for  everything  except  those  in  column  3;  loading, 

and  repairs  of  cart-road,)  45  cents  per  cubic  yard,  in  place;  for  loading,  8  cents,  per  cubic  yard,  in 

place;  and  for  repairs  of  road,  .2,  or  i  of  a  cent  for  each  100-feet  length  of  lead.     Contractor's  profit 

not  included. 

Hard  Rock,  by  Carts. 

Labor  $1  per  day,  of  10  working  hours. 


Length  of 
Lead,  or  dis- 
tance to 
which  the 
rock  is 
hauled. 

Number  of 
cubic  yards, 
in  place, 
hauled  per 
day,  bveach 
cart. 

Cost  per 
cubic  yard, 
in  place, 
for  hauling, 
and 
emptying. 

Total  cost 
per  cubic 
yard,  in 
place,  ex- 
clusive of 
profit  to 
contractor. 

Length  of 
Lead.ordis 
tance  to 
which  the 
rock  is 
hauled. 

Number  of 
cubic  vards, 
in  place, 
hauled  per 
day,  by  each 
cart. 

Cost  pet 
cubic  vard. 
in  place,  for 
hauling, 
and 
emptying. 

Total  cost 
per  cubic 
yard,  in 
place,  ex- 
clusive of 
profit  to 
contractor. 

Feet. 

Cubic  Yds. 

Cents. 

Cents. 

Feet. 

Cubic  Yds. 

Cents. 

Cents. 

25 

19.2 

6.51 

59.6 

1800 

5.00 

250 

81.6 

50 

18.5 

677 

59.9 

1900 

4.80 

26.0 

82.8 

75 

17.8 

7.03 

60.2 

2000 

4.62 

27.1 

84.1 

100 

17.1 

7.29 

60.5 

2250 

4.21 

29.7 

87.2 

150 

16.0 

7.81 

61.1 

2500 

3-87        i 

32.3 

90.3 

200 

15.0 

8.33 

61.7 

^  mile 

3.70 

33.7 

92.0 

300 

13.3 

9.37 

63.0 

3000 

3.33 

37.5 

96.5 

400 

12.0 

10.4 

64. 

3250 

3.12 

40.1 

99.6 

500 

16.9 

11.5 

65. 

3500 

2.92 

42.8 

1028 

600 

100 

12.5 

66 

3750 

2.76 

45.3 

105.8 

700 

9.23 

13.6 

68. 

4000 

2.61 

47.9 

108.9 

800 

8.57 

14.6 

69. 

4250 

2.47 

50.6 

112.1 

900 

8.00 

15.6 

70. 

4500 

2.35 

53.2 

115.2 

1000 

7.50 

16.7 

71. 

4750 

2.24 

55.8 

118.3 

1100 

7.06 

17.7 

72.9 

5000 

2.14 

58.4 

121.4 

1200 

,       6.67 

18.7 

74.1 

Imile 

2.04 

61.2 

124.8 

1300 

6.32 

19.8 

75.4 

lYt  " 

1.67 

75.0 

141.2 

1400 

6.00 

20.8 

76.6 

I}*" 

1.41 

88.8 

157.6 

1500 

5.71 

21.9 

77.9 

1%  " 

1.22 

102.5 

174.0 

1600 

5.45 

22.9 

79.1 

2      " 

1.08 

116.3 

190.4 

1700 

5.22 

24.0 

80.4 

2K" 

.962 

130.0 

206.8 

"  I^oose  rock  "  will  cost  about  30  cts  per  yd  less;  and  even  solid  rock  will 

average  about  10  cts  less  than  the  tables. 

442 


CENTER    OF    GRAVITY. 


CENTEE  OF  GEAVITT, 


The  cen  of  4;  rav  of  a  square,  rectangle,  rhombus,  or  rhom- 
boid, is  at  the  intersection  of  its  two  diagonals. 

Of  a  circle,  ellipse,  or  regular  polygon,  in  the  center  of  the  figure. 

Of  a  triangle,  at  the  intersection  of  lines  drawn  from  any  two  angles,  to  the 
middles  of  the  sides  respectively  opposite  said  angles. 

Or,  draw  a  line  from  any  one  of  the  angles,  to  the  middle  of  the  side  opposite  said 
angle ;  the  cen  of  grav  is  in  this  line  at  ^  of  its  length  from  the  side  which  it  bisects. 

Of  either  a  trapezium,  or  a  trapezoid, 

draw  the  two  diags  a  c  and  b  d.  Div  either  of  them,  say 
a  c,  into  two  equal  parts  as  at  in.  Take  the  longest  part 
d  s,  of  the  other  diag  d.  b,  atid  set  it  off  from  6  to  n.  Join 
M  m,  and  div  it  into  3  equal  parts.  The  cen  of  grav  will  be 
at  o,  the  first  of  these  divisions  from  m. 

Of  a  trapezoid  only.  Prolong  either  parallel 
side,  as  6  a,  in  eitiier  direction,  say  toward  g ;  and  make 
a  g  equal  to  the  opposite  side  d  c.  Then  prolong  the 
other  parallel  side  d  c,  in  the  opposite  direction  ;  make 
ing  c k  equal  to  the  side  6  «.  Join  y  h.  Find  the  centre 
e  of  a  6;  and  the  center  /  of  d  c.  Join  ef.  Then  o  is 
the  ceo  of  grav  dc  +  2aft  e/ 

Or  lo  =  — — • —  X  -5 — 

dc  +  a  b 

Of  a  semicircle.    Mult  the  height  or  rad  a  b  by  .4244;  the 
prod  will  be  a  c;  and  c  is  the  cen  of  grav.  • 


Of  a  sector  of  a  circle,  adbc.  Mult  twice  the 
chord  a  6,  by  the  rad  a  c.  Div  the  prod  by  3  times  the  length 
0  a  d  b  of  the  arc  of  the  sector ;  (sec  Lengths  of  Arcs,  p  21,  23.) 
The  quot  is  c  o;  and  o  is  the  cen  of  grav. 

Of  a  quadrant,  c  o  =  c  d  X  .6002. 

Of  a  segment  of  a  circle,  nob.  Cube  the  chord 
a  6.  Div  this  cube  by  12  times  the  area  of  the  segment ;  (see 
Areas  of  Segments,  page  24.)  The  quot  will  been;  and  n  is 
the  cen  of  grav. 

Of  a  circular  arc  n  o  6,  (the  line  alone,)  not  exceed- 
ing a  semicircle.  As  the  length  of  the  arc  is  to  its  chord,  so  is  rad  c  o  to 
c  n  ;  n  being  the  cen  of  gr.  Or,  quite  approx,  mult  the  rise  «  o  by  .65  for 
•  n.  Or  more  correctlv,  if  the  rise  s  o  is  .01  of  the  chord  a  b,  or  less,  mult 
it  by  .666  for  aw;  if  '.1,  mult  by  .665;  if  15  by  .663:  if  .2.  by  .660;  if  .25 
by  .657  ;  if  .3,  by  .653  ;  if  .35,  by  .649 ;  if  .4,  by  .645  ;  if  .45,  by  .641 ;  if  .5, 
by  .637. 

Of  a  parabola  a  b  <%  at  o  in  the  axis  x  6,  $ths  of  its  length 
from  x. 

Of  a  semiparabola  a  b  x.    Atn;  on  being  3-eighths  of 
the  half  base  a  x,  and  o  x  being  fths  of  z  6. 

The  common  center  of  gravity  g,  of  two  figs  a  and 
is  fauna  u.t.s  :  First  hud  tueti   separate  cen  of  grav  «  and  I; 


of  the  two  tiga  combiued.  The  longest  division  must  evidently 
be  next  to  the  smallest  body  b.  If  the  common  cen  of  grav 
of  the  three  bodies  a.  6,  and  c.  is  required,  begin  with  any  two 
of  them  as  has  just  now  been  done.  Then  draw  gc,  and 
measure  it.  Then  suppose  the  united  areas  of  a-and  6  to  be 
placed  at  a,  as  if  one  fig  at  that  point;  and  call  it  Fig  g.  Add 
the  aroaof  this  fig  to  that  of  Fig  c;  and,  as  before,  say  as 
a  of  c  (the  greatest  of  them)  to  the  longest  di 


solid  bodies  having  wt,  tne  process  •  1  ,      C •  1  V  V~.  x.,»in  «U 

instead  of  areas.    In  either  case  it  is  immaterial  which  two  w«  begin  wit 


MECHANICS.     .FdRCE   IN   RIGID   BODIES.  443 

IK  AN  IRREGULAR  PIG  whicb^Hiay  be  divided  into  triangles,  trapeziums,  trapezoids,  Ac,  the  process 
In  the  same  as  with  the  figj^dT  &.  c,  and  d. 

The  ceil  of  Jfrav  of  any  plane  fig  may  be  found  by  drawing  it  to  a 
scale  on  pasteboariKtneti  cut  out  the  figure  ;  balance  it  in  two  or  more  positions  over  the  edge  of  a 
table,  or  on  a  shdrp  knife-edge  ;  and  mark  on  it  the  directions  of  the  edge.  Where  these  directions 
intersect  each  faher,  will  be  the  reqd  point,  near  enough  for  most  practical  purposes.  The  paper  on 
which  the  fig  is  prepared,  must  be  so  stiff  that  the  fig  will  not  bend  when  balanced. 

Of  a  cube,  parallelepiped,  cylinder,  prism,  sphere,  sphe- 
roid, ellipsoid;  the  cen  of  grav  is  in  the  cen  of  the  body. 
Of  any  frustum  of  either  a  rig-lit  or  oblique  cylinder;  or  of  a 

right  or  oblique  prism;  whether  cut  parallel  to  its  base,  or  obliquely.    In  the  center  of  the  axis  of 
the  frustum. 

Of  a  r  igli  t  pyramid,  or  cone  ;  in  its  axis,  at  i^  of  its  length  from  the  base. 

Of  any  pyramid,  or  cone  ;  whether  right,  or  oblique.  In  a  line  from  its 
Vertex,  or  top,  to  the  cen  of  grav  of  its  base  ;  and  at  34  the  length  of  this  line  from  the  base. 

Of  a  frustum  of  a  right  cone,  cut  parallel  to  its  base.  Call  the  rad 
of  the  larger  end  R;  and  that  of  the  smaller  end  r;  and  the  height  of  the  frustum  nieasd  on  its 
axis,  h.  Then. 


R2  4-  2  R  r 
X    —  .-  ---  ---  =  dist  on  axis  from  greater  end,  or  base. 

Of  a  frustum  of  a  right  pyramid,  cut  parallel  to  its  base.    Call  the 

area  of  the  large  end  A  ;  that  of  the  small  end  a  ;  and  the  height  measd  on  the  axis,  h  ;  then, 

i  axis  from  greater  end,  or  base. 
A  +  V\a  +  a 

Of  any  frustum  of  any  pyramid,  or  cone;  whether  right,  or  ob- 

lique: or  whether  the  ends  of  the  frustum  are  parallel  to  the  base  or  not;  it  is  in  a  line  to,  drawn 

between  the  centers  of  grav  of  the  two  ends  of  the  frustum  j  but  its  height  y  c  above  the  lower  end 

an,  and  perp  to  it, 

must  first  be  found 

upon    the    line    t  y, 

drawn  from  the  cen 

of  grav  of  the  small 

end,  to,  and  at  right 

angles  with,  thelarge 

end  an.  Having  first 

found    the  line   t  y, 

whether  for  a  pyramid  or  a  cone,  use  it  instead  of  the  height  ft.  in  the  preceding  formula,  for  a  frus- 

tum of  a  right  pyramid.     The  result  will  be  y  c  ;  and  by  drawing  c  a  parallel  to  a  n,  we  find  the  reqd 

point  «,  in  the  line  t  o. 

Of  the  curved  or  slanting  surface  only,  of  a  right  cone;  in 

the  axis  ;  and  at  %  of  its  length  from  the  base. 

Of  a  hemisphere  ;  in  its  axis,  at  %  of  its  length  from  the  base. 

Of  a  segment  of  a  sphere.    From  twice  the  rad  of  the  sphere,  take  the 

height  of  the  segment.  Square  the  remainder.  Then  from  3  times  the  rad  of  the  sphere,  take  the 
height  of  the  segment.  Div  the  square  just  found  by  this  last  rem  ;  and  take  %  of  the  quot,  for  the 
dist  from  the  cen  of  the  sphere  to  the  cen  of  grav  of  the  segment. 

Of  the  curved  surface  only  of  a  hemisphere;  of  a  spherical  seg- 

ment; or  of  a  zone:  at  the  middle  of  its  axis,  or  height. 

Of  a  paraboloid  :  in  its  axis,  and  at  %  of  its  length  from  the  base. 

REM,  "We  must  not  confound  cen  of  grav  with  cen  of  weiqht.  It  does  not  follow  when  a  body  bal- 
ances on  a  knife-edee,  that  ther?  is  equal  wt  on  both  sides  of  it;  but  merely  that  the  wts  of  the  several 
particles  on  one  side,  when  mult  by  their  respective  leverages,  or  dists.  at  right  angles  from  the 
knife-edge,  have  a  united  moment  about  the  knife-edge,  equal  to  that  of  the  particles  on  the  other  side 
of  it,  when  mult  by  their  leverages.  See  Leverage,  ate,  in  Force  in  Rigid  Bodies,  p  473,  &c. 


MECHANICS.    FOECE  IN  RIGID  BODIES. 


Art.  1.  Mechanics  (the  very  foundation  of  civil  engineering)  is  that  brunch 
of  science  which  treats  of  the  effects  of  force  upon  matter.  This  broad  application 
of  the  term  necessarily  includes  hydrostatics,  pneumatics,  &c;  but  ordinarily  it  is 
restricted  to  the  effects  produced  upon  matter  by  the  application  of  outward,  extrane- 
ous, or  mechanical  force,  from  whatever  source  it  maybe  derived;  whether  from 
steam,  water,  air,  wind,  gravity,  animals,  &c.  Some  of  these  effects  consist  in  either 
deranging,  or  separating  the  particles  of  matter  which  comprse  a  body,  by  pushing 
them  close  together :  or  by  pulling  them  farther  apart;  and  in  nil  such  cases  the 
extraneous  force  is  to  be  considered  only  as  acting  against  the  natural  inherent 
forces,  or  strength  of  the  materials  of  which  the  bodies  consist.  This  class  of  effects 


444  FORCE    IN    RIGID    BODIES. 

is  comprised  under  the  distinct  head  of  Strength  of  Materials.  Another  and  very 
important  class  refers  only  to  the  action  of  force  upon  entire  bodies ;  either  to  move 
them  ;  or  to  keep  them  stationary.  Force,  when  it  moves  bodies,  is  called  Dyna- 
mic force;  and  when  it  keeps  them  stationary,  Static.  Hence,  we  have  the 
sciences  of  Statics  and  Dynamics ;  which  imply  merely  the  effects  of  force 
as  to  giving  motion  to  bodies ;  or  to  keeping  them  at  rest.  See  Note  p.  459. 

In  examining  the  abstract  static  and  dynamic  effects  of  force  thus  applied  to  entire 
bodies,  we  of  Bourse  have  to  assume  that  it  does  not  break,  bend,  penetrate,  or  alter 
in  any  way,  the  shape  of  the  body.  The  body  i,s  to  be  regarded  merely  as  something 
that  receives  the  force  ;  but  is  in  no  way  affected  thereby,  further  than  that  it  moves,  or 
stands  (or  rests)  when  the  force  does  so.  That  is,  the  forces  are  assumed  to  act  upon 
each  other  only,  when  there  are  more  than  one ;  the  body  being  only  the  field  of  their 
action.  This  assumption  is  of  course  not  strictly  true  in  any  case,  because  no  body 
is  so  perfectly  rigid  as  not  to  have  its  shape  somewhat  changed  by  the  application  of 
force  ;  still,  it  will  be  seen,  as  we  proceed,  that  it  is  not  liable  to  produce  error.  Thus, 
it  will  be  evident  that  the  illustrations  of  force  in  the  following  pages  would  not 
apply  to  walls,  arches,  beams,  &c,  of  snow,  cotton,  loose  sand,  or  soft  clay ;  and  in 
the  same  manner,  although  to  a  less  extent,  would  they  be  influenced  by  the  assump- 
tion of  any  yielding  whatever  of  the  particles  composing  bodies  of  metal,  stone,  or 
wood.  Such  yielding  has  of  course  to  be  taken  into  consideration  in  nearly  all  cases  in 
practice  ;  but  it  must  be  done  by  a  totally  distinct  process,  under  the  head  of  Strength 
of  Materials.  The  two  processes  do  not  interfere  with  each  other. 

It  is  so  absolutely  essential  in  the  study  of  Statics  and  Dynamics,  and  in  reading  the  following  articles, 
to  keep  this  assumption  constantly  in  mind,  that  we  repeat  it;  namely,  that  the  force  is  to  be  con- 
sidered only  as  acting  upon  entire  bodies;  or  upon  bodies  as  a  whole;  incapable  of  being  broken, 
bent,  or  affected  by  it  in  any  manner  whatever,  othec  than  in  merely  beiug  moved:  or  kept  station- 
ary by  it.  For  instance,  if  we  wish  to  ascertain  the  effects  of  a  given  force /A  to  overturn  or  upset  a 
stone.  S,  Fig  25,  around  one  of  its  edges,  n,  we  suppose  the  stone  to  remain  whole.  But  if  we  wish  to 
know  whether  the  edge  n  will  be  in  danger  of  being  fractured  when  the  whole  weight  of  the  stone 
comes  upon  it  during  the  process  of  upsetting,  we  resort  to  the  crushing  strength  of  stone.  It  is 
plain,  that  a  body  when  pushed,  or  pulled  in  several  directions  at  once,  may  not  as  a  whole,  have 
the  slightest  tendency  to  move  in  one  direction  rather  thiui  another;  yet  some  of  its  particles  must 
tend  to  move  in  one  direction ;  and  others  in  others.  Statics  and  Dynamics  regard  the  body  only 
in  the  first  point  of  view. 

RRM  1.  When  not  otherwise  stated,  or  apparent,  the  force  in  many  of  the  following 
examples,  is  supposed  to  be  imparted  to  the  body  in  a  direction  passing  through  its 
cen  of  grav :  so  as  to  move,  pull,  or  press  it,  without  tending  to  make  it  revolve,  or 
upset.  By  examining  first  the  action  of  forces  imparted  in  that  manner,  certain 
elementary  principles  become  much  more  easy  of  comprehension. 

Rein.  2.  If  the  direction  of  tlie  imparted  force  does  not 
pass  through  the  t^en  of  j;  rav  of  a  body  free  to  move,  the  body  will  still 

move  forward  in  that  direction  just  as  far  as  before;  but  while  so  doing,  it  will  also  revolve  around 
its  ceu  of  grav  to  the  same  extent  as  if  it  did  not  move  forward  at  all,  and  as  it  would  around  a  fixed 
axis,  and  under  the  full  action  of  the  force. 

REM  .1  Lastly,  when  not  otherwise  stated,  the  force  is  supposed  to  be  applied  to 
the  rigid  body  in  a  direction  at  right  angles  to  its  surf  at  the  point  where  it  isaj)- 
plied ;  otherwise  (except  as  per  Art  19)  only  a  part  of  the  force  will  be  imparted  to 
the  body  ;  that  is,  will  be  put  into  it,  or  enter  it;  and  produce  an  effect  upon  it. 

Art  2.  Matter  is  any  substance  whatever,  as  metal,  stone,  wood,  water,  air, 
steam,  gas,  &c.  A  body  is  any  quantity  ot  matter,  as  a  piece,  a  pound,  or  a. cub 
ft,  &c,  of  it.  The  weig-ht  of  a  body  is  the  amount  of  vert  pres,  or  pull,  which  the 
force  of  gravity  in  that  body  exerts  while  the  body  is  at  rest.  In  ordinary  practical 
mechanics,  the  quantity  of  matter  in  a  body  is  measured  by  its  wt ;  for  a  method 
which  is  the  only  theoretically  correct  one,  but  which  is  not  adapted  to  popular  use, 
see  note  to  p.  455.  Motion  is  change  of  place,  or  of  position.  Since  all  bodies 
are  constantly  in  motion  in  consequence  of  the  revolution  of  the  earth,  the  ordinary 
use  of  the  word  here  refers  to  the  motion  produced  by  the  extraneous  force  under 
consideration  at  the  time. 

The  base  of  a  body,  as  ordinarily  understood,  is  that  part  of  it  which  is 
underneath,  and  upon  which  it  stands  when  acted  upon  by  its  own  wt,  or  by  the  wt  of  other  bodies  also, 
which  may  be  placed  upon  it.  But  in  fact,  a  base  may  be  at  any  part  whatever  of  a  body  ;  at  its  top, 
on  one  side,  &c ;  as  when  we  press  a  block  of  wood  upward  against  the  ceiling  of  a  room  ;  or  horizon- 
tally against  a  wall ;  the  part  which  touches  the  ceiling,  or  wall,  is  the  base. 

Strain  is  the  effect  produced  by  the  action  of  tqual  forces  or  of  equal  parts  of 
unequal  forces,  against  each  other ;  or  in  opposite  directions.  It  may  consequently 
be  applied  to  the  effect  which  these  equal  parts  or  wholes  produce  among  the  par- 
tides  of  the  body  upon  which  they  act ;  as  tending  either  to  push  them  closer  to- 
gether ;  or  to  pull  them  farther  apart ;  in  other  words,  to  crush,  or  to  rend  the  body. 
But  as  before  stated,  this  view  of  strain  belongs,  not  to  pure  Statics  and  Dynamics, 
but  to  Strength  of  Materials.  Again,  it  may  refer  to  the  action  of  the  forces  upon 
each  other;  or,  as  usually  stated,  upon  the  body  as  a  whole. 


IN    RIGID   BODIES.  445 


Thns,  if  two  men  pu)K>r  push,  with  equal  forces,  a  body  that  is  between  them,  their  forces  merely 
react  against,  count»*fDalance,  or  destroy  (see  Rem,  Art  13)  each  other;  but  they  produce  no  effect 
whatever  upon  the'cody  as  a  whole  ;  it  remains  at  rest ;  and  has  no  more  tendency  to  move  in  any 
direction,  than  if  both  forces  were  absent.  The  forces  then  merely  strain  (pull  or  push  against, 
without  moviuaf each  other,  and  keep  each  other  at  rest;  but  no  practical  error  will  arise  from 
adopting  the  common  phraseology,  and  saying  they  act  upon  the  body  to  keep  it  at  rest. 

Strain  is  measured  by  weight;  as  by  pounds,  tons,  &c.  Its  amount  or 
quantity  is  equal  to  that  of  only  one  of  the  two  equal  opposing  forces.  Thus,  if  two 
men  pull  against  each  other  at  two  ends  of  a  rope,  each  with  a  force  of  30  ft>s.  the 
strain  on  the  rope  is  but  30  fts ;  and  it  is  equal  throughout  the  length  of  the  rope. 
The  two  30  R>  forces  strain  each  other,  30  Ibs ;  as  is  made  manifest  if  a  spring-balance 
is  inserted  at  any  part  of  the  rope.  If  a  rope  passes  over  a  pulley,  and  equal  \vts  be 
suspended  at  each  end  of  it,  then  the  two  equal  forces  of  gravity  of  the  two  wts  strain 
against  each  other;  and  also  strain  the  rope  ;  to  an  amount  equal  to  one  of  them. 

For  more  on  Strain,  see  Art  13,  p  449. 

Two  equal  opposing  forces  produce  strain  and  a  tendency  to  motion,  among  the 
particles  which  compose  a  body  to  which  they  are  imparted ;  but  exert  no  tendency 
to  motion  upon  the  body  as  a  whole;  because  the  particles  are  held  together  by 
their  internal  cohesive  force ;  and  this  cohesive  force  reacts  against  our  extraneous 
imparted  forces.  If  it  is  not  sufficient  to  do  so  completely,  the  body  is  broken ;  and 
the  remaining  portion  of  the  imparted  forces,  becoming  motion,  scatters  the 
fragments  in  all  directions. 

But  the  body  as  a  whole,  is  isolated,  detached  matter,  which  is  not  adhering  to  anything  by  any 
kind  of  force.  It  in  itself  opposes  no  resistance  to  force ;  it  is  inert.  Therefore,  so  far  as  the  imparted 
forces  are  concerned  with  it,  as  a  ivhole,  to  give  it  motion  in  any  direction,  it  is  merely  a  corpse,  over 
which  two  contending  forces  are  destroying  each  other,  in  their  struggles  to  get  possession  of  it.  If 
the  contending  forces  are  equal,  they  will  strain  against  each  other,  until  they  are  mutually  destroyed  ; 
but  the  body  will  remain  unmoved  by  either.  If  they  are  unequal,  the  surviving  portion  of  the 
greater  one  will  move  on  at  a  slackened  pace,  in  its  former  course;  moving  the  unresisting  body 
along  with  it.  See  Rem  3,  &c,  of  Art  28,  p  458. 

Art.  3.  Force  is  that  principle  of  which,  considered  simply  as  a  mechanical 
agent,  we  know  but  little  more  than  that  when  it  is  imparted,  that  is,  put  into,  a  body, 
it  produces  either  motion  alone  ;  or  strain,  with  or  without  motion.  This  is  all  that 
force,  mechanically  considered,  can  do  under  any  circumstances  whatever.  When  it 
produces  motion  alone,  it  is  called  moving  force.  When  it  produces  strain  alone,  or 
without  motion,  straining  force ;  or  simply  strain,  pull,  push,  tension,  or  pres,as  the 
case  may  be.  As,  for  instance,  when  a  body  rests  quietly  on  a  table,  or  on  a  post,  Ac; 
or  is  suspended  by  a  chain,  rope,  &c ;  its  effect  is  simply  strain ;  a  push  or  pres  on 
the  post  or  table,  and  a  pull  or  tension,  on  the  rope.  There  can  be  no  strain  except 
where  there  are  at  least  two  forces  to  strain  against  each  other.  When  it  produces 
strain  and  motion  at  the  same  time ;  or  in  other  words,  when  a  part  of  it  produces 
strain ;  and  a  part  of  it,  motion  ;  it  is  called  working  force  ;  and  its  effect  is  work. 
Thus,  when  a  man  is  lifting  a  wt,  a  part  of  his  force  is  straining  against  the  equal 
force  of  gravity  of  the  wt;  while  the  other  part  of  his  force  is  giving  motion  to  the 
wt ;  so  that  all  his  force  combined,  is  at  work ;  or  is  working  force ;  or  performs 
work.  See  Art  11,  p  448.  A  straining  force  is  often  called  a  stress. 

These  are  not  different  kinds  of  force ;  but  diff  effects  of  the  same  force ;  for  there  are  no  diff  kinds  of 
mechanical  force;  it  is  all  the  same:  no  matter  from  what  source  it  may  be  derived.  Indeed,  we 
might  generally,  and  without  fear  of  being  misunderstood,  call  it  simply  motion,  work,  or  strain, 
according  to  which  of  the  three  effects  it  is  producing  at  the  time;  and  we  shall  frequently  do  so 
in  the  following  pages.  The  only  means  we  have  of  measuring  it,  is  by  measuring  the  quantity  of 
these  its  three  effects. 

Not  hint?  font  force  can  resist  force.   When  all  the  force  imparted,  or 

put  into,  a  body,  is  unresisted  by  opposing  force ;  it  produces  motion  only  ;  when  it  is  all  resisted,  it 
produces  strain  only ;  when  a  part  of  it  is  unresisted,  and  a  part  resisted,  the  first  part  produces 
motion;  and  the  last  part  strain  ;  which  two  combined  constitute  work.  A  single  force  can  produce 
motion ;  but  at  least  two  are  required  to  produce  strain.  See  Art.  13,  p  449. 

Force  is  sometimes  defined  to  be  that  principle  which  either  does  produce,  or  tends  to  produce 
motion;  or  which  either  does  prevent,  or  tends  to  prevent  it.  Friction,  however,  which  is  classed 
among  forces,  always  tends  to  prevent  motion  ;  therefore,  when  the  word  force  is  used  alone,  it  must 
be  remembered  that  friction  is  not  included. 

REM  1.  We  infer  from  both  reason  and  observation,  that  force,  when  once  imparted, 
or  put  into  a  body,  would,  if  not  resisted  by  other  force,  remain  in  it,  and  move  it 
forever,  in  a  straight  line;  in  the  direction  in  which  the  force  was  imparted;  and  at 
a  uniform  velocity,  or  rate  of  speed. 

For  we  observe  in  practice,  that  motion  continues  longer  in  proportion  as  we  remove  resisting 
forces ;  hence  we  infer,  that  if  all  resistances  could  be  removed,  it  would  continue  forever.  But  in 
practice  we  cannot  remove  all  resisting  forces,  such  as  those  of  gravity,  wt,  friction,  the  air,  Ac. 
These  strain  against,  or  resist  the  forces  which  we  may  impart;  so  that  with  the  exception  of  a 
body  falling  by  the  force  of  gravity,  in  a  vacuum,  it  is  difficult  to  mention  a  case  of  motion  alone, 
without  strain. 

REM  2.  Matter,  in  itself,  cannot  resist  force.  See  Arts  5  and  11.  When 
force  produces  motion  alone,  without  strain,  it  is  often  called  Dynamic  force ;  when 

29 


446  FORCE   IN    RIGID   BODIES. 

continuous  strain  without  motion.  Static ;  when  a  sudden  strain  for  an  instant  only, 
as  a  blow  of  a  hammer,  Impulsive;  and  its  action,  an  impulse.   See  Art  9. 

REM  3.  The  Living  Force,  or  vis  viva  of  scientific  writers,  is  nothing  more 
than  an  expression  referring  to  the  quantity  of  work  (motion  and  strain  combined) 
which  the  force  in  a  body  at  any  given  instant,  could  perform,  if  left  to  itself,  with- 
out afterward  receiving  anjr  additional  force.  See  Art.  24,  p  455. 

Art.  4.  When  force  is  imparted  in  any  direction  which  passes  through  the  cen 
of  grav  of  a  homogeneous  rigid  body,  perfectly  free  to  move,  it  quickly  diffuses  itself 
equally  through  every  part ;  and  gives  to  every  atom  composing  the  body,  the  same 
tendency  to  move  onward  in  the  same  direction. 

But  although  this  diffusion  takes  place  rapidly,  especially  in  compact  bodies,  still 
it  requires  some  time ;  generally  so  short  as  not  to  be  appreciable  without  close  ob- 
servation.   If  we  put  the  body  B,  Fig  1,  into  motion  by  striking  it  a  blow  near  the  top 
D  or  bottom,  the  struck  end  will  for  an  instant  move  faster  than  the  other;  after  which 

D  both  will  move  with  equal  velocity ;  because  the  great  force  which  for  au  infinitely 

short  time  acted  upon  the  top  only,  becomes  equally  diffused  throughout  the  body. 
We  are  not,  however,  now  treating  of  cases  of  this  kind,  in  which  the  direction  of  the 
force  does  not  pass  through  the  cen  of  grav  of  the  body ;  for  the  body  may  then  ro- 
tate, or  whirl  around  as  it  moves  forward. 


•7.  This  equal  diffusion  of  force  is  owing  to  the  inherent  cohesive  force  of  bodies,  which 

-I- 111  1        holds  their  atoms  together,  somewhat  as  lime  holds  together  the  grains  of  sand  in  a 
</  piece  of  mortar;  and  will  not  permit  one  grain  to  move  unless  the  others  move  with 

it.  In  producing  this  diffusion,  the  cohesive  forces,  or  the  lime,  do  not  diminish  the 
quantity  of  the  imparted  force ;  but  merely  cause  that  instead  of  continuing  to  give  a  great  velocity 
to  that  part  only  at  which  it  was  imparted,  it  shall  give  a  slower,  but  equal  one,  to  every  part. 

When  a  body  breaks  on  the  application  of  outward  force,  it  indicates  that  the  cohesive  force,  or  ce- 
ment, was  not  sufficient  to  bear  the  rapid  passage  through  it  of  the  force,  from  one  part  of  the  body 
to  another ;  therefore,  nearly  all  the  force  continues  in  the  part  to  which  it  was  imparted,  and  moves 
it  away  from  the  remainder  with  great  vel.  See  Art  27,  p  456. 

But  if  the  rigid  body  to  which  we  impart  force  in  any  direction,  ig  not  free  to  move,  being  prevented 
from  so  doing  by  opposing  force,  then  our  imparted  force  strains  against  the  opposing  one ;  and  thus 
produces  strain,  instead  of  motion,  in  the  body. 

Art.  5.  We  must  clearly  distinguish  between  merely  mov- 
ing a  foody,  and  lifting  it.  The  smallest  imaginable  force  would,  if  un re- 
sisted by  other  force,  move  the  greatest  imaginable  body;  but  if  we  apply  force  to 
lift  a  body,  then  its  wt,  or  force  of  gravity,  resists.  Force  applied  merely  to  move 
things  without  lifting  them,  is  often  called  traction ;  as  in  the  case  of  horses  or  men 
towing  a  canal  boat,  or  hauling  a  vehicle  along  level  ground.  The  same  word,  how- 
ever, is  generally  used  for  hauling,  even  when  the  tiling  is  partially  lifted ;  as  when 
a  horse  is  hauling  up  hill ;  in  which  case  he  is  lifting  both  himself  and  the  vehicle 
with  its  load.  See  Traction,  pp  603,  605. 

Matter  always  appears  to  resist  force ;  but  this  deception  is  caused  by  its  gravity,  or  by  friction,  or 
some  other  resisting  force  acting  upon  it  at  the  same  instant. 

A  man  cannot  lift  a  wt  of  20  tons  ;  but  if  it  be  placed  nor  upon  proper  friction  rollers,  he  can  move 
it,  as  we  see  in  some  drawbridges,  turntables,  &c ;  and  if  friction  and  the  air  could  be  entirely  re- 
moved, he  could  move  it  by  a  single  breath  ;  and  it  would  continue  to  move  forever.  It  would,  how- 
ever, move  very  slowly,  because  the  force  of  the  single. breath  would  have  to  diffuse  itself  among  20 
tons  of  matter.  He  can  move  it,  if  it  be  placed  in  a  suitable  vessel  in  water,  or  if  suspended  from  a 
long  rope.  A  powerful  locomotive  that  may  move  2000  tons,  cannot  lift  10  tons  vertically. 

If  we  imagine  two  bodies,  each  as  large  and  heavy  as  the  earth,  to  be  precisely  balanced  in  a  pair 
of  scales  without  friction,  it  is  plain  that  since  all  of  their  two  opposing  gravities  strain  against  each 
other,  and  thus  neutralize  each  other's  motions,  a  single  grain  of  sand  added  to  either  scale-pan, 
would  give  motion  to  both  bodies.  It  would  of  course  be  very  slow,  but  still  it  would  be  in  exactly 
the  same  proportion  that  the  wt  of  the  grain  of  sand  bears  to  that  of  the  two  earths ;  or  in  exact  pro- 
portion to  the  unresisted  force  of  gravity  contained  in  the  grain.  If  two  grains  were'  added  instead 
of  one,  the  two  earths  would  move  twice  as  fast. 

The  inherent  cohesive  force  of  matter,  by  which  its  particles  are  held  in  close  union,  frequently 
causes  the  matter  itself  to  appear  to  resist  straining  force;  thus,  a  cake  of  ice  may  sustain  a  great 
pres;  but  if  we  destroy  its  cohesive  force  by  converting  it  into  water,  it  will  yield  readily.  So  with 
the  metals  and  stones  if  reduced  to  dust,  it  is  not  the  material  that  resists  being  broken  ;  but  the 
inherent  cohesive  force  which  holds  the  particles  of  the  material  firmly  together. 

The  fact  that  matter  in  itself  opposes  no  resistance  to  force, 
is  called  its  inertia,  or  inertness;  and  since  nothing  but  force  can  resist 
force,  it  follows  that  motion  (which  under  all  circumstances  is  the  effect  of  the  un- 
resisted  portion  of  force  in  a  body)  cannot  be  the  result  of  an  effort  of  that  force ; 
but  is  simply  the  manner  in  which  said  force  exhibits  itself,  or  indicates  its  presence. 
It  is  as  it  were  a  notice  hung  out  on  the  moving  body,  saying,  "  Unemployed  force 
stored  up  here;  work  wanted." 

Gravity  acting  in  a  body  falling  freely  in  a  vacuum,  and  consequently  unresisted,  exerts  no  effort 
upon  it ;  it  neither  goes  before,  and  pulls  it  along,  nor  behind,  and  pushes  it ;  for  there  can  be  no  pull 
or  push  except  when  there  is  some  force  to  pull  or  push  against.  But  it  simply  as  it  were  animates 
the  body,  or  endows  it  with  the  power  of  locomotion.  As  the  body  falls,  the  force  of  gravity  which 
gives  it  motion  all  remains  unimpaired,  and  stored  up  in  it,  ready  to  exert  an  effort  against  any  other 
force  which  it  may  chance  to  meet  with.  Therefore  a  body  falling  unresistedly,  has  no  weight;  for 
gravity,  which  gives  it  weight  alone  while  at  rest,  now  gives  it  motion  alone. 

In  most  kinds  of  machinery,  nearly  all  the  parts  are  supported  by  either  pivots 


FORCE   IN    RIGID   BODIES.  447 

or  gudgeon^which  resist  the  action  of  gravity  of  the  parts ;  so  that  the  working 
force,  or  jfower,  has  only  to  overcome  the  friction ;  and  not  to  lift  the  weight  of  the 
parts.  The  expression,  '"overcoming1  tlie  inertia  of  a  body."  is  founded 
upon  radical  error ;  for  thers  is  no  such  resistance  as  inertia  to  be  overcome.  Inertia 
is  the  name  of  a  fact ;  not  of  a  force.;  and  that  fact  is  simply,  that  matter  does  not 
resist  force.  What  is  called  overcoming  inertia,  is  simply  putting  in  force ;  as  when 
we  pour  water  into  a  glass,  we  do  not  overcome  any  resistance  of  the  glass  against 
being  filled  ;  we  simply  apply  the  water  to  the  glass,  which  receives  it,  and  retains 
it;  until,  by  applying  force,  we  take  it  out  again.  "Overcoming  the  inertia"  of  a 
railroad  train,  &c,  therefore,  means  nothing  more  than  that  if  we  wish  the  train  to 
move,  we  must  put  moving  force  into  it.  It  is  inert,  and  cannot  move  of  itself.  See 
Art  20,  p  453. 

Art.  6.  Care  must  be  taken  not  to  confound  motion  with  vel.  Velocity  is 
tlie  rate  of  speed  of  motion ;  or,  in  other  words,  it  is  the  dist  moved  over  in 
a  given  time,  without  any  regard  to  the  wt  of  the  body.  Thus  we  say  that  it  moves 
with  a  vel,  or  at  the  rate,  of  10  ft  per  sec ;  20  miles  per  hour,  &c ;  whether  it  weighs 
an  ounce  or  a  ton.  The  velocity  of  a  body  is  always  in  direct  proportion  to  the 
imparted  unresisted  force ;  and  in  inverse  proportion  to  the  wt  of  the  body ;  that  is, 
two  or  three  times  the  unresisted  force  will  give  to  the  same  body  two  or  three  times 
the  vel ;  or  if  one  body  has  two  or  three  times  the  wt  of  another,  the  same  amount 
of  unresisted  force  will  give  it  but  %OY%  the  vel.* 

Art.  7.  We  impart  force  to  a  body,  by  applying*  to  it  (that  is,  by 
bringing  into  close  contact  with  it)  a  second  body  in  which  the  force  already  is.  We 
must  discriminate  between  applied,  and  imparted  force;  and  the  precaution  is  the 
more  necessary  because  writers  generally  appear  not  to  recognize  the  distinction. 

Applied  force  is  merely  carried  to  a  body  ;  imparted  force  is  put  into  it.  A  heavy  body  containing 
great  force  may  be  applied  to  a  small  body  containing  none ;  and  yet  the  former  may  impart  but 
little  of  its  force  to  the  latter.  If  we  place  a  body  on,  that  is.  apply  it  to,  a  steep  inclined  plane,  we 
apply  also  the  wt  of  the  body,  or  its  force  of  gravity  ;  yet  only  a  portion  of  this  force  is  imparted,  or 
put  into  the  plane ;  while  another  portion  remains  in  the  body ;  and  being  unresisted  by  the  plane, 
produces  the  motion  of  sliding  down  it.  Force  may  be  applied  to  a  body  in  any  direction  whatever; 
but  can  be  imparted  to  it  only  at  right  angles  to  the  surface,  except  as  explained  Art  19. 

That  point  of  a  body  to  which  force  is  applied,  is  called  the  point  of  application ;  but  in  fact  we 
cannot  in  practice  apply  force  to  a  point,  according  to  the  scientific  meaning  of  that  word  ;  but  have 
to  apply  it  distributed  over  an  appreciable  area  'sometime!?  very  large)  of  the  surface  of  the  body; 
still,  as  explained  in  Art  57,  it  may,  so  far  as  regards  its  action  upon  the  entire  mass  of  a  rigid  body, 
be  considered  as  applied  at  one  point.  The  expression,  direction  of  a  force,  may  refer  either  to  the 
direction  in  which  it  is  applied,  or  to  that  in  which  it  is  imparted  ;  or,  in  a  word",  to  that  direction, 
•whatever  it  may  be,  in  which  it  may  happen  to  be  acting  at  the  moment  under  consideration. 

Art.  8.  When  diff  forces  act  upon  a  body,  it  is  absolutely  essential,  in  consid- 
ering their  effects  upon  it,  to  know  whether  they  all  act  in  the  same  plane ; 

for  if  they  do  not,  their  effects  become  totally  diff. 

A  flat  piece  of  paper  is  a  plane,  and  if  on  it  we  draw  any 
number  of  straight  lines,  in  any  direction  whatever,  they  will 
represent  so  many  forces  all  acting  in  that  same  plane  ;  that 
is,  the  same  flat  surface  coincides  with  the  directions  of  all  of 
them.  It  will  evidently  do  the  same  in  whatever  position  this 
plane  surf  may  be  placed,  whether  nor,  vert,  or  inclined. 
Straight  lines  drawn  on  the  floor  of  a  room,  will  represent 
forces  in  that  same  plane :  lines  on  the  ceiling,  forces  in  that 
same  plane;  which  of  course  is  not  the  same  plane  of  those 
on  the  floor;  so  with  lines  on  the  sides  of  the  room.  All  the 
lines  ot,  it,  at.  ct.  Pig  2,  are  in  the  same  plane  toic.  Although 
1 1  is  in  the  plane  toic:  and  t  e  at  the  same  time  in  the  plane 
tcge,  and  in  the  plane  tone;  and  se  in  the  plane  sneg;  and 
ta  in  the  plane  ites,  still  all  of  these,  namely,  it,  te.  se,  and 
ts,  are  evidently  in  the  same  plane  ites.  Any  two  lines  which  meet,  or  would  meet  or  intersect  each 
other  if  sufficiently  extended  in  either  direction,  are  in  the  same  plane;  as  ot,  it;  or  st,  it;  or  ts, 
es.  Still  two  lines  may  be  in  the  same  plane,  and  yet 'not  meet  if  extended;  as,  for  instance,  the 
parallel  lines  c  t,  g  e.  in  the  plane  tcge.  The  lines  a  t  and  g  e,  being  in  parallel  planes,  oict  and 
n  s  e  g,  cannot  meet  if  extended. 

REM.  We  must  not  confound  acting  in,  with  acting  on,  upon,  or  against  the  same  plane.  The 
floor  of  a  room  is  a  plane ;  and  upon  or  against  that  same  plane,  forces  in  a  thousand  diff  planes  may 
act.  The  distinction  is  so  self-evident,  that  a  bare  allusion  to  it  will  prevent  mistake.  See  Fig  53}$, 
Art  55,  p  481. 


*  Angular  Vel.  When  a  body  revolves  around  any  center,  it  is  plain  that  the  parts  which 
are  farthest  from  said,  center,  will  move  faster  than  those  nearer  to  it.  Therefore,  we  cannot  assign 
a  stated  vel  in  ft  per  sec,  or  miles  per  hour,  that  shall  apply  to  every  part  of  a  revolving  body.  But 
it  is  equally  plain  that  every  part  of  the  body  revolves  around  an  entire  circle  of  360°  of  angle,  in  the 
same  time.  Hence,  in  many  scientific  investigations,  the  number  of  degrees  per  sec  is  used,  (instead 
of  feet  per  sec,)  to  denote  the  vel  of  such  a  body  ;  and  is  called  its  angular  vel.  The  bub  of  a  wheel 
has  the  same  anqular  vel  as  the  tire ;  because  both  move  through  the  same  number  of  degrees  per 
sec ;  but  the  tire  has  a  greater  linear  vel  than  the  hub,  because  it  moves  through  a  greater  number 
of  feet  per  sec. 


448 


FORCE   IN    RIGID   BODIES. 


Art.  9.  The  quantity,  or  amount,  of  any  motion,  of  any  body,  is 
measd  by  mult  the  wt  of  the  body  by  the  dist  through  which  it  is  moved;  thus,  if  a 
body  of  4  fts  is  moved  3  ft ;  or  one  of  6  fts,  2  ft ;  or  one  of  120  fts,  y1^  of  a  ft ;  there 
is  in  each  c;ise  an  entire  motion  of  12  foot-pounds.  If  the  body  is  lifted,  there  still 
is  a  motion  of  12  ft -fts ;  but  there  is  also  work  of  12  ft-fts  performed.  See  Art  11. 
We  remind  the  student  that  he  must  carefully  avoid  confounding  motion  alone,  with 
motion  and  strain  combined;  that  is,  with  work.  Motion  alone  implies  that  the  force 
which  produces  it,  has  no  resistance  to  overcome.  It  merely  causes  the  body  to 
move,  instead  of  standing  still.  The  identical  force,  or  portion  of  a  force,  which  is 
producing  motion,  cannot  at  the  same  time  be  doing  anything  else. 

The  quantity  of  moving1  force  in  a  body,  is  estimated  or  measured, 
by  mult  the  wt  of  the  body  by  its  vel,  or  rate  of  motion ;  that  is,  by  the  dist  to 
which  the  force  could  move  it  in  a  certain  given  time;  as  in  one  sec,  hour,  &c. 

This  is  rendered  necessary  by  the  fact  that  the  force  would,  if  uuresisted,  move  the  body  for  ever, 
(see  Rem  1,  Art  3 ;)  hence  we  could  not  assign  to  it  any  total  dist.  If  a  body  of  4  fts  is  moving  with 
a  vel  of  3  ft  per  sec,  we  say  it  has  a  motion,  or  has  in  it  a  moving  force  of  4  X  3  :=  12  foot- fts  per  sec; 
that  is.  it  has  in  it  an  amount  of  force  sufficient,  if  unresisted  by  other  force,  to  keep  in  motion  for 
ever,  4  fts  at  3  ft  per  sec  ;  or  6  fts  at  2  ft ;  or  12  fts  at  1  ft  per  sec ;  or  any  other  number  of  fts,  at  any 
other  number  of  ft  per  sec,  provided  the  two  numbers  when  mult  together  make  12.  The  quantities 
of  motion  per  sec,  or  of  moving  force,  in  all  these  cases,  is  the  same;  although  the  vels  are  diff.  It 
never  happens  in  practice  that  all  the  force  imparted  to  a  body,  acts  as  motion  alone;  for  some  part 
of  it  has  always  to  act  as  strain  against  some  opposing  force  which  we  cannot  get  rid  of;  as  gravity, 
friction,  the  resistance  of  the  air,  &c;  and  since  these  do  not  cease  their  opposition,  they  gradually 
resist,  and  destroy  all  the  moving  force  also;  and  the  body  therefore  finally  comes  to  rest.  If  at  any 
moment  before  it  finally  stopped,  all  resistance  could  be  removed,  it  would  move  on  for  ever,  with  the 
vel  it  had  at  that  moment. 

Momentum  is  merely  another  name  for  moving  force.  Since  the  same  force 
that  will  move  a  body  of  I  ft  with  a  vel  of  1000  ft  per  sec,  will  impart  to  a  body 
of  1000  fts  a  vel  of  1  foot  per  sec :  and  since  the  quantity  of  motion  given  by  the 
force  is  the  same  in  both  cases,  it  follows  that  a  body  of  1000  fts  resists  moving  force 
no  more  than  does  a  body  of  1  ft.  Hence,  matter,  as  before  remarked,  does  not  re- 
sist moving  force  at  all;  but  any  given  amount  of  force,  unresisted  by  other  force, 
will  impart  to  an  infinitely  great  body  precisely  the  same  amount  of  motion  as  to  an 
infinitely  small  one ;  but  at  diff  vels. 

Art.  10.  An  impulse,  impact,  blow,  stroke,  or  collision,  is  when  force,  be- 
fore engaged  in  moving  a  body,  suddenly  encounters  opposing  force  in  another 
body ;  and  thus  instantly  becomes  changed,  either  wholly  or  in  part,  from  motion, 
into  strain. 

It  is  the  same  identical  force  as  before,  but  is  engaged  in  producing  a  diff  effect ;  in  the  first  case 
this  effect  was  motion;  in  the  second,  it  is  a  blow;  or  pres,  or  push,  for  an  instant.  We  cannot 
actually  measure  or  estimate  an  impulse  by  fts,  tons,  &c;  that  is,  we  cannot  assign  any  number 
of  pounds,  tons,  &c,  of  quiet  pres  or  pull,  which  would  produce  an  effect  equal  to  what  we  call  the 
force  of  the  blow.  Hence,  those  writers  are  in  error  who  give  rules  for  finding  the  number  of  tons 
of  force  with  which  a  pile-hammer  strikes  the  head  of  a  pile.  We  can  only  say  that  the  force  of 
the  stroke  is  that  due  to  a  moving  force,  or  momentum  of  so  many  foot-lbs  per  sec  ;  and  this  serves 
no  other  purpose  than  to  indicate  that  if  two  rigid  bodies,  whether  of  equal  or  unequal  wts,  have 
in  them  the  same  quantity  of  moving  force,  they  will  strike  equally  hard  blows;  but  how  hard  we 
cannot  say.  Thus,  if  a  body  of  100  fts  has  a  vel  of  3  ft  per  sec ;  and  another  body  of  5  fts  has  a 
vel  of  60  ft  per  sec,  each  has  a  moving  force  of  300  ft-fts  per  sec;  and  if  they  meet  each  other  from 
opposite  directions,  they  will  mutually  destroy  each  other's  moving  forces,  and  bring  each  other  sud- 
denly to  rest;  their  momentums,  moving  forces,  or  blow  striJting  capabilities  being  equal.  The  terms 
moving  force,  and  momentum,  may  be,  and  generally  are,  used  indifferently. 

"In  some  careful  experiments  made  at  Portsmouth  dock-yard,  England, a  man  of 
medium  strength,  and  striking  with  a  maul  weighing  38  fts,  the  handle  of  which  was 
44  ins  long,  barely  started  a  bolt  about  %  of  an  inch  at  each  blow ;  and  it  required  a 

Suiet  pres  of  107  tons  to  press  the  bolt  down  the  same  quantity ;  but  a  small  addi- 
onal  weight  pressed  it  completely  home." 

Art.  11.  Working-  force.  When  the  force  imparted  to  a  body,  not  only 
moves  it,  but  does  so  in  spite  of  resisting  forces;  or  in  other  words,  when  one  part 
of  it  strains  against,  and  balances  the  resistances  equal  to  itself;  while  the  other 
part  of  it  moves  the  body,  it  is  called  working  force.  The  quantity  of  working  force 
expended;  as  well  as  the  quantity  of  work  thereby  accomplished,  is  estimated  by 
mult  the  resistance  overcome,  (which  frequently  consists  of  wt  lifted,)  by  the  dist 
through  which  it  is  overcome.  Thus,  if  the  resistance  be  a  friction  of  4  fts,  over- 
come at  every  point  along  a  dist  of  3  ft;  or  if  it  be  a  wt  of  4  fts,  lifted  3  feet  high, 
then  either  the  working  force,  or  the  work  done  by  it,  amounts  to  4  X  3  =  12  ft-fts. 
The  lifting  of  wts;  and  the  friction  encountered  in  merely  moving  them  by  sliding, 
or  rolling,  constitute  the  principal  sources  of  resistance,  and  work,  in  practice.  The 
quantity  of  any  work  may  evidently  be  considered  by  itself,  without  regard  to  the 
time  reqd  to  perform  it ;  but  (as  in  motion)  we  generally  require  to  know  the  rate  at 
which  work  can  be  done;  that  is,  how  much  can  be  done  within  a  certain  time. 


FORCJXfN    RIGID   BODIES.  449 

Thus,  in  selecting  a  sterna-engine,  it  is  important  to  know  how  much  it  can  do  per 
minute,  hour,  or  dayx'We  therefore  stipulate  that  it  shall  be  of  so  many  horse- 
powers ;  which  mefrns  nothing  more  than  it  shall  be  capable  of  overcoming  resisting 
forces,  at  the  rjkte  of  so  many  times  33000  ft-fbs  per  min.  See  Arts  22  to  25. 

"Working  fonyr'is  often  called  power;  horse,  steam,  water  power,  &c.  Since  work,  in  a  strictly 
scientific  sens^ involves  motion  and  strain  combined,  a  man  who  is  standing  still  is  not  considered  to 
be  working,  any  more  than  a  post  or  a  rope  sustaining  a  heavy  load  ;  although  he  may  be  supporting 
uu  oppressive  burden ;  or  holding  a  car-brake  with  all  his  strength ;  for  iu  both  cases  his  force  is 
strain  only.  The  work  done  by  a  horse  drawing  a  heavy  load  on  a  level  road,  consists  entirely  in 
overcoming  the  friction  at  the  axles  and  rims  of  the  wheels  ;  but  in  drawing  it  itp  hill,  he  partly  'lifts 
the  load,  and  himself  ft\so.  In  the  first  case  his  work  (scientifically)  is  not  so  much  wt  of  load  moved 
a  certain  dist ;  but  so  many  pounds  of  rolling  and  axle-friction  overcome  through  said  dist.  Up  hill, 

the  wt  of  the  load,  vehicle,  and  himself,  lifted  to  another  dist ;  namely,  the  vert  height  of  the  hill.  Ou 
a  perfect  level  road,  and  without  friction,  the  horse  would  encounter  no  resistance  in  hauling  his 
load,  however  great  it  might  be.  He  would  have  nothing  to  do  but  impart  motion  to  unresisting 
matter.  Therefore,  in  such  a  case  his  hauling  would  not  be  work,  in  the  scientific  sense. 

Tlie  ordinary  unit  of  work,  or  standard  for  measuring  it,  is  one  R>  of  wt 

lifted  one  foot  high ;  or  one  ft>  of  any  kind  of  resistance,  overcome  in  any  direction 
whatever,  for  the  dist  of  one  foot ;  and  is  called  a,  font-pound.  Or,  when  more  conve- 
nient, we  may  use  foot-tons,  &c,  &c ;  as  we  use  a  two-ft  rule,  a  yardstick,  or  a  tape- 
line,  as  may  best  suit  our  purpose.  The  unit  of  KATE  of  work,  or  the  quan- 
tity done  in  a  given  time,  is  oneft-lb  per  sec. 

The  same  quantity  of  force  which  will  overcome  a  given  resistance  through  a  given  dist.  in  a  given 
time,  will  also  overcome  any  other  resistance  through  any  other  dist,  in  that  same  time,  provided 
the  resistance  and  dist  when  mult  together  give  the  same  amount  as  in  the  first  case.  Thus,  the 
force  that  will  lift  50  fts  through  10  ft  in  a  sec,  will  lift  500  fts,  1  ft;  or  25  fts,  20  ft;  or  5COO  fts 
•j-1^-  of  a  ft  in  a  sec ;  and  in  all  these  cases  the  amount  of  force  expended,  as  well  as  of  work  done,  is 
precisely  the  same.  In  practice,  the  adjustment  of  the  speed  to  suit  diff  amounts  of  resistance,  is 
usually  effected  by  the  medium  of  cog-wheels,  belts,  or  levers.  By  means  of  these,  the  engine, 
water-wheel,  horse,  or  other  motive  power,  may  be  made  to  overcome  small  resistances  rapidly  ;  or 
great  ones  slowly,  by  the  same  working  force. 

For  more  on  working  force,  see  Art  17,  18, 19,  &c.     Also  Note,  p  459. 

Art.  12.  When  vel  undergoes  no  change,  it  is  said  to  be  uniform  ;  so  with 
force,  motion,  strain,  and  work.  \Vhen  any  of  them  becornei  grachially  greater,  it  is 
said  to  be  accelerated;  when  gradually  less,  retarded.  If  the  acceleration,  or  retarda- 
tion is  in  exact  proportion  to  the  time ;  that  is,  when  during  any  and  every  equal 
interval  of  time,  the  same  degree  of  change  takes  place,  it  is  uniformly  accelerated, 
or  retarded.  \Vhen  otherwise,  the  words  variable,  and  variably  are  used.  Confusion 
arises  from  the  frequent  use  by  even  the  best  writers,  of  the  words  "  constant,"  and 
"  uniform,"  instead  of  uniformly  accelerating  force. 

Although  the  expressions  are  strictly  correct,  still  they  are  inexpedient ;  for  when  causes  and  effects 
are  so  intimately  connected,  as  moving  force  and  motion,  it  is  desirable  that  the  same  adjectives 
should  be  equally  applicable  to  both  ;  and  we  should  not  in  the  same  sentence  read  of  constant  or 
uniform  force,  producing  inconstant/or  uuuniform  motion. 

Gravity  is  a  uniformly  accelerating  force  when  it  acts  upon  a  body  falling  freely  ;  for  it  then  increases 
the  vel  at  the  uniform  rate  of  .322  of  a  foot  during  every  hundredtlTpart  of  a  sec  ;'  or  32.2  ft  in  every  sec. 
Also  when  it  acts  upon  a  body  moving  down  an  inclined  plane ;  although  in  this  case  the  increase 
is  not  so  rapid,  because  it  is  caused  by  only  a  part  of  the  gravity ;  while  another  part  presses  the 
body  to  the  plane;  and  a  third  part  overcomes  the  friction.  It  is  a  uniformly  retarding  force,  upon 
a  body  thrown  vert  upward;  for  no  matter  what  may  be  the  vel  of  the  body  when  projected  upward 
it  will  be  diminished  .322  of  a  foot  in  each  hundredth  part  of  a  sec  during  its  rise  ;  or  32  2  ft  during 
each  entire  sec.  At  least,  such  would  be  the  case  were  it  not  for  the  varying  resistance  of  the  nir 
at  diff  vels.  It  is  a  uniformly  straining  force  when  it  causes  a  body  at  rest,  to  press  upon  another 
body  ;  or  to  pull  upon  a  string  by  which  it  is  suspended.  The  fore'going  expressions,  like  those  of 
momentum,  strain,  push.  pull,  lift,  work,  Ac,  do  not  indicate  diff  kinds  of  force;  but  merely  diff  kinds 
of  effects  produced  by  the  one  grand  principle,  force.  See  footnote,  p  455;  also  p  587. 

The  above  32.2  ft  per  sec  is  called  the  acceleration  of  gravity ;  and  by 
scientific  writers  is  conventionally  denoted  by  a  small  g1 :  or,  more  correctly  speak- 
ing, since  the  acceleration  is  not  precisely  the  same  at  all  parts  of  the  earth,  g  de- 
notes the  acceleration  per  sec,  whatever  it  may  be,  at  any  particular  place.  See 
note  to  Art  25,  p  455. 

Art.  13.  Reaction.  Strain.  Strain,  as  before  said,  in  Art  2,  is  either  a 
pull  or  a  push.  The  term  may  be  applied  equally  to  the  act.  or  to  its  fffrctit;  that  is, 
it  may  be  said  to  be  either  the  action  of  opposing  forces  against  each  other:  or  tbe 
effects  which  that  action  produces  upon  the  particles  of  the  body  in  which  they  act. 
A  single  force  cannot  produce  strain  :  for  since  nothing  but  force  can  oppose,  resist, 
pull,  or  push,  against  force,  there  must  he  at  least  two,  to  strain  by  pulling  or  push- 
ing against  each  other.  This  mutual  opposition  or  straining  is  called  also  the  war- 
on  of  the  forces.  It  can  take  place  only  between  equal  forces,  or  equal  portions  of 
unequal  ones.  When  the  forces  are  equal,  and  meet  from  diametrically  opposite  direc- 
tions ;  that  is,  in  the  same  straight  line,  but  in  opposite  directions  along  it,  they 
become  entirely  converted  into  reaction,  or  strain.  If  they  are  unequal ;  or  are  not 


450 


FORCE    IN    RIGID   BODIES. 


in  the  same  straight  line ;  but  meet  obliquely ;  then  only  equal  portions  of  them  will 
re.ict  against  each  other;  while  the  remainder  will  continue  as  motion  ;  unless  soiae 
third  force  is  present  to  prevent  it;  as  when  friction  is  generated.  The  reaction  of 
the  equal  wholes  or  parts  consists  in  their  mutually  resisting,  opposing,  arresting, 
balancing,  equilibrating,  straining,  pulling,  pushing,  counteracting,  or  destroying 
eacli  other. 

All  these  words  are  equally  applicable.  As  a  matter  of  convenience  only,  we  often  say  that  the 
bodies  themselves  react.  If  a  cauuou-ball  in  its  flight  cuts  a  leaf  from  a  tree,  we  say  that  the  leaf 
has  reacted  against  the  ball  with  precisely  the  same  degree  of  force  that  the  ball  acted  against  the 
leaf.  That  degree  of  force  was  sufficient  to  cut  off  a  leaf,  but  not  to  arrest  the  ball ;  for,  after  a  smalt 
portion  of  the  moving  force  of  the  ball  had  been  converted  into  straining  force  to  react  against  the 
resistance  of  the  leaf,  the  remainder  was  sufficient  to  carry  it  onward  in  its  course.  It,  has.  how- 
ever, lost  precisely  as  much  force  as  that  which  the  leaf  opposed  to  it.  A  ship  of  war,  in  running 
against  a  canoe,  receives  as  violent  a  blow  as  it  gives ;  but  the  same  blow  that  will  upset  or  sink  a 
canoe,  will  not  appreciably  affect  the  motion  of  a  ship.  The  fist  of  a  pugilist  striking  his  opponent 
in  the  face  receives  as  severe  a  blow  as  it  gives;  but  the  blow  which  may  seriously  damage  a  nose, 
mouth,  or  eyes,  may  have  no  such  effect  upon  hard  knuckles.  The  paiu  received  in  the  one  case,  and 
not  in  the  other,  is  of  course  no  measure  for  force. 

BEM.  We  have  just  said  that  strain  is  the  mutual  destruction  of  two  equal  amounts  of  force.  This  may 
readily  be  conceived  in  cases  where  two  bodies  come  into  sudden  contact,  and  arrest  each  other's 
progress  by  a  mutual  blow;  for  we  then  see  that  their  forces  are  lost,  inasmuch  aa  they  no  longer 
produce  either  motion,  strain,  or  work.  But  in  continuous  strains  (pulls,  or  pushes)  the  principle  is 
not  so  evident,  at  first.  For  instance,  when  a  wt  rests  upon  a  table,  and  continues  to  strain  it  day 
after  day,  it  may  be  asked  where  is  the  loss  of  gravity  force  in  the  weight;  inasmuch  as  it  weighs  aa 
much  after  pressing  for  a  long  time,  as  it  did  at  the  beginning;  or  where  is  the  loss  of  inherent  cohe- 
sive force  in  the  material  of  the  table,  which  is  as  strong  aa  at  first? 

The  reply  is  that  gravity,  and  the  natural  strengths,  or  inherent  forces  of  matter,  are  being  inces- 
santly maintained  or  renewed,  by  unceasing  streams,  as  it  were,  of  those  forces.  The  gravity  of  a 
wt  which  presses  a  table,  or  pulls  on  a  rope,  at  one  moment,  is  not  identically  the  same  that  pressed 
or  pulled  it  the  moment  before;  and  so  with  the  cohesive  force  of  the  material  of  the  table.  If  it 
were  not  so,  a  post  or  a  rope  which  would  be  broken  by  a  single  force  of  10  tons,  would  also  be  broken 
by  sustaining  one  ton  10  consecutive  times  ;  for  the  one  ton  would  each  time  react  against,  or  destroy 
one  ton  of  the  inherent  resisting  force  of  the  post  or  rope;  and  in  ten  applications  would  destroy  it 
all.  We  must  therefore  conclude  that  these  natural  forces  are  being  unceasingly  supplied  from  that 
inexhaustible  source  of  power  "  by  which  all  things  are  upheld."  When  we  lean  forward  against  a 
strong  wind,  we  are  continuously  exerting  new  force  against  the  continuous  stream  of  force  which 
assails  us  ;  and  in  the  same  way  does  a  post  or  a  chain  sustain  its  load.  Continuous  strains  produced 
by  the  force  of  water,  steam,  animals,  &c,  we  well  know  can  only  be  maintained  by  a  continuous  sup- 
ply of  said  force :  to  be  as  continuously  reacted  against,  or  destroyed,  by  whatever  opposing  force  of 
grinding,  pumping,  Ac,  it  is  directed  against.  The  strain  of  an  impulse,  blow,  or  stroke,  lasts  only 
for  an  instant,  because  new  force  is  not  supplied  to  make  it  continuous.  Two  equal  forces,  straining 
against  each  other,  do  not  even  keep  a  body  at  rest;  but  the  body  rests  merely  because  the  two  forces 
destroy  each  other,  and  therefore  cannot  prevent  it  from  resting.  As  a  matter  of  convenience  only, 
we  may,  however,  say  they  keep  it  at  rest.  We  might  even  assume  strictly  that  force  produces  motion 
only  ;  and  that  the  destruction  of  force  produces  strain,  and  thus  restores  rest. 

Art.  14.  While  a  horse  is  hauling  a  boat  on  a  canal,  it  is  not  the  boat  and  ita 
load  which  react  against  his  force  ;  -because  matter  cannot  react  against  force;  it  is 
the  force  of  friction  of  the  boat  against  the  water ;  and  the  resistance  of  the  wa- 
ter in  front  of  the  boat,  produced  by  its  cohesive  force.  So  with  an  engine  and 
train  on  a  level  railroad ;  the  only  resistance  to  the  steam  force  of  the  engine,  is  the 
force  of  friction  at  the  axles  and  tires  of  the  wheels,  and  the  pres  force  of  the  air  in 
front.  But  on  an  up  grade,  the  engine  has  also  to  partly  lift  the  train ;  or,  in  other 
words,  to  react  against  its  forces  of  friction  and  gravity  combined. 

Neither  the  horse  nor  the  engine  exerts  any  more  force  upon  the  opposing  forces,  than  these  last 
exert  upon  them ;  but  both  the  horse  and  the  engine  possess  a  surplus  of  power  beyond  what  is  ne- 
cessary to  strain  against  or  destroy  the  forces  opposed  to  them :  and  this  surplus,  as  moving  force, 
enables  them  in  addition  to  move  forward,  as  in  the  case  of  the  cannon-ball  just  alluded  to;  and  since 
the  unresisting  matter  of  the  boat,  load,  and  train,  is  attached  to  them  by  the  tow-rope  and  coupling- 
links,  they  of  course  must  follow. 

The  resistance  which  an  abutment  opposes  to  the  pres  of  an  arch  ;  or  a  retaining-wall  to  the  pres 
of  the  earth  behind  it,  is  no  greater  than  those  pres  themselves ;  but  the  abut  and  wall  are,  for  the 
sake  of  safety,  made  capable  of  sustaining  much  greater  pressures,  in  case  accidental  circumstances 
•kould  produce  such. 

Art.  15.  The  mere  fact  that  a  body  is  subjected  to  g^reat 
Strains  from  equal  forces  reacting  upon  it  in  opposite  directions,  does  not  of  it- 
self render  the  body  more  difficult  to  move  than  if  it  were  free  from  strains ;  but 
in  the  cases  which  usually  present  themselves  in  practice  the  straining  forces  gener- 
ate friction,  which  does  oppose  motion. 

Thus  let  B,  Fig  8,  be  a  block  resting  on  a  nor  support,  and  acted  upon  by  a  downward  force  d  of  say 
100  tons,  produced  say  by  an  immense  block  of  granite  resting  upon  B.  Now  it  is  plain  that  this  100 
tons  downward  force  will  be  met  and  balanced  by  a  100  tons  upward  force  u,  being  the  resistance  of 
the  hor  support.  Heuce  these  two  equal  reacting  forces  produce  in  the  body  B  a  strain  of  100  tons ; 
but  evidently  do  not  impart  to  it  as  a  whole  any  tendency  to  move  in  any  direction  whatever  ;  nor 
do  they  tend  to  prevent  it  from  being  moved  in  any  direction.  The  body  therefore  remains  as  be- 
fore a  mere  inert  mass  incapable  of  resisting  the  slightest  moving  force. 

Now  suppose  no  friction  to  exist  at  either  the  base  or  the  top  of  B. 
Then  the  slightest  hor  force  A,  a  mere  breath,  would  slide  B  along  the  hor  support,  moving  it  from 


DRCE   IN    RIGID   BODIES. 


451 


under  the  100  top^olock  on  its  top.  No  matter  how  heavy  B  might  be,  the  same 
smallest  force  wfculd  slide  it,  the  only  difference  being  that  the  heavier  it  was  the 
less  would  beXts  velocity.  The  quantity  of  motion  (Art  9)  will  be  the  same  for 
any  wt. 

The  heaviest  bodies  resting  upon  the  surface  of  the  earth,  as  well  as  ourselves, 
would  be  swept  along  by  the  slightest  breeze  if  it  were  not  for  friction. 

If  the  screw  of  a  vise  be  worked  until  it  produces  a  great  strain  in  the  jaws  of 
the  vise,  the  vise  is  not  thereby  rendered  more  difficult  to  move. 

Again,  if  a  strain  of  thousands  of  tons  were  produced  by  the  jaws  of  a  vise,  in  a  body  weighing  an 
ounce,  this  immense  strain  would  not  prevent,  nor  even  tend  in  the  smallest  degree  to  prevent,  the 
ounce  body  from  falling  down  from  the  jaws  of  the  vise.  It  is  prevented  by  the  third  force,  friction, 
which  compels  the  one  ounce  of  gravity  force  to  become  vert  strain,  instead  of  motion.  The  two  forces 
of  thousands  of  tons  each,  which  produce  the  strain  of  the  vise,  are  thereby  entirely  destroyed,  aa 
regards  their  action  upon  the  body  as  a  whole.  Hence  they  could  not  prevent  the  one  ounce  from  pro- 
ducing motion  in  it;  nor  could  they  affect  it  as  a  whole  in  any  way  ;  for  all  their  action  is  actually 
against  each  other.  It  is  on  this  principle  alone  that  strains'do  not  interfere  with  motions. 

If  a  body  H,  Fig  4,  of  10  tons  weight,  is  suspended  from  a  long  rope,  its  reaction 
against  the  equal  opposing  force  at  the  other  end  of  the  rope,  produces  a  continuous 
strain  among  all  the  particles  which  compose  the  rope ;  but  which  does  not  in  the 
least  affect  the  rope  considered  as  a  whole ;  inasmuch  as  it  does  not  tend  to  move  it 
in  any  direction.  Now,  in  this  case,  there  is  no  friction  to 
be  overcome;  and  we  know  from  daily  experience  that  it  is 
therefore  easy  to  move  the  unresisting  body  a  little  dist,  by 
applying  a  very  small  hor  force/.  We  cannot  move  it  far, 
as,  for  instance,  to  m,  because  we  then  have  not  only  to  move, 
but  to  Lift  it,  or  overcome  its  gravity  force,  through  the  vert 
height  vc.  In  doing  this,  it  is  true  our  force  does  not  have 
to  sustain  the  entire  wt  of  the  body ;  because  most  of  it  is 
sustained  by  the  rope.  Still,  if  we  move  it  at  all,  we  have  to 
overcome  some  of  its  weight ;  otherwise,  a  mere  breath  would 
move  it,  although  very  slowly.  If  we  attempt  to  move  it  by 
an  upward  force  w,  we  shall  have  still  more  of  its  wt  to  re- 
sist us ;  and  if  by  a  downward  one  d,  we  shall  be  resisted  by 
the  cohesive  force  of  the  rope.  Therefore,  in  this  case,  we  can  move  it  more  readily 
by  the  hor  force  /. 

Art.  16.  If  two  unequal  forces,  which  for  illustration  we  will  call  10  and  12, 
are  imparted,  either  as  pulls  or  as  pushes,  in  precisely  opposite  directions,  to  a  rigid 
body  on  which  no  other  force  is  acting,  then  the  small  force  10,  and  10  parts  of  the 
large  one,  will  destroy  each  other  as  strain  ;  after  which,  of  course,  they  can  produce 
no  effect  of  any  kind ;  but  the  remaining  two  parts  of  the  large  one,  meeting  with 
no  opposing  force  to  react  against,  will  continue  onward  as  motion,  in  the  same  di- 
rection as  before;  taking  the  unresisting  body  along  with  them.  In  such  a  case, 
the  large  force  is  said  to  overcome  the  small  one ;  and  such  an  expression  is  very  con- 
venient, in  reference  to  the  entire  original  forces.  But  in  a  strictly  scientific  sense, 
one  force  cannot  overcome  another. 

Thus,  in  the  foregoing  case,  so  far  as  the  strain  Is  concerned,  the  large  force  must  be  considered  as 
separated  into  10  straining,  and  2  moving  parts.  Neither  of  the  two  10  forces  which  strained  against 
each  other  overcame,  or  gained  any  advantage  whatever  over  the  other ;  for  the  two  reacted  equally 
on  each  other,  and  mutually  arrested",  equilibrated,  and  destroyed  each  other.  And  this  is  plainly 
the  most  that  one  force  can  do  to  another.  The  2  force  of  the  large  body  took  no  part  whatever  in  the 
conflict;  but,  on  the  contrary,  moved  out  of  the  way  of  it.  As  it  had  lost  10  twelfths  of  its  previous 
moving  force,  it  now  moves  the  body  only  by  virtue  of  the  remaining  2  twelfths ;  and  consequently 
with  but  2  twelfths  of  its  former  vel. 

Since  two  equal' opposing  forces,  or  equal  portions  of  unequal  ones,  thus  bring  each  other  to  a 
stand-still,  or  equilibrate  each  other,  they  are  called  Static;  from  the  Latin  "Sto,  I  stand  ;"  and  that 
branch  of  the  science  of  force  which  treats  only  of  cases  in  which  all  the  applied  forces  keep  each 
other  at  rest,  is  called  "  Statics,"  or  "  Equilibrium." 

Art.  17.  When  force  has  once  been  put  into  a  body,  it  can  only  be  taken  out 
again  by  the  reaction  of  some  opposing  force.  The  same  identical  portion  of  any 
force  cannot  produce  both  motion  and  strain  at  the  same  time.  When  continuous 
force  is  applied,  as  in  mills,  Ac,  to  do  both,  (or,  in  other  words,  to  work,)  it  must  be 
considered  to  divide  itself  into  two  parts  for  those  separate  purposes.  If,  while  at 
work,  the  resistances  to  be  overcome  become  less,  the  strain  also  becomes  less,  and 
the  motion  becomes  greater;  and  vice  versa.  Motion  is  diminished  by  converting 
part  of  it  into  strain ;  and  strain,  by  converting  it  into  motion. 

Thus,  the  motion  or  moving  force  in  a  cannon-ball  is  gradually  converted  into  strain,  by  working 
against  the  resisting  force  of  the  successive  strata  of  air  through  which  it  passes.  These  gradually 
destroy  all  its  force  by  this  process,  and  thus  permit  it  to  rest.  The  moving  force  which  remains  in 
%  railroad  train  after  steam  is  shut  off,  is  gradually  destroyed  by  working  against  the  resistance  of 


452 


FORCE    IN    RIGID    BODIES. 


the  air  ;  of  friction  at  the  axles  and  rims  of  the  wheels  ;  up-grades,  curves.  Ac.  The  strain  produced 
in  a,  rope  by  two  men  pulling  against  each  other  at  its  opposite  ends,  is  converted  into  motion  if  the 
rope  breaks ;  throwing  both  men  backward.  So  with  the  strain  on  a  bent  bow,  if  the  string  is  cut- 
er if  we  cut  the  rope  that  holds  a  balloon,  part  of  the  force  with  which  the  balloou  before  strained 
the  rope  becomes  moving  force,  and  by  it  the  balloon  ascends.  The  other  part  balances  gravity. 
Art.  18.  If  force  /  be  imparted  to  any  rigid  body,  as  N,  Fig  5,  at  any  point  c ; 

and  if/o  represent  the  direction  in  which 
it  was  imparted,  whether  as  a  pull  or  a 
push,  then  the  force  would  produce  the 
same  effect  upon  the  body  considered  as 
an  entire  mass,  as  if  it  had  been  im- 
parted as  either  a  pull,  or  a  push,  in  the 
same  direction,  at  any  other  point  of  the 
body  in  said  line ;  as  at  t,  /,  $,  o,  &c. 

Under  Composition  and  Resolution  of 
Forces,  it  will  be  seen  to  be  sometimes 


IV  5 


necessary  to  consider  a  push  fc,  to  be 
changed  to  a  pull  oh\  and  vice  versa; 
when  we  wish  to  ascertain  the  joint 

effect  or  resultant  of  a  pull  and  push  imparted  to  a  body  at  the  same  time.  See 
Remark  1,  Art  29,  p  459. 

The  foregoing  important  principle  holds  good,  no  matter  how  many  diff  forces  may  be  acting  upoo 
the  body  at  the  same  time,  in  diff  directions ;  or  how  much  the  direction  of  their  joint  effect,  or  re- 
sultant, may  differ  from  that  of  any  one  of  them;  the  action  of  each  force,  considered  separately, 
may  be  regarded  as  just  stated.  The  tendencies  of  several  forces,  acting  at  the  same  moment,  may 
therefore  frequently  be  first  investigated  one  by  one;  and  these  tendencies  then  combined  into  one; 
or  the  forces  themselves  may  first  be  combined  into  one  or  more  resultants,  as  directed  under  Com- 
position and  Resolution  of  Forces,  and  the  effect  of  these  resultants  considered.  The  engineer  has 
generally  to  divide  all  the  forces  actipg  upon  his  structures  into  two  classes;  namely,  those  whose 
tendency  is  to  secure  the  stability  he  requires;  and  those  which  tend  to  impair  that  stability.  He 
therefore  first  finds  the  resultant,  or  joint  effort  of  each  class  separately ;  and  then  compares  these 
two  resultants  with  each  other.  The  mode  of  doing  this  will  be  shown  further  on.  See  Arts  35,  72.  <tc. 

It  is  plain  that  if,  instead  of  regarding  the  body  as  rigid,  we  considered  it  as  elastic,  or  as  breaka- 
ble, an  entirely  diff  course  would  be  necessary,  as  the  question  would  th<m  become  one  on  the  strength 
of  materials  .-"for  the  force/,  applied  at  c  or  t,  as  a  push,  might  break  off  the  pieces  c  and  t;  and  so 
with  the  same  force  as  a  pull  at  «  or  o.  Although  masonry,  iron,  timber,  and  other  building  materials 
are  by  no  means  absolutely  rigid,  yet  generally  they  may  be  assumed  to  be  so  when  we  are  investi- 
gating the  effect  of  force  to  overthrow,  or  derange  the  structure  as  a  whole. 

Art.  19.  The  full  amount  of  a  given  force  cannot  (theoreti- 
cally) be  put  into  a  resisting  body  B,  Fig  6.  except  when 
applied  in  a  direction  at  right  angles  to  the  surf  of  the  body 
at  the  point  of  application.  If  it  be  applied  in  a  direction  at  all  oblique 
to  the  surf,  then  only  a  portion  of  it  will  (according  to  theory)  enter  the  body,  and 
produce  any  effect  on  it.  The  remainder  will  continue  to  produce  motion  in  that 
other  body  in  which  the  applied  force  was  carried  to  the  body  B ;  unless  some  third 
force  be  present  to  receive  and  react  at  right  angles  to  that  remainder  also.  Thus, 
the  forces  m  o  and  n  e,  being  at  right  angles  to  the  surf  at  the  points  of  application 
i>  and  e,  will  all  enter  B,  if  the  resistance  ofR  is  as  great  or  greater  than  they ;  if  not, 
they  will  move  B.  But  the  force  F#  is  riot  at 
right  angles  torthe  surf  at  its  point  of  applica- 
tion g;  therefore,  only  a  portion  of  it  will  (by 
the  theory)  be  imparted  to  the  body  B,  however 
great  may  be  the  resistance  of  B. 

Under  the  head  Composition  and  Resolution  of  Forces, 
it  will  be  seeu  that  when  a  force  is  thus  applied  obliquely 
to  a  surf,  its  action  at  the  point  g  is  precisely  the  same  as 
that  of  two  entirely  separate  forces  ;  one  of  which,  v  g,  is 
at  right  angles  to  the  surf  at  g  ;  and  the  other.  s  g,  par- 
allel to  the  same  surf.  Only  vg  will  enter  the  body  H  • 
It1;  A  h  while  ag  will  remain  in  the  body  F  g,  which  carried  the 

-1-  *  V.  v  entire  force  to  B  ;  and  (if  F  g  also  is  rigid)  will  (by  theory) 

"T  cause  it  to  move  in  the  direction  g  t,  unless  some  third 

force  be  opposed  a.t  right  angles  to  it  also.     The  quantity 

of  each  of  the  forces  v  g  and  a  g,  is  very  readilv  found  thus  :  On  the  line  F  g  measure  off  by  any  con- 
venient scale  a  dist  gi,  to  represent  the  amount  in  fts.  tons,  <fcc,  of  the  force  F  g  ;  then  on  g  i  as  a 
diag  draw  a  parallelogram  j  v  g  a,  having  two  of  its  sides  perp  to  the  surf  at  g,  and  the  other  two 
parallel  to  it.  Then  is,  orvg,  measd  by  the  same  scale  as  g  i,  will  give  the  force  imparted  to  the  body 
B  ;  and  iv  or  sg  will  give  that  which  remains  in  the  body  F  g.  and  will  move  it  from  g  toward  t, 
unless  prevented  by  some  other  force.  If  F  g  is  not  thus  prevented  from  moving  toward  t,  it  is  plain 
that  the  force  vg,  which  is  transferred  to  the  body  B,  cannot  remain  stationary  at  g ;  but  must  move 
along  with  F  g,  and  thus  press  upon  the  surf  at  every  point  between  g  and  t. 

This  will  appear  in  a  clearer  light  on  referring  to  a  body  placed  upon  an  inclined 
plane  so  steep  that  the  body  will  slide  down.  We  know  that  a  part  of  the  force  of 
gravity,  or  wt  of  the  body,  presses,  as  strain,  upon  the  plane,  and  at  right  angles  to 
it;  while  a  part  remains  in  the  body,  as  motion;  and  causes  it  to  slide  down  the 


IN    RIGID    BODIES. 


453 


pkme;  see  Art  60.  As  it  slices,  it  is  evident  that  the  pressure  part  of  the  force  also 
moves  along  with  it ;  and^s thus  imparted  at  every  point  of  the  plane  along  the  dist 
slid  over.  The  case  is  identically  the  same  with  the  pres  force  v  g  in  Fig  6 ;  this  will 
elide  up  the  plane  from  g  toward  t ;  being  carried  along  by  the  sliding  force  sg. 

But  in  practice  this  theory  does  not  hold  good  because  it  ig- 
nores the  existence  of  friction,  which  pressures  always  generate  at  points  of  appli- 
cation. This  friction  always  acts  in  the  direction  of  the  pressed  surfaces,  and  con- 
sequently always  opposes  more  or  less  resistance  to  that  component,  sg,  which  tends 
to  produce  sliding  along  the  surface  st;  and  so  much  of  sg  as  is  thus  resisted  is 
thereby  changed' from  motion  to  strain,  and  enters  the  body  B.  If  it  were  not  for 
friction  a  body,  Fig  64,  p  487,  would  slide  down  an  inclined  plane  wx,  no  matter  how 
slight  its  inclination  might  be;  but  we  know  that  friction  often  prevents  such  slid- 
ing even  when  the  plane  forms  a.  considerable  angle,  yxw,  with  a  hor  line  yx. 
When  this  angle  becomes  so  great  that  the  body  is  just  on  the  point  of  starting  to 
slide  down  it,  it  is  called  the  angle  of  friction  ;  and  in  Fig  6,  if  the  force  V g 
does  not  form  with  the  perp  vg  an  angle  vg  F,  greater  than  this  angle  of  friction, 
then  friction  will  oppose  all  the  sliding  force  s  g,  no  matter  how  great  it  may  be  ; 
so  that  the  entire  force  F  </  will  then  enter  B  at  g,  and  in  its  original  direction  F  g. 
Study  Art  63,  p  487. 

This  remark  is  particularly  applicable  to  the  case  of  the  masonry 
joints  in  the  abuts  of  stone  arches  ;  especially  those  of  large  span,  with 
small  rise.  The  pres  which  such  an  arch  exerts  upon  its  abuts  is  very 
great;  and  its  line  of  direction  is  curved,  as  shown  by  the  dotted  line 
on  m,  Fig  7.  It  therefore  becomes  necessary  first  to  find  the  position 
of  this  line,  (see  Art  72.)  so  as  to  know  how  to  draw  the  varying  incli- 
nations of  the  joints  nearly  at  right  angles  to  it;  otherwise,  the  upper 
courses,  if  hor,  may  slide  outward  upon  the  lower  ones,  as  shown  by 
the  arrow.  In  small  arches  of  considerable  rise,  the  sliding  portion 
of  this  force  may  be  safely  resisted  by  good  mortar  or  cement,  if  suffi- 
cient time  be  first  allowed  for  it  to  harden  properly  ;  but  in  large  ones, 
the  direction  of  the  joints  must  be  relied  on,  unless  we  increase  the 
expense  by  making  the  abuts  unduly  thick. 

The  angle  o'  friction  of  masonry  on  masonry  (see  table,  p  599)  is  about  32°.  Therefore  if  at  any 
bed-joint  of  masonry,  as  in  the  abut  of  Fig  7,  the  resultant  that  cuts  said  joint  at  the  line  of  pressures, 
onm,  does  not  differ  more  than  32°  from  a  perp  to  said  joint,  there  will  be  no  tendency  to  slide  at  that 
joint.  See  Art  63,  p  487  ;  and  foot-note  p  348. 

Fig  8  is  added  merely  to  illustrate  more 
strikingly  the  necessity  for  clearly  distin- 
guishing between  applied,  and  imparted 
forces.  Here  the  great  force  a  o  is  applied 
to  the -body  B  B  at  the  point  o ;  but  all  of 
it  that  is  theoretically  put  into,  or  enters 
the  body  ;  or  produces  any  kind  of  effect 
upon  it,  is  the  very  small  amount  represented  by  co,  at  right  angles  to  the  surf  of 
B  B  at  o,  but  we  have  seen  that  in  practice  friction  makes  it  more. 

All  this  will  be  better  understood  after  studying  Comp  and  Res  of  Forces,  Art  28. 

Art.  2O.  It  rarely  happens  in  practice  that  we  impart  very 
l^reat  force,  or  velocity,  to  a  heavy  body  instantaneously ; 
or  by  a  single  effort.  In  the  machinery  of  mills,  in  railroad  trains,  in  steam- 
ships, &c,  it  is  done  very  gradually :  and  indeed,  few  water  or  steam  powers 
possess  sufficient  force  to  do  it  otherwise.  The  principle  explained  in  the 
following  example,  applies  to  the  others.  Thus,  no  locomotive,  or  marine  en- 
gine, has  sufficient  power  to  start  a  heavy  train,  or  a  steamship,  at  once  at  a  rapid 
speed.  The  first  stroke  of  the  piston  imparts,  or  puts  into  the  train,  a  quantity  of 
force  sufficient  not  only  to  react  or  strain  against,  balance,  or  destroy  (and  be  de- 
stroyed by)  all  the  resisting  forces  of  friction,  the  air,  grade,  curvature,  &c,  which 
present  themselves  along  the  short  portion  of  the  road  passed  over  during  the  first 
stroke;  but  a  small  excess  besides,  which,  being  unresisted,  is  not  destroyed ;  but 
continues  in  the  train,  as  motion  ;  giving  a  slow  movement  to  the  unresisting  matter 
of  which  it  is  composed.  This  last  portion  would  remain  in  the  train,  and  move  it 
at  the  same  very  slow  vel  forever,  if  only  we  could  remove  all  resisting  forces  from 
before  it.  The  next  stroke  in  the  same  manner  furnishes  a  new  instalment  offeree  ; 
which,  like  the  first,  divides  itself  into  two  parts ;  one  of  them  as  strain  to  destroy, 
and  be  destroyed  by  the  new  resistances  met  with  in  the  next  short  dist  along  the 
track ;  and  the  other  as  motion,  to  remain  stored  up  in  the  train,  united  with  the 
motion  put  there  by  the  first  stroke ;  thereby  imparting  increased  vel  to  the  unre- 
sisting matter  of  the  train  ;  and  this  process  is  repeated  during  a  great  number  of 
strokes  ;  the  train  moving  faster  and  faster.  But  after  a  while  it  is  found  that  no 
matter  how  powerful  the  engine  may  be,  or  how  light  its  train,  the  speed  no  longer 
increases. 


fin-  R  7? 

*  L 


454 


FORCE   IN    RIGID    BODIES. 


The  reason  of  this  is  that  the  resistances  of  the  air,  friction,  &c,  increase  with  the  speed,  but  at  a  more 
rapid  rate ;  so  that  after  a  certain  vel  has  been  attained,  they  require  all  the  power  the  eugiue  is  capable 
of,  in  order  as  strain  to  react  against,  balance,  or  destroy  them  alone;  without  leaving  any  more  sur- 
plus to  be  stored  away,  as  motion,  iu  the  train.  After  this  point  is  attained,  the  power  of  the  engine 
no  longer  draws  the  train  ;  but  barely  suffices  to  remove  all  the  resistances  which  would  otherwise  im- 
pede it;  and  thus  permits  the  moving  force  (which  has  previously  been  stored,  and  accumulated  in  the 
train,  by  small  instalments)  to  continue  to  move  the  unresisting  matter  of  the  train  at  an  undiminished 
vel.  The  train  might  now  be  said  to  "  move  itself;"  and  it  would  do  so  forever,  without  requiring 
any  additional  moving  force,  if  the  engine  would  only  continue  to  destroy  the  resistances  in  its  way. 
Should  a  stiff  up-grade,  sharp  curvature,  or  other  resistance  present  itself,  so  that  the  opposing  forct?s 
actually  exceed  the  entire  power  of  the  engine,  for  a  short  time,  the  train  will  actually  come  to  its 
assistance,  and  by  converting  part  of  its  own  motion  into  strain,  will,  as  it  were,  lend  pressure-force 
to  the  engine,  to  help  it  push  through  its  increased  work  ;  performing  in  fact  the  part  of  a  fly-wheel 
in  machinery.  Frequently,  when  an  engine  appears  to  he  pulling  a  heavy  train  up  a  short  sharp 
grade,  it  is  actually,  so  to  speak,  the  train  that  is  pushing  itself  up;  the  engine  probably  could  not 
do  it.  This  state  of  affairs  can,  however,  continue  for  but  a  short  time;  otherwise  all  the  motion 
of  the  train  would  be  converted  into  strain  ;  and  being  thereby  destroyed,  the  whole  would  come  to  rest. 

When  a  train  is  going  at  speed,  and  it  becomes  necessary  to  stop  at  a  station  some  dist  ahead, 
•team  is  shut  off,  so  that  the  steam  force  of  the  engine  shall  no  longer  counterbalance,  or  destroy 
the  resisting  forces  in  front  of  it;  and  the  number  of  the  resistances  themselves  is  increased  by  add- 
ing to  them  the  friction  of  the  brakes.  Against  all  these  combined,  the  train  has  now  to  "work" 
its  way  unaided;  and  it  does  so  by  the  gradual  conversion  of  its  previously  accumulated  motion,  or 
moving  force,  into  straining  force.  Work,  as  before  stated,  consists  in  motion  and  strain  combined. 
When  the  conversion  has  been  completely  effected,  the  train  stops.  Thus,  we  see,  that  up  to 
the  time  of  its  stopping,  the  force  which  had  gradually  accumulated  in  it  as  motion  before  it  had 
reached  its  greatest  vel  is  gradually  taken  out  of  it  as  work,  after  steam  is  shut  off.  We  may  there- 
fore speak  of  it  as  well  under  the  head  of  accumulated,  or  stored-up  work;  as  of  accumulated  "motion, 
when  we  intend,  in  any  kind  of  machinery,  to  convert  such  motion  into  work.  Thus,  the  motion 
gradually  accumulated  in  a  fly-wheel,  ia  also  accumulated  work,  held  in  reserve  until  some  extra 
strain  on  the  machinery  calls  for  its  aid. 

Art.  21.  Up  to  the  time  that  the  vel  ceased  to  increase  because  all  the  power 
of  the  engine  became  required  as  strain  to  react  against  resisting  forces,  and  conse- 
quently could  no  longer  spare  any  as  motion  to  the  train,  the  work  of  the  engine  was 
what  is  termed  variable,;  being  gradually  accelerated.  Also,  when  steam  was  shut 
off,  the  work  of  the  train  was  variable ;  being  gradually  retarded.  Such  is  the  case 
in  almost  every  kind  of  machinery,  during  the  interval  between  starting,  and  ulti- 
mately attaining  its  maximum  speed.  When  the  latter  point  is  reached,  supposing 
the  resistances  afterward  to  remain  the  same,  the  work  is  termed  uniform,  or  steady. 
All  these  remarks,  as  well  as  those  of  Art  20,  apply  alike  to  all  kinds  of  heavy  ma- 
chinery; no  matter  by  what  kind  of  force  or  power  it  is  driven;  the  machinery 
takes  the  place  of  the  train  just  spoken  of;  and  the  friction  of  the  cog-wheels,  gud- 
geons, pivots,  the  grinding,  sawing,  or  whatever  the  work  may  be,  takes  the  place 
of  the  grades,  curves,  and  friction  of  the  train.  All  alike  are  simply  cases  of  fores 
in  its  various  shapes  of  motion,  strain,  and  work. 

Art.  22.  The  quantity  of  any  work,  considered  by  itself,  without 
reference  to  the  time  reqd  to  perform  it,  is  plainly  to  be  measd  by  mult  together  the 
resistance  in  Ibs,  by  the  dist  through  which  it  is  overcome  in  ft,  as  stated  in  Art  11. 

After  work  becomes  uniform,  that  is,  when  neither  its  strain  nor  its  motion  un- 
dergoes any  change;  its  rate  is  measured  by  mult  the  resistance,  or  strain, 
in  fes,  tons,  &c,  by  the  vel,  or  dist,  in  feet,  £c,  through  which  the  resistance  is  over- 
come in  a  given  time,  as  a  sec,  min,  hour,  Ac. 

Thus,  if  the  resistance  is  3300  Ibs,  and  is  overcome  through  a  dist  of  10  ft  in  every  min  ;  or  if  the 
resistance  is  33  tt>s,  and  is  overcome  through  a  dist  of  1000  ft  per  min,  the  rate  of  the  work  is  in  each 

Ibs        vel     tts       vel 

case  the  same,  namely,  33000  ft-tts  per  min,  or  one  horse-power;  for  3300  X  10  —  H3  X  1000  —  33000 
f't-ft>s  per  min.  The  quantity  of  motion  of  a  body  (Art  9)  is  also  estimated  in  ft-tbs  ;  and  under  the 
head  Levers,  it  will  be  seen  that  the  tendencies  (called  moments)  which  the  power  and  the  weight  re- 
spectively have  to  commence  motion  about  the  fulcrum  as  a  center,  are  measured  in  the  same  term. 
Work,  mere  motion,  and  moments,  are,  however,  effects  of  force  so  diff  from  each  other,  that  confu- 
sion is  no  more  likely  to  occur,  than  in  applying  the  same  measure,  one  foot,  to  materials  as  diff  ad 
cloth,  bar  iron,  hoards,  &c. 

In  scientific  phraseology,  work  is  either  useful  or  prejudicial,  the 

latter  being  the  quantity  of  force  lost  by  friction,  by  the  resistance  of  the  air,  &c.  Thus,  in  pump- 
ing water,  part  of  the  applied  force  or  power  is  lost  in  the  friction  of  the  diff  parts  of  the  pump:  so 
that  a  steam  or  water  power  of  100  tbs,  moving  6  ft  per  sec,  cannot  raise  100  tts  of  water  to  a  height 
of  6  ft  per  sec.  Therefore  machines,  so  far  from  paining  power,  according  to  the  popular  idea,  ac- 
tually lose  it,  in  one  sense  of  the  word.  In  the  practical  application  of  all  machinery,  the  object  is 
twofold;  namelv.  to  enable  us  conveniently  to  apply  straining  force,  to  balance,  react  against,  or 
destroy,  the  resisting  forces  of  friction,  and  the  cohesive  forces  of  the  material  which  is  to  be  operated 

operated  on,  after  the  resisting  forces  which  had  acted  upon  them  have  thus  been  rendered  ineffectiva. 

Art.  23.  The  total  quantity  of  work  that  will  be  performed  by  the 
moving  force  that  is  in  a  body  at  any  given  moment,  provided  that  after  changing 
from  mere  moving  force  into  working  force,  it  is  left  to  expend  itself  in  uniformly 
retarding  work,  without  receiving  any  additional  force  to  aid  it,  (as  in  the  case  of  the 
moving  force  in  the  locomotive  in  Art  20,  after  steam  is  shut  off;  and  when  said 
force  begins  to  work  against  the  resistances  of  the  road,)  is  as,  or  in  proportion  to, 


FORCE   IN   RIGID   BODIES.  455 

(not  equal  to,)  th/wt  of  the  body,  mult  by  the  square  of  its  vel  at  the  moment  it 
begins  to  work.'  For  example,  if  a  train  at  the  time  steam  is  shut  off,  has  in  it  an 
accumulated  or  stored-up  moving  force  of  10  miles  an  hour ;  and  if  that  force  will 
by  itself  work  the  train  against  the  resistances  of  the  road  for  a  dist  of  one-quarter 
of  a  mile,  before  coming  to  a  stop;  then,  if  steam  is  shut  off  while  the  train  is 
moving  at  2,  3,  or  4  times  that  vel,  and  consequently  with  2,  3,  or  4  times  the  moving 
force,  it  will  work  through  4,  9,  or  16  times  the  dist  of  the  first  case,  before  coming 
to  rest.  If  bullets  of  equal  wt  be  fired  with  vels  proportioned  to  each  other  as  1,  2, 
8,  they  will  respectively  penetrate  a  plank  to  depths  as  1,  4,  9.  If  an  engine,  water- 
wheel,  &c,  works  steadily  in  a  mill,  grinding  at  2,  3,  or  4  revolutions  per  min,  it  per- 
forms only  2,  3,  or  4  times  as  much  uniform  work  per  min,  as  when  at  but  1  rev  per 
min.  But  if  steam  or  the  water  be  suddenly  shut  off  at  2,  3,  or  4  revs  per  mm,  then 
the  2,  3,  or  4  times  quantity  of  moving  force  accumulated  in  the  machinery  at  that 
moment,  will,  as  ivorking  force,  run  the  mill  through  4,  9,  or  16  times  as  many  revs 
before  stopping,  as  if  shut  off  at  1  rev.  If  a  rolling  ball,  started  against  a  row  of 
bricks,  will  overcome  their  resistances,  and  knock  them  down  for  a  distance  of  4  ft; 
then,  if  it  be  started  at  a  vel  3,  4,  or  5  times  as  great,  it  will  overcome  and  knock 
them  down  for  dists  of  9,  16,  and  25  times  4  ft ;  and  in  but  3,  4,  or  5  times  the  time. 
But  in  all  these  cases  the  rate  of  the  work  done,  that  is,  the  quantity  done  t'n  any  given  time,  as 
one  sec,  is  directly  as  the  vels.  Thus,  the  locomotive  whose  steam  Is  shut  off  at  20,  30,  or  40  miles 
an  hour,  will  require  but  2,  3,  or  4  times  as  many  seconds  for  running  its  4,  9,  or  16  dist  before  it 
comes  to  a  stop ;  in  other  words,  when  its  moving  force  is  2,  3,  or  4  times  as  great,  it  will  overcome 
but  2,  3,  or  4  times  the  resistances  in  the  same  time ,  although  the  total  amount  of  resistances  over. 
come  will  be  as  4,  9,  and  16.  And  so  with  the  other  examples. 

Rem.    We  know  that  the  dist  through  which  a  body  must  fall  by  the  unif  ac- 

cel  force  of  gravity  in  order  to  acquire  any  given  vel  and  moving  force,  is  as  the  square  of  said  vel ; 
but  directly  as  the  time  of  falling.  Also  if  a  body  is  thrown  vert  upwards  with  any  given  vel  or  force, 
grav  will  retard  it  unif,  and  the  height  to  which  it  will  rise  by  the  time  that  grav  destroys  all  the 
force  with  which  it  was  thrown,  will  be  as  the  square  of  said  vel ;  but  the  time  will  be  directly  as  said 
vel.  And  so  with  anybody  moving  in  any  direction,  and  acted  upon  by  any  unif  accel  or  retard  force 
whatever.  It  will  either  acquire  or  part  with  its  moving  force  within  dists  proportionate  to  the  square 
of  its  vel,  and  in  times  proportionate  to  its  simple  vel.  See  Caution*  Gravity,  p  587. 

Art.  24.  Vis  viva,  or  living*  force.  The  preceding  article  serves  as  an 
introduction  to  this  subject ;  of  which  we  shall  endeavor  to  give  some  idea  in  plain 
language.  The  term  itself  is  merely  one  of  those  absurdities  to  which  savants  re- 
sort, in  order  to  impart  an  air  of  mystery  to  their  writings.  We  might  with  the 
same  propriety  speak  of  a  brickbat  viva,  or  a  living  hod  of  mortar. 

We  have  seen  in  Art  23,  that  if  that  portion  of  force  in  a  body  which  is  occupied  in  giving  motion 
alone  to  the  body,  be  suddenly  converted  into  working  force,  the  quantity  of  work  which  it  would 
perform  against  uniformly  retarding  resistances,  before  being  entirely  destroyed,  or  coming  to  rest, 
would  be  in  proportion  to  the  square  of  its  vel  at  the  time  of  beginning  to  work.  Now,  if  "  vis  viva," 
or  "  living  force,"  were  merely  the  name  given  to  this  force;  or  to  the  quantity  of  work  done  by  it, 
(as  measured  in  ft-Ibs,  by  mult  the  resistances  in  Ibs,  by  the  dist  in  ft  through  which  they  were  over- 
come,) the  expressions,  although  silly,  would  still  convey  an  idea  readily  understood  by  practical 
men.  We  could  then  say,  for  instance,  of  a  moving  body,  that  its  vis  viva  was  100  foot-fibs  ;  mean- 
ing thereby  that  it  would  overcome  a  uniform  resistance  of  I  ft  through  a  dist  of  100  ft;  or  a  resist- 
ance of  100  Ibs  through  a  dist  of  1  ft,  &c.  But  scientiflc  writers  apply  the  terms  to  a  purely  imagi- 
nary quantity,  equal  to  twice  this  :  and  which  does  not  exist  in  any  body,  under  any  circumstances. 
The  reason  they  do  so  is  that  it  facilitates  some  of  their  calculations.  But  the  practical  engineer 
need  not  concern  himself  with  either  this  reason,  or  vis  viva  itself;  the  simple  statement  of  facts 
contained  in  the  preceding  and  following  articles,  probably  contains  all  that  it  is  essential  for  him  to 
know  on  the  subject  of  moving  force,  converted  into  uniformly  retarded  working  force. 

Art.  25.  The  actual  total  amount  of  worlc,  in  ft-lbs,  that  can 
be  accomplished  by  a  given  moving  force,  when  converted  into  unaided  working 
force,  is  found  by  div  the  square  of  the  vel  of  the  body  in  ft  per  sec,  by  64.4 ;  and 
then  mult  the  quot  by  the  wt  of  the  body  in  pounds ;  or,  in  shape  of  a  formula, 

Uniformly  retarded  _  wt  of  tJie  working  v  S1  of  it*  vel  infi  Per  sec 

working  force  body,  in  Ibs        *  64.4. 

Or  to  wt  X  fall  in  ft  reqd  to  give  the  vel.    See  "  Falling  Bodies,"  p  587. 
An  imaginary  force  equal  to  double  this,  will  be  the  vis  viva,  or  living  force  of 
the  savants :  or 

Vis  viva  =  Wei9f't  °f  body  v  s?  «/*'<*  wl  ™  ft  per  sec* 
in  Ibs  32.2. 

*  For  the  purposes  of  abstract  science,  it  is  not  sufficiently  exact  to  measure  the  quantity  of  matter 
by  its  wt;  because  the  wt  or  gravity  of  a  body,  as  shown  by  a  spring  balance,  varies  somewhat  in 
diff  latitudes,  and  at  diff  heights  above  the  fevel  of  the  sea.  Therefore,  the  vel  with  wnich  it  will 
fall  by  that  gravity,  of  course  becomes  a  proper  measure  of  the  gravity  ;  because  it  also  varies  in  the 
same  proportion;  all  moving  force  being  in  proportion  to  the  vel  it  "imparts.  Therefore,  if  the  wt 
of  a  body  at  any  place,  be  div  by  the  vel  imparted  by  gravity  in  one  sec  at  the  snme  place,  (and  called 
the  acceleration  of  gravity  of  that  place.)  the  quot  will  be  the  same  at  nil  places.  And  ?ince  the 
quantity  of  matter  undergoes  no  change  at  diff  places,  the  measure  of  that  quantity  should  likewise 


456 


FORCE   IN    RIGID   BODIES. 


By  way  of  practical  application  of  the  first  formula,  snppose  that  a  railroad  train  of 
500  tons,  is  moving  at  the  rate  of  15  miles  an  hour ;  and  that  steam  is  suddenly  shut 
off;  into  how  much  working  force  will  this  moving  force  be  converted?  Here, 
500  tons  =  1,120,000  ft>s ;  and  15  miles  per  hour,  =  22  feet  per  sec;  and  the  square  of 
22  =  484 ;  Hence,  by  the  formula,  we  have, 

Uniformly         .„,.•„;.*      r>d*  484 

retarded  =  JJjjS;  X  ,-7—  =  1,120,000  X  -7-  =  1,120,000  X  7.5  =  8,400,000  ft-ft>s. 
working  force  °J  body  b4-4  &A 

Now,  how  far  will  this  working  force  work  the  train  before  its  coming  to  rest,  sup- 
posing the  resistance  of  friction,  air,  &c,  to  be  uniform,  and  to  amount  to  10  Jbs  per 
ton  wt  of  train  ;  or  to  a  total  resistance  of  500  X  10  =  5000  Ibs.  This  resistance  being 
assumed  to  present  itself  equally  at  every  point  along  the  road,  the  reqd  dist  evi- 

8400000  ft  H*8 

dently  becomes  =  1680  feet;  for  1680  X  5000  =  8,400,000  ft-Ibs. 

In  practice,  the  resistance  of  the  air  certainly  would  not  be  uniform  ;  but  if  we 
were  to  introduce  its  variableness  into  the  question,  the  solution  would  become  very 
difficult.  The  unif  retard  force  div  by  the  dist  run  gives  the  total  fric. 

Art.  26.  If  an  engine  has  not  sufficient  power  to  overcome  the  friction  of  a  train, 
and  to  impart  some  (no  matter  how  little)  motion,  by  the  first  stroke  of  the  piston,  it 
will  not  be  able  to  do  so  by  any  greater  number  of  strokes.  For  the  motion  of  the 
first  stroke  was  entirely  converted  into  strain  by  the  resisting  force  of  friction,  &c,  of 
the  train;  and  by  the  time  a  second  stroke  can  apply  a  second  instalment  of  force, 
renewed  friction  is  also  ready  to  react  against  it,  and  thus  destroy  it ;  and  so  with 
any  number  of  strokes. 

The  friction  of  the  train  is  caused  by  its  gravity  or  wt:  and,  since  gravity  acts  as  an  unceasing 
stream  of  force,  continually  pouring  into  every  body  ;  and  being  continually  destroyed  by  the  reaction 
of  whatever  the  body  is  resting  upon  ;  so,  in  like  manner,  does  it  maintain  a  constant  stream  of  fric- 
tion between  the  body  itself,  and  what  it  rests  upon.  It  is  plain  that  the  rolling  friction  of  the 
wheels,  and  the  sliding  friction  of  their  journals,  along  one  mile  of  road,  are  not  identically  the 
same  as  that  on  the  next  mile ;  although  it  may  be  precisely  the  same  in  amount.  So  with  quiescent 
friction;  that  of  one  instant  is  not  identically  the  same  as  that  of  the  next  instant;  for  if  it  were, 
then  the  strain  from  the  first  stroke  of  the  piston  would  destroy  at  least  a  portion  of  it;  and  a  few 
more  strokes  the  whole  of  it. 

Art.  27.  If  a  body  were  perfectly  rigid,  any  force  imparted  to  it  at  one  end 
would  at  the  same  instant  reach  the  other  end,  no  matter  how  long  it  might  be ;  and 
moreover,  no  amount  of  force  could  break  it,  no  matter  how  thin  it  might  be.  But 
no  bodies  are  perfectly  rigid;  and  hence  their  inherent  tensile  or  compressive  forces, 
or  strengths,  will  yield  to  any  extraneous  force  applied  in  excess.  Therefore,  if  we 
wish  to  transmit  a  great  amount  of  force  through  a  body  which  would  otherwise 
crush,  or  pull  apart  under  it  we  must  take  time,  and  transmit  it  by  degrees. 
Thus,  the  coupling  links  of  a  long  train  of  cars,  would  snap  instantly  under  a  pull 
sufficient  to  throw  back  into  the  train,  at  one  effort,  a  moving  force  of  20  miles  an 
hour;  if  an  engine  of  such  power  existed.  In  practice  we  are  therefore  compelled  to 
transmit  the  force  from  the  engine  to  the  train  in  instalments  so  small  as  not  to  ex- 
ceed the  tensile  force  of  the  links.  Rapid  speeds  are  produced  by  the  force  thus 
accumulated  in  the  train  by  degrees.  See  Art  20,  p  453. 

The  cogs  of  wheels  which  transmit  force  from  the  motive  power,  to  the  working  points  in  ma- 
chinery, are  frequently  broken  if  the  power  is  applied  too  suddenly ;  that  is,  too  much  of  it  at  once. 
If  a  pistol -ball  be  thrown  with  very  little  vel,  against  a  pane  of  glass,  its  force  will  have  time  to  dif- 
fuse itself  ovar  the  whole  pane,  and  will  probably  crack  and  shatter  it  in  every  direction  ;  but  if  it 
be  shot  with  great  vel  from  a  pistol,  it  will  frequently  pass  through  the  glass  so  quickly  as  not  to  give 
its  force  time  to  spread  over  the  whole  pane ;  but  it  will  merely  allow  it  time  to  act  upon  the  small 
circular  piece  which  comes  into  immediate  contact  with  it;  and  it  will  therefore  cut  out  this  small 
piece  neatly,  and  carry  it  away.  A  person  may  safely  skate  across  a  thin  piece  of  ice,  which  would  break 
under  his  wt  at  rest ;  for,  before  the  ice  has  time  to  bend  sufficiently  to  break,  the  load  is  removed 
from  it. 

undergo  none.    Therefore  scientific  men  adopt  - 

body  ;  and  they  call  the  resulting  quot  the  "mass  "  of  the  body,  to  distinguish  it  from  mere  wt.  Thus 
at  any  place  where  the  acceleration  of  gravity  is  32.2  feet  per  sec,  and  where  a  body  weighs  20  Ibs  by 

•20 

a  spring  balance,  the  body  s  mass,  or  scientific  quantity,  is  equal  to  — —  rr  .621.  To  the  prac- 
tical man,  this  mass  or  quantity  conveys  no  idea  whatever:  but  it  is  plain  that  the  ordinary  measure 
by  weight  cannot  be  perfectly  correct,  because  the  weight  changes  at  diff  places,  while  the  quantity 
remains  the  same;  and  the  measure  by  size  would  be  equally  incorrect,  because  the  size  varies  with 
the  temperature. 

Since,  therefore,  the  only  absolutely  correct  measure  of  quantity  in  a  body  is  the  scientific  mass  ; 
and  since  the  imaginary  vis  viva  is  the  quantity,  mult  hv  the  square  of  the  vel.  we  have  vis  viva 
represented  strictly  by.  Mass  X  Vel2;  or  the  M.V2  of  scientific  writers.  In  science,  the  mass  of  100 
Ibs  of  iron  is  equal  to  that  of  100  Ibs  of  cotton.  The  greatest  discrepancy  that  can  occur  at  various 
heights  and  latitudes,  by  adopting  wt  as  the  measure  of  quantity,  would  not  be  likely  to  exceed  1  in 
300;  or,  under  ordinary  circumstances,  1  in  1000. 


FORCI 


RIGID   BODIES. 


457 


A  string  may  safelv^dstain  a  wt  of  1  ft)  suspended  from  our  hand ;  and  if  we  wisb 
to  impart  a  great  u^rward  vel  to  the  wt,  we  evidently  can  do  so  only  by  imparting  t$ 
it  a  great  force  ixand  we  may  do  this  by  jerking  the  string  violently  upward.  But  if 
it  has  not  tensile  force,  or  strength,  sufficient  to  transmit  this  force  all  at  once  from 
our  hand  to  the  wt,  it  will  break.  It  plainly  is  not  broken  by  the  wt,  but  by  the 
excessive  force  which  we  endeavored  to  pass  along  it.  We  might  safely  give  the  wt 
all  the  vel  or  force  we  desire,  by  simply  raising  the  string  slowly  at  first,  and  more 
and  more  rapidly  by  degrees ;  thus  putting  the  force  into  the  body  gradually,  in 
instalments  too  small  to  break  the  string.  Some  imagine  that  the  string  is  broken 
by  the  so-called  inertia  of  the  wt,  which  they  say  causes  it  to  resist  moving  force; 
and  that  we  actually  feel  its  resistance;  and  that  it  is  made  apparent  by  a  spring 
balance,  if  we  hold  the  balance  in  our  hand,  with  the  string  and  wt  attached  to  it. 
Tnere  is  no  doubt  that  when  we  jerk  the  string  upward,  the  balance  will  indicate 
that  there  is  an  increased  strain  upon  its  spring.  But  this  strain  arises  not  from 
resistance  of  the  wt,  but  from  the  direct  action  of  the  force  which,  as  motion  only,  we 
have  imparted  to  the  string,  to  be  by  it  conveyed,  as  work,  to  the  wt.  Work,  because, 
when  it  reaches  the  wt,  it  has  not  only  to  impart  motion  to  it ;  but  also  to  strain 
against  its  gravity  force.  The  wt  cannot  receive  great  vel,  unless  we  impart  to  it 
great  force;  this  force  is  plainly  imparted  through  the  medium  of  the  string:  and  if 
we  attempt  to  impart  too  much  at  once,  the  string  must  break.  The  breaking  of  the 
string  by  the  pull,  is  the  same  as  the  breaking  or  bending  of  a  nail  under  too  heavy 
a  blow  of  a  hammer:  in  both  c;»ses  the  failure  is  caused  by  the  attempt  to  transmit 
at  once  a  quantity  of  force  which  the  inherent  strength  of  the  body  is  insufficient 
to  sustain.  Whether  that  force  is  motion  or  strain  makes  no  difference. 

Springs  ease  the  force  of  blows  because  their  elasticity  gives  them  time, 

by  gradually  yielding,  to  receive  the  whole  action  of  the  force,  and  react  against,  or  destroy  it,  by 
degrees,  or  part  at  a  time.  The  imparting,  and  the  receiving,  of  force  are  often  attended" by  the 
same  effects.  If  we  hold  our  hand  still,  and  let  a  hard  play-ball  strike  it,  the  hand  will  experience 
the  same  sensation  as  if  we  first  throw  the  ball  upward ;  and" afterward  strike  it  with  our  open  hand, 
at  the  moment  it  is  turning  to  fall,  and  is  consequently  still,  or  at  rest.  In  the  first  case  the  ball  im- 
parts its  moving  force,  as  strain,  to  the  hand  at  rest;  in  the  second,  the  hand  imparts  its  moving 
force  as  strain,  to  the  ball  at  rest.  In  the  first,  we  received ;  in  the  second,  we  gave  away  force ;  and 
in  both  cases,  the  effect  on  the  hand  was  the  same. 

Art.  28.  Composition  and  resolution  of  forces.  We  hnve  already 
said  that  when  diff  forces  are  imparted  *  (whether  so  applied,  or  not)  in  the  same 
direction  to  a  rigid  body  free  to  move  unresistedly,  they  all  act  as  motion  alone,  in 
that  same  direction.  If  two  equal  forces  are  imparted  in  diametrically  opposite 
directions,  they  mutually  destroy  each  other  entirely,  as  strain  (pull  or  push)  against 
each  other  r  thereby  producing  strain  among  the  particles  of  the  body;  but  having 
no  tendency  to  move  the  body,  as  a  whole,  in  either  direction.  If  unequal,  and  in 
diametrically  opposite  directions,  the  whole  of  the  small  one,  and  a  part  of  the  large 
one,  equal  to  the  small 
one,  destroy  each  other 
as  strain;  while  their  diff 
remains  as  motion,  (or 
a  moving  of  the  whole 
body,)  in  its  original  di- 
rection. But  if  t w< 

"*m 

— d 


m. 


-  .n 


forces,  a  o  and  b  o,  Figs 
9,  whether  equal  or  un- 
equal, are  imparted  at 
the  same  time  to  an  un- 
resisting rigid  body  o, 
in  directions  either  con- 
verging toward ;  or  di- 
verging from,  the  same 

point  o,  at  any  angle  whatever ;  then  the  body  o  cannot  possibly  be  kept  at  rest  by 
them ;  or  in  other  words,  equilibrium  cannot  exist  between  them  ;  or  they  cannot 
balance,  or  completely  react  against  each  other  ;  the  body  must  move.  Equal  parts 
of  each  of  the  two  forces  will  mutually  destroy  each  other  as  strain  among  the  par- 
ticles of  the  body;  while  the  remaining  portions  will  unite  to  constitute  a  single 
force  r  o,  which  will  move  the  whole  body  in  a  direction  o  d,  in  the  line  r  o  extended  : 
and  which  direction  o  d  will  always  be  somewhere  bftvjeen  those  in  which  the  separate 
forces  would  have  moved  it. 

If  we  lay  off  c  o  and  t  o  by  any  convenient  scale,  to  represent  respectively  the  amount  of  the  forces 
a  o  and  b  o.  and  then  complete  the  parallelogram  o  c  r  t ;  the  diag  r  o,  measured  by  the  same  scale,  will 
represent  both  the  direction  and  the  amount,  of  the  single  remaining  force. 

*  Tt  is  absolutely  necessary  to  keep  distinctly  in  mind  the  diff  between  applied  and  imparted  force. 
Writers  carelessly  confound  the  two  very  frequently.  See  Art  19. 


458 


FORCE   IN   RIGID   BODIES. 


The  same  procesa  will  answer  also  for  forces  which  instead  of  motion,  produce  strain,  not  only  in 
the  particles  of  the  body,  but  in  the  body  itself  considered  as  a  whole  ;  or,  in  other  words,  a  tendency 
to  press  or  pull  the  entire  body  in  a  certain  direction.  Thus,  suppose  that  two  men  were  either  pull- 
Ing  or  pushing  with  the  forces  c  o  and  t  o ;  trying  in  vain  to  detach  a  piece  o  of  rock,  from  a  cliff  of 
which  it  forms  a  portion  ;  and  which,  by  its  inherent  force  of  cohesion  to  the  cliff,  defies  their  efforts. 
Here  we  have  a  case  of  extraneous  forces,  resisted,  or  reacted  against,  or  balanced,  by  strength  o/ 
material. 

As  in  the  case  of  motion,  the  two  forces  partly  destroy  each  other  as  strain  among  the  particles  of 
the  body;  and  the  remainders  combine  to  forni  the  single  force  ro,  which  tends  to  move  the  whole 
body  toward  d.  The  rock  resists  this  single  force,  by  a  cohesive  force  precisely  equal,  and  diametri- 
cally opposite  to  it;  and  so  long  as  it  does  so,  there  is  strain  but  no  motion.  The  piece  of  rock  may 
have  strength  enough  to  oppose  a  much  greater  resistance ;  but  cannot  actually  exert  it  unless  the 
men  also  exert  more  force. 

In  the  matter  of  comp  and  res  of  forces,  it  must  be  remembered  that  when  force  i»  applied  to  a 
body  in  order  to  produce  motion,  care  must  be  taken  that  there  is  no  other  force  to  prevent  it ;  but 
when  the  force  is  intended  to  produce  strain,  it  is  equally  necessary  that  other  force  should  be  present 
to  oppose  it ;  for  strain  is  the  opposition  of  forces. 

The  fig  ocrt,  Figs  9,  is  called  the  parallelogram  of  forces.  The  two 
original  forces  co,  to,  are  called  the  components  of  the  force  ro;  which  results  from 
their  joint  action ;  and  the  force  r  o  is  called  the  resultant  of  the  original  ones  which 
compose  it.  The  principle  of  the  parallelogram  of  forces,  than  which  there  is  none 
more  important  in  the  whole  range  of  mechanical  science,  may  be  expressed  thus : 
If  any  two  forces,  (both  motions,  or  both  strains,)  whose  directions  either  converge 
toward,  or  diverge  from,  the  same  point,  be  represented  both  in  quantity  and  in  di- 
rection by  two  adjacent  sides  of  a  parallelogram :  then  will  their  resultant  be  simi- 
larly represented  by  the  diag  of  the  parallelogram,* 


REM.  1.  If  one  of  the  forces,  as  c,  upper  Fig  !%,  is 
a  pull,  and  the  other  a  push,  then  to  find  their  result- 
ant o  t  we  must,  before  drawing  the  parallelogram  of  forces,  move  (or  imagine 
to  be  moved)  one  of  the  forces  to  the  opposite  side  of  the  point  o,  so  as  to  cbange 
it  from  a  pull  to  a  push,  or  vice  versa,  so  that  both  shall  be  pulls,  or  both  pushes, 
as  shown  by  the  two  lower  figs.  Otherwise  we  should  obtain  a  wrong  resultant 
no  of  the  top  fig.  Either  a  push  or  a  pull  equal  to  ot,  if  applied  at  o,  would  be 
equal  in  effect  to  the  push  a  and  the  pull  c.  The  remark  is  of  frequent  use  when 
finding  strains  in  bowstring  and  crescent  trusses  ;  as  in  many  other  cases. 

Item.  2.  When  any  three  forces  as  a,  b,  c,  form- 
ing only  two  angles  axb  and  bxc,  balance  each  other 
at  any  point  x,  then  a  straight  line  as  oe  can  be 
drawn  through  that  point  so  that  all  three  forces 
shall  be  on  one  side  from  it ;  then  also  a  parallelogram  x  n  can  be 
drawn  on  the  three  lines  a,  6,  c,  having  the  middle  line  b  for  its 
diagonal ;  and  this  diagonal  will  be  of  a  different  character  from 
the  two  outer  forces  a  and  c;  that  is,  if  they  are  pulls,  it  will  be 
a  push,  and  vice  versa.  But  if  as  in  the  tnree  balancing  forces 
/,  i,  s,  three  angles  as  s  x  t,  txi,  sx  i,  are  formed,  neither  such  a  line, 
nor  such  a  parallelogram  can  be  drawn ;  and  the  three  forces  will 
all  be  alike,  all  pulls  or  all  pushes.  All  this  is  evident  from  the  two 
figures. 

REM.  3.  We  have  alluded  to  equal  parts  of  each  component  as  being  lost,  or  de- 
stroyed, by  reacting  against  each  other;  thus  producing  within  the  body  a  straining 
of  its  particles  ;  and  therefore  having  no  tendency  to  move,  push,  or  pull,  the  body 
os  a  whole,  in  any  direction. 

Let  6  a  and  c  a  be  any  two  components,  and  na  their  resultant.  From 
the  two  angles  b  and  c,  opposite  to  the  diagonal,  draw  bo  and  ct  at  right 
angles  to  the  diagonal;  or  to  the  diagonal  extended,  if  necessary,  as  in 
Fig  9%.  These  two  lines,  bo,  tc,  will  always  be  equal  to  one  another; 
whatever  may  be  the  lengths  and  directions  of  the  components  b  a,  ca. 
When  two  forces,  as  5  a,  ca,  are  imparted  at  a.  there  occurs  a  loss  offeree 
equal  to  what  would  result  from  the  reaction  of  two  forces  equal  to  ft  o  and 
c  i.  It  is  lost  by  becoming  strain  against  the  cohesive  forces  of  the  parti- 
cles which  compose  the  body  a.  In  anticipation  of  what  is  said  in  Art  31, 
we  will  state  that  the  force  6  a  may  be  regarded  as  made  up  of  the  forces 
60,  oa;  and  the  force  ca,  of  ct,  ta;  which  act  also  in  those  directions, 
when  ft  a  and  c  a  converge  toward  a,  as  in  Fig  9V6  ;  or  in  the  directions 
a  o,  o  5,  and  a  i,  i  c,  when  the  forces  diverge  from  a,  as  in  Fig  9%.  In  either  case,  however,  these 
forces,  ft  o,  ao,  ct,  »  a,  &c,  must  be  considered  as  being  imparted  at  a.  This  being  supposed,  It  be- 
comes plain  that  when  6  a  and  c  a  meet  at  a,  inasmuch  as  ft  o  and  ct  destroy  each  other  as  strain 
against  the  internal  cohesive  forces  of  the  body,  there  remains  nothing  to  act  upon  the  body  consid- 
ered as  a  whole,  except  oa  and  ta;  which,  being  together  equal  to  na,  (as  seen  in  the  fig.)  are,  in 
other  words,  equal  to,  or  actually  compose,  the  resultant  n  a  of  the  two  components  6  a,  ca.  See  RetnS. 

*  Components  and  Resultants  may  be  calculated  by  the  form- 
ulas in  Art  45,  p  472,  when  a  diagram  is  not  considered  sufficiently  accurate. 


^Qfim  ] 


RIGID   BODIES. 


459 


We  conceive  that  ea«fiof  the  original  forces  endeavors  as  it  were  to  compel  the  other  to  leave  its 
own  course,  aud  fpWow  that  of  its  antagonist;  and  the  struggle  continues  until  they  have  succeeded 
in  forcing  each  other  into  the  same  direction.  This  is  of  course  effected  by  their  reactions  against 
each  other;  and,  as  occurs  in  all  cases  of  reaction,  they  expend  equal  parts  of  their  forces  on  each 
other.  When  the  two  forces  act  in  diametrically  opposite  directions,  where  there  is  no  neutral  diag 
direction  that  can  be  adopted,  there  is  no  alternative  but  for  the  larger  force  to  react  against  or  de- 
stroy the  smaller  one  entirely  ;  thereby  losing  an  equal  amount  of  its  own  force.  Its  remains  totter 
«n  slowly  in  their  former  unchanged  direction.  The  writer  can  see  no  difference  of  principle  between 
the  reaction  of  opposite  forces;  that  of  oblique  ones;  and  that  of  those  at  right  angles  to  each  other. 

REM.  5.  When  the  direction  a&.  Fig  9%,  of  one  of  the  forces,  forms  an  angle  ft  a  n, 
greater  than  90°,  with  the  diagonal,  the  shape  of  the  parallelogram  of  forces  becomes 
such  that  the  two  equal  lines  bo  and  ct,  cannot  be  drawn  at  right  angles  to  the  diag 
a  n  itself;  or  within  the  parallelogram ;  in  which  case  the  diag  must  be  extended 
each  way,  as  to  o  and  t;  and  the  lines  bo,  ct,  must  be  drawn  at  right  angles  to  the 
extensions.  .  . 

When  this  occurs,  the  component  forces  a  o,  a  t,  cannot  as  in  Fig 
9J^  be  measured  on  the  diag  a  n  of  the  parallelogram  ;  because  they 
will  be  greater  than  it ;  but  must,  like  bo,  ci,  be  measured  outside 
of  the  fig.  And  here  it  must  be  remembered  that  oo  and  ai  no 
longer  measure  forces  acting  (like  those  in  Fig9«^)  in  the  same  di- 
rection. Thus  the  strain  along  a  b  may  be  considered  (see  Comp 
and  Res  of  Forces.  Art  31)  to  be  made  up  of  two  forces  imparted 
at  a  ;  namely,  a  hor  force  equal  to  o  b,  and  a  vert  one  equal  to  a  o, 
acting  upward.  And  the  strain  along  a  c,  as  made  up  of  one  hor 
force  equal  to  i  c,  and  a  vert  one  ai.  (greater  than  the  whole  diag,) 
acting  downward ;  both  of  them  imparted  at  o.  Hence,  the  re- 
sultant on  we  find  is  equal  to  the  diff  between  the  two  vert  compo- 
nents a  o  and  a  i.  Thus  it  is  seen  that  this  shape  of  the  parallelo- 
gram in  no  way  affects  the  principle  laid  down  in  Remark  3. 


Art.  29.    According  to  Art  18,  the  force  w  e,  Fig  10,  may  be  considered  as  im- 
parted to  the  rigid  body  B  at  any  point  whatever  in  its  line  of  direction  we;  also, 
the  force  xi,  at  any  point  in  its  direction  x  d ;  conse- 
quently, both  of  them  may  be  considered  as  imparted  at 
the  same  point  a;  inasmuch  as  it  is  situated  in  both  these 
lines.    Hence,  it  is  immaterial,  so  far  as  regards  the  effect 
of  those  two  converging  forces  upon  the  body  considered 
as  one  entire  rigid  mass,  whether  they  are  actually  im- 
parted like  zo  and  yn,  at  the  same  point  o;  or  like  we  and 
25 1,  at  diff  points  i  and  e.    For  in  either  case  their  result- 
ant, or  joint  effect  upon  the  body  as  a  whole,  is  precisely 
i  the  same ;  namely,  a  tendency  to  move  the  body  in  the 
i  same  line  of  direction  oat.    This  tendency  will  actually        _        /( 
\  produce  motion  if  no  opposing  force  prevents;  otherwise     JmlOL 
1  it  will  produce  strain  in  the  body.  J    ~ 

REM.  1.    Hence  the  resultant  R,  of  two  converging  forces  F/,  Fig  10J4;  or  of  two 
diverging  ones  F/,  Fig  10%,  acting  in  the  same  plane,  but  imparted  at  diff  points 

£. 


i  NOTE.  The  savants  now  call  "Dynamics"  Kinetics;  "Motion"  they  call 
^Kinematics;  and  u  Working  Force,"  Energy.  They  subdivide  Energy  into 
Potential  and  Kinetic.  Potential  energy  is  that  whose  work  is  measured  by 
Resistance  X  Distance,  as  in  Arts  11  and  22.  Kinetic  Energy  performs  work  meus- 

.     wt  X  vel^in  ft  per  sec 
uredby  —       — _ — -     — ;  as  jn  Arts  23  and  25.    The  employment  of  such 

terms  in  the  study  of  mechanics  merely  muddles  the  mind  and  memory  of  the 
young  engine^,-. 


460 


FORCE   IN   RIGID    BODIES. 


\ 


Fiq  11 


of  a  rigid  body  W,  may  be  found  as  readily  as  when  imparted  at  the  same  point;  as 
at  o,  Figs  9,  or  Fig  10. 

Thus,  produce  their  lines  of  direction,  either  forward  as  in  Fig  10>£  :  or  backward  as  in  Fig  10%  ; 
as  the  case  may  require,  until  they  meet,  as  at  b.  Make  b  a  by  any  scale,  equal  to  the  force  /;  and 
b  c  equal  to  the  force  F.  From  a  and  c,  draw  lines  respectively  parallel  to  6  c  and  b  a  ;  thus  complet- 
ing the  parallelogram  of  forces,  baic.  The  diag  6  i  of  this'parallelogram,  measured  by  the  same 
scale,  will  represent  the  reqd  resultant  R  both  in  quantity,  and  in  direction.  It  is  thus  seen  that  it 
is  not  necessary  that  the  point  b  shall  be  in  the  body  itself. 

REM.  2.  It  is  perhaps  almost  useless  to  again  remind  the  young  student  that  the  bodies  are  all  along 
assumed  to  be  rigid ;  or  inelastic,  and  incapable  of  being  broken  or  bent  by  the  imparted  forces.  For 
otherwise  the  force/,  in  Fig  10>$,  might  split  off  the  top  of  the  body  ;  or  F  might  crush  to  dust  its 
toe  t;  or  both  might  penetrate  it.  But,  assuming  that  the  material  is  sufficiently  strong  to  resist 
such  splitting,  crushing,  and  penetration,  we  at  present  confine  ourselves  to  the  effect  of  the  forces, 
whether  as  motion,  push,  or  pull,  upon  the  body  as  a  whole.  The  splitting,  crushing.  &c,  is  a  mat- 
ter  that  must  be  considered  under  the  head  of  Strength  of  Materials.  It  is  of  course  quite  as  neces- 
sary in  practice  to  pay  attention  to  these  effects  as  to  the  others,  but  it  must  be  done  by  a  separate 
process. 

.T1.  Art.  3O.     Since  the  effect  produced  upon  a  rigid  body  (con- 

sidered as  a  whole)  by  the  resultant  (ac,  Fig  11)  of  any  two  forces 
(6  c,  dc)  tending  to  or  from  the  same  point,  is  the  same  as  the  joint 
effect  of  those  two  forces  themselves,  it  follows  that  if  we  oppose 
to  those  two  forces  a  third  one  (nc)  equal  to  the  resultant  (ac), 
and  diametrically  opposite  to  it,  that  this  third  force  will  com- 
pletely react  against,  balance,  or  destroy  said  two  forces;  or  rather 
their  remains.  It  is  frequently  necessary  to  consider  such  a  third 
force,  (n  c,)  equal  and  opposite  to  a  resultant  (a  c);  and  inasmuch 
as  we  do  not  know  that  any  specific  name  has  been  applied  to  it, 
although  one  is  needed,  we  suggest  anti-resultant.  Re- 
sultant (a  c)  may  be  defined  to  be  a  single  force  which  will  pro- 
duce upon  a  body  considered  as  a  whole,  the  same  result  that  its 
components  (6  c,  dc)  produce.  Or  as  a  force  which,  if  its  direction 
were  reversed,  (thus  making  an  anti-resultant,)  would  balance  its  components. 

In  the  preceding  Figs,  the  arrows  represent  pressures;  if  all  the  arrows  be  reversed,  thus  indi- 
cating pulls,  the^principle  and  processes  remain  precisely  the  same;  for  force  is  still  only  force  ;  and 
its  effect  upon  a  rigid  body,  considered  as  a  whole,  is  the  same  whether  it  act  as  a  pull,  or  as  a  push. 
Bee  Art  18. 

When  the  forces  diverge  from  the  same  point,  their  strain  is  a  pull,  or  a  tension  ;  when  they  con- 
verge toward  it,  a  push,  or  pres,  or  compression. 

Art.  31.  By  a  process  the  reverse  of  that  in  Art  28,  any  single  force,  od,  Fig  12, 
may  be  resolved  into  two  component  ones,  n  d,  m  d,  one  on  each  side  of  it,  and  in 

the  same  plane  with  it ; 
which  would  produce  the 
same  effect  as  it  upon  a 
rigid  body,  d,  (considered 
as  a  whole,)  by  merely 
drawing  from  d,  2  lines 
dg,  dt,  showing  the  di- 
rections of  the  two  forces; 
and  then,  drawing  from 
o  two  other  lines  o  n,  o  m, 
respectively  parallel  to  dg,  d  t ;  thus  completing  the  parallelogram  (dnom)  of  forces, 
upon  od  as  its  diag.  Then  measure  dn,  and  dm,  by  the  same  scale  as  od;  and  they 
will  give  the  amount  of  each  of  those  forces.  See  Rem,  p  260. 

It  is  plain  that  an  infinite  number  of  differently  proportioned  parallelograms,  such  as  d  n  o  m,  d  $  o  a, 
Ac,  may  be  drawn  upon  any  line  o  d  as  a  diag;  and  in  any  one  of  them,  two  adjacent  sides  will  rep- 
resent components  equal  in  effect  to  the  single  force  o  d,  represented  by  the  diag.  Thus  the  forces 
nd,md,  are  equal  to  o  d,  as  regards  their  effect  upon  a  rigid  body  d,  as  a  whole.  So  are  also  the 
forces  s  d  and  a  d  ;  consequently  the  effect  of  s  d,  and  a  d,  is  equal  to  that  of  n  d,  and  m  d.  It  will  be 
observed  that  the  longer  any  two  components  on  the  same  diag  are,  (as  nd,md,  longer  than  s  d,  a  d,) 
the  more  nearly  in  a  straight  line,  and  more  directly  opposed  to  each  other,  do  they  become :  and 
consequentlv  the  more  nearly  do  they  mutually  destroy  each  other ;  leaving  smaller  portions  of  each 
to  act  upon  the  body.  Thus  the  portion  of  the  great  forces  nd.  md,  left  to  act  upon  the  body  d,  is 
no  greater  than  that  of  the  small  forces  ad,  a  d  ;  this  remainder  being  in  both  cases  represented  by 
the  resultant  o  d. 

RKM.  Hence,  if  we  have  two  forces,  as  the  two  pulls 
a  b,  ac,  Fig  12^,  whose  amounts  and  directions  both  are 
given;  and  which  are  counteracted,  or  held  in  equilibrium, 
by  two  otner  forces  such  as  the  two  pulls  a/,  ae,  whose 
directions  alone  are  known,  it  becomes  easy  to  find  the 
amounts  a  d  and  a  o  of  these  last,  thus :  Complete  the 
.-  ,.  parallelogram  b  a  c  t ;  and  draw  its  diag  a  t.  Make  a  t 

fl;rt  1Q  1       /  \  equal  to  at,  and  in  a  line  with  it,     Complete  the  paral- 

-TICJ  J,C~9~    /  lelogram  adio;  then  plainly  a  d  will  be  the  amount  of 

**         **  f  the  force  in  the  direction  a/;  and  ao  that  in  the  direc- 

tion ae. 


FORCE   IN 


SID   BODIES. 


461 


Art.  32.  It  follows  from  the  fefregoing  articles,  that  a  single  force  cannot  be 
resolved  into  two  components,  onp^of  which  only  is  in  the  same  direction  as  that 
Jorce  itself;  for  if  a  line  representing  that  force  be  taken  as  a  diag,  it  is  self-evident 
that  no  parallelogram  can  he  drawn  upon  it  which  shall  have  any  of  its  sides  par- 
allel to  said  diag. 

Therefore  a  rope,  as  a  b.  F*£  15,  sustaining  a  wt  to,  so  long  as  it  remains  perfectly  vert,  that  is,  pre- 
cisely in  the  direction  of  the  force  of  gravity  of  the  wt,  will  receive  no  assistance  in  upholding  the 
wt  by  having  added  to  it  a  single  rope  as  06,  or  by;  or  one  extending  from  the  wt  itself  in  any  in- 
clined direction.  In  other  words,  a  perfectly  vert  rope  cannot  sustain  one  part  of  a  load,  and  one  in- 
clined rope  another  part.  All  this,  indeed,  is  a  result  of  the  fact  stated  in  Art  15;  that  any  force, 
however  great,  (as  the  vert  force  of  an  immense  suspended  weight  w,  Fig  15.)  will  be  turned  out  of 
its  direction  by  any  other  force,  however  small,  (as  a  slight  pull  from  a  rope  06,  or  by,)  unless  there 
i>e  some  third  force  Co  prevent  it.  In  the  present  instance,  this  third  force  might  be  a  third  rope ; 
for  the  rope  a  b  will  be  relieved,  and  still  remain  vert,  if  we  employ  two  oblique  ones  to  assist  it,  pro- 
vided they  be  exactly  opposite  each  other;  or,  in  other  words,  that  all  three  ropes,  or  forces,  be  iu 
%ue  plane. 


W 


So  also  in  the  case  of  a  vert  post  sustaining  a  load ;  the  pres  from  the  load  cannot  pass  vert  through 
the  axis  of  the  post,  if  the  load  at  the  same  time  is  partly  sustained  by  a  single  oblique  brace  pressing 
against  the  post.  Indeed,  such  a  brace,  by  turning  away  the  direction  of  the  strain  from  the  axis 
of  the  post,  may  very  materially  diminish  the  power  of  the  latter  to  sustain  the  load;  for  it  will  be 
found  under  Strength  of  Materials,  that  if  the  strain  along  a  post  or  column  does  not  pass  directly 
through  its  axis,  the  column  may  in  some  cases  lose  two-thirds  of  its  strength.  The  principle  of 
course  applies  to  force  in  any  other  direction,  as  well  as  vert. 

A  resultant  may  be  greater  or  less  than  either  one  of  its  two  oblique  components: 
bnt  it  can  never  be  greater,  or  even  quite  equal,  to  both  of  them :  on  the  plain  prin- 
ciple that  any  two  sides  of  a  triangle  are  greater  than  the  third  side.  If  the  com- 
ponents are  equal,  and  inclined  to  each  other  at  an  angle  of  120°,  the  resultant  will 
be  equal  to  one  of  them;  therefore,  the  same  weight  that  would  bnjak  a  single  vert 
rope,  or  post,  would  break  two  ropes  each  of  the  same  strength  as  the  single  one.  or 
two  posts,  inclined  120°  to  each  other.  If  the  angle  o  a  b,  or  y  a  fc,  which  either  of 
the  forces  form  with  the  diag  a  6,  exceeds  90°,  see  Rems  5,  of  pp  459,  465. 

Art.  33.  The  principle  of  the  parallelogram  of  forces  is  of  constant  applica- 
tion in  constructions  of  every  kind;  for  instance,  bridges,  centers,  roofs,  retaining- 
walls,  &c.  Figs  13, 14,  15, 16,  show  a  few  of  the  most  simple  cases  of  force  (the  load 
w)  applied  to  produce  strain  ;  by  reacting  against  opposing  forces  ya,  oa,  presented 
by  the  walls.  In  all  these,  the  load  w,  applied  at  a,  is  a  single  force  of  gravity ;  and 
consequently  acts,  in  a  vert  direction  downward.  It  is  to  be  resolved  into  two  com- 
ponent forces  in  the  direction  am,  an,  in  order  that  we  may  find  the  strains  which 
it  produces  (according  to  the  ordinary  phraseology)  along  the  pieces  am,  an,  so  that 
we  may  proportion  their  dimensions  to  resist  those  strains ;  which  strains  are  in 
fact  produced  by  the  reactions  of  the  three,  forces,  of  the  load,  and  the  two  walls.  To 
do  this,  in  all  the  figs,  from  a  draw  a  vert  line  a  6,  to  represent  the  direction  of  grav, 
or  of  the  force  in  the  load  w.  On  this  line,  lay  off  by  any  convenient  scale,  the  dist 
a  b  to  represent  the  amount  in  fbs,  tons,  &c,  of  the  load  w.  Also,  from  a  draw  the  two 
lines  am,  an,  in  the  directions  of  the  reqd  component  forces.  Then  complete  the 
parallelogram  of  forces,  by  drawing  lines  b  o,  b  y,  from  b,  respectively  parallel  to 
am,  a  n.  Then  will  a  o,  measd  by  the  same  scale  as  a  b,  give  the  amount  of  strain, 
whether  push  or  pull,  which  the  load  w  produces  along  the  piece  am;  and  in  like 
mariner  will  ay  give  the  amount  which  it  produces  along  the  piece  a  n. 

It  must  be  especially  borne  in  mind,  that  we  here  speak  only  of  the  amounts  and  directions  of  the 
strains  produced  by  the  extraneous  load  w  alone ;  without  reference  to  those  produced  by  the  weight 
of  the  pieces  themselves.  If  the  force  acting  at  a  is  not  vert,  but  oblique,  then  the  direction  of  «  b 
must  of  course  be  drawn  oblique  ;  but  if  the  force  at  a  is  gravity  or  wt,  it  mutt  be  vert. 

30 


462 


FORCE   IN  RIGID   BODIES. 


m-- 


This  mode  of  finding-  strains  does  not  apply  unless  a  w,  a  n  pass 

straight  to  supports  m,  n  able  to  react  against  them  in  the  same  straight  lines.  Thus  in  Fig  9,  p  251, 
with  a  load  at  Z  only,  the  parallelogram  would  not  give  the  strain  along  Z  W  because  at  W  there  is  no 
reacting  force  in  a  direction  from  H  to  W.  Footnote,  p  252. 

REM  1.  Fig  16^  will  explain  what  we  mean  in  saying  that  the  strains  e  t,  e  s,  are 
in  reality  produced  by  the  walls;  although  they  are  usually  ascribed  to  the  load  lt 
which  is  represented  by  the  diag  e  i.  We  have  said  that  a  force  cannot  produce  strain 
unless  there  is  opposing  force  to  strain  against.  Now,  when  we  place  the  force  of  the 
load  I  at  the  point  e,  it  is  evident  that  it  is  upheld  by  the  walls  at  A  and  B;  or  in 
other  words,  that  it  reacts  against  these  walls ;  and  the  walls  against  it.  The  wall  A 
furnishes  the  force  indicated  by  the  arrow  A  ;  and  which  may  be  considered  as  the 
resultant  of  the  hor  force  c;  and  of  the  vert  one  o.  So  also  the  force  B;  as  the  re- 
sultant of  m  and  n. 

Now  these  forces 
A  and  B  are  ap- 
plied at  the  point 
«,  just  as  well  as 
the  load  I  is ;  for 
they  pass  up  as 
pushes,  along  the 
rafters;  as  the 
force  of  I  passes  up 
as  a  pull,  along  the 
rope.  The  rafters 
and  rope  are  mere- 
ly the  mediums 
through  which  the 
three  forces  reach 

e ;  and  the  forces  in  passing  through  them  from  end  to  end,  of  course  produce  in 
them  strains  respectively  proportionate  to  the  forces.  Now,  the  forces  e  t  and  e.  s, 
which  are  usually  said  to  be  produced  by  the  load,  are  nothing  more  or  less  than  the 
two  forces  A  and  B,  produced  by  reaction  of  the  walls;  and  which,  for  convenience 
of  drawing  the  parallelogram  of  forces  in  practice,  are  laid  off  each  way  from  e.  We 
have  then  three  forces  t  e,  s  e,  and  e  i,  all  acting  at  e,  to  produce  strain  alone;  and 
this  they  must  do  by  straining  against  each  other. 

The  following  is  the  manner  in  which  they  do  so.  The  two  hor  components  m  and  c,  (which  will 
always  be  equal  to  each  other;  no  matter  how  ditfereut  the  slopes  of  the  two  ratters  may  be,)  being  dia- 
metrically opposite  in  direction,  react  or  strain  against,  or  balance,  each  other  ;  thereby  producing 
a  hor  strain,  equal  to  one  of  them,  throughout  every  part  of  each  rafter.  The  two  vert  components  o 
and  »,  (however  unequal  they  may  be,)  will  together  be  equal  to  the  load  I;  or  to  its  representative 
e  i ;  and  having  a  direction  exactly  opposed  to  it,  they  react  against,  or  balance  it ;  thereby  producing 
in  every  part  of  the  rafter  e  8,  a  vert  strain  equal  to  n;  and  in  the  rafter  e  t,  one  equal  to  o.  There- 
fore, since  n,  is  here  greater  than  o,  the  rafter  e  s  bears  more  of  the  load  I,  than  the  rafter  e  t  does; 
and  in  the  same  proportion. 

Thus,  we  see  that  every  part  of  each  of  the  three  forces  e  i,  et,es,  produces  strain,  by  balancing 
an  equal  part  of  one  of  the  others.  The  walls  really  oppose  to  the  load  no  force  greater  than  its  own  ; 
namely,  o  and  n.  against  e  i.  With  the  hor  components  TO  and  c,  the  walls  react  only  against  each 
other.  Hence  is  seen  the  error  of  saying  that  the  load  /.  produces  the  forces  e  s,  e  t. 

As  it  is  difficult,  however,  to  introduce  a  new  phraseology,  in  place  of  one  which,  although  errone- 
ous, is  in  universal  use,  we  also  shall  speak  of  component  strains  like  e  t,  c  s,  as  if  they  were  really  pro- 
duced by  the  resultant,  or  load,  e  i.  And  in  alluding  to  resultant  motion,  we  shall  probably  often  say 
they  are  the  effects  of  components,  instead  of  effects  of  their  remainders,  after  the  components  have 

Sartially  destroyed  each  other's  moving  forces  by  straining  against  each  other  to  produce  change  of 
irection. 

BBM  2.  The  truth  of  such  examples  as  Fig  14,  with  a  rope  or  string,  may  easily  be 
shown  by  means  of  two  spring  balances,  to  which  the  ends  m  and  n  of  the  string  may 
be  fastened.  Suspend  a  weight  w  from  the  string,  and  the  balances  will  show  the 
strains  along  a  m  and  a  n.  The  balances  must  be  held  in  inclined  positions. 

The  student  should  try  all  such  experiments.  This  one  will  show  that  in  proportion  as  the  two  parts 
am,  an,  of  the  rope,  approach  nearer  to  one  straight  line,  the  greater  will  be  the  strain  produced 
upon  them  by  any  given  load,  or  force  w;  and  so  great  will 
this  be,  that  if  the  weight  w  be  only  one  pound,  two  of  the 
strongest  men  cannot  strain  the  rope  perfectly  straight  be- 
tween them.  Or  if  they  stretch  the  rope  alone  to  as  nearly 
a  straight  line  as  possible,  and  if  then  a  weight  of  a  few  IDS 
be  suspended  from  it,  this  small  weight  will  pull  the  men 
closer  together.  Or  if  the  rope  be  stretched  nearly  straight 
.between  the  two  spikes  so  firmly  driven  as  to  require  a  great 
force  to  draw  them,  it  will  be  found  that  a  much  smaller 
force  applied  as  at  w,  will  draw  them  readily.  In  ether 
words,  a  rope  so  situated,  and  with  force,  or  power  w,  applied 
to  it  in  this  manner,  between  its  ends,  and  oblique  to  its  di- 
rection, becomes  a  machine;  for  by  it  power  may,  (to  use  the 


*P 
16J 


5CE   IN    RIGID   BODIES. 


463 


ordinary  incorrect  expression,)  be  gained.  It  is  called  the  fuilidllar  Iliactlilie  ;  or  some- 
times simply  the  COTCi.  Fig  16>6  shows  the  principle  on  which  this  machine  is  frequently  employed 
for  overcomingj/great  resistance,  r,  througa  a  short  distance,  by  a  small  power  p.  One  end  c,  of  a 
rope  c  d  r.  is  -orrnly  fixed.  The  rope  passes  over  a  pulley  d  ;  and  its  other  end  is  tied  to  the  resist- 
ance, or  load  r.  By  applying  a  small  downward  force  p,  at  the  center  of  the  rope,  drawing  it  down 
to  s,  the  load  r  is  thereby  raised  a  short  dist ;  for  the  same  great  strain  which  the  small  force  p  pro- 
duces from  «  to  d,  extends  also  down  the  rope,  from  d  to  r ;  except  a  slight  loss  produced  by  the 
friction  of  the  pulley.  Thus,  the  strain  along  the  back-stays  of  a  suspension  bridge,  is  equal  to  that 
on  the  main  chains  just  inside  of  the  suspension  piers;  supposing  the  cables  to  rest  upon  rollers.  In 
the  theoretical  consideration  of  ropes  and  chains,  they  are  in  most  cases  assumed  not  to  stretch ;  to 
be  perfectly  flexible  ;  without  weight;  and  infinitely  thin. 

In  such  a  machine  the  two  parts  s  c,  s  d,  Fig  16}^,  are  to  be  considered  as  two  entirely  distinct  ties  , 
in  the  same  manner  as  a  m  and  a  n,  Figs  13  and  16,  are  two  distinct  struts  Each  of  'these  ties  may 
have  to  sustain  a  different  amount  of  strain,  depending  on  their  respective  inclinations  to  s  p.  Thus 
if  the  load  p.  Fig  16>6,  be  suspended  from  a  perfectly  frictiooless  pulley  or  slip  knot  resting  on  the 
perfectly  flexible  cord  c  s  d  r,  and  if  this  pulley  or  knot  be  at  first  placed  near  c  or  d,  it,  with  its  load  p 
will  descend  by  gravity  along  the  cord  until  it  comes  to  rest  at  a.  which  is  the  lowest  point  that  the 
cord  admits  of  its  attaining  and  at  which  alone  the  angles  of  Inclination  of  8  C  and  s  d  to  s 
P  become  equal;  and  the  strains  on  the  two  parts  will  then  ue  equal.  But  it  us  in  Fig  14  the 
short  string  which  sustains  W  is  tied  fast  to  the  cord  (so  as  not  to  move  as  the  pulley  did)  at  any 
point  a,  such  that  the  angles  of  inclination  of  a  m  and  a  n  to  the  diagonal  a  b  shall  be  different,  then 
the  strains  or  pulls  along  a  m  and  a  11  will  also  bp  different. 

It  is  Immaterial  whether  m  and  n,  Fig  14,  or  c  and  d.  Fig  16  i,  arc  at  the  same 
height  or  not. 

For  more  on  the  funicular  machine  see  p  662 

Let  the  end  g  of  the  rope  g  c  on  be  fixed  ;  a  power  of  9  tons  at  n ;  the  rope  passing  over  a  pulley 
at  P ;  and  bent  out  of  line  at  c  by  a  fixed  pin.  Make  c  g  and  c  o  by  scale  each  equal  to  the  power  9 
atn;  and  complete  the  parallelogram  ;  the  diagonal  ex  of  which  is  then  found  to  be.  say  6;  or  a  result- 
ant of  6  tons.  Now,  in  this  case,  theoretically  the  strain  lengthwise  of  the  rope  is  everywhere  equal  to 
the  power  n,  or  9  tons ;  and  we  have  found  that  it  produces  also  a  strain  ex,  against  the  pin  at  c,  of  6 
tons.  It  also  produces  a  pushing  strain  on 
the  pulley  P.  Its  amount  may  be  found  in  the 
same  way,  by  measuring  9  tons  by  scale  each 
way  from  o  toward  c  and  n ;  completing  the 
parallelogram;  and  measuring  its  diagonal 
resultant.  But  now  let  us  use  this  rope  as  a 
funicular  machine  ;  and  apply  a  power  x  c  of 
6  tons  at  c.  We  find  that  this  6  tons  produces 
a  strain  eg  or  c  o,  of  9  tons  along  the  rope; 
and  this  strain  along co  will  pass  along  to  n; 
and  thus  the  power  of  6  balances  a  resistance 
of  9  tons  acting  at  n  in  the  direction  n  o. 

The  diagonal  c  x  or  any  other  will  plainly  be  vertical  only  when  the  angles  of 
Inclination  of  c  g  and  c  o,  with  the  horizon  are  equal.  If  they  differ,  both  the  di- 
rection and  the  length  of  the  diagonal  will  change. 

All  will  remain  the  same  if  the  end  g  instead  of  being  fixed,  is  passed  over  a 
pulley  as  at  P,  and  a  load  or  a  pull  equal  to  that  at  the  other  end  is  applied  to  it. 

Rem.  3.  The  surfaces  of  contact  of  pieces  used  in  construction,  are  called 
joints.  When  a  piece  is  intended  to  resist  compression,  or  push,  it  is  called  a 
strut;  or  if  inclined,  it  is  often  called  a  brace;  or  if  vertical,  a  post,  pillar, 
or  column.  When  to  resist  tension  or  pull,  a  tie.  When  to  resist  both  tension 
and  pull  alternately,  a  tie-strut,  or  a  strut-tie.  A  strut  should  be  stiff  or 
inflexible ;  but  a  rope,  chain,  or  thin  rod,  may  answer  for  a  tie. 

REM  4.  To  distinguish  a  tie  from  a  strut  at  a  glance  is  sometimes 
difficult;  but  it  may  be  done  thus.  From  the  point  a,  Figs  1*%,  at  which  the  force 
acts,  draw  a  line  a  c,  in  the  direction  in  which  the  force,  if  at  liberty,  would  move 
away  from  that  point.  On  any  part, 
a  o,  of  that  line  as  a  diag,  draw  a  paral- 
lelogram offerees.  Through  the  point 
a  draw  a  line  i  i,  parallel  to  the  other 
diag  1 1.  Then  all  the  pieces  which 
are  on  the  same  side  of  that  line,  that 
a  c  is,  are  struts ;  while  those  on  the 
opposite  side,  are  ties.  We  may  also 
frequently  determine,  by  imagining 
the  piece  to  be  a  rope  or  chain,  in- 
stead of  a  beam  ;  and  seeing  whether  it  would  then  bear  the  strain.  If  it  would  it  is 
a  tie ;  if  not,  a  strut. 

When  a  piece  of  material  is  used  to  resist  forces  which  tend  to  bend  or  break  it  crosswise  or  trans- 
versely of  its  length,  as  in  Figs  47.  48,  49,  50,  it  is  called  a  beam;  such  as  joists,  girders,  &c.  The 

beam  transversely ;  but  in  our  present  illustrations  of  comp  and  res  of  forces,  this  strain,  although 
frequently  the  most  important  one,  could  not  be  well  considered  at  the  same  time. 


464 


FORCE   IN    RIGID   BODIES. 


R 


Art.  34.  Since  any  single  force  may  be  resolved  into  two  oblique  ones  in  the 
same  plane  with  it,  and  which  shall  produce  the  same  effect  upon  a  rigid  body  con- 
sidered as  a  whole,  it  follows  that  the  single  strain  along  any  piece  a  m  or  a  w,  of  the 
four  figs  on  p  461,  may  be  thus  resolved.  Jn  practice,  it  13  frequently  necessary  to  do 
this ;  and  especially  so  for  finding  components  at  right  angles  to  each  other,  in  hor 
and  vert  directions. 

For  instance,  the  joint  o  d.  Fig  17,  at  the  foot  of  the  beam  A,  if  made 
at  right  angles  to  the  resultant  r  r  of  all  the  pressures  along  the  beam, 
of  course  receives  the  whole  of  these  pressures;  which  consequently  are 
all  imparted  to  the  abutment ;  leaving  no  portion  unresisted,  so  as  to  pro- 
duce sliding;  or  even  a  tendency  to  slide  along  the  joint  o  d.  Conse- 
quently, this  joint  is  perfectly  adapted  to  its  duty.  See  Art  19.  But  a  joint 
of  the  form  of  6  i  c,  which  is  equally  effective,  is  sometimes  reqd  for  re- 
ceiving a  single  strain  (like  that  along  A)  along  a  piece  E ;  and  in  order 
to  properly  proportion  the  vert  and  hor  faces  6  i.  and  c  t,  of  the  joint,  we 
must  flnd  the  proportion  existing  between  the  vert,  and  the  hor  compo- 
nents equal  to  the  single  strain  r  r  along  E.  To  do  this  is  very  easy ;  foi 
•we  have  only  to  lay  off  by  scale,  any  length  e  n  along  r  r,  to  represent 
the  single  strain  in  that  direction;  and  on  it  as  a  diag,  from  e  and  n 
draw  vert  and  hor  lines  e  t,  n  t,  meeting  in  t.  Then  e  t  measured  by  the 
same  scale,  will  give  the  vert  strain  ;  while  n  t  will  give  the  hor  one.  The 
parts  6  i,  i  c  of  ttie  joint,  must  consequently  have  the  same  proportion  as 
•p.  *,|— •  these  two  components  have  to  each  other ;  bearing  in  mind,  however,  that 

JjlCI  _lf  since  joints  should  be  at  right  angles  to  the  forces  they  have  to  sustain, 

«*  the  vert  part  b  i  must  bear  the  hor  strain ;  and  the  hor  part  i  c,  the 

vert  one. 

When,  by  Art  33,  we  are  finding,  by 
means  of  the  parallelogram  of  forces 
o  n  y  g.  Fig  18,  the  total  strains  o  n.  o  g, 
•which  an  extraneous  load  F  produces 
along  two  beams.  FR,  Fg.it  is  easy  at  the 
same  time  to  find  the  vert  and  hor  compo- 
nents also ;  by  drawing  the  two  hor  lines 
n  t,  gj,  and  measuring  them  by  the  same 
scale  used  for  the  diag  oy.  Likewise 
measure  o  t.  and  oj,  for  the  correspond- 
ing vert  forces  at  the  joints;  because 
when  n  t  and  (jj  may  be  drawn  inside 
of  the  parallelogram,  (which  is  not 
always  the  case;  as  see  Fig  18^.)  the 
component  forces  in  the  direction  of  any 
diag,  \vhethervertornot.  are  measured 
respectively  from  the  point  o,  where  the 
extraneous  force  F  is  imparted  to  the  beams ;  to  those  points  t  and  j,  where  the  diag  is  met  by  the 
equal  lilies  n  t,  gj.  Sot  R-'rn,  p  2(50. 

REM.  1.  It  ia  an  important  fact  that  however  diff  may  be  either  the  inclinations, 
or  the  lengths  of  the  two  beams;  or  how  diff  the  total  strains  in  the  directions  of 
their  respective  lengths;  the  hor  s; ruins,  caused  both  by  the  extraneous  load  and 
by  the  weights  of  the  beams  themselves,  will  always  be  equal  on  both  of  them. 
Thus,  in  Fig  18,  n  t  is  equal  to  gj ;  and  in  Figs  13  to  10,  if  hor  lines  be  drawn  from  o 
and  y,  to  a  6,  those  in  any  one  fig  will  be  equal  to  each  other. 

HEM.  2.  The  beam  o  R,  Fig  18,  is  not  to  be  considered  as  acted  upon  at  the  same 
time  by  three  distinct  forces  on,  ot,  and  t  n ;  nor  the  beam  og  by  three  forces  og,  ojt 
gj\  but  each  is  acted  upon  by  one  force ;  thus  o  R  is  acted  upon  by  o  n ;  which  pro- 
duces upon  it  precisely  the  same  effect  as  would  be  produced  by  its  two  components 
o  t,  n  t.  And  so  with  the  other  beam.  Each  beam  may  be  considered  as  receiving 
from  the  load  F,  either  one  force  or  its  two  components. 

The  vert  component  oj,  of  the  triangle  o  gj,  being  longer  than  o  t,  of  the  triangle  o  tn,  shows  that 
the  beam  o  </  bears  more  of  the  vert  force  or  weight  of  the  load  F,  than  o  R  does  ;  and  in  the  sume 
proportion  as  oj  is  to  o  t.  The  two  components  on  the  diagonal,  (when  inside  of  the  parallelogram,) 
•will  always  together  equal  the  length  of  the  diag,  or  the  weight  F.  But  as  n  t  and  gj  are  of  the  same 
lengths,  they  indicate  that  both  beams  are  pressed  sideways,  or  hor,  to  the  same  extent. 

When  we  come  to  treat  on  trusses,  we  shall  find  this  method  of  obtaining  vert  components,  by 
means  of  ot  and  oj,  very  useful.  The  lengths  of  the  beams  o  R,  0/7,  do  not  affect  the  amount  of 
strains  produced  upon  them  hy  the  load  F  at  their  summits ;  but  as  their  own  wts  must  increase  with 
their  lengths,  the  strains  arising  from  them  must  increase  also  ;  but  we  have  not  yet  taken  their  own 
wts  into  consideration ;  neither  are  we  yet  prepared  to  do  so.  See  Trusses,  p  247  &c. 

REM.  3.  If,  as  in  Fig  18^,  one  of  the  beams,  as  n  o,  is 
hor,  it  is  plain  that  all  of  the  diag  o  y,  that  is,  all  of  the 
weight  F  or  W,  is  borne  by  the  other  beam  og ;  and  n  o 
sustains  hor  strain  only.  The  beam  og  of  course  bears 
an  equal  hor  strain  also,  as  shown  by  y  g,  equal  to  n  o. 

REM.  4.  It  is  immaterial  (Art  18)  whether  the  load 
rests  on  top,  as  F ;  or  is  suspended  below,  as  W ;  for  in 
either  case  it  is  simply  vert  force  imparted  at  o. 


TORCE    IN    RIGID    BODIES. 


465 


HEM.  5.  When  one  of  the  forces,  as 
ft  o,  makes  an  angle  n  o  ?/,  greater  than 
90°,  with  the  diag  o  y,  the  positions  of 
the  beams  o  n,  oy,  become  as  in  Fig  18^, 
and  we  have  a  case  like  fc  ig  8% ;  that  is, 
the  hor  lines  n  6,  <ja,  from  the  angles  n 
and  <;,  and  at  right  angles  to  the  diag, 
cannot  be  drawn  inside  of  the  parallelo- 
gram. Therefore  we  must  extend  the 
diag  both  ways,  to  a  and  b.  If  we  wish 
to  consider  each  ef  the  forces  on  and  og 
as  made  up  of  two  components;  then 
for  those  of  o  n,  we  have  6  ?v,  and  o  b ; 
and  for  those  on  og,  we  have  ag  and 
o  a.  Hence  when  the  angle  no  a  ex- 
ceeds 9i)  the  vert  strain  on  o  g  is  greater  than  the  load  wf 

which  (according  to  the  ordinary  phraseology)  produces  it.  But  the  part  a  y  of  the  vert  o  a,  has 
no  reference  to  the  load;  but  represents  an  upward  vert  force  produced  by  the  wall  M,  to  balance  a 
downward  vert  one  equal  to  &  o  from  the  wall  P.  This  excess  a  y  over  the  diag,  occurs  only  when 
one  of  the  beams  forms  an  angle  greater  than  90°  with  the  diag.  We  call  the  attention  of  the  student 
to  this  cn«e.  because  we  do  not  remember  to  have  met  with  it  in  any  book.  It  will  perhaps  be  a  new 
Idea  to  many,  that  the  vert  pres  on  the  wall  M,  can  be  greater  than  the  entire  load.*  See  Rem  5,  p  459. 

Art.  35.  As  a  simple  practical  example  of  very  common  occurrence,  of  the  ap- 
plication of  the  foregoing  principle  of  finding  the  resultant  of  two  forces  in  the  same 
plane,  and  tending  to  one  point;  let  S,  Fig  19,  represent  a  block  of  stone  weighing 
3  tons ;  and  standing  on  a  hor  base  m  n ;  but  not  attached  to  it  in  any  way  by  ce- 
ment, Ac.,  but  with  a  stop  at  n,  merely  to  prevent  sliding  toward  b. 

In  this  case  there  will  be  no  force  acting  upon  the  body  in  such 
a  way  as  to  prevent  its  being  overturned  around  its  toe  n  as  a 
turning  point,  except  its  wt,  or  force  of  gravity,  which  always 
acts  vert  downward.  By  Arts  56,  57,  all  this  force  may  be  con- 
sidered to  be  concentrated  at  the  cen  of  grav  i  of  the  stone ;  and 
as  acting  at  any  point  whatever  in  its  vert  line  of  direction  Ig. 

Now  suppose  a  pres  fh,  of  2  tons,  (which  may  be  either  one 
simple  force,  or  the  resultant  of  many  forces.)  to  be  imparted  to 
the  stone,  in  the  same  plane  with  the  force  of  gravity;  which  it 
will  evidently  be  only  if  its  direction  fc  meets  the  direction  of 
gravity  Ip,  at  some  point;  as,  for  instance,  at  a;  because  then  a 
plane  surface  would  coincide  with  both  directions.  Art.  8.  The 
question  is,  which  of  these  two  forces  will  prevail ;  the  two  tons 
of  fh,  to  overturn  the  stone  ;  or  the  3  tons  of  gravity  to  prevent 
its' being  overturned?  By  Art  29,  both  forces  may  be  considered 
to  he  imparted  to  the  rigid  stone,  at  the  point  a,  where  their  lines 
of  direction  meet;  and  we  may  make  a  c  equal  to  2  inches,  ft,  <fec, 
to  represent  the  amount  and  the  direction  of  the  2  tons  of  pres  of  the 
force  fh  :  and  a  v,  by  the  same  scale,  for  the  direction,  and  3  tons 
of  pres  of  the  weight  of  the  stone.  From  c  draw  a  line  c  d  paral- 
lel to  a  v ;  and  from  v  draw  v  d  parallel  to  a  c  ;  these  will  meet  at 
d  :  thus  completing  the  parallelogram  of  forces  a  c  d  v.  The  diag 
ad  of  this  parallelogram,  measd  by  the  same  scale,  will  give  about  2*f  tons  for  the  single  resultant 
force,  which  would  bv  itself  produce  upon  the  rigid  stone  the  same  effect  as  gravity  and  fh  com- 
bined. This  force  by  Art  18,  may  be  considered  as  imparted  to  the  stone  at  any  point  in  the  line  of 
its  direction  (w  o)  through  the  stone  ;  as  a  push  at  w,  as  shown  by  the  arrow  zw;  or  as  &pull  at  o, 
by  means  of  a  rope  6  o.  fastened  to  the  stone  at  o.  Since  this  resultant  is  supposed  to  take  the  place 
both  of  gravity  and  of  fh,  the  two  last  must  of  course  be  considered  as  annihilated  ;  so  that  the  stone 
becomes  as  it  were  an  unresisting  body  of  matter  without  weight;  and  acted  upon  by  the  force  ad, 
or  zw,  which  must  of  course  move  it,  and  thus  compel  it  to  overturn  around  n  as  a  pivot. 

ItaM.  1.  ITad  the  direction  of  the  resultant  a  d  struck  the  base  of  the  stone  at  n, 
instead  of  striking  outside  of  the  base  as  at  6,  the  stone  would  barely  have  stood; 
because  then  the  resultant,  on  leaving  the  body  at  n,  would  have  encountered  the 
resisting  force  of  the  ground  on  which  the  stone  stood,  acting  upon  the  body  at  that 
point.  Had  the  direction  struck  within  n,  that  is,  between  n  and  m,  the  stone  would 
stand  still  more  firmly;  (spe  Kern  2,  p  492;)  and  the  more  firmly  in  proportion  as 
it  strikes  nearer/;,  where  the  direction  Iff  of  the  gravity  of  the  stone  meets  the  base. 
The  direction  of  a  resultant  may  strike  within  the  base ;  and  the  body  remain  firm, 
BO  far  as  regards  overturning ;  but  yet  may  slide.  See  Art  63 ;  very  important. 

This  example  shows  also  the  necessity  for  assuming  at  times  that  bodies  are  rigid,  or  unbreakable. 
For  in  the  c.ase  of  stones  of  but  little  strength,  the  application  of  the  great  force  of  the  resultant  so 
near  to  »,  would  break  the  body  at  that  point ;  and  might,  besides,  mash  n  into  the  yielding  earth  on 

*  If  the  writer  is  mistaken  in  this,  he  wishes  to  be  corrected, 


V 


11 


19 


466 


FORCE   IN   RIGID   BODIES. 


•which  It  stood.  A  knowledge  of  the  direction  of  the  resultants  of  forces  acting  on  bridge  abutment*, 
retaining- walls,  &c,  is  therefore  of  use  also  by  enabling  us  to  guard  against  such  accidents,  by  select- 
ing the  strongest  stones  for  the  most  strained  parts  of  the  structure ;  as  well  as  by  adopting  extra 
precautions  in  preparing  those  portions  of  the  earth  foundations,  upon  which  those  parts  rest. 

It  must  be  remembered,  however,  in  such  cases  as  the  foregoing,  that  with  the  exception  of  the  point 
o,  at  which  the  resultant  leaves  the  body,  ad  is  not  the  direction  which  the  resultant  actually  follows 
in  the  body  ;  but  is  one  which  we  may  assume  it  to  have,  so  long  only  as  we  assume  the  body  to  be 
practically  rigid:  that  is,  that  it  cannot  be  in  any  way  broken,  bent,  or  have  its  form  changed,  by 
the  forces  actually  imparted.  Frequently  we  cannot  safely  assume  a  mass  of  masonry  to  be  thus 
rigid ;  for  it  may  be  composed  of  many  separate  pieces  merely  placed  in  contact  with  each  other 
without  mortar,  as  in  dry  masonry  ;  or  even  if  mortar  or  cement  be  used  to  unite  these  pieces,  it  may 
not  have  time  to  set  or  harden  properly,  before  the  deranging  forces  are  brought  to  bear  upon  it.  la 
that  case,  although  the  resultant  might  fall  entirely  within  the  body,  and  within  the  base,  thus  de- 
noting perfect  security  to  a  rigid  body;  yet  the  structure  might  be  completely  destroyed  by  the 
sliding  or  other  derangement  of  its  parts  among  one  another,  under  a  force  much  less  than  would  be 
required  to  overturn  it.  On  this  account,  if  we  wish  to  obtain  security  at  the  least  expense,  we  must 
frequently  trace  the  actual  curved  direction  of  the  resultant  through  its  entire  course;  so  that  we 
may  at  every  point  of  it  place  the  joints  of  our  masonry  at  right  angles  to  it,  as  in  Fig  7;  or  adopt 
other  precautions  to  prevent  the  parts  of  the  structure  from  separating.  See  Art  72,  p  491. 

RKM.  2.  If  in  Fig  19  we  suppose  strong  mortar  or  cement  to  exist  between  the  base  of  the  stone 
and  a  rigid  foundation  of  masonry  or  rock,  upon  which  we  may  assume  it  to  stand,  then  it  may  not 
be  overthrown,  although  the  direction  of  the  resultant  a  d  falls  outside  of  the  base  mn.  For  then  a 
third  force,  namely,  the  cohesive  strength  of  the  mortar,  is  brought  to  act  upon  the  stone  ;  and  the 
resultant  of  all  three  forces  may  fall  within  the  base.  In  all  cases  where  a  body  remains  at  rest,  not- 
withstanding that  the  resultant  of  the  forces  falls  outside  of  its  base,  (whether  the  base  be  hor,  vert, 
or  inclined,)  we  may  be  certain  that  it  is  because  some  other  force,  which  we  have  neglected,  is  acting 
upon  it  at  the  same  time ;  for  when  the  direction  of  all  the  forces  passes  beyond  its  base,  and  is  con- 
sequently force  unresisted,  the  body  munt  move.  See  Rem,  Art  65 ;  also  see  Art  72.  When  one  body 
is  thus  cemented  to  another,  the  two  become  in  fact  one  body,  so  long  as  the  cement  does  not  give 
way  under  the  imparted  forces :  so  that  a  problem  which  is  one  in  Statics,  if  there  is  no  cement,  may 
become  one  in  Strength  of  Materials,  when  there  is  cement.  The  cement  takes  the  place  of  natural 
cohesive  force  between  the  bodies  which  it  unites. 

Art.  36.  When  the  number  of 
forces  in  the  same  plane,  whether 
tending:  to  or  from  the  same  point 
or  not,  is  greater  than  two,  their  result- 
ant may  be  found  in  the  manner 
already  given  for  two.  Thus,  with  the 
three  forces  b  a,  ca,  oa,  Fig  19^,  first  find  the 
resultant  of  any  two  of  them  ;  as,  for  instance, 
the  resultant  n  a,  of  o  a,  and  c  a.  Then  consider 
oa  and  ca  as  removed  and  na  as  taking  their 
r  place ;  and  then  find  the  resultant  ra  a,  of  n  a  and 
a 6;  then  is  ra  a  the  single  resultant  that  will 
produce  upon  the  rigid  body  W,  the  same  effect 
as  the  three  forces  oa,  ca,  6 a.  If  the  three 
forces  are  imparted  at  diff  parts  of  the  body,  proceed  as  in  Figs  10^  and  10%.  If 
the  number  of  forces  be  greater  than  3,  the  process  is  precisely  the  same ;  find  the 
1st  resultant  of  two  of  them ;  then  the  resultant  of  the  1st  resultant  and  3d  force; 
then  the  resultant  of  the  2d  resultant  and  4th  force ;  and  so  on  to  the  end. 


Fig:  19%  illustrates  a  case  in 
which  three  forces  af  b,  and  cf  in 
the  same  plane,  do  NOT  tend  to- 
wards the  same  point.  We  may  be- 
gin with  any  two  of  the  forces  at  pleasure.  We 
will  take  b  and  c;  and,  as  at  Fig  10%,  p  459, 
prolong  them  backwards  to  fc,  and  find  their 
resultant  hi.  Then  prolonging  /u,  and  the 
third  force  a  to  meet  at  n,  we  lay  off  from  n 
two  sides  of  the  parallelogram  equal  respec- 
tively to  a  and  to  h  i,  and  complete  the  paral- 
lelogram no.  Then  the  diagonal  on  is  the  re- 
quired resultant,  to  be  applied  to  the  body 
J  at  e,  as  shown  by  x  (..  This  resultant  would 
then  by  itself  produce  upon  the  rigid  body 
considered  as  a  whole,  the  same  effect  as  would 
the  three  forces  a,  6,  c.  Or  if  its  direction  were 
inverted,  so  as  to  pull,  instead  of  push  at  e,  it 
would  become  an  antiresultant  to  the  three 
forces,  and  thus  hold  them  in  equilibrium. 

The  same  process  applies  to  any  number  of 
forces. 


FORCE  IN    RIGID   BODIES. 


467 


Ktf20 


Art.  37.  It  sometimes  happens,  after  having 
found  the  resultant  of  all  the  forces  except  the  last 
one,  that  said  resultant  and  remaining  force  are  in 
the  same  straight  line.  Thus,  with  the  forces  u,  r,  w, 
Fig  20;  the  resultant  r  of  u  and  w,  is  in  the  same 
straight  line  \vith  the  last  remaining  force  v ;  and  of 
course  no  parallelogram  of  forces  can  be  drawn  which 
shall  have  v  and  r  for  two  of  its  sides.  When  this 
happens,  if  v  and  r  are  of  diff  lengths,  we  have  the 
case  given  in  Art  16,  of  two  unequal  forces  meeting 
in  the  same  straight  line ;  but  in  opposite  directions 

along  it.    Consequently,  the  small  one,  and  an  equal  part  of  the  large  one,  mutually 
destroy  each  other ;  and  the  remainder  of  the  large  one  is  the  resultant  of  the  two. 

But  if  v  and  r  are  of  the  same  length,  then  we  have  the  case  of  two  equal  opposing  forcei,  which 
mutually  destroy  each  other  entirely ;  and  the  body  remains  at  rest.  Consequently,  there  is  no  result- 
ant in  this  case ;  for  no  single  force  can  have  the  effect  of  keeping  a  body  at  rest ;  but  will  always 
move  it.  In  other  words  u,  v,  and  w,  are  then  in  equilibrium. 

Art.  38.  The  polygon  of  forces.  The  resultant  of  any  number  of 
forces  in  the  same  plane ;  and  acting  through  one  point  only,  may  be 

found  thus :  Let  a,  6,  and  c,  be  three 
such  forces ;  whose  resultant  R  is  to  .   n 

be  found.  Begin  with  any  one  of 
them,  as  a,  and  draw  a',  parallel,  and 
equal  to  it ;  and  place  an  arrow-head 
at  the  proper  end  of  it,  to  show  its 
direction.  From  this  arrow-head, 
draw  &',  equal  and  parallel  to  6; 
placing  an  arrow-head  at  its  end. 
From  this  second  arrow-head,  draw 
c',  equal  and  parallel  to  c ;  and  so  on 
with  any  number  of  forces ;  taken  in  any  order.  Finally,  from  the  arrow-head  d  of 
the  last  of  the  forces,  draw  a  line  A  to  the  butt-end,  n,  of  the  first  one ;  thus  closing 
the  figure ;  and  place  an  arrow-head  as  on  the  others.  Now,  this  closing  line  A,  or 
dn,  with  its  arrow,  represents  both  in  quantity  and  direction  the  antiresultant  of 
the  three  given  forces;  or  if  its  arrow  be  reversed,  it  will  represent  their  resultant. 
Consequently,  we  have  only  to  draw  from  o  a  line,  o  R,  parallel  to  A ;  and  to  make 
R  equal  to  A;  but  pointing  in  the  opposite  direction.  Then  is  R  the  resultant. 
This  process  will  give  different  figures,  according  to  which  force 
we  begin  with ;  or  whether  we  take  the  forces  in  right,  or  left-hand  order ;  still,  A 
will  always  come  out  the  same  in  all  of  them.  If  the  three  given  forces  (or  any 
greater  number,  as  the  case  may  be)  had  been  in  equilibrium  with  each  other,  that 
is  had  mutually  destroyed  each  other's  tendency  to  cause  motion,  they  of  course 
could  have  no  resultant,  or  single  force  that  would  produce  an  equal  effect;  because 
a  single  force,  if  the  only  one  acting  on  a  body,  must  produce  motion.  When  this 
is  the  case  the  forces  will  of  themselves  form  a  closed  polygon.  In  either  case  some 
of  the  lines  may  cross  each  other  as  do  a'  and  c'  at  A  (forming  what  is  called  a  gauche 
polygon) ;  or  not,  as  at  N  below. 

The  foregoing  depends  upon  the  following, 
known  as  the  principle  of  the  polygon 

of  forces.  If  any  number  of  forces  as  a,  6,  c, 
d,  Fig  22,  in  the  same  plane,  whether  acting 
through  one  point  s,  or  not,  keep  each  other  in 
equilibrium,  then  if  drawn  consecutively  in 
their  proper  directions,  and  in  any  order  what- 
ever, they  will  form  a  closed  polygon ;  either 
gauche  as  at  A  Fig  21,  or  plane  as  at  N  Fig  22. 

But  it  does  not  follow  because  a  num- 
ber of  forces  may  thus  form  a  polygon  that  they 
must  be  in  equilibrium ;  unless  when  they  all 
act  through  one  point.  This  is  proved  near  end 
of  Rem,  Art  50,  p  476. 

It  is  plain,  that  in  any  number  of  such  forces,  any  one  of  them  is  the  antiresultant  of  all  the  rest, 
because  it  keeps  them  all  in  equilibrium  ;  it  is  also  equal,  of  course,  to  their  resultant ;  but  acts  ia 
the  opposite  direction.  Also,  if  any  straight  line  as  tt  be  drawn  through  the  point  *,  the  forces  on 
one  side  of  it,  will  balance  those  on  the  other  side ;  thus,  d'  and  c',  will  balance  a'  and  &'. 

Any  diagonal  across  a  polygon  of  forces  represents  both  the  re- 
sultant and  the  ante  resultant  of  all  the  forces  on  either  one  side  of  it. 


Tig  22 


468 


FORCE    IN    RIGID    BODIES. 


AKM.  1.  It  must  not  be  inferred  because  forces  balance  each  other,  that  therefore  they  balance  the 
tody  to  which  they  are  imparted ;  for  the  body  might  move  under  the  iutiueuce  of  other  forces.  Thus, 
the  forces  a',  b',  c',  d\  may  be  supposed  to  be  the  balancing  forces  of  several  persons  holding  a  body 
at  rest  in  a  railroad  car  moving  with  great  speed.  Their  forces  prevent  each  other  from  giving  motion 
to  the  body  ;  but  do  not  prevent  the  steam  force  of  the  engine  from  doing  so.  It  is  only  when  all  the 
forces  imparted  to  a  body,  including  its  own  weight,  are  iu  equilibrium,  that  the  body  itself  is  also  at 
rest.  Or,  if  we  hold  a  book,  ruler,  &c,  vert  between  our  thumb  and  forefinger;  the  opposite  and 
equal  pressures  of  the  thumb  and  finger,  hold  each  other  in  equilibrium,  so  that  they  cannot  move 
the  book  either  to  the  right  or  left,  and  the  friction  between  them  and  the  book,  holds  the  gravity  or 
weight  of  the  book  in  equilibrium,  so  that  it  does  not  fall.  So  that  so  far  as  these  forces  are  concerned, 
they  produce  no  motion  in  the  book;  but  we  can  move  it  vertically  up  and  down,  by  introducing 
the  third  force  of  our  wrist;  or  hor  by  stretching  out  our  arm;  or  by  walking;  none  of  which  will 
interfere  with  the  equilibrium  of  the  other  forces;  they  only  prevent  each  other  from  producing  any 
motion. 

HEM.  2.  A  triangle  being  a  polygon  of  3  sides,  if  any  3  forces  which  form  a  triangle, 
be  applied  in  one  plane,  to  a  body,  and  in  directions  parallel  to  the  sides  of  the  tri- 
angle ;  and  tending  either  to  or  from  one. point;  they  will  hold  each  other  in  equili- 
brium. And,  vice  versa,  when  we  see  a  body  kept  at  rest  solely  by  the  action  of  3 
forces  which  are  not  parallel  to  each  other,  we  may  be  sure  that  those  forces  are  pro- 
portional to  the  sides  of  a  triangle  drawn  parallel  to  them;  that  they  are  in  one 
plane;  that  they  all  tend  either  to  or  from  one  point;  and  that  any  one  of  them 
acts  in  the  direction  of  an  antiresultant  to  the  other  two.  Moreover,  each  of  the 
forces  is  proportionate  to  the  sine  of  the  angle  included  between  the  other  two;  so 
that  if  we  know  one  of  the  forces,  we  can  readily  find  the  others  if  we  have  the 
angles.  This  is  very  often  of  use  in  practice ;  as  in  finding  by  calculation  alone,  the 
line  of  pressures  through  an  arch ;  the  pres  of  earth  against  retaining-walls,  <tc.  It 
must  be  remembered  that  the  wt  of  a  body  usually  constitutes  one  of  the  forces  to 
be  considered  as  acting  upon  it.  This  rem  is  very  important. 

Ex.  1.  Let  a  c,  Fig  22^,  be  a  beam ;  its  foot  rest- 
ing on  o  i ;  and  its  head  c  merely  leaning  against  a. 
smooth  vert  wall ;  and  whether  a  c  be  unloaded  ;  or 
whether  it  supports  a  load  placed  in  any  manner 
upon  it,  or  suspended  from  it;  let  the  vert  line 
which  passes  through  the  cen  of  grav  of  the  beam 
and  its  load  (both  of  which  are  supposed  to  be 
known)  be  represented  \->y  pg.  The  beam  and  its 
load  may  be  regarded  as  a  single  body,  acted  upon, 
and  kept  at  rest,  by  three  forces  ;  namely,  its  owa 
gravitv  or  wt;  the  force  ft,  at  c;  and  the  force/,  at 
a.  No  other  forces  act  on  it.  Now,  gravity  acts  vert 
only ;  and  in  the  case  before  us  it  may  all  be  re- 
garded as  acting  in  the  line  p  g.  The  force  at  c  can 
act  only  at  right  angles  to  the  surf  or  joint  at  that 
place,  (see  Art  19;)  and  since  the  joint  is  vert,  the 
force  A  must  be  hor,  or  along  ft  p.  The  question 
now  is,  how  to  find  the  direction  of  the  third  force  /. 
To  do  this  we  must  avail  ourselves  of  the  principle 
that  when  three  forces,  not  parallel  to  ea^h  other, 
hold  a  body  at  rest,  or  in  equilibrium,  as  these 
three  forces  hold  the  beam  a  c,  their  directions  all  tend  to  or  from  one  point;  which  is  either  at  the 
cen  of  grav  of  the  body  ;  or  in  a  vert  line  passing  through  said  cen.  Hence,  since  the  vert  direction 
p  g  of  the  force  of  gravity  of  the  body ;  and  the  direction  ft  p  of  the  force  ft,  meet  at  p,  therefore,  the 
direction  fp,  of  the  force  /,  must  also  meet  there.  Hence  we  have  only  to  draw  a  line  fp,  in  order 
to  find  the  reqd  direction.  A  post  intended  to  support  the  end  a  of  the  beam,  should  have  the  position 
fa;  and  the  joint  o  i  should  be  at  right  angles  to  fa;  and  not  to  a  c,  as  mfekt  at  first  be  supposed 
from  Figs  13  and  16,  of  Art  32;  in  which  the  wt  of  the  beams  is  not  considered. 

Having  found  the  directions  of  the  three  forces  in  Fig  22^,  it  only  remains  to  find  their  amounts. 
To  do  this,  we  already  have  one  of  them  given,  namely,  gravity,  or  the  wt  of  the  beam  and  its  load  ; 
and  we  know  that  they  must  be  in  proportion  to  the  sides  of  the  triangle  drawn  parallel  to  their  di- 
rections. Consequently,  if  on  the  vert  direction  p  g,  we  lay  off  by  scale  any  portion  whatever,  as  p  d, 
to  represent  the  force  of  gravity,  then  will  the  hor  side  of  the  triangle,  pdb,  represent  by  the  same 
scale  the  hor  pres  at  c :  and  the  side  b  p,  the  oblique  pres  at  a.  The  hor  pres  at  the  foot  is  equal  to 
that  at  the  head  of  the  beam.  It  is  of  course  included  in  the  oblique  pres  /;  which  is  compounded 
of  said  hor  force,  and  of  the  vert  force  at  a.  The  vert  force  is  equal  to  the  weight  of  the  beam  and 
its  load;  none  of  which  is  sustained  at  c ;  nor  can  be,  so  long  as  the  joint  at  the  head  and  wall  is 
vert. 

Ex.  2.  This  is  very  similar  to  the  preceding.  Let  a  6  ij\  Fig  22%,  be  the  half  of 
any  arch  bridge,  loaded  or  unloaded  equally  throughout;  and  of  which  we  know  in 
either  case  the  total  wt ;  and  that  the  cen  of  grav  of  said  wt  is  somewhere  in  the  vert 
line  gg.  Now  this  half  bridge  is,  like  the  preceding  beam,  kept  in  equilibrium,  or 
at  rest,  by  three  forces  only;  namely,  the  wt;  a  hor  pres  A,  at  the  crown,  arising 
from  the  other  half  of  the  arch;  and  an  oblique  pres  o,  at  the  springing,  or  skew- 
back  j  i.  To  find  the  directions  and  amounts  of  these  forces,  from  a  draw  a  hor  line, 
meeting  the  vert  gg  at  c.  From  c  draw  a  line  to  the  center  of ./ 1 ;  this  is  the  direc- 
tion of  the  oblique  force  o.  From  c  measure  down  by  scale  any  dist  cs,  on  the  vert 
direction  g  q,  to  represent  the  weight ;  and  from  *,  draw  s  t  hor.  Then  s  t,  measd  by 
the  same  scale,  will  be  the  hor  pres  7i;  and  c  t,  the  oblique  one  o.  The  joint  j  i,  at 
the  spring  of  the  arch,  bears  all  of  c  s ;  that  is,  all  the  wt  of  the  half  arch,  and  half 


FO 


IN    RIGID    BODIES. 


469 


load.    No  vert  pres  or  wtfis  sustained  at  the  center  In  of  the  arch  ;  nothing  but  the 
hor  pres.    Butj>  i  also  sustains  this  hot  pres,  for  c  f  is  composed  ol  c  s  and  s  t. 

The  oblique  force  ct  constitutes  the  total  thrust  exerted  by  the  entire  arch, 
against  each  of  its  two  abuts;  and  the  line  ct  shows  the  direction  in  which  this 
thrust  enters  the  abut  at  the  skewbackji.  Alter  entering  at  that  point,  it  begins  to 
curve  downward,  on  the  principle  explained  in  Art  72.  Since  ctia  the  hypothenuse 
of  a  right-angled  triangle,  of  which  the  leg  cs  represents  the  half  wt;  and  st  the 
hor  pres  of  the  arch,  it  follows  that  the  total  thrust  c  t  of  an  arch  may  be  found  thus : 
add  together  the  square  of  half  its  wt ;  and  the  square  of  the  hor  pres ;  and  take  the  sq 
rt  of  the  sum.  This  applies  also  to  arches  of  iron  or  wood. 

The  joints  of  any  arch  which  is  a  portion  of  a  circle,  are 
usually  drawn  toward  the  center  of  the  circle;  and,  practi- 
cally, this  answers  every  purpose;  but  it  is  plain  that  strict 
theory  would  require  the  joint  ij  to  be  at  right  angles  to  co. 
Bo  also  the  other  joints  of  the  archstones  would  be  reqd  to 
be  perp  to  the  pres  which  they  have  to  sustain. 

Moreover,  since  the  pres  ct,  upon  the  joint  ji,  is  much 
greater  than  the  pres  st,  upon  the  joint  In,  theory  would 
require  the  joint \ji  to  be  proportionally  deeper  than  In; 
whereas  in  practice  they  are  usually  made  the  Mine,  except 
in  very  large  arches.  See  Stone  Bridges.  P  349,  footnote. 

The  last  few  lines  of  Ex  1,  respecting  the  her  pres  at  the 
foot,  and  at  the  head,  apply  equally  here.  The  young  stu- 
dent should  familiarize  himself  thoroughly  with  the  princi- 
ple illustrated  by  these  two  examples,  as  it  is  one  of  very 
frequent  application  in  practice ;  as  in  retaining- walls,  abut- 
meats,  &c. 

Here,  as  in  the  preceding  example  of  the  beam,  we  do  not 
consider  the  strains  produced  along  the  length  of  the  arch 
Inji  itself ;  but  merely  the  two  forces  which,  acting  at  its  center  In,  and  at  its  foot  ji,  keep  each 
other,  and  the  wt  of  the  half  arch  and  its  load,  in  equilibrium.    For  the  others,  see  Stone  Bridges. 

Art.  39.    A  third  mode  of  finding1  the  resultant  R,  Figs  23, 

of  any  number  of  forces  E,  F,  G,  in  one  plane;  and  acting:  through  one 
point,  x. 

Draw  two  lines,  H  H,  and  V  V,  at  right  angles  to  each  other.  From  their  point  of  intersection  o, 
draw  lines  by  any  convenient  scale,  to  represent  the  directions  and  amounts  of  the  forces.  By  Art 
34,  resolve  each  of  these  forces  into  two  component  ones  parallel  to  H  H  and  V  V.  Thus  F  o  is  re- 
solved into  «o  and  eo  ;  Go,  into  m  o  and  to;  E  o,  into  to  and  no.  Then  measure  by  the  scale,  and 
add  together,  those  components  io  and  to,  parallel  to  H  H,  and  which  tend  to  move  the  point  o 


toward  the  left  hand.  Also  add  together  those  (in  this  case  only  one)  «  o,  which  tend  to  move  o 
toward  the  right  hand.  Subtract  the  least  snm  from  the  greatest;  their  diff,  equal  to  s  o,  will  be  the 
resultant  of  the  two  sets  of  forces  which  respectively  tend  to  move  o  to  the  right,  and  to  the  left.  In 
this  instance,  this  so  must  evidently  be  placed  to  the  right  of  o,  because  the  components  on  that  side 
give  the  greatest  sum. 

Next,  add  together  those  components  eo,  no,  parallel  to  W,  which  tend  to  move  the  point  o  up- 
ward. In  like  manner  add  together  those  (in  this  case  only  one)  components  mo,  parallel  to  V  V, 
which  tend  to  move  o  downward,.  Subtract,  as  before,  the  least  sum  from  the  greatest;  their  diff, 
equal  to  ao,  will  be  the  resultant  of  the  two  set*  of  forces  which  respectively  tend  to  move  o  upward 
and  downward.  In  this  instance,  a  o  must  be  measd  off  below  o,  because  the  upward  tendency  Is  the 
greatest.  By  this  process,  then,  we  first  reduce  all  the  original  forces  to  t  wo  components,  so  and  ao. 
This  being  done,  we  have  only  to  complete  the  parallelogram  of  forces  osca,  and  draw  its  diag  co; 
which  will  be  the  flnal  single  resultant  of  all  the  original  forces.  From  x  draw  xy  parallel  to  co, 
and  make  by  equal  to  co ;  then  is  by.  or  R,  th«  reqd  resultant;  and  6  the  point  fnr  imparting  it  to 
;he  body  P,  ao  that  its  effect  may  be  equal  to  that  of  the  three  original  forces  combined. 


470 


FORCE   IN   RIGID   BODIES. 


Art.  4O.  Even  when  any  number  of  forces  in  the  same 
plane  do  NOT  tend  to  or  from  the  same  point,  the  principle  of 
the  polygon  offerees,  or  of  Art  39,  may  be  used  in  precisely  the  same  manner  as  at 
Figs  21  and  23,  for  finding  the  length  and  direction  of  their  resultant.  Or  if  they 
are  in  equilibrium,  and  hence  can  have  no  resultant,  they  will  still  form  a  closed 
figure  as  A  Fig  21,  or  N  Fig  22,  as  well  as  if  they  acted  through  one  point.  There 
will  however  be  this  difference,  that  when  all  the  forces,  as  a,  6,  and 
c.  Fig  21,  tend  to  or  from  one  point  o,  we  know  that  their  resultant,  as  R,  must  be 
applied  parallel  to  A,  and  must  tend  to  or  from  that  same  point  o.  In  other  words, 
-we  know  where  its  point  of  application  must  be.  And  so  with 
the  resultant  co,  Art  39.  But  when  the  forces  do  not  tend  to  or  from  one  point,  and 
we  find  their  resultant  by  Art  38  or  39,  we  know  only  its  amount  and  direction ; 
but  do  not  know  where  to  apply  it.  In  such  cases  we  may  use  Art 
36,  p  466,  Fig  19^. 


Art.  41.  Forces  in  different  planes  ;  but  tending:  to  or  from 
the  same  point.  Such  forces  cannot,  like  those  in  one  plane,  be  correctly  rep- 
resented together  on  one  flat  surf,  such  as  a  sheet  of  paper. 
Thus,  let  Fig  27  be  a  cube;  and  tx,  ex,  ix,  three  forces 
acting  in  the  directions  of  its  edges  ;  and  all  tending  to 
the  same  point  x.  It  is  plain  that  the  relative  positions 
of  these  forces  are  not  correctly  represented;  for  txc,txi, 
and  csct,  are  in  reality  right  angles;  whereas,  in  the  fig, 
t  x  c  appears  to  be  an  acute  one  ;  c  x  t,  a  right  angle  ;  and 
t  x  i  an  obtuse  one. 


On  this  account  the  resultant  of  such  forces  cannot  be  had  by 
measurement  from  a  drawing.  Recourse  must  therefore  be  had  to 
calculation  ;  which,  however,  will  be  facilitated  by  a  drawing.  The 
theoretical  principle  is  very  simple  ;  being,  in  fact,  the  same  as  when 
the  forces  are  all  in  one  'plane;  namely,  first  find  by  Art  28  the  re- 
sultant of  any  two  of  them,  (for  any  two  are  really  in  one  plane;) 
then  find  the  resultant  of  this  resultant  and  the  third  force;  and 
80  on  to  the  end.  It  is  easy  to  find  the  first  resultant;  but  the 
others  are  more  troublesome.  Instances  are  comparatively  rare,  in 
which  the  resultants  of  such  forces  are  reqd  to  be  found:  the  attention  of  the  engineer  being  gen- 
erally confined  to  those  in  one  plane  ;  as  when  proportioning  bridges,  roofs,  retaining-  walls,  &c. 


FORCE   IN^IGID   BODIES. 


471 


Art.  42.  To  find 
but  all  tendin 


ult  ant  of  forces  in  different  planes, 
rough  one  point. 


In  cases  where -mathematical  accuracy  is  not  necessary,  and  the  number  of  forces 
only  three,  or  four,  the  writer  will  venture  to  propose  a  method  by  models ;  which, 
if  open  to  the  objection  of  empiricism,  has  the  ad  vantage  of  requiring  less  time  than 
other  processes ;  is  sufficiently  correct  for  most  practical  purposes;  and  shows  the 
resultant  in  its  actual  position,  which  ie  done  by  no  method  of  calculation. 

Let  ao,  bo,  co,  Fig  30,  be  the  three  forces,  meeting  at  o ;  their  angles  with  each  other,  a  o  b,  6  o  c, 
Co  a,  (which  alone  are  necessary  in  this  method,)  being  of  course  known.  Draw  on  pasteboard  the 


JTi£30 


three  forces  ao,bo,co,  as  in  Fig  31,  with  their  actual  angles  ao  b,  b  o  c.  c  o  a.  By  Art  28,  draw 
the  parallelogram  of  forces  for  the  middle  pair  bo,  co;  and  draw  its  diag  w  o,  which  will  be  the  re- 
sultant of  those  two;  leaving  the  resultant  of  it,  and  a  o.  yet  to  be  found.  Cut  away  neatly  the 
•whole  fig,  aoacwb  a.  Make  deep  knife-scratches  along  ob,  o  c,  so  that  the  two  outer  triangles  may 
be  more  readily  turned  at  angles  to  the  middle  one.  Turn  them  until  the  two  edges  o  a,  o  a,  meet; 
and  then  paste  a  piece  of  thin  paper  along  the  meeting  joint,  to  keep  them  in  place.  Stand  the  model 
upon  its  side  o  b  w  c  as  a  base  ;  and  we  shall  have  the  slipper  shape  aobw,  Fig  32 ;  o  w  being  the  sole, 
and  a  o  ft  the  hollow  foot. 

We  new  have  the  first  resultant  w  o,  and  the  third  remaining  force  a  o,  in  their  actual  relative,  po- 
sitions. Now,  to  find  their  resultant,  also  in  its  actual  position,  cut  a  separate  triangular  piece  of  paste- 
board of  the  size  and  shape  of  w  a  o.  Find  the  center  i,  of  the  edge  w  a,  and  draw  a  line  i  o  on  each 
side  of  it.  Finally,  by  means  of  tbe  edges  ao,  wo,  paste  this  piece  to  the  inside  of  the  model,  along 
its  center-Hue  wo.  This  done,  io  represents  one  half  of  the  reqd  resultant,  in  its  actual  position. 

The  reason  why  it  represents  but  one-half  of  it  is  plain  ;  for,  as  be- 
fore stated,  we  now  have  a  o  and  w  o  in  their  actual  positions  in  the 
model;  consequently,  if  we  complete  the  parallelogram  of  forces 
wo  an,  and  draw  its  diagonal  no,  this  last  will  be  their  resultant. 
But  since  the  two  diags  of  every  parallelogram  divide  each  other  into 
two  equal  parts,  the  diag  aw;  thus  divides  the  resultant :  consequently 
t  o  is  one-half  the  resultant. 

If  there  be  four  forces,  as  an,  bn,  en,  dn,  Fig  34,  draw  them  as  in 
the  fig,  with  their  actual  angles  anb,  bnc,  &c.  Draw  also  the  re- 
sultants n  v,  of  an  and  6  n  ;  and  n  w,  of  n  c  and  n  d.  Then  cut  out 
the  entire  fig,  as  before;  and  paste  together  the  two  edges  an,  an. 
Then  we  have  the  two  resultants  av,  aw,  Fig  35,  forming  two  simple 
forces,  in  their  actual  relative  positions ;  and  we  have  only  to  measure 
their  dist  apart  from  v  to  w;  and  thence  find  their  resultant  ar, 
which  will  evidently  be  that  of  the  four  original  forces. 

Or,  as  in  the  preceding  case,  cut  out  a  separate  piece  of  pasteboard, 
avw.  Fig  35,  and  having  drawn  on  each  side  of  it  a  line  from  a  to  the 
center  o  of  vw,  paste  it  inside  of  the  model.  Then  will  ao  represent 
one-half  of  the  resultant  of  the  four  forces,  in  its  actual  position. 

Should  the  model  be  exposed  to  hard  usage  by  workmen,  it  should  be 
made  of  wood ;  the  triangles  anb,  bnc,  <fec,  being  cut  out  separately  ; 
the  joining  edges  bevelled  ;  and  then  glued  together.  See  also  Art  43. 

Art.  43.    The  parallelopiped  of  forces.    If  any  three  forces,  ao,  60, 
co,  Figs  36,  in  diff  planes,  meet  at  one  point  o,  whether  they  all  be  strains,  or  all 
motions,  their  resultant  or  joint  ef- 
fect  will  be    represented,   both  in     cV -[, 

quantity  and  in  direction,  by  the       |\    0       ~~N. 
diag  y  o,  of  a  parallelopiped  act  hi,  *.f       "  8    Vr 

of   which   three  converging   edges  *  "~- 

may  be  assumed  to  represent  the 
three  converging  forces. 

This  suggests  another  mode  of  showing 
the  resultant  of  three  such  forces  by  a 
model;  for  it  is  only  necessary  to  prepare 
a  box  A  or  B,  as  the  case  may  be ;  and  y  o 
will  represent  the  reqd  resultant. 


472 


FORCE   IN   RIGID    BODIES. 


Mo  three  forces  in  different  planes  can  be  in  equilibrium. 
Art.  44.  Forces  in  different  planes;  and  not  tending  to  or 
from  one  point.  It  is  but  rarely  that  such  lorees  have  a  resultant,  or  auti- 
resultant;  that  is,  no  single  force  can  usually  be  found  either  to  produce  an  equal 
effect,  or  to  balance  them.  It  is  so  seldom  that  they  present  themselves  to  the  engi- 
neer's attention,  and  their  solution  is  so  tedious,  except  in  very  simple  cases,  that 
we  shall  confine  ourselves  to  one  of  that  kind.  As  in  Art  41,  the  resultants  cannot 
be  had  by  measurement  from  a  drawing. 

Let  ao,  oo,  a  o,  Fig  37,  be  three  such  forces;  and  suppose 
them  all  to  act  against  (see  Remark,  Art  8)  the  same  plane 
ppp;  and  against  the  same  side,  or  surf  of  it:  that  is,  none 
of  them  pointing  upward  against  the  under  side  of  the  plane 
in  the  fig.  Having  the  points  o  of  application,  and  the  rela- 
tive positions  of  the  forces  themselves,  as  well  as  the  angles 
ooc  which  they  form  with  the  plane  ppp,  resolve  each  of  tne 
forces  into  two  components  ;  one  of  which,  co,  coincides  with 
the  plane  ;  while  the  other  (parallel  and  equal  to  a  c,  but  meet- 
ing c  o  at  o)  is  at  right  angles  to  c  o,  or  to  the  plane.  We  then, 
have  two  sets  of  forces ;  one  set  in  the  plane,  and  the  other  at 

P<W  \  U        $  rl8ht  angles  to  it.     Since  those  in  the  plane  do  not  tend  to  or 

^*Ufer^  „     \     •     ff  from  one  point,  their  resultant  must  be  found  by  Fig  19&,  p 

466;  while  that  of  the  several  parallel  components  (equal  to  a 
c,  but  applied  at  o)  may    be   obtained  by    Arts  56  and  59. 

-_^        rt—    -sjj,     fj/y  These  two  resultants  will  rarely  be  in  the  same  plane  with 

•<Tfi*   "VY       ^Htw(7  each  other,  and  consequently  can  have  no  joint  resultant.    If 

-LL5J,  *•*•  they  should  chance,  however,  to  fall  in  the  same  plane,  use 

Art  28  fo-  finding  their  resultant.     In  simple  cases,  where  the 

forces  act  against  one  plane,  as  in  our  fig,  pieces  of  wire,  cut  to  lengths  to  represent  the  forces  ;  and 
•tuck  into  a  piece  of  smooth  board,  in  their  proper  relative  positions,  will  greatly  facilitate  the  find- 
ing of  the  resultant  approximately  enough  for  most  practical  oases. 

The  same  general  process  must  be  used,  no  matter  how  great  may  be  the  number  and  directions  of 
the  forces.  A  plane  must  be  assumed  to  pass  somewhere  through  the  system :  and  the  directions  of 
all  the  forces  must  be  conceived  to  be  so  extended  as  to  terminate  at  points  of  application  in  said 
plane.  Each  force  must  then  be  resolved  into  two,  as  in  the  foregoing  example;  and  the  resultants 
of  the  two  sets  of  forces,  as  well  as  their  joint  resultant,  if  they  have  one.  must  be  found  as  before. 
If  any  of  the  forces  should  be  parallel  to  the  assumed  plane,  but  not  in  it,  it  evidently  cannot  be  re- 
solved into  two,  one  of  which  shall  be  in  the  plane ;  for  (Art  32)  no  force  can  have  one  of  its  compo- 
nents para'lel  to  itself.  Hence,  in  such  a  case,  the  resultant  cannot  be  found  by  this  process. 

Art.  45.  It  is  comparatively  seldom  that  strict  mathematical  accuracy  is  reqd 
in  finding  the  resultants  of  forces  in  engineering  practice;  therefore,  the  foregoing 
easy  methods  by  measurement  from  a  drawing,  or  model,  will  usually  answer  every 
purpose.  Moreover,  they  appeal  to  the  eye ;  and  are  therefore  much  less  liable  to 
serious  errors  than  methods  involving  numerous  calculations.  But  when  more  cor- 
rect results  are  needed,  they  may  be  had  by  means  of  a  table  of  nat  sines,  tangents, 
&c.  Thus,  in  the  case  of  two  components  and  their  resultant,  calling  the  components, 
Fig  38,  C  and  c;  and  the  resultant  R,  then 

If  the  angle  m  b  11  between  the  compo- 
nents^, c  is  9O°,  R  will  =  yc*  +  c*;  and  C  = 
y'ffl  _c2 ;  and  c  =  j/R  2—  C2.  Or  R  will  =  C  X  secant 
of  a  b  n  =  c  X  secant  of  a  b  m.  And  C  will  =  R  X  co- 
sine of  a  b  n  ;  and  c  =  R  X  cosine  of  a  b  m. 

Or  whether  the  angle  between  the  com- 
ponents €  and  c  be  9O°,  or  more,  or  less,  as 
m  Fig  39, 

R  X  sine  of  v  R  X  sine  of  x 

=    sineofCbc   '  and  c  =  sine  of  C  be  ' 
Observe  that  v  is  used  for  finding  C;  and  x  for  find- 
ing c. 

And  R  will  =  c  X  sine  of  Cbc  '     C  X  sipe  of  cbc. 
sine  of  x       or       sine  of  v 

If  the  angle  Cbc,  or  either  of  the  others  exceeds 
90°,  subtract  it  from  180°,  and  use  the  sine  of  the 
remainder. 


Eg  38 


FORCE--IN   RIGID   BODIES. 


473 


Art.  46.  Monr€nts.  Leverage.  If  a  b, 

Fig  40,  represenymy  force  acting  in  any  direction 
whatever;  and/If  o,  or  i,  be  auy  point  whatever, 
whether  in  or  out  of  the  body  on  which  the  force 
is  acting;  and  if  from  said  point  a  line  o  s,  or  i  c, 
be  drawn  at  right  angles  to  said  direction  of  the 
force,  then  said  line  o  «,  or  i  c,  is  called  the  arm. 
or  leverage  of  the  force  a  6,  about  said  point. 
And  if  the  amount  of  the  force  in  Bbs,  &c,  be  mult 
by  the  length  of  the  arm  or  leverage  in  feet,  &c, 
the  prod  in  ft-fts,  <fec,  is  called  the  moment  of 
the  force  about  that  point.  Thus,  if  the  force  a  b 
be  8  fbs,  or  tons;  and  the  line  os,  6  feet,  then  the 
moment  of  a  6  about  o  is  8  X  6  =  48  ft-fts ;  or  ft- 
tons.  A  force  whose  direction  passes  through  a  point,  has  no  moment  about  that  point. 

This  moment  represents  the  total  tendency  of  the  force  to  produce  motion  about  the  given  point. 
We  cannot  hold  bor,  between  the  ends  of  a  thumb  and  forefinger,  a  piece  of  stick  a  foot  long,  which 
has  a  3  ft  wt  at  tbe  other  end  of  it ;  because  the  tendency  of  the  wt  to  produce  motion  is  too  great  for 
the  force  of  our  lingers  to  resist;  but  we  can  in  that  manner  hold  a  stick  two  feet  long,  with  a  3  ft  wt 
at  each  end,  if  we  take  it  at  the  center.  For  although  in  this  case  there  is  twice  as  much  moment  as 
before  exerted  at  our  fingers  ;  yet  it  is  not  now  exerted  against  them  ;  because  we  now  have  two  equal 
moments  in  opposite  directions,  reacting  against  each  other ;  and  leaving  nothing  for  the  fingers  to 
react  against,  except  the  mere  vert  wt  of  6  fts. 

Since  the  moment  about  o  tends  to  produce  motion  at  that  point  in  the  direction  in  which  the  hands 
of  a  watch  move,  or  from  the  left  hand,  toward  the  right,  it  is  called  a  right-hand  moment.  But  the 
moment  of  the  same  force,  about  the  point  i.  tends  to  produce  motion  at  that  point,  from  right  to  left, 
as  shown  by  the  arrow-head  on  the  small  circle  ;  hence  it  is  called  a  left-hand  moment.  The  moment 
of  the  force  d  y,  with  its  leverage  y  i,  about  the  point  t,  is  a  left-hand  one;  as  is  also  that  of  x  w 
with  its  leverage  e  i. 

When  the  arm  o  s,  or  i  c.,  instead  of  being  merely  an  imagined  dist,  is  a  rigid  bar,  at  one  end  of 
•which,  as  8  or  c,  the  force  is  imparted  ;  thus  giving  the  bar  a  tendency  to  move  around  the  point  o  or 
t  as  a  fixed  center,  it  is  frequently  called  a  LEVER;  and  the  point  o  or  i,  the  FULCRUM  of  the  lever. 

If  the  lever,  instead  of  being  like  c  r,  at  right  angles  to  the  direction  a  m  of  the 
force,  should  be  oblique  to  it,  as  in  i  w,  or  i  a  ;  or  should  be  curved,  or  bent  in  any 
way,  as  i gt-  this  in  no  way  affects  the  leverage,  or  moment  of  the  force;  for  the 
leverage  is  always  tlie  perp  dist,  or  in  other  words;,  the  shortest  dist 
from  the  fulcrum,  to  the  direction  of  the  force;  and  is  entirely 
independent  of  the  length  of  the  lever  itself.  This  is  a  grand  funda- 
mental principle  of  all  levers,  and  leverages;  and  the  young 
student  should  carefully  impress  it  upon  his  memory,  inasmuch  as  it  is  of  constant 
application  in  practice. 

The  fulcrum  is  not  always  at  one  end  of  the  lever,  but  may  be  between  the  two 
ends ;  so  that  there  are  two  arms.  Cog-wheels  are  merely  continuous  circular  levers, 
with  the  fulcrum  at  the  center. 

Art.  47.  As  a  further  illustration, let 
a/fe,  Fig  41,  be  a  bent  lever,  turning  on 
its  fulcrum/;  and  m  and  n  two  wts  sus- 
pended from  its  ends,  constituting  two 
forces  acting  in  the  vert  directions  a  w, 
6  c.  Now,./  c,  at  right  angles  to  the  di- 
rection b  c;  and  /a,  at  right  angles  to 
the  direction  a  w,  are  the  arms,  or  lever- 
ages of  the  forces  m,  and  n,  about  the 
point  /. 

In  the  fig  these  leverages  are  equal,  say  each 
is  6  ft;  and  let  each  wt,  m  and  n.  be  100  Ibs:  then 
the  right-hand  moment  of  m,  and  the  left-hand 
one  of  n.  about  /,  are  each  6X10(t600  ft-fts  ;  and 
since  both  the  forces,  and  /,  are  all  in  the  same 
plane,  the  two  forces,  or  the  two  opposite  mo- 
ments, balance  each  other;  although  the  actual 
lever  fb,  is  much  longer  than /a. 

Now  suppose  the  wt  m  to  be  removed ;  and  that  instead  of  it.  a  person  pulls  in  the  direction  b  s,  by 
means  of  a  string  fastened  at  b.  With  what  force  must  he  pull  in  order  to  balance  the  wt  n?  First 
measure  the  leverage  ft,  from  the  fulcrum,  and  at  right  angles  to  the  direction  b  s  of  the  new  force ; 
and  mippose  it  is  found  to  be  9  ft.  Now  we  have  already  found  the  moment  of  n  about  /to  be  600  ft- 
fts ;  and  we  require  the  same  moment  on  the  opposite  side  ;  so  that  all  that  is  reqd  is  to  find  what 

number  the  9  feet  leverage  must  be  mult  by,  in  order  to  make  600.    This  is  plainly  —    =  66.66  fts, 

for  the  pull  which  the  person  must  exert ;  because  »  X  66.66  —  600. 
Bo  it  is  seen  that  with  the  same  length  of  lever,  fb,  we  can  have  diff  powers,  (BO  called,)  or  lever. 


474 


FORCE   IN   RIGID   BODIES. 


agos,  according  to  the  direction  in  which  we  apply  our  force  to  the  lever.  This,  however,  evidently 
has  its  limit;  for  the  greatest  power  is  gained  (to  use  the  popular  expression)  when  we  apply  our 
force  in  the  direction  by,  at  right  angles  to  a  line/ b.  drawn  from  the  fulcrum  to  the  outer  end  of  the 
lever.  If  we  apply  it  in  the  direction  6  d,  we  get  only  the  leverage  fh. 

On  the  same  principle  as  in  the  foregoing  example,  if  o  t  and  .9  a,  Fig 
42,  be  two  beams  of  equal  scantling,  but  of  diff  lengths ;  with  one  end 
of  each  firmly  fixed  in  a  vert  wall,  and  both  sustaining  equal  suspended 
wts  w,  x;  the  moments  of  the  wts  about  the  points  o  audy  will  be  equal, 
because  the  arms  or  leverages  oe,ga,  are  equal.  Therefore  the  wt  w 
will  have  no  more  tendency  to  break  off  the  long  beam  at  o,  than  x  has 
to  break  the  short  oii€  at  g.  The  wts  of  the  beams  themselves  are  not 
here  taken  into  consideration ;  and  this  is  always  the  case  iu  speaking 
of  levers,  unless  otherwise  expressed.  In  very  many  cases,  the  wt  of 
levers  of  two  arms  does  not  affect  the  result  aimed  at,  provided  the  arms 
are  so  proportioned  as  to  balance  each  other  when  unloaded ;  no  matter 
what  their  comparative  lengths  or  wts  may  be. 

If,  in  Fig  42,  we  apply  pulls  to  the  beams,  in  the  parallel  directions 
tm,  en,  at  right  angles  to  ot,  then  the  leverages  become  changed  from 
o e  and  g  a,  to  ot  and  g  c  ;  and  since  o  t  measures  6  times  the  length  of 
gc,  it  follows  that  the  beam  ot  would  be  broken  off  at  o,  by  i  part  as 
much  force  in  this  new  direction,  as  ga  would;  for  the  leverage  being 
6  times  as  great,  must  be  mult  by  only  i  as  much  wt,  in  order  to  have 
an  equal  breaking  moment. 

Art.  48.  In  ordinary  phraseology,  the  load,  or  resistance  of  any  kind,  which 
we  wish  to  move,  overcome,  or  balance,  by  means  of  a  lever,  is  called  the  weight; 
while  the  force  of  whatever  kind  which  we  apply  to  accomplish  this,  is  called  the 
power.*  Usually,  but  not  always,  the  power  is  applied  to  the  longer  arm. 
Equilibrium,  or  balance,  or  equal  momeuts  in  opposite  directions,  will  plainly  always 
take  place,  when  the  long  leverage  (not  lever)  has  the  same  proportion  to  the  short 
one,  that  the  wt  has  to  the  power;  because  then  only  can  the  long  leverage  mult  by 
the  small  power,  have  the  same  moment  as  the  short  leverage  mult  by  the  great  wt. 
This  is  seen  in  the  common  steelyard,  Fig  43 ;  which  is  merely  an  iron  lever,  turning  on  a  fulcrum 
/,  and  having  the  wta  of  its  two  arms  fa,fb,  so  proportioned  as  to  balance  each  other  when  un- 
loaded. Here  the  power,  PI,  at  the  dist  of  two  divisions  from  the 
fulcrum ;  balances  the  wt,  W  2,  at  the  dist  of  one  division.  If  the 
wt  W  2  were  suspended  at  y,  only  half  a  division  from  /,  it  would 
balance  the  power  P  1,  suspended  where  W2  is  in  the  fig,  at  a  whole 
division  from  /.  If  the  power  is  reqd  to  move,  or  overcome  the  wt,  it 
is  plain  that  either  the  power  itself,  or  the  length  of  its  arm,  must  be 
greater  than  when  mere  equilibrium  is  to  be  effected;  in  other  words, 
besides  the  two  straining  forces,  which  by  their  mutual  action  balance 
or  equilibrate  each  other,  we  need  some  unresisted  force  to  impart 
motion  to  the  inert  matter. 

In  the  two  levers  of  the  same  length,  Fig  44,  the  leverage  fw  of  the 
wt  w,  is  of  the  same  length  in  both ;  namely,  one  division  ;  but  the 
leverage  fp,  of  the  power  p,  is  but  two  divisions  long  in  the  upper  one, 
and  three  divisions  in  the  lower.  Therefore  a  power  of  1  ft  will  balance 
only  a  wt  of  2  fts  in  the  upper  one,  and  of  three  Ibs  in  the  lower.  In  the 
upper  one,  the  power  will  move  twice  as  fast  as  the  wt;  in  the  lower  one, 
three  times  as  fast.  When  the  fulcrum  is  between  the  wt  and  the  power, 
as  in  the  upper  one,  the  lever  is  said  to  be  of  the  first  class;  when  the  ful- 

f  I         I          I         l"P     crum  is  at  one  end,  and  the  power  at  the  other,  second  class;  fulcrum  at 
J  A iS  ibvi      one  end,  and  wt  at  the  other,  third  class.     In  all  cases  it  is  assumed  that 

the  fulcrum  is  in  the  same  plane  (Art  8)  with  the  directions  of  both  the  wt 
and  the  power ;  otherwise  the  principles  do  not  apply.  When  two  weights 
balance  each  other  on  two  arms  of  a  lever,  as  a  steelyard,  or  common 
weighing  scales,  &c,  their  directions  are  the  vert  lines  passing  through 
their  centers  of  grav  ;  and  the  same  imaginary  vert  plane  which  coincides 
•with  those  directions,  coincides  with,  or  passes  through,  the  fulcrum  also.  When  this  is  not  the 
case,  no  equilibrium  can  exist.  This  may  be  readily  proved  by  experiment ;  for  we  cannot  balance  a 
bow-shaped  piece  of  stick  or  wire,  so  long  as  the  bow  is  hor;  for  it  will  turn  on  the  fulcrum,  of  its 
own  accord,  until  the  bow  becomes  vert;  so  that  the  same  vert  plane  that  passes  through  the  fulcrum 
shall  pass  also  through  the  cen  of  grav  of  each  half  of  the  bow.  If  all  the  forces  acting  on  the  lever 
are  hor,  or  oblique,  the  imaginary  plane  must  be  so  too. 


id  respecting  the  lower  Fig  44,  it  follows  that  when  a  load  w  is  borne 
rted  at  both  ends,  then  the  portion  of  the  load  supported  by  each  of 


and/  supports  %.    Or  as/p  :  w  :  ifw  :  load  &tp     And 


*  The  fact  that  by  means  of  leverage  a  small  power  can  be  made  to  move  a  great  wt,  is  in  common 
parlance  styled  a  gain  of  power.  In  a  scientific  sense  the  expression  is  absurd,  yet  in  practice  it 
has  by  its  universal  use  become  very  convenient ,  and  we  shall  therefore  employ  it.  When  the  lever, 
instead  of  merely  balancing  the  power  and  the  weight,  has  to  be  put  into  motion,  it  is  plain  that  there 
must  be  s»me  excess  of  force  applied  at  the  power  end,  to  produce  the  motion. 


IN  KIGID  BODIES. 


475 


Art.  49.  TfTis  example  is  the  same  as  in  Fig  19,  p  465, 
where  the  question  is  solved  independently  of  leverage  and 
moments. 

Let  S  be  a  stone  of  3  tons ;  standing  on  a  hard  hor  base  n 
m  ;  let  i  be  its  cen  of  grav ;  and  let  the  dist  ng  from  the  toe 
n,  and  at  right  angles  to  the  vert  direction,  ig,  of  the  gravity 
of  the  stone,  be  2  feet.  Also  let  /  h  be  a  force  of  2  tons,  im- 
parted to  the  stone  at  h  ;  and  let  the  dist  n  o  from  the  toe  n, 
and  at  right  angles  to  the  direction  fa  of  the  force  fh  be  5 
ft.  Will  the  force  fh  upset  the  stone  around  the  toe  n,  as  a 
turning  point? 

Here  n  g,  or  2  ft,  is  the  leverage  of  the  force  (3  tons)  of  gravity ;  consequently,  the  moment  of  the 
stone  about  the  point  n,  is  equal  to  2  ft  X  3  tons  =  6  foot-tons  ;  and  this  moment  (which  is  called  the 
moment  of  stability  of  the  stone,  about  n)  alone  tends  to  prevent  the  stone  from  overturning  about 
n-  Again,  n  o,  or  5  ft,  is  the  leverage  of  the  force,  (2  tons,)  of  /  h  ;  consequently,  the  moment  of  fh 
about  the  point  n,  is  equal  to  5  ft  X  2  tons  =  10  foot-tons  ;  and  this  moment  alone  tends  to  overturn 
the  stone  about  n.  Since  the  overturning  moment  is  the  greatest,  the  stone  will  of  course  upset. 
The  foregoing  case  resembles  that  of  an  abutment  resisting  the  thrust  of  an  arch ;  or  that  of  a  re- 
taining -  wall,  sustaining  the  thrust  of  earth  against  its  back ;  said  thrust  be! 


being  supposed  to  be  con- 

cu 


centrated  at  its  center  of  pressure.  ("Art  57.)  It  is  analogous  to  a 
simple  bent  lever  efo,  Fig  255^,  supported  at  its  fulcrum/; 
around  which  it  may  revolve.  The  short  arm  fe,  of  2  ft,  is  acted 
upon  by  the  3  tons  wt  of  the  stone;  moment  =  2X3  =  6  ft-tons. 
The  long  arm  fo,  of  5  ft,  is  acted  upon  by  the  2  ton  force  ft ;  mo- 
ment 5  X  2=10  ft-tons;  which,  being  greater  than  the  6  ft-tons 
of  the  stone,  equilibrium  cannot  exist;  and  motion  must  ensue. 
The  wt  of  the  body  »,  in  Pig  25,  constitutes  one  of  the  forces  act- 
ing upon  it ;  while  its  inert  matter  constitutes  the  two  lever  arms 
at  Fig  25K-  It  frequently  thus  happens  that  a  body  is  at  the 
same  time  the  resistance  to  be  overcome ;  and  the  lever  with 
which  to  overcome  it.  It  is  plain,  that  in  the  same  manner  as 
above,  we  may  lind  separately  the  moments  of  any  number  of 
forces  acting  in  the  same  plane,  upon  a  body  ;  and  may  afterward 
ascertain  their  united  effects,  by  adding  into  one  sum  those  which 

tend  to  overturn  it;  and  into  another  sum,  those  which  tend  to  prevent  its  overturning;  the  diff 
between  these  sums  will  be  their  joint  effect. 

REM.  In  Fig  25  we  have  supposed  the  body  S  to  turn  about  a  single  point,  n; 
but  in  practical  cases  they  turn  about  edges,  as  d  y,  Fig  25%,  the  assumed  turning 
edge  of  the  body  B  ;  which  may  be  re- 

farded  as  a  retaining- wall,  or  abutment, 
c.  In  making  calculations  for  the 
strength  of  such  structures,  it  is  usual 
to  restrict  ourselves,  for  convenience,  to 
a  supposed  vert  slice,  one  foot  thick, 
like  the  shaded  end  of  the  fig ;  in  which 
case  the  turning  edge  is  one  foot  long 
instead  of  extending  along  the  toe,  dy, 
of  the  entire  structure;  (see  Art  70.) 
This,  however,  causes  no  change  in  the 
calculations,  which  remain  the  same  as 
for  Fig  25 ;  for  we  suppose  all  the  wt  of 
the  slice  to  be  concentrated  at  its  cen 
of  grav ;  and  the  forces  to  be  imparted 
in  the  same  vert  plane  with  the  direction  of  the  gravity  ;  so  that  it  amounts  virtually 
to  the  same  thing  as  if  we  assumed  our  one-foot  slice  to  be  infinitely  thin ;  but  still 
to  have  the  same  wt  as  if  it  were  one  foot  thick. 

But  in  fact,  it  is  not  absolutely  neces 
same  vert  clane  with  the  gravity  of  the 
which  the  edge  d  y  bears,  and  revolves,  may  b 
action  of  those  forces.   For,  if  there  be  no  yielding  whatevi 

so  far  as  regards  overturning,  whether  the  force  be  applied  at  o,~or  at  t  f  for  d  y  then  becomes  analo- 
gous to  the  rigid  axle  c  t,  Fig  25%.  with  two  lever-arms  s  e,  and  c  a.  The  wt  r,  may  be  supposed  to  be 
that  of  the  structure ;  and  from  that  common  machine,  the  wheel  and  axle,  we  know  that  it  is  imma- 
terial whether  the  applied  force,  arid  other  lever-arm,  are  attached  to  the  axle  at  c ;  or  at  any  other 
point  along  its  length  j  so  long  as  the  axle  is  equally  unyielding  at  every  point.  Art  53  will  p'erhaps 
make  this  more  clear. 


iary  in  such  cases,  to  suppose  our  applied  force?  to  be  in  the 
all,  provided  that  both  our  structure,  and  the  baxe  against 
egarded  as  practically  rigid,  or  unyielding,  under  the 
'  along  the  edge  dy.  then  it  is  immaterial, 


476 


FORCE   IN    RIGID   BODIES. 


;n 


Art.  5O.  Equilibrium  of  forces,  and  of  moments.  The  stu- 
dent must  distinguish  clearly  between  those  cases  in  which  forces  hold  each 
other  in  equilibrium,  and  those  in  which  the  moments  of  forces  do  so.  Thus,  if 
two  equal  forces  n  and  R,  Fig  Y,  act  against  each 
other  in  the  same  straight  line  n  R,  but  in  op- 
posite directions,  we  correctly  nay  that  these  two 
forces  are  in  equilibrium,  or  balance  each  other, 
or  prevent  each  other  from  giving  motion  to  the 
body.  But  also  in  the  case  of  a  lever,  one  arm  of 
which  is  say  10  ft  long,  and  the  other  only  2  ft,  we 
usually  say  that  2  ros  of  force  at  the  end  of  the 
long  arm,  will  balance  or  hold  in  equilibrium  10 
ros  at  the  end  of  the  short  one.  But  this  is  not 
scientifically  correct ;  for  a  force  of  2  ros  cannot 
possibly  balance  one  of  10  ros.  It  is  actually  the 
moment  of  the  2  ros  that  balances  the  mo- 
ment of  the  10  ros.  That  is,  2  ros  X  10  ft  leverage  =  20  ft-Bbs,  the  moment  of 
th«  2  ros,  balances  the  10  ros  X  2  ft  leverage  =  20  ft -ros,  the  moment  of  the  10  ros, 
As  to  the  two  forces  2  and  10,  they  are  balanced  by  the  upward  12  ft)  reaction  of  the 
fulcrum. 

REM.  When  any  number  of  forces  as  m,  n,  o,p,  Fig  Y,  in  the  same  plane, 
•whether  acting  through  one  point  or  not,  hold  each  other  in  equilibrium,  then  any  one  of  them,  as  n, 
is  equal  to  the  resultant  R  of  all  the  rest;  and  will  be  in  the  same  straight  line  with  it,  but  will  act 
in  the  opposite  direction.  And  this  is  the  most  ready  method  that  suggests  it- 
self to  the  writer  for  determining-  whether  several  given  forces 
are  in  equilibrium  or  not.  Art  30,  p  466,  may  be  used  for  finding  the 
required  resultant  R.  It  is  true  (Art  38,  Fig  22)  that  if  the  forces  m,  n,  o,  p,  are  in  equilibrium, 
they  will  form  a  closed  polygon,  as  at  P :  but  they  would  evidently  form  the  same  polygon  if  the 
points  of  application  of  any  of  them  at  Y  were  changed  so  that  they  would  no  longer  be  in  equilib- 
rium. Therefore  their  mere  forming  of  a  closed  polygon  is  no 
proof  that  they  are  in  equilibrium;  although  their  not  forming  one 
is  a  proof  that  they  are  not  so. 

Art.  51.    Equality  of  moments.    This  principle  consists  in  the  follow- 
ing :  If  any  number  of  forces  as  a,  6, 

.,  c,  d,  Fig  26,  all  in  the  same  plane,  and 

&  acting  upon  a  body  N,  in  any  directions 

whatever  in  that  plane,  hold  each  other  in 
equilibrium,  then  if  any  point  i  be  taken 
in  that  same  plane,  whether  within  the 
body,  or  out  of  it,  the  moments  of 
the  forces  will  hold  each  other  in  equi- 
librium around  that  point;  that  is,  the 
moments  of  all  those  forces  (b  and  d) 
which  tend  to  turn  the  body  N  in  a  right 
hand  direction,  (or  like  the  hands  of  a 
watch,)  around  the  point  i,  will  together 
be  equal  to  the  moments  of  all  those  (a  and  c)  which  tend  to  turn  it  around  i  in  a 
left  hand  direction. 

If  the  four  forces  a,  6,  c,  c?,  mutually  prevent  each  other  from  imparting  to  the 
body  N,  any  tendency  to  move,  as  a  whole,  in  any  straight  direction  whatever,  they  are  themselves 
in  equilibrium:  and  this  being  the  case,  their  right  and  left  hand  moments 
around  the  point  i  will  also  be  found  to  be  in  equilibrium,  as  shown  below.  And  so  also  with  any 
other  point  in  the  same  plane.  But  in  that  case  the  arms  or  leverages  would  be  changed. 


Forces. 

Arms. 

Right-hand  Momenta. 

Left-hand  Moments. 

a=6 
c  =  3 
b=4 

d=7.2 

3.9 
5.8 
4.6 
3.111 

1*8.4 
22.4 

23.4 
17.4 

40.8  Total. 

40.8  Total. 

But  moments  around  a  point  may  hold  each  other  in  equi- 
librium even  when  the  forces  themselves  do  not  balance 
each  other;  as  in  the  case  of  the  lever  in  Art  50. 

REM.  If  any  number  of  forces  as  «,  6,  c,  d,  Fig-  26,  in  the  same 
plane,  whether  acting-  through  the  same  point,  or  not.  hold 
each  other  in  equilibrium,  they  will  do  the  same  if  they 
all  be  made  to  act  in  the  same  directions  through  one 
point ;  as  A,  B,  C,  D,  through  p. 

But  it  dnes  not  follow,  that  because  forces  may  balance  each  other  when  applied  at  one  point  p,  they 
will  do  so  if  applied  to  a  body  N  at  any  point*  whatever,  in  the  same  plane. 


IN   RIGID   BODIES. 


477 


Art.  52.  Jrfrtnal  velocities.  Whenever  the  power  and  the  wt  balance 
each  otlieiytiither  in  a  single  lever,  or  in  a  connected  system  of  levers,  or  leverages 
of  any  kind  whatever,  then  if  we  suppose  them  to  be  put  into  motion  about  the  ful- 
crum, their  respective  vels  will  be  in  the  same  proportion  or  ratio  as  their  leverages; 
that  is,  if  the  leverage  of  the  power  is  2,  5,  or  60  times  as  long  as  that  of  the  wt,  the' 
power  will  move  2,  5,  or  50  times  as  fast  as  the  wt.  Therefore,  by  observing  these 
vels,  we  may  determine  the  ratio  of  the  leverages.  The  wt  and  the  power  are  to 
each  other,  therefore,  inversely  as  their  vels,  as  well  as  inversely  as  their  leverages; 
and  this  is  based  upon  the  principle  of  virtual  velocities;  and  is  very  important. 

Art.  53*  Neither  the  amount,  nor  the  effect  of  leverage  is  changed,  if  the  arms 
of  the  lever,  (whether  straight  or  crooked,  or  in  whatever  relative  positions  they  may 
be,)  instead  of  being  in  one  piece,  and  supported  by  a  single  point  or  edge  as  a  ful- 
crum, as  in  Fig  44 ;  should  consist  of  two  separate  pieces  m  and  n,  Fig  45,  firmly 
united  to  a  straight  rigid  axle  a  a, 
of  any  length;  (usually  placed  at 
right  angles  to  the  levers  m,  n,)  and 
supported  at  two  or  more  points 
/,/.  The  moments  are  then  about  -«r 
the  axis,  or  longitudinal  center-line 
of  the  axle ;  any  point  of  which 
may  be  regarded  as  the  fulcrum.  It 
is  moreover  immaterial  at  what 
points  along  the  axle, the  lever-Hrms 
TH,  n,  may  be  attached  to  it ;  both  "p 
may  be  between  /,/;  or  both  out- 
side  ;  or  one  in  each  position.  This 
is  illustrated  by  the  common  hoisting  wheel  and  axle.  The  rad  of  the  wheel, 
and  that  of  the  axle,  measd  from  their  common  axis,  constitute  two  continuous 
levers.  Also  by  series  of  cog-wheels,  which  we  see  placed  indifferently  at  any  points 
along  extended  shafts,  or  axles',  whether  vert,  hor,  or  inclined. 

If  the  levers  m,  n,  are  not  at  right  angles  to  the  length  of  the  axle,  then  their  lever- 
ages, and  not  their  actual  lengths,  (measd  from  the  center  line  of  the  axle,)  must  evi- 
dently be  used  in  calculating  the  moments  of  forces  acting  upon  them.  See  Remark, 
Art  49,  p  475. 

REM.  The  assumption  of  an  imaginary  axle,  or  axis,  as  in  Rem  to  Art  49,  enables  to  solve  such 
cases  as  this.  The  entire  load  being  known  which  is  sustained  by  three  vert  supports ;  also  the  posi- 
tion of  its  cen  of  grav ;  to  find  how  much  of  it  is 
sustained  by  each  of  the  supports :  Draw  by  scale, 
a  triangle  a  b  c,  Fig  46,  showing  in  plan  the  correct 
position  of  the  supports ;  and  of  the  cen  of  grav  g, 
of  the  sustained  load.  Assume  any  side,  as  a  c,  to 
be  an  axis  ;  and  from  it  draw  the  two  perps :  i  b,  to 
the  opposite  angle  b ;  and  e  g.  Jfow,  it  is  plain  that 
e  g,  and  i  b,  may  be  considered  as  two  leverages 
from  the  supposed  axle  a  c.  At  g  is  placed  the  en-  Si  \ 
tire  load,  whose  moment  about  the  axis  of  the  axle, 
is  equal  to  the  load,  (say  5  tons,)  mult  by  the  meas- 
ured (by  scale)  length  of  e  g,  say  8  ft.  And  this 
moment  (5  X  8  =  40  ft-tons)  is  balanced  by  that 
of  an  upward  force  at  6  :  which  upward  force  must 
be  equal  to  that  share  of  the  load  which  rests  upon 
6,  inasmuch  as  the  two  are  in  equilibrium.  To  find 
the  amount  of  the  force  at  b,  we  have  only  to  div 
the  moment  (40  ft-tons)  of  g,  by  the  leverage  i  6,  of  the  reqd  force  b.  Suppose  i  6  is  found  by  the  scale 
to  be  25  feet;  then  the  upward  force  at  b,  is  4£:sU  tons ;  and  we  have  its  moment  about  the  axis 
=  1.6  X  25=- 40  ft-tons,  the  same  as  that  of  g.  Therefore  b  supports  1.6  tons  of  the  load.  In  pre- 
cisely the  same  manner  we  may  assume  each  side  in  turn  to  be  an  axis ;  and  find  how  much  is  sus- 
tained at  the  opposite  angle.  The  pressure  on  each  of  four  legs  cannot  be  calculated. 

The  Fig  T,  in  which  a  represents  one  end  of  the  axis  ;  a  g  the  leverage  e  g  of  the  load  g;  and  a  8 
that  (i  b)  of  the  force  5,  will  make  the  principle  more  apparent. 

Art.  54.  Ex.  1.   The  condition  of  a  beam 
a  b,  Fig-  47,  may  often  be  examined  on  the        « 
principle  of  a  lever.    Suppose  it  to  be  of  uniform    '*%% 

depth  and  thickness ;  its  length  a  b,  in  the  clear  between  its  supports,  C  < 
20  ft ;  its  wt  600  fts  ;  and  its  position  hor.  In  this  case  we  know  that 
one-half  its  wt.  or  300  fts,  is  borne  by  each  support.  To  prove  this,  we 
mav  consider  its  entire  weight  to  be  concentrated  at  its  cen  of  grav, 
which  in  this  case  will  be  at  its  center  t.  See  Arts  56, 57.  Then  we  suppose 
one  of  the  supports,  as  o,  to  be  removed  ;  and  an  upward  force /to  be 
acting  upon  a  lever  a  b,  20  ft  long,  without  wt ;  and  sustaining  a  load  of  600  fts  at  10  ft  from  the  ful- 
crum a.  Since  the  force  and  the  load  both  act  vert,  and  the  beam  is  hor,  or  at  right  angles  to  their 
directions,  therefore  the  dist  a  t  and  a  b  from  the  fulcrum,  are  the  true  leverages  of  said  force  and 
load.  Now,  the  moment  of  the  load,  about  the  fulcrum,  is  600  fts  mult  by  10  ft  =  6000  ft-lbs ;  and  to 

31 


o 


478 


FORCE   IN   RIGID   BODIES. 


Cl 


6000 
i  upward  force /must  DC  equal  t 

the  proof  reqd. 

If,  in  addition  to  its  own  wt,  the  beam  had  actually  sustained  a  load  w  at  its  center,  we  must  add 
this  load  to  the  wt  of  the  beam ;  and  then  proceed  as  before.  So  also  if  it  sustains  a  load  uniformly 
distributed  over  its  length. 

Ex.  a.    If  the  cen  of  grav  of  the  beam  be  at  any  point 

y,  Fig  48,  not  at  its  center,  we  use  the  leverages  a  y  and  a  s,  instead  of 
r^  at  and  a  b  ;  and  if  in  addition  it  sustains  a  load  z  at  any  point  n  what- 
ever, we  first  find  as  before  the  force  reqd  at  F  for  the  beam  alone ;  and 
afterward,  by  using  a  n,  and  a  s  as  leverages,  we  find  the  force  reqd  by 
the  load;  and  add  the  two  forces  together  for  the  total  F. 

>_.        Rem.    To  find  the  portions  of  z  borne  by  a  and  by  * 

p      say,  as  the  whole  span  s  a  is  to  the  whole  load  z,  so  is  n  a  to  the  load 
on  s ;  and  as  8  a  is  to  z  so  is  n  «  to  the  load  on  a. 

Ex.  3.    If  the  beam  sustains  several  loads  at  diff  points, 

as  in  Fig  49,  calculate  for  each  of  them  separately,  using  the  leverages 
a  f,  a  c,  a  o,  &c  ;  and  add  all  together  for  total  F.  For  portions  of  each 
of  these  loads  borne  by  a  and  v  see  above  Rem. 

For  more  on  this  subject  see  p  218. 

Ex.  4.    If  the  beam  in  any  such  case,  in  inclined,  as  in  Fig  50,  the 

hor  dist  a  o,  a  g,  &c,  must  be  taken  as  measured  from  the  fulcrum  a, 
instead  of  at,  a  i,  &c  5  because,  since  all  the  forces  are  vert  in  direction, 
only  a  hor  line  can  be  at  right  angles  to  them ,  and  serve  to  measure 
their  leverages  from  th«  fulcrum  a.  If  the  beam  be  rigid,  and  its 
ends  cut  hor,  as  shown  in  this  fig,  it  will  have  no  tendency  to  slide; 
because  all  the  forces  which  through  it  are  applied  to  the  bodies  m 
and  p  are  vert;  and  since  the  joints  are  at  right  angles  to  those 
bodies  at  those  points,  the  entire  forces  will  be  imparted  also ;  no  por- 
tion of  them  remaining  unresisted,  to  act  as  motion,  so  long  as  the 
beam  remains  rigid,  and  consequently  straight.  But  if  it  bends  un- 
der either  its  own  wt,  or  that  of  its  load,  new  forces  come  into  action, 
which  will  tend  to  push  the  supports  outward  from  each  other;  so 
also  in  the  foregoing  cases. 

It  is  only  where  we  may  practically  regard  a  beam  as  rigid,  or  unchangeable  under  the  forces,  that 
the  foregoing  concentration  of  entire  weights  or  forces  at  the  cen  of  grav,  can  be  safely  assumed. 
It  will  not  apply  when  we  are  investigating  the  strength,  and  deflections  of  beams ;  see  Art  58. 

After  having  thus  obtained  F,  in  any  of  these  cases,  or  in  other  words,  having  found  how  much 
«f  the  entire  wt  of  beam  and  load  bears  upon  one  support,  we  have  only  to  subtract  it  from  the  eutire 
wt,  to  obtain  that  on  the  other  support.  It  is  plainly  immaterial  which  end  of  the  beam  is  assumed 
to  be  the  fulcrum  in  any  of  these  cases. 

Ex.  5.  Let  a  o,  Figs  51,  be  a  hor  beam  10  ft  long,  projecting  from  a  vert  wall  a  c ;  and  resting  at  one 
end  on  a  step  a;  the  other  end  being  sustained  by  either 
a  strut,  or  a  tie  p  c,  12&  ft  long.  The  beam,  and  its  uni- 
form load,  weighing  together  3  tons,  what  will  be  the  push- 
ing strain  along  the  direction  of  the  strut;  or  the  pulling 
strain  along  the  tie  ?  Draw  the  Fig  to  scale ;  and  meas- 
ure a  i  (which  will  be  found  to  be  6  ft)  at  right  angles 
to  c  o.  Now,  the  weight  of  a  rigid  body,  when  considered 
only  with  regard  to  its  effect  in  moving  the  entire  un- 
altered body,  or  in  straining  it  bodily  against  another 
body,  acts  the  same  as  if  it  were  all  concentrated  at  its 
cen  of  grav;  and  since  we  are  now  about  to  consider  it 
in  that  light,  and  not  as  tending  to  lend  or  break  the 
beam  a  o  (in  which  case  only  half  its  uniform  load,  and 
wt  must  be  assumed  to  be  concentrated  at  its  cen  of  grav ;) 
we  consider  the  3  tons  wt  to  act  at  g,  5  ft,  or  half  the 
length  of  the  beam  from  a.  Now,  the  3  tons,  being  a  force 
of  grav,  will  act  in  a  vert  direction ;  and  since  the  beam 
is  hor,  a  g  is  at  right  angles  to  this  direction  of  the  force 
exerted  by  the  beam  and  its  load.  Consequently,  if  we 
assume  the  beam  to  be  a  lever,  movable  about  a,  as  a  fulcrum,  a  g  is  the  leverage  of  that  force  of 
grav  ;  and  the  moment  of  that  force  about  a,  consequently,  is  3  X  5  —  15  foot-tons.  But  this  moment 
is  reacted  against  by  that  of  another  force  in  the  direction  c  o ;  which  acts  at  the  point  o  of  the  lever 
o  o,  to  uphold  the  beam  and  its  load.  The  leverage  a  i  of  this  force,  that  is,  the  dist  from  the  fulcrum 
a,  and  at  right  angles  to  the  direction  c  o  of  the  force,  has  already  been  found  to  be  6  ft ;  consequently, 
the  force  itself,  in  order  to  have  a  moment  of  15  ft- tons  alxmt  a  (as  the  beam  and  its  load  have)  must 

evidently  be  •—  =  2.5  tons,  the  reqd  strain  along  the  strut,  or  along  the  tie,  o  c;  for  2.5  X  6  —  15. 

Ex.  6.  The  following-  is  very  important  in  its  application 
to  arches  of  any  material. 

Let  end  rj,  Fig  52,  represent  one  half  of  a  bridge  arch.  If  this  half  were  not  prevented  by  the 
hor  pres,  ha,  of  the  opposite  half,  it  would  evidently  fall,  as  in  the  shaded  fig.  by  turning  about  the 
point  r  as  a  fulcrum.  (See  Rem  1.)  Let  us  find  what  this  hor  pres  amounts  to  in  any  case.  To  do 
this,  we  may  consider  the  half  bridge  cndrj  to  be  a  lever.  Suppose  its  wt  to  be  80  tons  ;  and  t« 
be  concentrated  at  its  cen  of  grav  g ;  'the  vert  line  g  s  being  of  course  the  direction  in  which  it  would 
act.  And  let  its  leverage  r  t,  about  r,  be  \  ft;  r  t  of  course  being  at  right  angles  to  the  direction 


5RCE   IN   RIGID   BODIES. 


479 


g  a.  TheLKisits  moment  about  r  equal  to  80  X  6  =  480  ft-tous.  (Art  46.)  Now,  whatever  may  b<» 
the  anvsruiu  of  the  hor  force  h  a,  which  acts  at  the 
end  a  of  this  lever,  to  counteract  this  moment  of 
480  ft-tons,  its  leverage  (Art  46)  is  plainly  equal  to  re, 
measured  from  the  fulcrum  r,  and  at  right  angles  to 
the  direction  A  i  of  said  force.  Suppose  we  find  by  mea- 
surement from  the  drawing  that  r  e  is  8  feet.  Then  the 


force  itself  must  necessarily  be  —  =  60  tons 


which 


is  the  hor  pres  which  the  opposite  half  of  the  bridge 
exerts  against  the  keystone  a,  of  the  arch  ;  for 
60  X  8  —  480  ft-tons  of  moment. 

KKM.  1.  But  so  long  as  an  arch  is  not  deranged, 
but  remains  firmly  in  position,  the  half  arch,  in- 
stead of  tending  to  revolve  about  the  point  r,  presses 
equally  over  the  entire  surf  r  of  of  its  skewback. 
Therefore,  the  leverage  with  which  the  hor  force  A 
acts  upon  the  skewback,  is  actually  y  o,  measured 
from  the  center  of  rd,  and  in  practice  it  must  be 
used  instead  of  r  e.  In  the  same  manner,  ym  be- 
comes the  leverage  for  the  wt,  instead  of  r  t. 

HEM.  2.  The  cen  of  g-rav  of  the 
half  arch,  can  be  found  by  making 
a  drawing  end  rj,  about  4  to  6  ins  long,  on  pasteboard,  or  on  a  stiff  drawing-paper,  to  a  scale. 
Cut  out  the  fig;  and  balance  it  flatways  on  a  sharp  straight-edge,  or  over  the  edge  of  a  table,  in 
two  directions  or  positions.  Where  these  two  directions  intersect  each  other  is  the  cen  of  grav.  It 
is  not  indeed  this  cen  itself  that  is  needed,  but  the  line  g  s,  of  its  direction  ;  which  may  be  found 
«t  once  by  taking  care  that  the  straight-edge  is  parallel  to  the  back  n  d,  while  balancing  the  fig. 

RKM.  3.  Under  the  head  leverage,  may  be  classed  the  tread-wheel;  windlass  and  lever;  capstan 
and  lever;  and  all  axles  turned  by  a  winch  or  by  a  crank  ;  such  as  the  drum  and  winch  with  which  a 
water-bucket  is  raised  from  a  well,  &c.  They  are  all  merely  continuous  simple  levers,  of  which  the 
axis  is  the  fulcrum  ;  the  rad  of  the  circle  described  by  the  power  is  one  arm,  and  the  rad  of  that  de- 
scribed by  the  shaft,  drum,  <fec,  is  the  other. 

REM.  4.  Compound  levers,  a  a,  bb,  cc,  Fig  52^,  may  be  used  where 
there  is  not  space  for  the  arms  of  a  single  lever  of  sufficient  power.  They  need  not 
extend  in  one  line  ;  but  may  be  placed 
one  over  the  other  ;  or  in  such  other  po- 
sitions as  may  be  convenient.  Their 
effect  is  much  greater  than  the  combined 
effects  of  the  three  simple  levers,  and  is 
found  thus  :  As  the  product  of  the  weight- 
arms,  2  X  1  X  3  «  6;  is  to  that  of  the 
power-arms,  10  X  8  X  7  =  560  ;  so  is  the 
power  to  the  weight  ;  or,  as  6  :  560  :  : 
P  1  :  W  93i/£.  These  arms  are  measd  in 
all  cases  from  the  fulcrum;  which  is 
sometimes  at  the  end  of  one  or  more  of 
the  levers,  when  compounded  ;  see  Fig  44. 
The  combined  effects  of  the  three  simple 


ar 


levers  would  be  but  5  -f-  8  +  2%  =  15%;  or  P  1, 


A  series  or  train  of  toothed  pinions  and  wheels,  working  into  each 

other,  is  merely  a  series  of  continuous  compound  levers.  These  are  generally  set  in  motion  by  a 
winch-handle,  the  rad  of  which  is  the  first  leverage  of  the  series  ;  while  at  the  other  end  of  the  train 
the  wt  is  usually  suspended  from  a  drum,  the  rad  of  which  is  the  last  leverage.  To  find  the  effect, 
mult  into  one  prod  the  radii  of  the  winch  and  of  the  wheels,  and  into  another  prod  the  radii  of  the 
pinions  and  of  the  drum  ;  then,  as  the  last  of  these  prods  is  to  the  first,  so  is  the  power  to  the  wt, 
as  in  the  preceding  case. 

In  both  the  foregoing  cases  of  compound  leverage,  as  in  all  other  cases  whatever  of  leverage,  the 
vel  of  the  wt  is  to  that  of  the  power,  as  the  power  itself  is  to  the  wt;  thus,  in  Fig  52%,  the  wt  will 

move  only  -—  —  part  as  fast  as  the  power  ;  or  the  power  must  descend  93#  inches,  in  order  to  raise 

••»• 
the  weight  1  inch  ;  on  the  principle  of  virtual  vels.     See  Art  52,  p  477. 

REM.  5.  The  Screw  is  a  combination  of  leverage,  with  an  inclined  plane;  a  spiral  inclined 
plane  being  formed  by  the  threads  of  the  screw.  While  the  power  applied  to  the  lever  which  turns 
the  screw  moves  around  an  entire  circle,  the  body  moves  only  the  dist  between  the  centers  of  two 
threads  ;  and  since  in  all  mechanical  contrivances,  the  wt  is  to  the  power  as  the  vel  of  the  power  is 
to  that  of  the  wt,  so  in  this  case,  theoretically,  the  wt  is  to  the  power,  as  the  entire  circumf  of  the 
circle  described  by  the  power,  is  to  the  dist  between  the  centers  of  two  threads;  but  in  practice,  the 
friction  of  the  screw  (which  under  heavy  loads  becomes  very  great)  has  also  to  be  overcome  by  the 
power  ;  and  this  fact  makes  the  calculations  of  but  little  use. 

The  Pulley,  also,  when  a  fixed  one,  is  referable  to  leverage.  In  the  fixed  pulley  A,  Pig 
525>£,  there  is  no  gain  of  power  ;  for  here  the  diam  a  b  is  a  lever  of  two  equal  arms,  revolving  around 
its  fulcrum  at  the  center  of  the  pulley.  Consequently,  the  wt  and  the  power  have  equal  leverages; 


480 


FORCE   IN    RIGID   BODIES. 


Fio53 

o 


each  equai  to  the  rad  of  the 
circle ;  aud  in  order  to  balance 
a  wt  W  of  say  1  ton.  the  power 
F  must  also  be  1  ton ;  for  if  one 
of  them  moves,  the  other  must 
plainly  move  with  the  same 
rel.  To  raise  the  wt,  the  power 
must  exceed  the  wt;  because 
it  has  also  to  overcome  the 
friction  of  the  axle  around 
which  the  pulley  revolves,  and 
the  friction  of  the  rope  in  the 
groove  around  its  circumf. 
These  frictions  become  so 
great  when  many  pulleys  are 
combined,  that  theoretical  cal- 
culations of  the  power  are  of 
little  value.  Although  a  fixed 

Sulley  gives  no  gain  of  power, 
t  is  very  convenient  for  al- 
lowing change  of  direction  in 
applying  the  power;  so  that 
by  pulling  downward,  or  hor, 
&c,  we  can  cause  the  wt  to 
rise  vert.  It  is  plain  that  the 
rope  in  this  pulley  is  equally 
strained  at  all  points.  Theo- 
retically, this  is  the  case  with 
any  one  single  rope,  as  rcdf 
ge.  Fig  52%,  passing  around 
any  numberof  pulleys, whether 
fixed,  as  A  or  D,  or  movable, 
as  B ;  and  all  the  theoretical 
calculations  of  the  power  may 
be  based  upon  this  principle 

T/iT//  alone.      They  will,   however, 

be  incorrect  in  practice,  on 
account  of  the  friction  just 

alluded  to.  In  Fig  52%,  where  only  one  rope  is  used,  the  lower  pulley-block  B  «,  to  which  the  wt  W 
is  attached,  is  directly  upheld  by  the  two  parts  df  and  eg  of  the  rope.  Consequently  each  of  these 
parts  is  equally  strained  by  a  force  equal  to  one  half  the  wt ;  and  since  the  whole  single  rope  is  theo- 
retically strained  to  the  same  extent,  that  part  of  it  to  which  the  power  is  applied  must  be  strained 
equal  to  half  the  wt  W ;  or,  in  other  words,  the  power  itself  must  be  equal  to  half  the  wt,  and  will 
move  twice  as  fast,  aud  twice  as  far. 

In  Fig  53,  the  lower  pulley-block  ty  is  sustained  directly  by  the  4  parts  cccc  of  the  single  rope; 
therefore,  each  part  of  the  rope,  and  consequently  the  whole  o'f  it,  is  equally  strained  by  a  force  equal 
to  %  of  the  wt  W ;  and  the  power  P  must  be  equal  to  the  same  }^,  and  will  move  4  times  as  fast,  and 
4  times  as  far,  as  the  wt.  It  is  immaterial  whether  the  two  pulleys  in  the  lower  block  of  Fig  53,  be 
placed  one  above  the  other,  as  shown,  or  (as  usual,  and  more  convenient)  side  by  side;  so  also  with 
those  in  the  upper  block.  If  there  were  3,  4,  or  5.  &c,  pulleys  in  each  block,  then  there  would  be  6, 
8,  or  10  sustaining  parts  c  c,  &c,  of  rope,  each  stretched  equal  to  -^,  |-,  or  y1^-  of  the  wt  W ;  and  the 
power  would  also  be  in  the  same  proportions  to  the  wt;  in  other  words,  to  find  the  theoretical  pro- 
portion of  the  power  to  the  wt,  divide  1  by  the  number  of  parts  c  ccc  of  rope  which  directly  sustain 
the  lower  block.  In  our  fig  it  is  %.  The  same  rule  applies  to  Fig  52%. 

When  more  than  one  rope  is  used  in  a  system  of  pulleys,  the  strains 
become  diff  from  the  foregoing,  on  the  principle  illustrated  by  Fig  53J4. 
Here  the  lower  block  y  b,  with  its  attached  wt  of  say  4  tons,  is  directly 
sustained  by  the  two  parts  a  and  c  of  one  rope.  Consequently,  each 
part  has  a  strain  of  %  the  wt,  or  '2  tons ;  which  is  uniform  throughout 
that  rope.  But  all  the  2  tons  strain  on  the  part  c  is  sustained  by  the 
hook  s ;  while  that  on  the  part  a  is  sustained  bv-the  two  parts  n  and  m 
of  the  other  rope :  each  of  which  plainly  sustains  one  half  of  it.  or  1 
ton,  which  is  uniform  throughout  this  second  rope  to  its  very  end. 

ir  @fo  \  Therefore  the  power  also  is  1  ton,  or  y±  of  the  wt  W.     The  mode  of 

V .    /    /  proceeding  is  the  same,  whatever  may  be  the  number  of  movable  pul- 

VI    I/  leys.     To  find  the  theoretical  proportion  of  the  power  to  the  wt.  mult 

together  continuously  as  many  2s  as  there  are  movable  pulleys,  and 

"P|  div  1  by  the  prod.     Thus,  here  we  have  two  movable  pulleys,  and  2  X 

2  n  4 ;  and  %  =  the  answer.     If  there  were  4  movable  pulleys,  we 
should  have  2X2X2X2  =  16;  and  y1^  =  answer. 

In  all  our  figs,  that  end  of  the  rope  to  which  the  power  P 1  is  applied 
is  represented  as  hanging  vertically,  and  parallel  to  the  other  parts  of 
the  rope  ;  but  this  was  done  merely  because  the  power  is  supposed  to 
be  a  weight,  and  of  course  acting  vert.  But  if  the  power  is  muscular 
force,  or  any  other  kind  that  may  act  in  any  direction  whatever,  then 
the  power  end  of  the  rope,  as  mn.  Fig  53,  may  have  that  direction  in 
Tfhich  it  is  most  convenient.  The  amount  of  power  required  will  not 
be  thereby  changed ;  for  it  is  plain  that  leverage  from  the  center  of  the 
pulley  o  s  to  m,  when  the  power  is  at  n.  and  acting  in  the  inclined  di- 
rection of  the  rope,  is  equal  to  that  from  the  same  center  to  o,  when  the 
pow«er  is  at  P,  and  acting  vert. 

The  parts  of  the  ropes,  except  the  power-ends,  (as  m  n,  Fig  53)  are  sup- 


tL  Us 


RIGID   BODIES. 


481 


posed  to  be 
53)  it  will ' 


_.flel;  for  if  a  part  be  inclined  (as  is  the  end  attached  to  the  fixed  pulleys  in  Fig 
trained  more  than  the  other  parts,    See  Art  4,  p  668. 


Tlie  Wedg'C  is  a  kind  of  double  inclined  plane,  (Art  60.)  and  is  a  very  powerful  machine 
But  inasmuch  as  it  is  usually  worked  by  blows,  the  effect  of  which  cannot  be  calculated  ;  and  in*<- 
much  as  its  theoretical  action  is  in  practice  totally  changed  by  friction,  no  serviceable  rules  can  be 
given  respecting  it. 

Art.  55.  Parallel  forces  are  those  whose  directions,  as 
in  Fig  53*^,  (whether  opposite  to  one  another,  or  not;  or  whether 
in  the  same  plane,  or  not,)  are  parallel.  In  Fig  53*/£,  the  forces, 
although  acting  upon  one  plane,  oooo,  are  not  in  one  plane,  but 
in  several. 

It  is  a  peculiarity  of  parallel  forces  in  one  plane,  that  all  their 
arms,  or  leverages  with  respect  to  any  given  point  in  the  same 
plane  are  in  the  same  straight  line.  Thus,  if  /,  TO,  n,  o,  Fig  54, 
he  in  the  same  plane,  then  their  leverages  p  q,  p  r,  p  a.  about  the 
point  p.  are  all  in  the  same  straight  line  p  a.  The  pointy 
is  supposed  to  be  in  the  same  plane  as  the  forces. 


Two  parallel  forces  are  evidently  always  in  the  same  plane:  that  is, 
the  same  flat  surf  could  coincide  with  'both  of  them  ;  and  their  re- 

to  saying:,  in  other  words,  that  if  3  parallel  forces  hold  each  other  in 
equilibrium,  they  are  in  the  same  plane.     See  Rem  2,  page  468. 


Fitf  54 


Art.  56.    The  resultant  of  any  number  of  parallel  forces, 

whether  in  the  same  plane,  or  in  the  same  direction,  or  not,  is  always  parallel  to 
them.  If  they  all  act  in  the  same  direction,  whether  in  the  same  plane  or  not,  their 
resultant  is  equal  in  amount  to  their  sum :  or,  in  other  words,  an  autiresultant 
force  sufficient  to  balance  them,  must  be  equal  to  all 
the  forces  added  together.  But  if  they  are  in  oppo- 
site directions,  their  resultant  will  be  equal  to  the 
diff  between  those  which  act  in  one  direction,  and 
those  which  act  in  the  opposite  one  ;  and  its  direction 
will  be  that  of  the  greater  sum.  Thus,  in  Fig  53^, 
if  the  forces  pointing  to  the  left  amount  to  10  tons, 
arid  those  to  the  right  4  tons;  then  the  resultant  will 
be  10  —  4  =  6  tons  ;  and  it  will  point  to  the  left. 

The  parallel  vert  downward  forces  of  gravity,  upon  the  innu- 
merable separate  particles,  situated  in  the  infinite  number  of 
imaginary  vert  planes,  in  any  body,  as  W,  Fig  55,  is  an  illustra- 
tion of  this.  If  any  such  body  be  suspended  by  a  string  from  a 
spring-balance  B,  the  vert  upward  pull  of  the  string  will  balance 
or  equilibrate  all  these  innumerable  forces.  Consequently,  the 
string  represents  their  antiresultant.  which  is  equal  to  their  re- 
sultant. We  know  that  the  vert  pull  on  the  string,  as  shown  by 
the  spring-balance,  is  equal  to  the  wt  of  the  body  ;  which  wt  is  made  up  of  the  innumerable  parallel 
vert  forces  alluded  to.  Thus  we  see  that  when  any  number  of  parallel  forces,  whether  in  the  same 
plane  or  not,  net  in  the  same  direction,  their  antiresultant  is  parallel  to  them,  and  equal  to  their  sum  ; 
consequently  their  resultant  must  be  so  also.  The  same  principle  applies  to  parallel  forces  in  any 
direction  whatever. 

When  a  body  thus  acted  on  by  gravity  is  kept  at  rest,  or  balanced,  as  in  the  fig,  then  the  direction 
of  the  resultant  or  antiresultant,  or  of  the  string  in  the  fig,  passes  through  a  certain  point,  called  the 

center  of  gravity  of  the  body.    This  is  a  certain  point,  upon  which  when  acted 

upon  by  gravity  only,  the  body  will  balance  itself,  in  whatever  position  it  may  be  placed :  and  if  the 
entire  wt  or  grav  of  the  body  could  be  concentrated  into  that  single  point,  its  effect,  whether  regarded 
as  moving  the  entire  rigid  body,  or  as  producing  strain  (pnll  or  push)  between  it  and  another  rigid 
body,  would  remain  precisely  the  same  as  it  actually  is  with  the  grav  diffused  throughout  the  entire 
mass.* 

*  In  some  bodies  the  cen  of  grav  is  also  the  center  of  the  wt  of  the 
body  ;  but  very  frequently  this  is  not  the  case.  Thus,  in  a  body 
a  b  c.  Fig  55>£,"  with  its  cen  of  grav  at  c.  there  is  more  wt  on  the  side 
a  c.  than  on  the  side  c  6. 

If  a  body  W,  Fig  55,  suspended  freely  from  any  point  n,  is  at  rest, 
its  cen  of  grav  is  directly  under  said  point.  If  the  body  W  be  pushed 
a  little  to  one  side,  and  then  left  to  itself,  it  will  plainly  tend  of  itself 
to  swing  back  to  its  first  position :  and  when  this  is  the  case,  it  is 
said  to  be  in  stable  equilibrium.  But  if  the  body,  instead  of  being 
suspended,  be  balanced  on  top  of  a  slim  rod,  and  if  we  then  push  it 
a  little  to  one  side,  it  will  not  tend  to  return,  but  will  fall  over ;  and 
therefore  the  equilibrium  of  a  body  so  balanced  is  said  to  be  unstable.  Also 


482 


FORCE   IN   RIGID   BODIES. 


K356 


Art.  57.  That  point  through  which  the  direction  of  a  single  antiresultant 
force  must  pass,  in  order  to  balance  several  other  ibices  acting  at  diff  points ;  or,  in 
other  words,  that  point  through  which  the  direction  of  the  resultant  of  those  forces 
must  pass,  is  called  the  center  of  pressure,  or  of  force,  or  of  strain, 

of  those  forces,  as  the  case  may  be. 

For  instance,  let  S,  Fig  56,  be  a  common  wooden  box;  but  having 
one  side,  as  o  o,  looselv  fitted,  so  as  barely  to  allow  of  pushing  it  back- 
ward and  forward.  Fill  the  box  with  dry  sand,  (clean  small  gravel 
will  be  better,)  aud  it  will  be  found  that  there  is  but  one  single  point, 
i,  at  which  we  can.  by  holding  to  it  a  thin  rod  r  i,  balance  the  pres  of 
the  gravel  against  the  opposite  side  of  o  o.  If  we  apply  the  rod  at  any 
other  point,  o  o  will  give  way  before  the  sand ;  if  the  rod  is  held  above 
»,  the  bottom  of  o  o  will  be  pushed  outward ;  if  held  below  i,  the  top  of 
oo  will  move  outward.  This  point  i  is  dist  above  the  bottom  of  the 
sand  %  of  the  depth  of  the  sand ;  in  other  words,  the  cen  of  pres  of 
sand  of  any  depth  is,  like  that  of  water,  at  %  of  that  depth  from  the 
bottom.  In  the  case  before  us.  the  depth  is  supposed  to  be  uniform,  so 
that  the  cen  of  pres  is  at  the  same  height  above  the  bottom,  clear 
across  the  box. 

Now  the  balancing  force  applied  through  the  rod  at  i,  is  the  antiresultant  of  all  the  pressures  re- 
sulting from  the  several  particles  of  gravel  against  the  opposite  side  of  oo;  and  its  effect  upon  the 
rigid  body  o  o,  (omitting  of  course  any  tendency  to  bend  or  break  it,  which  comes  under  the  head  of 
Strength  of  Materials,)  is  precisely  the  same  as  that  of  all  those  forces  combined;  except  that  it  is 
in  the  opposite  direction.  Its  tendency  to  push  oo  bodily,  or  as  an  entire  mass,  toward  the  right 
hand,  is  precisely  the  same  as  that  of  the  gravel  to  push  it  to  the  left  hand;  or  it  is  the  same  as 
would  result  were  we  to  heap  up  sand  in  front  of  oo,  so  as  to  balance  the  sand  behind  it. 

RKM.  It  is  this  important  principle  of  the  cen  of  pres,  that  enables  us  to  adopt  the  convenient 
practice  of  representing,  by  a  single  line,  the  effect  of  force  actually  distributed  over  a  considerable 
surf.  Thus,  in  Fig  52,  the  hor  force  ha,  by  which  each  half  of  the  arch  mutually  prevents  the  other 
half  from  falling,  is  actually  distributed  over  an  area  whose  depth  is  the  depth  cj  of  the  keystone; 
and  its  breadth,  that  of  the  whole  bridge,  as  measd  acres*  the  roadway.  Yet  the  arrow  ft  a,  when 
drawn  to  a  scale,  perfectly  represents  the  effects  of  this  distributed  force  in  upholding  the  half  arch, 
considered  as  an  entire  rigid  mast.  So  far  as  regards  splitting  or  cracking  the  stone  immediately  at 
a,  the  effects  would  of  course  be  diff;  but  as  the  whole  force  is  only  supposed,  for  convenience,  to  be 
applied  at  a,  this  diff  is  merely  ideal  in  this  instance.  See  Arts  5  and  8,  p  525,  526. 

It  is  evident  from  the  foregoing  that  "cen  of  grav"  means  nothing  more  than  "cen  of  force;  "  ex- 
cept only  that  the  former  is  a  convenient  term  for  denoting  that  the  force  is  that  of  grav  alone. 

Art.  58.  It  may  be  well  here  to  direct  particular  attention  to  the  fact  alluded 
to  in  Ex  4,  Art  54,  that  when  either  grav  or  other  forces  are  to  be  considered  with 
regard  to  their  eifects  in  bending,  or  breaking  bodies;  instead  of  moving  or  straining 
them  as  entire  masses,  supposed  to  be  rigid,  or  incapable  of  change  of  form,  they 
cannot  be  assumed  to  be  concentrated  at  either  the  cen  of  force,  or  the  cen  of  grav. 

In  the  last  case  our  applied  extraneous  forces  are  supposed  to  be 
brought  to  bear  only  upon  other  extianeous  forces  acting  upon  the 
bodies  at  the  same  time ;  the  bodies  themselves  being  regarded  mere- 
ly as  unalterable  mediums  through  which  said  forces  are  enabled  to 
act  upon  each  other;  but  in  bending,  breaking,  twisting,  shearing, 
&c,  our  extraneous  or  mechanical  forces  must  be  considered  to  act 
against  the  inherent  cohesive  forces  of  the  bodies   themselves; 
therefore  these  forces,  which  before  were  entirely  neglected,  now 
acquire  a  primary  importance ;  the  question  of  strength  of  materials 
comes  in,  and  the  assumption  of  perfect  rigidity  must  be  altogether 
discarded.     Thus,  in  Fig  56*4,  so  far  as  regards  either  moving  a 
rigid  body  c  o,  or  straining  it  against  the  force  n,  it  is  immaterial 
whether  we  employ  the  two  equal  parallel  forces  a,  ft,  or  a  single 
force  m,  equal  to  both  of  them,  and  acting  at  their  center  of  force. 
But  since  no  bodies  are  absolutely  rigid,  but  may  all  be  bent 
or  broken,  it  is  plain  that  the  two  forces  a,  b,  straining  the 
body  c  o  against  the  force  n.  would  bend  or  break  it  much  more 
readily  than  the  force  m  would. 


An  absolutely  rigid  hor  beam  *  s,  would  sustain  any  amount 
of  load  I,  without  bending;  and  consequently  would  always 
press  vert  upon  its  upright  supports  u,  u,  without  any  tendency 
to  press  them  sideways.  But  an  actual  beam  n  «,  if  overloaded, 
will  bend ;  thereby  generating  at  its  ends  forces  which  are  not 
vert,  but  which  will  tend  to  overthrow  the  supports  *  t. 


Art.  59.    To  find  the  point  of  impartation  of  the  resultant 
Of  parallel  forces.     Case  1.    Two  parallel  forces,  1  ton,  and  3  tons,  Fig  57,  in 

in  such  cases  as  that  of  a  grindstone  supported  by  its  hor  axis  passing  through  its  cen  of  grav,  if 
•we  cause  it  to  revolve  a  short  dist,  and  then  leave  it  to  itself,  it  will  have  no  tendency  either  to  re- 
turn, or  to  keep  on  revolving ;  and  its  equilibrium  is  called  indifferent.  Bee  p  635. 


FORCE 


RIGID    BODIES. 


483 


-4TONS 
3  TONS 


Fig  57 


the  same  direction ;  and  cprtiequently  in  the  same  plane.    Draw  any  straight  line 
a  o,  uniting  their  direc£kms  1  w,  3  n. 

Measure  this  Hue,  and/div  it  into  two  cl     x~— "\-&. 

parts,  io,  ia,  proportioned  like  the  TTt — '^"^~"~'~"j^T. — '1TON 
forces;  hut  plaora  inversely  to  the 
forces ;  that  is,  place  the  longest  part 
near  the  smallest  force,  and  vice  versa. 
Through  the  point  i  draw  the  direc- 
tion of  the  resultant  R,  parallel  to  the 
forces.  Then  is  t  the  point  of  imparta- 
tion.  Make  the  resultant,  £4,  equal  to 
the  sum  of  the  two  forces. 

Ex.  Let  one  force  be  1  ton;  and  the  other,  3  tons;  and  let  ao  be  8  ft.  Then,  as 
the  sum,  4  tons,  of  the  two  forces ;  is  to  the  length,  8  ft,  of  a  o ;  so  is  the  large  force, 
3  tons ;  to  the  long  part,  i  cr,  of  a  o.  Or,  4  :  8  : :  3  :  6  =  t  a.  Consequently,  i  o  =  8  — 
6  =  2  ft ;  as  shown  in  the  fig. 

The  foregoing,  as  well  as  some  other  facts  connected  with  parallel 
forces,  will  be  more  easily  recalled  to  mind,  by  associating  them  with 
the  idea  of  the  common  steel-yard,  Pig  58.  Her*  the  two  forces  in  Pig 
57  are  represented  by  the  wt  1  ft  at  c,  and  the  wt  3  tts  at  d ;  one  of  them 
3  times  as  great  as  the  other.  We  know  that  these  weights,  when  sus- 
pended at  dists  fa,fb,  one  of  them  3  times  as  great  as  the  other,  from 
the  fulcrum/,  are  balanced  by  their  antiresultant//»,  which  is  equal  to 
3  +  l~4:fl>s,  or  the  sum  of  the  two  forces.  This  antiresultaut  has 
precisely  the  same  tendency  to  pull  the  steel-yard  upward,  that  the  two 
weights  have  to  pull  it  downward.  Also/*,  equal  to  fh,  but  acting  in 
the  opposite  direction,  is  the  resultant  of  the  two  forces;  its  effect  to 
pull  the  steel-yard  bodily  downward,  when  applied  to  it  at/,  is  precisely 
equal  to  that  of  the  two  wts  applied  at  a  and  6.  The  effect  of  the  two 
wts  at  a  and  5,  to  bend  or  break  the  steel-yard,  would  plainly  be  very 
diff  from  that  of  their  resultant  */,  applied  at/.  See  Art  58. 

Case  2.  Two  unequal  parallel  forces,  da 
of  3  tons,  and  hf  of  4  tons,  Fig  59.  imparted 
to  a  rigid  body  in  opposite  directions,  but 
not  in  the  same  straight  line. 

Starting  from  the  line  of  direction  dm,  of  the 
small  force,  draw  any  line  mn,  passing  through 
and  beyond  the  direction  hf  of  the  large  one.  Now 
find  the  amount  of  the  resultant  R.  This,  by  Art 
56,  is  equal  to  the  diff  between  hf  and  da,  since  they 
are  in  opposite  directions  ;  and  has  the  same  direc- 
tion  as  the  large  one ;  therefore  it  is  equal  to  4  —  3  = 
1  ton;  and  its  direction  is  the  same  as  that  of  hf. 
Then  say,  as  the  resultant  is  to  the  small  force  d  a, 
so  is  the  dist  om,  to  the  dist  o«,  along  the  line  m  n. 
Through  «  draw  «  c,  parallel  to  the  two  forces  ;  and 
c  will  be  the  reqd  point  of  impartation  of  the  resultant  R  ;  the  tendency  of  which  to  move  the  entire 
rigid  body,  will  be  equal  to  the  joint  tendency  of  d  o  and  hf.  An  antiresultant  at  t  would  of  course 
balance  the  two  forces  da,  hf. 

This  case,  like  the  preceding  one,  is  illustrated  by  the  steel-yard,  Fig  58;  where  ad,  and/ft,  repre- 
sent the  two  forces  in  the  last  fig ;  while  6  c  represents  their  balancing  antiresultant,  corresponding  to  t. 

Case  3.  Couples.  Two  equal  parallel  forces,  a,  and  &, 
Fig  59  a,  imparted  to  a  rigid  body  in  opposite  directions,  but  not 
in  the  same  straight  line,  are  called  a  couple.  The  force  of  a 
couple,  means  simply  one  of  the  forces  ;  the  perp  dist  c,  between 
the  directions  of  the  two  forces,  is  called  the  arm,  or  leverage 
of  the  couple.  If  one  of  the  forces  be  mult  by  this  arm,  the 
prod  is  the  moment  of  the  couple,  in  foot-Ibs,  Ac.  A  couple  has 
no  tendency  to  move  the  entire  body  forward  in  the  direction 
of  either  force ;  but  merely  to  make  it  rotate  around  a  point  o,  half-way  between 
the  points  at  which  the  two  forces  are  imparted.  A  couple  (as  is  also  the  case  with 
two  equal  opposing  forces  in  the  same  straight  line)  has  no  single  resultant ;  only 
another  couple  can  hold  it  in  equilibrium. 

Case  4.  Any  number  of  parallel  forces,  whether  in  the  same  plane,  or  in  the  same 
direction,  or  not.  The  process  in  this  case  consists  in  a  mere  repetition  of  that  in 
Case  1,  Fig  57,  as  follows. 

Namely,  commencing  with  those  forces  which  point  in  the  same  direction,  as  a,  b,  c.  Fig  60,  which 
all  point  downward ;  between  the  directions  of  any  two  of  them,  as  a  and  6,  draw  any  straight  line 
oi,  anddiv  it  into  two  parts  jo,  ji,  proportioned  like  the  forces,  but  placed  inversely  to  the  forces. 
Through  ./draw  ms,  parallel  and  equal  to  the  forces  a  and  b.  Then  is  ma  the  resultant  of  those  two 
forces.  Next  find  the  resultant  of  this  resultant  m  «  and  any  third  force,  as  c,  in  the  same  manner 
That  is,  draw  any  line  t  w,  uniting  their  directions;  div  it  into  two  parts  It  and  Iw,  proportioned 
like  the  forces,  but  placed  inversely;  Through  I  draw  ng,  and  make  it  equal  to  m  «,  and  c.  Then  is 


Pi.,59 


484 


FORCE    IN    RIGID    BODIES. 


n  g  the  resultant  of  the  three  forces  a,  b,  c;  and  g  is  the  cen  of  force  of  those  forces,  aud  consequently 
is  the  point  of  application  of  their  resultant.  So  do  with  any  number  in  the  same  direction.  Then 
proceed  in  the  same  manner  with  those  which  point  in  the  opposite  direction,  as  d,  e,  x,  y;  and 
having  found  their  resultant,  find  by  Case  2,  the  resultant  of  the  two  resultant  forces,  now  obtained 
in  opposite  directions. 

It  is  not  at  all  necessary  that  the  forces  be  supposed  to  act  upon  a  plane  surf,  as  in  Fig  60;  the  pro- 
plane.  This  is 'a  consequence  of' the  principle  laid  down  in  Art  18,  Fig  5  ;  namely,  that  the  effect  pro- 
duced upon  a  rigid  body  by  an  imparted  force,  remains  the  same,  no  matter  at  what  point  of  the  uody 
it  be  imparted,  so  long  as  that  point  is  in  the  line  of  the  direction  of  that  force. 


I    e  ErfGO  ] 


at 

Although  Figs  60  and  61  serve  to  illustrate  the  principle,  they  plainly  do  not  give  the  actual  posi- 
tions of  the  forces  and  resultant;  because  they  are  necessarily  drawn  in  a  kind  of  perspective,  so 
that  all  the  parts  cannot  be  measd  by  a  scale.  The  amounts  of  the  resultants  are  easily  fouud  by 
calculation ;  inasmuch  as  they  are  equal  to  the  sums  of  the  forces. 


The  points  for  imparting  them  can  be  found  correctly  from 
a  drawing  in  plan,  like  Fig  62 :  where  the  stars  represent  by 
scale  the  actual  dists  apart  of  the  directions  of  the  forces. 
The  quantities  of  the  forces,  instead  of  being  shown  by  lines, 
are  to  be  written  in  figures,  as  shown  in  the  fig.  This  being 
done,  it  is  easy  to  tiiid  the  points  a,  g,  &c,  of  the  resultants. 


Art.  60.  The  Inclined  Plane  is  a  rigid  straight  plane  surf,  as  a  &,  Fig 
63,  not  hor.  If  a  vert  line  b  c  be  drawn  from  the  top  b  of  the  plane,  to  meet  a  hor 
line  ttc,  drawn  from  its  bottom  a,  then  be  is  called  the  height  of  the  plane;  ac  its 
fca.sv ;  and  a  b  its  length.  The  angle  b  ac,  which  the  plane  forms  with  the  hor  line 
a  c,  is  called  its  inclination,  slope,  or  steepness ;  which,  however,  is  frequently  ex- 
pressed also  by  the  proportion  which  th*» 
base  bears  to  the  height ;  thus,  if  the  length 
ac  of  the  base  be  1,  1%,  2,  &c,  times  that  of 
the  height  be,  the  inclination  or  slope  is 
said  to  be  1  to  1, 1%  to  1,  2  to  1,  &c.  The 
angle  6  ac  is  the  angle  of  inclination  of  the 
plane. 

It  follows  from  Arts  56,  57,  that  when  one  rigid 
body  as  N  or  M,  Fig  63,  is  placed  loosely  upon  an- 
other, as  upon  the  rigid  plane  a  b,  the  effect  produced 
\  by  its  wt  is  the  same  as  if  all  that  wt  were  concen- 

\"1       trated  at  its  cen  of  grav  g,  and  acted  in  the  direction 

iV     of  a  vert  line  git;  drawn  through  said  center.    When 

we  assume  the  wt  to  be  thus  concentrated  at  the 
point  g,  we  must  remember  that  all  other  parts  of 
the  body  must  be  considered  to  be  without  weight ; 
although  still  retaining  their  inherent  cohesive  force,  or  strength. 

If,  as  in  N,  this  vert  line  gv,  which  now  represents  the  direction  of  the  entire  wt  of  the  body, 
passes  beyond,  or  outside  of  the  base,  the  body  must  fall;  because  this  wt  meets  with  no  opposing 
force,  in  the  direction  vgr  to  react  against  it;  and  thus  prevent  it  from  producing  motion.  See  Re- 
mark, Art  65. 

But  if,  as  in  the  body  M,  the  line  g  v  falls  within  the  base  r  8,  the  body  will  not  upset :  but  we  shall 
have  (Art  19)  a  force  gv  equal  to  the  wt  of  the  body,  and  applied  obliquely  to  a  rigid  surface  aft,  *t 


1W63 


IN    RIGID   BODIES.  485 


the  point  vjaarfconsequently  resolvable  into  two  components;  namely,  iv,  perp  to  the  surf  ao,  and 
therefore  istfp arted  to  it  as  a  pressure  ;  and  x  v,  parallel  to  the  surf,  and  consequently  not  imparted 
to  it.  All  these  lines  may  be  drawn  by  scale,  to  represent  their  respective  forces.  When  we  consider 
a  single  force  as  y  v  to  be  thus  resolved  into  two  components,  with  a  view  to  ascertaining  their  effects, 
it  is  plain  that  said  single  force  must  then  be  considered  as  no  longer  existing;  but  as  being  replaced 
by  its  components.  Now  the  component  force  xv  being  parallel  to  the  plane,  it  follows  (Art  15)  that 
thepreasure  or  strain  i  v,  no  matter  how  great  it  may  be,  cannot  in  the  slightest  degree  oppose  the  cross 
action  of  the  moving  force  x  v,  no  matter  how  small  it  may  be  ;  and  x  v  must  therefore  produce  motion  in 
the  body,  causing  it  to  slide  down  the  plane ;  unless  some  third  force,  not  yet  spoken  of,  shall  present 
itself,  opposed  to  xv.  But  any  forces  which  press  bodies  together,  always  produce  a  new  force,  fric- 
tion, at  the  joint,  or  surfs  of  contact  of  the  bodies  ;  and  this  friction  acts  in  direct  opposition  to  any 
force,  in  any  direction  whatever  that  is  in  the  plane  of  that  joint  or  surfs.  See  Friction.  Therefore 
the  force  i  v  produces  fric  between  the  surfs,  r  s,  of  the  body  M  and  of  the  inclined  plane ;  and  this 
fric  acts  in  the  direction  r  a,  or  diametrically  opposite  to  that  of  the  force  xv.  The  amount  of  fric 
depends  upon  that  of  the  pres ;  as  also  upon  "the  nature  of  the  bodies  at  whose  surfs  it  is  produced, 
upon  the  degree  of  smoothness  of  those  surfs,  and  upon  whether  they  are  lubricated  or  not.  If  the 
fric  is  greater  than  the  force  xv  in  the  opposite  direction,  the  body  of  course  cannot  move ;  but  if  less, 
it  will  move,  under  the  action  of  a  force  equal  to  the  excess  of  xv  over  the  fric.  It  must  be  remem- 
bered that  the  pres  component  t  v,  which  produces  fric  on  an  inclined  plane,  is  not  equal  to  the  wt 
of  the  body,  but  is  less  than  it.  It  is  equal  only  when  the  surf  is  hor,  so  that  the  vert  force  g v,  rep- 
resenting the  entire  wt  of  the  body,  is  at  right  angles  to  the  joint,  and  when,  consequently,  it  all  acts 
as  pres.  Therefore,  the  steeper  the  plane  becomes,  the  less  is  the  fric;  because  then  less  of  the  wt 
of  the  body  acts  as  pres,  and  more  of  it  as  moving  force.  Hence,  a  locomotive  has  less  adhesion  on 
an  inclined  grade,  than  on  a  level ;  for  the  so  called  adhesion  is  in  reality  nothing  but  fric.  But 
although  both  the  perp  pres  and  the  fric  become  less  in  amount  as  the  plane  becomes  steeper,  yet  they 
constantly  retain  the  same  proportion  to  each  other;  until  pressures  become  so  great  that  abrasion 
of  the  surfs  of  contact  takes  place;  the  proportion  of  the  fric  to  the  pres  then  increases. 

REM.  It  is  evident  that  when  we  wish  to  push  a  body  up  an  inclined  plane,  we  must  overcome 
both  the  Trie,  and  the  parallel  force  x  v ;  but  in  pushing  it  down,  we  are  opposed  only  by  the  fric  ;  for 
the  parallel  force  assists  us. 

Art.  61.  Experiment  has  determined  the  amount  of  fric  which  takes  place  be- 
tween the  surfs  of  such  materials  as  are  employed  in  construction  ;  that  is,  it  has 
determined  the  proportion  (or,  more  correctly,  the  ratio)  between  the  pres  and  the 
fric.  Any  person  may  easily  do  this  for  himself,  thus:  A  body  r.s  M,  Fig  63,  is 
placed  upon  the  plane  surf  ab;  of  which  one  end,  as  fc,  is  gradually  raised  until  the 
body  is  barely  about  to  begin  to  slide.  When  this  takes  place,  we  know  that  the 
force,  x  v  has  become  barely  equal  to  the  fric ;  and  the  angle  b  a  c,  which  the  plane  then 
makes  with  the  hor  a  c,  is  called  the  angle  of  friction,  or  limiting  angle  of  resistance, 
or  angle  of  repose,  for  the  particular  kind  of  surf  experimented  on. 

Now,  a  little  reflection  will  show  that  whatever  may  be  this  angle,  feac,  of  fric.  the  line  xv,  which 
measures  not  only  the  parallel  force,  but  also  the  fric  existing  at  that  moment,  (and  at  no  other  one,) 
is  to  the  line  i  v,  which  measures  the  perp  pres,  (not  the  wt  of  the  body  ;)  as  the  vertical  height  ft  c 
of  the  plane  at  the  same  moment,  is  to  its  hor  base  a  c.  That  is,  at  the  point  of  sliding,  as  Fric- 
tion :  Perp  Pressure  :  :  Height  :  Base;  or,  as  Base  :  Ht  : :  Perp  Pres  :  Friction.  Therefore,  when 
a  body  barely  begins  to  slide,  measure  a  c  hor.  and  b  c  vert;  div  the  last  by  the  first,  and  the  quot 
will  be  the  proportion  which  the  fric  of  the  bodies  experimented  upon,  bears  to  the  pres  which  causes 
it.  Or,  measure  the  angle  b  ac  in  degrees.  &c  ;  the  nat  tang  of  this  angle  will  be  that  same  propor- 
tion. This  proportion  is  called  the  coefficient  of  friction  for  those  bodies;  a  table  of  which  will  be 
found  under  Friction.  A  hor  line  dg.  drawn  from  g,  and  terminating  in  vi  extended,  will,  when 
measd  by  the  same  scale  as  gv.iv,  xv,  give  a  hor  force  which,  without  the  aid  of  friction,  would 
react  against  the  force  xv,  and  prevent  it  from  moving  the  body  down  the  plane.  Or  if  the  length 
a  b  of  the  plane  be  taken  by  scale  to  represent  the  wt  of  a  body,  then  6  I,  perp  to  a  5,  to  meet  a  c  pro- 
duced at  I,  will  give  that  same  hor  force. 

Art.  62.     If  the  length  m  n,  Fig  63%,  of  an  inclined  plane,  be  taken  by  a  scale, 
to  represent  the  wt  in  Ibs,  tons,  &c,  of  any  body  placed  upon  it ;  then  the  base  o  n 
will,  by  the  same  scale,  give  the  perp  pres  in  ft»s, 
tons,  &c,  which  the  body  imparts  to  the  surf  of 
the  plane ;    and  the  height  m  o  will   give  the 
amount  of  force  parallel  to  m  n,  and  which  tends 
to  move  the  body  down  the  plane,  either  by 
sliding  or  rolling.    If  the  pres  on  be  mult  by 
the  proper  coeflf  of  fric,  the  prod  will  plainly  be 

the  actual  amount  of  fric  in  ft>s,  &c.    If  the  fric  «  ~~ ; - 

thus  obtained  proves  to  be  greater  than   the  I  res.  perp   to  plane, 

sliding  force  m  o,  then  the  body  will  remain  at  -„  j 

rest  on  the  plane;    but  if  less,  then  sliding  or  .tlCI  6  3  IT 

rolling  down  the  plane  will  be  the  result ;  and  J  & 

the  amount  of  force  which  starts  or  begins  the 
motion,  will  be  equal  to  the  excess  of  mo  over  the  fric. 

As  the  motion  continues,  it  will  be  accelerated  by  the  accumulation  of  gravity.     See  p  172,  p  449. 

When  a  body  is  placed  upon  an  inclined  plane,  whether  it  slides  or  not,  the  pres  which  it  pro- 
duces at  right  angles  to  the  surf  of  the  plane,  is  equal  to  the  wt  X  nat  cosine  of  angle  of  slope; 
the  sliding  force  parallel  to  the  surface  of  the  plane,  =  wt  X  nat  sine  of  angle  of  slope;  the  actual 
amount  of  fric  =  wt  X  nat  cosine  angle  of  slope  X  coeff  of  fric.  For  the  Trie  does  not  vary  as  the 
angle  of  slope  of  the  plane,  but  as  the  cosine  of  that  angle ;  in  the  same  manner  as  the  perp  pres 
varies.  Coeffs  of  fric  are  given  on  pp  599,  600,  602. 

Ex.  1.    Suppose  we  wish  to  slide  a  wooden  box  M,  Fig  63,  filled  with  stone,  and  weighing  in  all 


486 


FORCE   IN    RIGID   BODIES. 


1200  9>s,  up  the  iron  rails  of  aa  inclined  plane,  sloping  5° ;  what  force  must  we  use,  parallel  to  to* 
plane;  assuming  the  coeff  of  wood  on  iron  to  be  .4,  or  ^^  of  the  perp  pres?  Here  we  have  to  over- 
come the  parallel  force  x  v,  and  the  fric.  Now,  as  just  stated,  this  parallel  force  x  v  is  equal 
to,  wt  X  nat  sine  of  slope,  =  1200  X  .087  —  104.4  Ibs.  The  fric  is  equal  to,  wt  X  nat  cos  of  slope  X 
coeff  of  frm;  =  1200  X  .996  X  .4  =  478.  Consequently.  104.4  -}-  478  —  582.4  Ibs,  is  the  force  reqd.  In 
fact,  however,  this  force  merely  balances  the  downward  tendency  of  the  box,  together  with  its  fric ; 
thus  rendering  them  incapable  of  resisting  any  additional  upward  force;  but  it  is  plain  that  we  must 
apply  some  additional  force,  in  order  to  impart  motion  to  the  now  unresisting  box. 

Now,  suppose  we  wish  to  slide  the  box  down  the  plane,  what  force  must  we  use?  Here  nothing  re- 
sists us  but  the  fric,  just  found  to  be  478  Ibs.  The  parallel  force  helps  us  to  the  amount  of  104.4  Ibs; 
therefore  we  need  only  to  add  478  —  104.4  =  373.6  S>3. 

For  acceleration  on  inclined  planes  see  p  172. 

The  following  table  will  facilitate  calculations  respecting  the  draft  required  on  grades,  inclined 

planes,  Ac.    In  practice,  allowance  for  friction  must  be  made 

ill  tne  last  2  cols  ;  see  p  172,  and  near  foot  of  485.  Original. 


Pres.  on 

Tendency 

Inclination  or  Slope  of  the  Plane. 
For  the  nat  sine  of  slope,  divide  the 
vert  height  by  the  sloping  length. 

Plane,  in 
parts  of  the 
wt.    Or,  nat. 
cos.  of  angle 
of  Plane. 

Pres.  on 
Plane,  in  fts 
per  ton. 

down  the 
Plane,  in 
parts  of  the 
wt.    Or.  nat. 
sine  of  angle 
of  Plane. 

Tendency 
down  the 
Plane,  in  Ibs 
per  ton. 

Ft.  per  mile. 

Deg.  Min. 

1     n       3. 

1760.00 

18      26 

.9437 

2125 

.3162 

708. 

n        4. 

1320.00 

14        2 

.9702 

2173 

.2425 

543. 

a       5. 

1056.00 

11       19 

.9806 

2196 

.1962 

439. 

a       6. 

880.00 

9      28 

.9864 

2210 

.1645 

368. 

n       8. 

660.00 

7        8 

.9923 

2223 

.1242 

278. 

u       9. 

588.66 

6      20 

.9939 

2226 

.1103 

247. 

a      10. 

528.00 

5      43 

.9950 

2229 

.0996 

223. 

n      11.4 

461.94 

5      00 

.9962 

2231 

.0872 

195. 

n      12. 

440.00 

4      46 

.9965 

2232 

.0831 

186. 

n      14.3 

369.23 

4      00 

.9976 

2232 

.0698 

156. 

n      15. 

352.00 

3      49 

.9978 

2233 

.0666 

149. 

n      19.1 

276.73 

3      00 

.9986 

2237 

.0523 

117. 

n      20. 

264.00 

2      52 

.9987 

«• 

.0500 

112. 

n      23.1 

229.04 

2      30 

.9990 

i« 

.0436 

97.7 

n      25. 

211.20 

2      17 

.9992 

2238 

.0398 

89.2 

u      28.6 

184.36 

2      00 

.9994 

ii 

.0349 

78.2 

n      30. 

176.00 

1      55 

<« 

.0334 

74.8 

n      32.7 

161.47 

1      45 

.9995 

2239 

.0305 

68.4 

n      35. 

150.86 

1      38 

.9996 

.0285 

63.8 

n      38.2 

138.22 

1      30 

.9997 

2240 

.0262 

58.6 

n      40. 

132.00 

1       26 

« 

.0250 

56.0 

n      45.8 

115.29 

1      15 

it 

ti 

.0218 

488 

n      50. 

105.60 

1        9 

.9998 

" 

.0201 

45.0 

n      57.3 

92.16 

1        0 

'i 

.0175 

39.1 

Q      60. 

88.00 

0      57>$ 

.9999 

it 

.0167 

37.4 

n      70. 

75.43 

0      49 

«• 

«• 

.0143 

32.0 

n      76.4 

69.12 

0      45 

ii 

ii 

.0131 

29.3 

n      80. 

66.00 

0      43 

ii 

ii 

.0125 

28.0 

n      90. 

58.67 

0      38 

«• 

«• 

.0111 

24.9 

n    100. 

52.80 

0      34 

1.0000* 

•  i 

.0100 

22.4 

n    114.6 

46.07 

0      80 

•  I 

M 

.0087 

19.6 

n    125. 

42.24 

0      27% 

u 

•  « 

.0080 

17.9 

n    150. 

35.20 

0      23 

ei 

n 

.0067 

15.0 

n    175. 

30.17 

0      19% 

i< 

14 

.0057 

12.8 

n    200. 

26.40 

0      17 

« 

il 

.0050 

11.2 

n    229.2 

23.04 

0      15 

«i 

.0044 

9.77 

n    250. 

21.12 

0      14 

n 

.0041 

9.18 

n    300. 

17.60 

0      11^ 

•  i 

«l 

.0033 

7.39 

n    343.9 

15.35 

0      10 

u 

<l 

.0029 

6.52 

n    400. 

13.20 

0        8% 

ii 

II 

.0025 

5.60 

n    500. 

10.56 

0        7 

«i 

.0020 

4.48 

n    600. 

8.80 

0        6 

ii 

II 

.0017 

3.81 

n    800. 

6.60 

0        4% 

«i 

(1 

.0013 

2.91 

n  1000. 

5.28 

0        3% 

it 

II 

.0010 

2.24 

n  3437. 

1.54 

0        1 

ii 

«i 

.0003 

0.65 

Level. 

0.00 

0        0 

H 

.0000 

0.00 

F  c  differs  from  ordinary  mechanical  force,  inasmuch  as  it  exerts  no  tendency  to  produce  motion, 
but  nly  to  prevent  or  reduce  it ;  and  moreover,  it  acts  equally  in  all  directions  that  are  in  its  own 
plan  .  Thus,  the  fric  at  the  hor  surf  or  joint  of  one  block  of  masonry  laid  upon  another,  resists 
alik  all  forces  acting  hor  on  the  plane  of  the  joint;  whether  to  push  the  block  backward  or  forward, 
or  to  the  right  or  left,  &c.  It  resists  the  motion  of  a  body  up  an  inclined  plane,  or  across  it.  as  well 
as  down  it.  The  resistance  to  moving  forces,  which  fric  imparts  to  a  body,  is  sometimes  called  the 

frictional  stability  of  the  body. 

Art.  63.    The  following  principle  is  one  of  great  practical  importance.    When 

*  Near  enough  for  practice ;  actually  .99995,  or  less  hy  1  part  in  20000,  or  about  1  ft  in  9  tons. 


FORCE   IN    RIGID   BODIES. 


487 


a  plane  w  x,  Fig  64,  has  just  that  inclination,  y  x  w,  at  which  the  fric  of  any  given 
body  is  balanced ;  and  sliding  is  about  to  commence,  from 
the  action  of  any  force  h  o,  applied  to  the  plane,  through 
the  body,  in  any  direction  /to,  not  perp  to  it;  if  from  the 
point  o  of  application,  we  draw  a  line  op,  at  right  angles  to 
the  surf  of  the  plane,  then  the  angle  hop  will  always  be 
equal  to  the  angle  of  fric  y  x  w  of  the  body.  If  the  plane 
is  so  steep  that  sliding  must  take  place,  then  the  angle 
formed  between  the  force  ho,  and  a  perp  op  to  the  plane,  be- 
comes greater  than  the  angle  of  fric ;  but  if  the  steepness  is 
so  slight  that  the  body  rests  firmly  on  the  plane,  then  said 
angle  is  less  than  the  angle  of  fric. 

The  practical  applications  of  this  principle  are  very  numerous ;  they 
extend  to  pressures  in  any  directions  whatever ;  and  apply  to  plain 
surfs  in  any  position  whatever,  whether  inclined,  vert,  or  hor ;  for  any 
given  pres  produces  precisely  the  same  amount  of  fric,  whether  we  im- 
part it  to  the  ceiling,  the  floor,  or  the  wall  of  a  room  ;  provided  they  all 
be  of  the  same  material.    The  angle  of  fric  of  cut  stone 
upon  cut  stone  is  about  32° ;  that  is,  one  block  of  cut  stone 
will  not  slide  upon  another  at  a  less  slope  than  about  32° ; 
the  fric  then  being  full  y6^  of  the  pres.     Therefore,  if  the 
floor  /,  Pig  65,  ceiling  c,  and  walls  w,  w,  of  a  room  be  of 
cut  stone  ;  and/>,  p,  p,  p,  lines  at  right  angles  to  them  ;  we 
may  press  a  piece  «  of  cut  stone  against  them  with  any 
force  whatever,  applied  in  the  direction  of  the  stone  itself, 
without  danger  of  its  sliding ;  provided  only  that  the  di- 
rection of  the  pres  along  s  does  not  form  with  the  perp  p 
an   angle  exceeding  32°.      But  sliding  will  take  place, 
whether  the  pres  be  great  or  small,  if,  as  at  o,  o,  o,  o,  said 
angle  exceeds  32°.     The  angle  of  fric  is,  by  some  writers, 

called,  in  such  cases,  the  limiting:  angle  of 
resistance. 

Kem.  The  friction  at  the  feet  of 
rafters  when  highly  inclined  diminishes  very 
much  their  horizontal  pressure  and  tendency  to 
split  off  the  ends  of  the  tie-beams. 

The  angle  of  fric  of  oak  endwise  against  hard  limestone,  is,  according  to 
Moriu,  20%°;  therefore,  if  the  walls,  Ac,  of  a  room  consisted  of  such  lime- 
stone, we  could  not  press  a  piece  of  oak  endwise  against  it  without  sliding, 
if  the  angle  withp  exceeded  209£°;  and  the  legs  of  a  wooden  trestle,  Fig  66, 
would  not  spread,  on  the  level  surf  of  such  limestone,  under  any  wt  w,  if 
the  angle  abcbe  less  than  20%° ;  but  certainly  would  if  it  be  greater,  unless 
other  preventives  besides  fric  at  the  feet  be  depended  on.  In  this  case  the 
fric  amounts  to  very  nearly  -j&y  of  the  pres :  that  being  the  proportion  cor- 
responding to  '20%°.  These  two  illustrations  show  how  wide  is  the  applica- 
tion of  this  principle :  for  the  announcement  of  which  we  are  (the  writer 
believes)  indebted  to  Moseley. 


EigGG 


Art.  64.  To  find  the  effect  of  an  extraneous  force  (fg,  Fig-  67,) 
imparted  in  any  direction,  to  a  rigid  body  (B)  on  an  inclined 
plane,  ip ;  when  we  know  the  angle  of  fric,  and  the  wt  of  the  body.   The  prin- 
ciple laid  down  in  the  preceding  Art  } 
enables  us  to  do  this. 

Through  the  cen  of  grav  c,  of  the  fl-i 3* 

body,  draw  avert  line  a  w ;  and  extend  j        /' ! 

the 'direction  fg  of  the  force,  to  meet 
this  line,  as  at  o.  Make  o  a  by  scale,  to 
represent  the  wt  of  the  body ;  and  o  z 
to  represent  the  amount  of  the  force 
fg.  Then  is  o  a  point  at  which  we 
may  assume  both  these  forces  to  be  im- 
parted to  the  body.  (Art  29.)  Complete 

the  parallelogram  of  forces  a  x  z  o,  by     _^ 

drawing  a  x,  and  z  x,  parallel  and     "  "C"^  f*rt 

equal  to  o  z,  and  o  a.    Draw  the  diag  tlO  b  I 

x  o,  and  extend  it  to  meet  the  plane,  as 

at  t.    Make  the  line  t  v  perp  to  the  surf  of  the  plane.    This  done,  we  have  a  single 

force  x  o,  equal  in  effect  upon  the  rigid  body,  to  its  wt,  and  fg  combined. 

This  single  force  may  (Art  18)  be  considered  as  imparted  to  the  body  at  any  point  that  lies  in  its 
line  of  direction  x  t ;  therefore,  we  will  assume  it  to  be  imported  at  t,  where  it  encounters  the  force 
of  fric  acting  in  the  direction  8  e,  of  the  joint  formed  between  the  body,  and  the  plane.  Now,  if  t 
strikes  within  the  base  te,tv  being  at  right  angles  to  this  joint,  it  follows  from  the  last  Art,  that 
if  the  angle  xtv  is  less  than  the  angle  of  fric  corresponding  to  the  nature  of  the  materials  which 


488 


FORCE   IN   RIGID   BODIES. 


compose  the  body  and  plane,  then  the  body  will  remain  at  rest  on  the  plane.  But  if  the  angle  *  t  9 
be  greater  than  said  angle  of  fric,  the  body  will  slide  up  or  down  the  plane,  (according  to  circum- 
stances, stated  in  the  next  paragraph ;)  if  the  angles  be  equal,  the  body  will  be  just  on  the  point  of 
beginning  to  slide  either  up  or  down. 

When  the  angle  x  t  v  is  on  the  down  hill  side  of  v  t,  as  in  the  fig,  the  tendency  of  the  body  will  evi- 
dently be  to  move  up  the  plane ;  but  if,  in  consequence  of  a  diff  direction  of  the  force  /  g,  (and  conse- 
quently of  the  resultant  x  o,)  the  angle  x  t  v  is  on  the  up  hill  side  of  v  t,  then  the  tendency  will  be 
down  the  plane. 

REM.  1.  If  the  direction  of  the  resultant  x  o,  or  the  point  t,  falls  outside  of  its  base  s  e,  the  body, 
instead  of  sliding,  will  upset.  It  will  fall  up  hill,  if  t  strikes  p  i  up  hill  from  the  base;  and  down  hill, 
if  t  strikes  down  hill  from  the  base.  See  Remark,  Art  65. 

REM.  2.  In  order  to  draw  the  parallelogram  of  forces  a  xz  o,  and  its  resultant  diag  x  o,  the  line  a,  o, 
which  represents  the  wt,  may  sometimes  have  to  be  regarded  as  pulling  instead  of  pushing  down- 
ward at  the  point  o,  where  the  other  force  meets  it.  See  Art  28,  Fig  9J4  ;  and  Fig  69,  Art  65. 


,  68 


REM.  3.  It  follows  from  the  foregoing,  that  when  at  the  joints  p  q,  r  *, 
Fig  68,  of  a  mass  of  masonry  ;  or  at  the  joints  of  timbers  in  carpentry, 
iron  work,  &c,  the  fric  alone  is  depended  on  to  prevent  sliding,  the  re- 
sultant AS  me,  co,  o  n,  &c,  of  all  the  forces  acting  at  any  joint,  must  not 
form  an  angle  m  c  i,  c  o  a,  o  n  e,  with  a  perp  c  i,  o  a,  n  e,  to  the  joint, 
greater  than  the  angle  of  fric  corresponding  to  the  nature  of  the  ma- 
terials whose  surfaces  constitute  the  joint. 

REM.  4.  The  extraneous  force  reqd  to  move  a  body  up  a  plane,  will 
be  the  least  when  its  direction,  i  n.  Fig  67.  makes  with  surf  ip,  of  the 
plane,  an  angle,  nip,  equal  to  the  angle  of  fric. 


Art.  65,  To  find  the  force  required  to  prevent  a 
body  S,  Fig  69,  from  falling;  when  the  direction,  o  w,  of  its  wt, 
strikes  outside  of  its  base  1  1,  Thus,  suppose  we  wish  to  impart 
a  pulling  force  at  e,  and  in  the  direction  e  a,  to  prevent  the  body 
from  upsetting  down  the  plane.  Through  the  cen  of  gray  c,  draw 
a  vert  line  x  w;  and  continue  the  line  of  direction  of  a  e  to  meet 
it  at  o.  From  o  draw  oy  at  right  angles  to  the  plane  t  p.  By  scale 
make  o  w  equal  to  the  wt  of  the  body  ;  and  from  w  draw  w  y  par- 
allel to  o  a.  Make  e  a  equal  to  w  y  ;  then  is  e  a  the  reqd  force, 
which  will  resist  all  tendency  of  the  body  to  fall.  For  in  the  par- 
allelogram of  forces  o  w  y  z,  we  have  the  force  o  w  tending  to  make 
the  body  fall;  and  the  force  o  z  (equal  to  e  a)  tending  to  prevent 


it  from  falling;  and  the  resultant  o  y,  of  these  two  forces,  equal  to 
ir  joint  effect,  is  at  right  angles  to  the  surf  of  the  plane  ;  and 


their 


is  consequently  (Art  19)  all  imparted  to  it  as  pres ;  no  part  being 
left  unresisted,  to  produce  motion  in  any  direction.  For  as  before 
said,  when  two  forces,  as  o  w,  o  z,  are  compounded  into  one  result- 
ant force  o  y,  those  two  forces  must  be  considered  as  no  longer  ex- 
isting; thus,  in  this  case,  so  long  as  we  regard  the  joint  effect  of  ow  and  o  z  as  being  concentrated 
in  their  resultant  o  y,  we  cannot  of  course,  consider  them  as  acting  in  other  directions  at  the  same 
time  ;  so  that  there  is,  as  it  were,  no  longer  any  wt,  o  w,  tending  to  make  the  body  fall ;  nor  any  force 
c  a,  tending  to  uphold  it;  but  only  the  single  force  o  y,  which  presses  the  inert  body  against,  and  at 
right  angles  to,  the  surf  i  p ;  imparting  to  it  a  tendency  to  move  only  in  the  direction  o  y ;  which  ten- 
dency is  reacted  against  by  the  inherent  cohesive  force,  or  strength,  of  the  plane. 

REM.  If  the  resultant  of  all  the  forces  of  any  kind,  acting  upon  a  body,  does  not  strike  inside  of  its 
base,  no  matter  whether  the  base  be  inclined  or  hor,  the  body  must  evidently  move  ;  on  the  same  prin- 
ciple as  when  grav  is  the  only  force  acting  upon  it,  in  a  direction  which  strikes  outside  of  the  base. 
See  Art  35,  Fig  19.  Or,  in  other  words,  in  all  cases,  if  the  direction  of  the  resultant  of  all  the  forces 
acting  upon  a  body,  meets  with  no  resisting  force  as  it  passes  out  of  the  body,  the  body  must  move  in 
the  direction  of  that  resultant.  Thus,  suppose  that  the  only  forces  acting  upon  S,  were  three,  in 
the  directions  ao,yo.  and  wo,  or  the  reverse  of  what  they  are  in  the 
II  fig;  and  that  y  b  was  their  resultant.  Then,  as  this  resultant  meets 

with  no  opposing  force  at  v,  the  body  must  move  in  the  direction  v  b. 
Or,  the  body  A,  Fig  70,  rests  on  a  hor  base.  Being  acted  upon  only  by 
grav,  the  direction  of  the  resultant  is  gj ;  which,  as  it  leaves  the  body 
at  3,  encounters  an  equal  opposing  force  from  the  resistance  of  the 
ground  ;  and  consequently,  no  motion  takes  place.  But  now  suppose  an 
upward  force  tj,  to  be  imparted  to  the  body  ;  and  let  it  be  four  tin;es  as 
great  as  the  grav.  Then  the  resultant  will  be  a  single  force,  three  times 
as  great  as  grav ;  and  acting  in  the  direction  t  n.  As  this  direction 
leaves  the  body  at  o,  it  meets  no  opposing  force;  hence  the  body  must 
move  in  the  direction  o  n,  since  unopposed  force  always  produces 
motion. 

Art.  66.  Stability.  The  stability  of  a  structure,  or  of  any  body,  is,  strictly 
speaking,  that  resistance  which  its  wt  alone  enables  it  to  oppose  against  forces  tend- 
ing to  change  its  position.  Such  resistance  may  be  assisted  by  extraneous  wts,  or 
by  other  forces  properly  applied;  but  such  must  be  distinguished  from  the  stability 
inherent  in  the  structure,  or  body  itself.  To  insure  the  stability  of  a  structure,  the 
disposition  of  its  parts,  as  well  as  that  of  the  entire  mass,  tfmst  be  such  that  neither 
of  them  shall  move,  either  by  sliding,  or  by  overturning,  under  the  action  of  the  im- 
parted forces.  Stability  is  therefore  a  branch  of  Statics;  or  of  forces  at  rest,  or  in 
equilibrium  with  each  other ;  Art  16,  p  451. 


RCE   IN    RIGID   BODIES. 


489 


Stability  ijtfiist  not  be  confounded  wi tit  strength.    A  structure 

may  be  very  strong;  and  yet  very  unstable.  A  block  of  stone  is  quite  as  strong  while  sliding  down 
a  smooth  plane,  or  rolling  down  a  steep  bank,  as  when  resting  on  a  firm  hor  base  ;  but  it  has  stability 
only  in  the  last  case.  A  pyramid  of  weak  chalk  may  have  great  stability  :  while  a  globe  of  granite 
or  cast  iron,  has  very  little".  We  generally  have  to  examine  into  the  strength,  as  well  as  the  stability 
of  our  structures  :  but  it  must  be  done  by  diff  processes.  The  stability  has  reference  to  the  structure 
considered  as  consisting  of  one  or  more  rigid  bodies,  which  may  be  moved  as  entire  masses,  but  not 
broken,  or  changed  in  form,  by  the  applied  forces.  See  Remark  2,  Art  29,  p  460. 

Those  forces  which  tend  to  impair  the  stability  of  a  structure,  are  called  acting  ones ;  and  those 
which  tend  to  maintain  it.  resisting  ones.  This  distinction  is  merely  a  matter  of  convenience ;  for 
all  the  forces  act,  and  resist. 

The  forces  which  affect  the  stability  of  a  rigid  structure  considered  as  one  mass,  are  its  wt ;  extra- 
neous wts,  or  strains  .  and  the  foundation,  or  support;  which  last  reacts  as  an  antiresultant  (Art  30; 
against  the  others.  When  these  three  balance  each  other,  the  structure  is  stable.  When  the  struc- 
ture is  to  be  considered  as  composed  of  several  rigid  bodies,  then  the  joints  or  surfaces  of  contact  be- 
tween these  bodies  must  also  be  regarded  as  so  many  secondary  foundations,  and  these  also  must  re- 
spectively balance  the  forces  acting  upon  them  ;  otherwise  these  parts  may  slide,  or  overturn,  while 
other  parts  may  remain  firm. 

Art.  67.  In  order  to  guard  against  accidents,  a  structure  must  generally  be  so  designed  as  to 
be  capable  of  resisting  much  greater  forces  than  those  which  it  sustains  under  ordinary  circum- 
stances. The  proportion  which,  with  this  object,  we  give  to  the  resisting  forces,  in  excess  of  the  act- 
ing ones,  is  called  the  coefficient  of  stability ;  or  simply  the  stability,  or  the  safety,  of  the  structure. 
Thus,  if  we  make  it  capable  of  resisting  2,  3,  or  6  times  the  amount  of  the  ordinary  acting  forces,  we 
say  it  has  a  stability,  or  a  coeff  of  stability,  or  a  safety,  of  2,  3,  or  6. 

Art.  68.  Since  the  stability  of  a  structure,  considered  apart  from  its  founda- 
tion, consists  entirely  in  the  resistance  which  its  several  parts,  as  well  as  the  entire 
mass,  can  present  against  both  sliding  and  overturning,  it  follows  that  two  precau- 
tions, already  adverted  to  in  previous  articles,  must  be  resorted  to.  Namely,  1st, 
against  sliding,  take  care  that  the  resultant  of  all  the  forces  acting  upon 
any  joint,  (including  that  between  the  base  and  the  foundation,)  shall  act  either  at 
right  angles  to  said  joint ;  so  as  to  be  entirely  imparted  to  it  as  strain,  (press  or  pull,) 
leaving  no  part  unresisted  to  tend  to  produce  motion ;  or  else  that  it  shall  not  de- 
viate from  a  right  angle,  to  a  greater  extent  than  the  angle  of  fric  corresponding  to 
the  materials  which  compose  the  joint;  so  that  the  portion  of  it  which  is  not  im- 
parted at  right  angles,  shall  be  resisted  by  friction ;  and  thus  be  prevented  from 
producing  a  motion  of  sliding. 

Otherwise,  instead  of  relying  upon  the  position  of  the  joints,  resort  must  be  had  to  the  cohesive 
strength  of  joint-fastenings,  such  as  bolts,  spikes,  cramps,  joggles,  mortises  and  tenons,  mortar, 
cement,  &c,  to  prevent  sliding.  As  to  mortar  and  cement,  however,  it  is  important  to  remember  that 
frequently,  and  especially  in  very  massive  work,  they  have  not  time  to  harden,  or  acquire  their  full 
strength,  before  the  acting  forces  are  brought  to  bear  upon  them  :  therefore,  great  care  is  necessary, 
when  we  use  them  as  substitutes  for  position.  On  this  account  we  frequently  cannot  consider  a  mass 
of  masonry  to  be  a  single  rigid  body,  but  must  regard  it  as  composed  of  several  detached  rigid 
bodies  ;  the  stability  of  each  of  which  must  be  separately  provided  for,  before  we  can  secure  that  of 
the  whole.  Therefore,  in  large  massive  structures  of  importance,  we  should,  as  far  as  possible,  omit 
all  consideration  of  the  strength  of  the  mortar,  and  rely  for  stability  chiefly  upon  placing  the  joints 
at  or  nearly  at  right  angles  to  the  forces  acting  upon  them. 

In  the  2d  precaution,  against  overturning ;  we  must  take  equal  care  that 
the  resultant  of  all  the  forces  acting  upon  any  single  part,  or  upon  the  whole 
structure,  shall  fall  within  the  base-joint  of  that  part,  or  whole.  See  Remark,  Art 
35;  Art  60;  Remark  1,  Art  64;  Remark,  Art  65;  Remark  2,  Art  72. 

Art.  69. 

and  49)  thai 

overturned  about  any  given  point  a,  is  equal  to  that 
which  would  be  produced  if  the  entire  wt  of  the 
body  were  concentrated  at  its  cen  of  grav  g ;  and 
acted  at  the  end  i  of  a  straight  lever  a  i,  of  which  a 
is  the  fulcrum ;  or  at  the  end  o,  s,  or  «,  of  any  straight 
lever  (Art  49)  ao,  ax.  an:  or  of  any  bent  lever  ai  n, 
a  i  s,  at  o,  a  so,  provided  that  in  every  case  there  is 
the  same  leverage  at,  measured  from  the  fulcrum  a, 
and  at  right  angles  to  the  direction  mn  of  the  force 
of  grav  of  the  body.  So  far  as  regards  tendency  to 
resist  overturning,  it  is  immaterial  (Art  18)  at  what 
point  of  the  body,  in  this  line  of  direction  m  n,  we 
conceive  the  grav  to  act :  or  whether  as  a  push  at  o, 
or  a  pull  at  t,  as  denoted  by  the  arrows.  We  have 
also  said  that  the  tendency,  or  moment,  of  this  force 


.    Moment  of  stability.    We  have  already  stated  (see  Arts  46 
t  the  resistance  which  any  rigid  body  as  B,  Fig  71,  opposes  against  being 

Fig  71 


TTL 


,  , 

of  grav,  or  wt,  to  produce  or  to  resist  motion  about  the  fulcrum  a,  through  the  me- 
dium of  any  of  these  levers,  is  found  by  mult  the  force  or  wt  in  fbs,  by  the  leverage 
ai  in  feet.  The  prod  in  ft-fbs  is  generally  called  the  moment  of  the  force  about  the 
point  a ;  but  in  cases  like  that  before  us,  in  which  this  moment  becomes  the  measure 
of  the  stability  of  the  body,  it  is  called  the  moment  of  stability  (or  simply  the  star 


490 


FORCE   IN   RIGID   BODIES. 


bility)  of  the.  body,  about  that  point.  Therefore,  if  bodies  of  the  same  size  and  shape 
have  diff  wts,  or  sp  gravities,  their  respective  stabilities  will  be  in  proportion  to  their 
wts,  or  sp  gr. 

A  body  may  have  diff  moments  of  stability,  about  diff  points.    Thus  it  would  be 

far  more  difficult  to  overturn  B  about  the  point  6,  than  about  a  ;  because  the  lever- 

age bi  is  2>£  times  as  great  as  at  ;  and  since  the  wt  and  the  point  of  the  cen  of  grav 

remain  unchanged,  the  moment  about  6  is  2%  times  as  great  as  about  a. 

BBM.  1.  Let  a  b  c  o,  Fig  72,  be  a  squared  block  of  stone  6  feet  long  ;  on  a  nor  base  ;.  and  weighing  12 

tons  ;  and  h,  a  force  applied  to  overturn  it  about  the  toe  c.   Since  its 

A  Ti  length  o  c,  is  6  feet,  its  cen  of  grav  t,  will  be  dist  o  g,  or  3  ft  back  from 

T~  o.  Consequently,  the  moment  with  which  the  block  resists  being 
overturned,  is  12  (tons)  X  3  (ft  leverage)  =  36  ft-tous.  Now.  suppose 
the  upper  half  a  ft  o,  to  be  removed  ;  the  remainder  o  b  c  will  weigh 
but  6  tons.  But  its  cen  of  grav  »,  is  farther  from  o,  than  that  of  the 
whole  block  was.  Being  now  triangular  in  shape,  the  dist  o  y  will  be 
%  of  o  c;  or  will  be  4  ft.  Consequently,  the  resisting  moment  will  be 
6  (tons)  X  4  (ft  leverage)  —  '24  foot-tons.  So  that  although  the  block 
has  but  half  the  wt  of  the  first  one,  it  has  %  as  great  resisting 
power.  It  is  on  this  principle,  that  in  order  to  save  masonrv,  the 
faces  of  retaining-  walls,  Ac,  are  sloped,  or  battered  back. 

REM.  2.  Of  two  bodies,  as  A  and  B,  Fig  72^,  of 
precisely  the  same  size,  wt,  and  position;  and 
having  the  same  moment  of  stability  ;  one  may  re- 
quire the  expenditure  of  a  greater  amount  of  the 
same  degree  of  force,  than  the  other,  to  overturn  it. 
Thus,  let  the  upper  part  en,  of  A  ;  and  the  lower 
part  o  y,  of  B,  be  made  of  lead  ;  and  the  remain- 
ing part  of  each,  of  cork.  Then  the  cen  of  grav  of 
the  body  A,  will  be  near  the  dot  t  ;  and  that  of  B, 
near  the  dot  «.  The  wt  of  both  bodies  being  the 
eame;  and  the  cen  of  grav  of  both,  being  at  the 
same  hor  dist,  a  o,  from  the  fulcrums  o,  o,  around 
which  the  bodies  are  to  be  overturned  ;  their  mo- 
ments of  stability  must  also  be  the  same.  Conse- 
quently, both  will  require,  in  order  to  begin  to 
overturn  them,  precisely  the  same  amount  of  force, 
applied  in  any  same  given  direction  ;  and  at  any 
same  given  point;  as,  for  example,  the  equal  forces 


, 

/and  g,  applied  in  a  hor  direction,  at  the  points  i 
and  i.  But  the  body  B  will  plainly  require  this  force 


.  requi 

to  be  continued  for  a  longer  time,  (or  in  other 
words,  will  require  the  expenditure  of  a  larger 
amount  of  force,)  in  order  to  actually  overthrow  it, 
than  the  body  A  will  ;  for  the  body  A  will  be  overthrown  when  the  force  has  acted  for  only  the  short 
time  necessary  to  move  it  into  the  position  of  the  dotted  lines  J.  The  cen  of  grav  being  then  carried 
to  e,  which  is  beyond  the  base  at  o,  the  body  must  necessarily  fall.  But  in  the  body  B,  the  force  must 
act  at  least  long  enough  to  move  it  into  the  position  N  ;  for  not  until  then  will  its  cen  of  grav  «,  be 
moved  to  the  position  I,  so  as  to  be  beyond  the  base.  When  the  stability  of  a  body  is  considered  only 
with  regard  to  the  degree  of  force  necessary  for  its  resistance  to  beginning  to  move,  (which  degree  is 
the  same  in  both  A  a»d  B  :  and  is  the  force  with  which  engineers  are  most  concerned,)  it  is  called  the 
static  stability  of  the  body  ;  and  when  considered  with  regard  to  the  total  expenditure  of  that  same  de- 
gree of  force,  necessary  to  complete  the  overthrow  of  the  body,  it  is  called  its  dynamic,  or  moving  sta- 
bility. The  engineer  rarely  need  concern  himself  about  the  last;  his  object  being  to  secure  his 
structures  against  beginning  to  move. 

RKM.  3.  It  is  not  alone  the  wt  of  the  body  itself,  which  contributes  to  its  stability  in  all  cases  ;  for 
this  may  be  assisted  by  extraneous  wts  or  loads.  Thus,  the  wt  of  a  pier  P,  Fig  73,  gives  it  in  itself  a 
certain  degree  of  stability  ;  but  when  we  add  the  wt  of  the  two  equal  arches, 
its  stability  is  thereby  increased,  supposing  the  foundation  to  be  secure.  And 
a  passing  load,  when  it  is  directly  over  the  pier,  increases  it  still  more.  It  is 
true  that  the  wt  of  the  arches  might  crush  the  pier  to  fragments,  if  the  stone 
be  soft;  but  this  is  a  matter  of  Strength  of  Materials  ;  not  of  stability  ;  and 
must  be  examined  into  by  itself.  If  the  two  arches  be  of  unequal  sizes,  or  if 
there  be  but  one  arch,  the  stability  of  the  pier  may  become  either  increased 
or  diminished,  according  to  circumstances;  as  will  appear  farther  on. 

Whether  the  force  acting  for  or  against  the  stability  of  a  structure,  be 
gravity,  or  pushes,  or  pulls,  produced  from  other  sources,  is,  as  in  other  cases, 
a  matter  of  no  importance;  for  force  is  simply  force,  no  matter  whence  de- 
rived. We  have,  therefore,  only  to  look  at  the  diff  forces  acting  upon  our 

structures,  as  so  many  tendencies  to  produce  motions  in  certain  directions.  If  these  tendencies  are 
reacted  against,  or  destroyed  by  others,  they  will  not  produce  it  ;  but  if  they  meet  no  resistance,  mo- 
tion must  take  place.  The  only  peculiarity  we  need  assign  to  grav,  is  that  the  direction  of  its  actio* 
i*  always  vert  downward  ;  while  other  forces  may  be  imparted  in  either  that,  or  any  other  direction. 
The  resultant  of  grav  combined  with  other  force  or  forces,  may  be  in  any  direction. 

Art.  7O.  Fig  73%.  In  the  principal  cases  of  stability  that  present  themselves 
to  the  civil  engineer,  both  the  acting  and  resisting  forces  w  x,f,f,f,  &c,  may  all  be 
considered  to  be  imparted  and  acting  in  the  same  plane  ;  which  is  a  vert  one,  el  o  c, 
passing  through  the  cen  of  grav,  t?,  of  the  structure,  m  np  q  r  s  t  u;  and  of  course, 
coinciding  with  the  line  of  direction  w  x,  of  its  wt,  or  force  of  gravity.  The  plane 
f.  I  o  c,  and  all  the  forces,  therefore,  may  be  considered  as  coinciding  with  a  leaf  of 
paper  standing  vert  on  one  edge.  This  renders  the  calculations  much  more  simple 


.]<V73 


IN    RIGID    BODIES. 


491 


in  difif  planes ;  in  which  case  diagrams  alone  would  not  suf- 

m  n 


fice  for  determining  the  resultants,  leverages,  mo- 
ments, &c.  See  Art  44.  Whereas,  when  they  are 
in  the  same  plane,  such  diagrams,  neatly  drawn 
to  a  convenient  scale,  will  usually  possess  all  the 
accuracy  reqd  in  practice. 


Thus,  -n  examining  the  strains  on  the  diff  pieces  composing  the 
truss  of  a  roof,  or  of  a  bridge,  &c,  not  only  their  wts,  but  all  the 
strainiug  forces,  may  be  assumed  to  act  in  a  vert  plane  passing 
lengthwise  of  the  truss  from  end  to  end  :  splitting  it  into  two  equal 
parts.     In  the  case  of  a  structure  of  masonry,  such  as  a  retaining- 
wall,  or  a  stone  bridge  B,  Fig  74,  we  base  our  calculations  upon  a 
thin  vert  slice  of  it,  having  a  length  a  o,  i  i,  or  y  t,  of  only  one  ft ; 
no  matter  what  may  be  the  height  y  s  ;  or  the  thickness  y  c,  of  the 
structure.    Through  the  center  of  this  one  ft  of  length,  we  suppose 
a  vert  plane  to  pass ;  splitting,  as  it  were,  the  body  B  into  two  pre- 
cisely similar  parts ;  and  all  the  forces  actually  diffused  equally  J 
throughout  the  whole  one  ft  of  length,  are  supposed  to  be  couoen-         *D '     "7  A. 
trated,  and  to  act  upon  one  another,  in  this  plane.    See  Remark,         XlM  •   r 
Arts  49  and  57. 

Art.  71.  In  order  to  ascertain  the  effect  of  diff  forces  to 
produce  either  sliding-,  or  overturning-,  of  either  an  entire  rigid 
body,  or  of  one  composed  of  several  rigid  bodies  placed  together  without  joint- 
fastenings;  we  first  find  the  direction  of  the  resultant  of  those  forces.  If  the  body 
or  structure  is  not  composed  of  diff  parts;  or  if,  being  so  composed,  these  parts  are 
so  firmly  united  together  by  joint  fastenings,  as  to  constitute  virtually  but  a  single 
rigid  mass,  then  we  need  do  nothing  more  than  (as  in  Art  35)  find  the  resultant  ad, 
Fig  19,  of  all  the  forces.  If  the  direction  of  this  resultant  strikes  inside,  of  the  base, 
the  body  will  not  overturn;  see  Remark  2,  Art  72;  and  if,  besides  striking  inside  of 
the  base,  it  forms  with  a  line  bx,  at  right  angles  to  the  base,  an  angle  abx,  less  than 
the  angle  of  fric  between  the  body  and  its  support  n  m,  then  it  cannot  slide. 

At  Fig  67,  Art  64,  the  direction  of  the  resultant  x  o  strikes  at  t,  inside  of  the  base  «  e ;  therefore  the 
body  B  cannot  upset ;  whether  it  will  slide  or  not,  depends  upon  whether  the  angle  o  tv  is  greater  or 
less  than  the  angle  of  fric  corresponding  to  the  materials  composing  the  body,  and  the  plane.  If  both 
are  of  ordinary  dressed  granite,  this  angle  must  not  exceed  about  32°.  See  other  angles,  under  head 
Friction. 

Questions  on  overturning,  may  also  be  solved  on  the  principle  of  leverage.    See  Art  49,  p  475, 

Art.  72.  We  will  now  consider  a  case  in  which  the  structure  is  assumed  to  be 
composed  of  several  rigid  bodies,  merely  placed  together  without  joint-fastenings  of 
any  kind,  such  as  cramps,  bolts,  mortar,  &c;  but  depending  entirely  upon  their  wt 
and  positions,  for  securing  their  stability.  The  process  in  this  case  is  the  same  as 
in,  p  465,  Fig  19,  except  that  instead  of  assuming  the  body  to  have  but  the  one  joint 
nm\  and  finding  the  effect  of  the  resultant  ad  with  reference  to  this  joint  alone; 
we  consider  it  to  have  several  joints,  as  P  Z,  F  L,  W  X,  &c,  Fig  75 ;  and  then  examine 
the  effect  of  the  diff  resultants  which  they  must  respectively  sustain  in  consequence 
of  the  diff  wts  of  the  several  parts  NMPZ,  NMFL,  NMWX,  resting  on  them. 

L,et  II  !%'  .1  T  be  one  half  of  a  stone  arch ;  or  rather  a  vert  slice  of 
it,  1  ft  thick ;  and  let  NMWX  be  a  similar  slice  of  a  dressed  stone  abut  which  has 
been  designed  to  sustain  the  thrust  of  the  arch ;  and  the  fitness  of  which  for  the 
purpose,  we  wish  to  ascertain. 

Suppose  the  thrust  of  the  slice  of  the  arch,  (that  is,  the  resultant  of  its  wt,  and  of  its  nor  pres,)  to 
have  been  previously  ascertained  by  Ex  2,  Art  38,  to  be  30  tons ;  that  this  is  concentrated  at  o,  (the 
center  of  the  skewback  ;)  and  that  its  direction  is  o  b.  Also,  suppose  the  wt  of  the  part  NMPZ  be 
found  to  be  10  tons,  and  to  be  concentrated  at  its  cen  of  grav  G ;  and,  consequently,  to  act  in  the 
vert  direction  G  a.  Now  (Arts  18  and  35)  we  may  suppose  the  30  tons  resultant  of  the  arch,  and  the 
10  tons  grav  of  the  part  N  M  P  Z,  to  act  at  the  same  point  c,  at  which  their  directions  G  a  and  o  6  meet. 
Make  c  d  by  scale  equal  30  tons,  and  c  a  10  tons :  complete  the  parallelogram  of  forces  cdya;  and 
draw  its  diag  cy,  which  by  the  same  scale  will  give  the  resultant  of  all  the  forces  acting  upon  the 
part  NMPZ. 

Now  we  see  that  the  direction  cy  of  this  resultant  strikes  at  i,  or  within  the  base  P  Z ;  consequently 
(Art  60,  &c>  NMPZ  cannot  upset,  no  matter  how  great  may  be  the  pres  cy ;  see  Remark  2.  From  i 


492 


FORCE   IN    RIGID   BODIES. 


draw  if,  at  right  angles  to  the  line  P  Z  ;  and  measure  the  angle  cit,  which  the  resultant  cy  forma 
with  it.  This,  we  find,  is  greater  than  32°;  -that  is,  it  exceeds  the  angl;  of  fric  between  surfaces  of 
dressed  stone.  Therefore,  the  part  NMPZ  must  slide  along  the  joint  PZ.  This  might  possibly  be 
prevented  by  good  mortar,  if  time  be  allowed  it  to  solidify  properly,  before  the  centers  are  eased  so 


M  to  bring  the  pres  of  the  arch  upon  the  abut;  or  by  iron  cramps,  stone  joggles,  &c;  but  these  are 
expensive.  The  most  obvious  remedy,  as  well  as  the  least  expensive,  is  simply  to  incline  the  joint 
PZ  into  a  direction  somewhat  like  from  R  to  Z :  so  as  to  receive  the  pres  of  the  resultant  cy  more 
nearly  at  right  augles  ;  at  least  so  nearly  as  to  be  fairly  within  the  limits  of  the  angle  of  fric.  If  this 
is  done,  stability  is  secured  ;  for  the  part  NMPZ,  being  now  safe  against  both  sliding  and  overturn- 
ing, can  move  in  no  other  way  ;  unless  the  strength  of  the  stone  composing  the  masonry  is  insufficient 
to  bear  the  pres,  and  may  therefore  crush  to  pieces  under  it.  But  this  is  a  question  of  Strength  of 
Materials.  See  Remark  2.  Art  35.  The  point  i,  of  Fig  75,  comes  much  nearer  to  Z  than  would  be 
desirable  in  practice  ;  for  it  might  cause  crushing  at  Z.  See  Rem  2,  following. 

Having  thus  provided  for  both  the  sliding  and  the  overturning  stability  of  the  abut  as  far  down  as 
the  joint  P  Z,  we  will  now  examine  as  far  down  as  the  joint  F  h.  Taking  the  entire  part  N  M  F  L  of 
the  abut,  we  first  find  its  weight,  say  25  tons ;  and  this  we  assume  to  act  at  its  cen  of  grav  K,  and  in 
the  vert  direction  v  K  I.  The  amount  and  direction  of  the  thrust  of  the  arch  at  o,  of  course,  remain 
as  before.  Therefore,  from  the  point  v,  where  the  two  directions  meet,  lay  off  ve  to  represent  as 
before  the  30  tons  thrust  of  the  arch  ;  and  v  I,  the  25  tons  wt  of  the  abut.  Complete  the  parallelogram 
of  forces  v  e  s  I,  and  draw  its  diag  v  s ;  which,  measd  by  the  same  scale,  will  give  the  resultant  of  all 
the  forces  acting  upon  the  part  N  M  F  L.  Now  we  see  that  the  direction  of  this  resultant  does  not 
fall  within  the  base  F  L  ;  but,  on  the  contrary,  passes  out  of  the  body  atj  ;  outside  of  which  it  meets 
no  force  to  resist  it.  Consequently,  (Remark,  Art  65,)  since  this  resultant  must  be  considered  as  an 
only  force  acting  upon  an  inert  body  or  abut,  NMF  L,  without  wt,  (Art  35,)  that  body  must  upset 
around  the  point  L ;  or  around  the  nearest  joint  in  the  masonry  between  L  and./;  and  cannot  con- 
tinue to  stand  of  itself,  unless  its  base  be  above  j.  It  is  true  that  by  placing  earth  behind  it.  espe- 
cially if  well  compacted  by  ramming,  the  abut  of  a  small  arch  might  be  made  to  stand  safely  even 
upon  the  base  W  X ;  and  in  the  case  of  arches  of  moderate  spans,  this  aid  may  be  resorted  to  for 
strengthening  the  abuts,  when  there  is  no  danger  that  the  earth  may  be  washed  away  by  floods  or 
rains,  and  thus  expose  them  to  ruin  ;  and  this  is  generally  and  properly  done. 

RKM.  1.  If  in  the  same  manner  that  the  point  i  was  found  in  the  joint  P  Z.  others  between  P  Z  and 
W  X  be  determined  also :  then  a  curve,  commencing  at  the  akewback  o,  and  drawn  through  them, 
will  represent  the  line  of  pressures  or  of  resistance,  or  of  thrust,  (see  footnote,  p  348)  through  the 
abut.  At  any  point  whatever  in  this  line,  say  at  i,  the  entire  pres  above  said  point  may  be  supposed 
to  be  concentrated  ;  while  the  entire  length,  as  cy,  of  that  resultant  which  cuts  said  point,  gives  the 
amount  of  said  pres  at  that  point ;  and  the  direction,  as  c  i,  of  the  same  resultant  ia  also  the  direc- 
tion in  which  said  pres  acts  upon  said  point.  See  Art  15  of  Hydrostatics. 

REM.  2.  The  line  of  pres  enables  us  to  determine  another  very  important  point  connected  with  the 
stability  of  a  structure.  It  is  not  sufficient  in  practice  that  this  line  should  strike  merely  within  the 
base;  it  must  strike  at  a  considerable  dist  within.  If  the  structure  and  its  foundation  were  ofeso- 
lutely  rigid,  so  that  no  conceivable  force  could  bend  or  break  them,  this  would  not  be  necessary ;  but 
all  materials  are  more  or  less  weak,  so  that  if  great  pressures  come  too  near  to  their  edges,  there  ia 
danger  of  splitting  or  crushing  at  those  points  ;  or  if  near  the  edge  of  a  base,  an  unequal  settlement 
-»f  the  soil  beneath  may  take  place.  Therefore,  even  in  structures  of  but  small  size,  the  dist  iZ,  Fig 
75,  of  the  line  ot  pres,  from  the  outer  point  Z.  should  never  be  less  at  any  joint  than  y±  of  the  width 
•f  that  joint ;  except,  perhaps,  in  a  case  like  that  of  a  small  arch  in  which  the  earth  filling  is  depoa- 


FORCE   IN   RIGID   BODIES. 


493 


iited  behind  the  abuts  before  the  centers  are  removed.  In  important  works,  it  should  not  be  less  than 
about  %  of  the  width  of  the  joint;  and  it  is  still  better,  when  possible,  at  H  ;  or,  in  other  words,  at 
the  center  of  each  joint.  When,  as  at  Q,  a  footing  U  is  added  at  a  base,  W  X  should  be  taken  as  the 
joint;  not  W  Q. 

REM.  3.    The  line  of  pressure  in  an  arch  itself,  as  H  J  N  T,  Fig  75, 

may  also  be  found  much  in  the  same  way,  thus :  First  divide  the  half  arch  H  J  N  T, 
and  the  filling  above  it,  by  vert  lines  ru,  wx,  &c,  which  need  not  be  at  equal  dists 
apart.  Four  such  lines  will  suffice  for  a  flat  arch,  and  about  six  for  a  semicircular 
one.  We  then  consider  in  turn,  and  separately,  each  part,  as  r  u  H  J,  w  x  H  J, 
N  D  T  H  J,  &e,  which  extends  from  these  lines  to  the  center  H  J  of  the  half  arch  ; 
the  last  of  these- being  the  entire  half  arch.  The  cen  of  grav,  and  the  wt,  of  each 
of  these  parts,  must  be  found ;  also,  (Ex  6,  p  478,)  the  hor  pres  at  the  keystone. 

Now  each  of  these  parts,  like  the  beam  in  Ex  1,  or  the  half  arch  in  Ex  2,  p  469,  is  acted  upon,  and 
kept  in  equilibrium,  by  three  forces  ;  namely,  the  hor  pres  at  the  keystone,  (see  Ex  6,  p  478  ;)  its  own 
wt,  acting  vert;  and  the  reacting  force  of  the  part  next  behind  it.  We  proceed  with  each  part  sepa- 
rately, as  we  did  with  the  entire  beam  alluded  to,  thus  :  Beginning  with  the  part  r  u  H  J,  from  its 
cen  of  grav,  ra,  draw  a  vert  line  TO/.  From  the  center  E  of  the  keystone,  draw  a  hor  line  E  n,  to  meet 
mf.  From  n  lay  off  nf  by  scale,  to  represent  the  wt  of  the  part  r  u  H  J ;  and  from  /  lay  off  fg,  hor 
by  scale,  to  represent  the  hor  pres  at  the  key.  Draw  the  diag  ng;  which  will  give,  by  scale,  the  re- 
sultant of  these  two  forces.  The  point  b,  at  which  the  diag  ng  intersects  the  vert  ru,  is  a  point  in 
the  line  of  pres  reqd.  Next  go  to  the  part  tvxlIJ;  and  in  the  same  manner  find  another  point  in 
the  line  w  x,  using  the  cen  of  grav  and  the  wt  of  that  part.  The  hor  pres  will  be  the 
Same  in  each  part.*  Finally,  treat  the  entire  half  arch  N  D  T  H  J  in  the  same  way.  The 
resultant  diag  of  this  last  will  pass  through  o,  the  center  of  the  skewback,  if  the  archstones  have  the 
same  depth  throughout.  A  curve  drawn  by  hand  through  the  points  thus  found,  will  be  the  reqd 
line  of  pres.  These  points  will  not  all  fall  equally  well  within  the  thickness  of  the  archstones.  In- 
deed, if  the  intrados  of  the  arch  is  a  full  semicircle,  or  semi-ellipse,  some  of  them  will  even  fall 
entirely  below  the  archstones,  as  shown  at  a  and  u,  in  the  dotted  line  of  pres  in  Fig  4>£,  p  348. 
When  this  is  the  case,  the  tendency  of  the  line  of  pres  is  to  bend  the  arch  still  more  upward  at  a  and 
u;  and  thus  allow  the  parts  about  the  crown  o  to  descend.  Each  half,  o  e  and  o g,  of  the  arch  is  then 
in  the  same  condition  that  a  crooked  pillar  or  column  would  be  if  a  heavy  load  were  placed  on  top  of 
it.  The  line  of  pres  of  the  load  would  then  not  paas  through  the  axis  of  the  pillar,  but  outside  of  it, 
or  as  the  string  of  a  bow;  so  that  instead  of  being  borne  in  safety,  the  load  would  bend  the  pillar 
still  more;  as  the  tightening  of  the  string  would  bend  the  bow.  If  the  concave  side  of  the  pillar,  or 
of  the  bow,  should  crush  under  the  undue  strain  upon  it,  the  whole  would  fail.  In  like  manner,  if 
the  stone  of  the  arch  along  the  concave  intrados  near  a  and  u,  is  not  strong  enough  to  resist  the  undue 
crushing  strain  thus  brought  against  it,  it  will  crumble,  and  the  arch  will  fail,  by  rising  at  its 
haunches,  and  falling  at  its  crown.  See  footnotes  to  Art  6  of  Stone  Bridges,  p  349. 

Art.  73.  The  stability  of  bodies  on  inclined  planes,  as  regards 
overturning,  is  measd  in  the  same  way  as  when  the  base  is  hor ;  namely,  by  mult 
their  wt,  by  the  perp  dist  (ao,  or  c  (,  at  A,  B,  and  D,  Fig  76,)  from  the  fulcrum,  or 
turning-point  a  or  c,  to  the  vert  line  of  direction  (g  o)  drawn  from  the  cen  of  grav 


of  the  body.  Hence,  it  is  evident  that  the  body  B  has  less  overturning  stability 
about  its  toe  a,  than  the  similar  body  A  has,  when  the  force,  w,  tends  to  upset  it 
down  hill.  But  it  has  more  than  A,  when  the  force  tends  to  upset  it  up  hill,  or  about 
the  toe  c;  for  the  leverage  t  c  of  B  is  greater  than  that,  o  c,  of  A. 

The  body  C,  which  would  overturn  upon  a  level  base,  because  the  line  g  o  strikes  outside  of  th« 
base ;  would  be  stable  against  overturning,  if  placed  as  D  upon  an  inclination,  where  the  vert  g  o 

#  That  the  hor  pres  throughout  every  part  of  the  arch  is  equal  to  that  at  its  center,  may  be  under- 
stood by  supposing  one  half  of  the  arch  to  be  changed  into  a  bent  column  having  the  skewback  for 
its  base :  and  the  kevstone  for  its  top ;  and  to  be  loaded  on  its  top  by  a  wt  equal  to  the  hor  pres  at 
the  center  of  the  arch.  Now  it  is  plain  that  every  part  of  this  column  has  equally  to  bear  the  vert 
pres  of  the  load ;  on  the  same  principle  that  every  part  of  a  long  bent  hook  has  equally  to  bear  the 
vert  pull  of  any  load  upheld  by  it.  And  on  the  same  principle  every  part  of  the  arch  sustains  a  bori- 


„,„.!  ^  u~r  pres  at  the  same  poin1. , „ 

thus  producing  the  curved  line  of  pres  or  thrust  of  the  arch.  In  other  words,  the  hor  pres,  although 
equal  at  every  part  of  the  arch,  is,  as  it  were,  pushed  downward  as  we  approach  the  skewback,  by 
the  increasing  vert  pres  of  the  greater  height  of  masonry  and  earth, 

32 


494 


CENTRIFUGAL   FORCE. 


strikes  within  the  base.  Inasmuch  as  the  leverage  a  o,  is  greater  than  the  one  c  t,  D  would  present 
more  resistance  to  a  force  tending  to  upset  it  down  hill,  than  up  hill. 

Structures  built  upon  slopes  are,  however,  liable  to  slide  t 

that  is,  they  are  deficient  in  frictional  stability.  In  practice  this  is  remedied  by  cutting  the  slope 
into  hor  steps,  as  at  E.  But  works  so  constructed  are  not  as  strong  as  if  the  base  were  a  continuous 
hor  line;  because  the  vert  faces  of  the  steps  break  the  bond  of  the  masonry;  and  because  the  mortar 
in  the  higher  portions  a  d,  being  in  greater  quantity  than  that  in  the  lower  portions  e  y,  necessarily 
allows  more  settlement  of  the  masonry  in  the  former  ;  and  thus  renders  the  work  liable  to  crack,  or 
split  open  vertically.  The  case  is  analogous  to  that  of  a  foundation,  firm  in  some  parts,  and  com- 
pressible in  others.  Therefore,  when  circumstances  permit,  the  foundation  should  be  levelled  off  as 
at  d  v,  or  if  the  masonry  has  to  sustain  down-hillward  pressures,  v  should  be  lower  than  d;  and  the 
courses  of  masonry  be  laid  with  a  corresponding  inclination. 


CENTBIFTJGAL  FOECE, 


Art*  1.  CENTRIFUGAL  force  is  that  with  which  a  body  while  revolving  around  a  cen- 
ter, tends  to  fly  off  in  a  radius,  or  direct  line  drawn  through  said  center  and  the  body  :  and  is  entirely 
diff  and  distinct  from  the  force  which  the  body  has  in  the  curved  direction  in  which  it  is  moving. 
Centripetal  force,  on  the  contrary,  (ahvays  precisely  equal  to  the  centrifugal,  but  in  an  opposite  direc- 
tion,) prevents  the  body  from  so  flying  off;  and  keeps  it  in  the  circle.*  They  are  both  called  central 
farces.  It  is  centrifugal  force  that  pulls  the  string  with  which  we  whirl  a  stone  :  or  which  tends  to 
burst  a  millstone,  or  a  fly-wheel  when  revolving  rapidly  ;  or  which  presses  a  railway  train  against  the 
outer  rails  in  curves ;  -while  the  cohesive  force  of  the  string,  millstone,  or  fly-wheel ;  or  the  spikes  at 
the  rails,  furnish  the  centripetal  force.  Both  forces  vary  directly  as  the  wt  of  the  body ;  and  also 
as  the  square  of  its  vel ;  and  inversely  as  the  rad  of  the  circle  in 
which  it  moves.  Let  Fig  1  represent  a  uniform  homogeneous  circular 
ring  like  a  fly-wheel,  revolving  by  means  of  its  arms,  around  its  cen- 
ter of  motion,  c;  and  Fig  2,  any  isolated  body  n,  revolving  by  means 
of  a  string,  around  its  center  of  motion,  c.  In  either  case,  if  the  di- 
meusion  a  t  of  the  body,  in  the  direction  of  a  rad  a  c,  does  not  exceed 
about  %  t  part  of  the  entire  rad  a  c,  we  may  near  enough,  for  ordi- 
nary practice,  consider  en  as  the  rad  infect  to  be  used  in  calculating 
the  centrifugal  force ;  n  being  the  cen  of  grav  of  the  entire  body  in 
Fig  2 ;  and  the  cen  of  grav  of  its  cross-section  at  t  a  in  Fig  1. 

In  either  case,  if  we  know  the  wt  of  the  body  in  ft>s  or  tons,  &c ; 
the  rad  c  n,  in  feet;  and  either  the  vel  in  ft  per  sec  with  which  the 
point  n  revolves;  or  else  the  number  of  revs  which  the  body  makes 
per  rain;  then 

The  centrif  force,  in          Wt  of    ^  Square  of  vel  at  n, 
Ibs  or  tons,  Ac,  as  the  _  the  body  *      in  ft  per  sec, 
case  may  be,  jj^c  n  in  ft  x  32.2. 

The  centrif  force  _      Wt  of    v  Square  of  number  v  R    ,        .     f.± 
in  Iba  or  tons, Ac.  ~  the  body  X   of  revs  per  min.     X  .Barf  c  n  in  ftl 


Ex.   Suppose  the  body  in  either  Fig  1,  or  Fig  2,  to  weigh  1.2  tons  ;  the  rad  c  n  to  be  2.5  ft;  (in  which 
case  the  circumf  will  be  15.7  ft;)  and  suppose  the  body  to  make  100  revs  per  min.     In  this  case  its 

vel  will  plainly  be  15.7  X  100  =  1570  ft  per  min  ;  or  ^  =  26.17  ft  per  sec.    Then  by  the  first  for- 
aiula, 

Centrif  force 
Centrif  force 


L,  x  10000  x  «.s  =  .oog=10LgtBBB. 


*  Centrifugal  and  centripetal  force  being  always  exactly  equal  to  each  other,  it  may  be  asked  why  do 
rapidly  revolving  bodies  sometimes  break;  and  the  fragments  fly  off?  Does  not  this  prove  that  the 
centrifugal  force  has  become  greater  than  the  centripetal  one?  The  answer  is,  No.  When  the  centrifu- 
gal force  becomes  precisely  equal  to  the  ultimate  strength  of  the  string,  or  of  the  millstone.  &c.  so  that 
the  string,  stone,  &c,  can  furnish  no  greater  resistance,  then  that  force  which  is  moving  the  body  for- 
ward, causes  the  breakage  ;  and  at  that  instant  both  the  centrifugal  and  centripetal  forces  ce.ase  en- 
tirely ;  and  the  fragments  of  the  body  do  not  fly  off  in  the  direction  which  the  centrifugal  force  had  ; 
but  in  a  tangent,  or  at  right  angles  to  it  :  being  carried  in  that  direction  by  said  other  force. 

t  When  a  t  is  equal  to  %  of  a  c,  the  rad  thus  found  will  be  but  -fa  part  too  short  ;  when  %  of  a  c, 
about  J  part  too  short  ;  and  when  %  of  a  c,  about  %  part  too  short  ;  and  the  resulting  centrif  forces 
will  consequently  be  too  small  in  the  same  proportion.  But  when,  as  is  usually  the  case,  in  fly-wheels, 
&o,  a  t  is  much  less  than  even  X  of  a  c,  the  error  is  not  worth  notice  in  prnctice. 

t  These  are  not  strictly  correct,  not  only  because  n  is  not  in  reality  at  the  cen  of  grav  of  t  a  ;  but 
because  the  ring,  Fig  I,  or  the  body,  Fig  2,  must  be  united  to  the  center  c,  either  by  arms,  or  by  a 
string,  or  wire,  Ac-  and  the  weight  of  these  arms,  &c.  will  slightly  shorten  the  rad  of  gyration.  Se« 
Art  2.  But  in  practice  this  effect  is  usaally  too  small  to  be  regarded  ;  or  a  trifling  allowance  is  made 
for  it  by  guess. 


IFUGAL   FORCE. 


495 


Al*t«  2.     But  in  the  cas^of  a  solid  homogeneous  uniform  circular  tody,  like  a  millstone  of  uni- 
form thickness,  Fig  3,  rejurfving  around  its  center  c  either  vert  or  hor,  the'rad  to  be  used  is  what  is 
called  the  rarijsla  of  gyration,  c  h.     In  such  a  body 
the  parts  nearer.Ane  circumf  move  with  greater  vel  and  force  than  those 
nearer  thecentelC    But  there  is,  in  such  cases,  an  imaginary  interior  circle  ~pj  ,  Q 

h  m  s  y,  caned  the  circle  of  gry ration  ;  such  that  if  all         ^^^^^   •' 

the  weight  of  the  entire  body  be  supposed  to  be  concentrated  at  the  cit  cunif 
of  said  circle,  then  th«  force  of  the  revolving  body  would  produce  the  same 
effect  as  it  actually  does  while  distributed  over  the  whole  body  Any  point 

as  h,  in  said  circle,  where  it  is  cut  by  any  rad  c  w,  is  the  cen- 
ter of  g-y  ration  of  that  rad.  In  a  circular  body  of  uni- 
form thickness,  like  a  millstone,  it  is  very  easy  to  find  the  rad  c  A  of  gyra- 
tion ;  for  we  have  only  to  mult  the  actual  ouf-to-out  rad  c  w,  by  the  deci- 
mal .707.  This  being  done,  we  use  the  resulting  rad  c  A  of  gyr/instead  of 
the  rad  c  n  of  Figs  1  and  2;  and  in  precisely  the  same  manner;  using  also 
the  vel  at  A,  instead  of  that  at  n  in  Figs  1  and  2.  See  "  center  of  gyration,"  p  617 

pe: 
rad 

of  the  circle  of  gyr  is  1.296  X  2  =  2.592  ft ;  and  its  circumf  A  m  s  y  is  2.592  X  3.1416  =  8.143  ft.    And 
since  the  stone  makes-  326  revs  per  miu,  the  vel  of  the  circle  of  gyr  is  8.143  X  326  =  2655  ft  per  min  ; 

or,    — —  =  44.25  ft  per  sec.    Hence  the 

Wt  of      v    Square  of  vel  at  h 
Ceil  trif  forces   the  body   A         in  ft  per  sec 


Ex.     Let  Fig  3  be  a  millstone,  3  ft  8  ins,  or  3.666  ft  diam  ;  its  wt  .5  of  a  ton  ;  and  making  326  revs 
T  min.     Its  circumf  will  then  be  11.52,  and  will  move  3755.5  ft  per  min  ;  or  62.592  ft  per  sec.    Its 

6  =  1.833  ft;  consequently,  its  rad  of  gyr  cA  is  1.833  X  .707  =  1.296  ft;  and  the  diam 


.5  (on  X  44.252 


rad  ch, 
i  X  1958 


i  ft  X  32.2; 
979 


Ceiitrif  force  = 


Wt  of      ~,    Sq  of  number    v 
the  body  X   of  revs  per  min  X 


.5  ton  X  3262  X  1 .296         .5  X  106276  X  1.296         68867 


Al*t«  3.  When,  in  a  homogeneous  revolving  ring,  Fig  4,  of  uniform 
thickness,  the  dimension  a  t  becomes  so  great,  that  it  is  not  sufficiently  ac- 
curate to  assume  the  point  h,  of  the  rad  of  gyr  ch,  to  be  at  the  cen  of  grav 
of  at,  as  was  done  in  Figs  1  and  2,  the  true  rad  c/t  may  be  found  thus  : 
Add  the  square  of  the  inner  rad  c  t,  to  the  square  of  the  outer  rad  c  a.  Div 
their  sum  by  2.  Take  the  su  rt  of  the  quot. 


Ex.    Let  c  t  be  3  ft ;  and  c  a,  7  ft.    Then, 


Bad  of  *yr,  ch- 


/S2  +  72    _  /9  +  49    _  / 

~V"~  V"1"       V' 


- 


~  5.38  ft. 


The  rad  of  gyr,  or  dist  from  the  cen  c 
be  found  thus : 


•olution,  to  the  circle  of  gyr,  of  a  few  other  bodies,  may 


solid  globe 

thin  circular  ring 

thin  hollow  globe 

solid  cone 

thin  straight  rod 


ar«und  its  centre.       Rad  X  -707 

its  diameter,  Rad  X  .5 
"      its  diameter,  Rad  X  .633 
"      its  diameter,  Rad  X  .707 
"       its  diameter,  Rad  X  .817* 
"      its  axis,  Rad  X  .548 1 

(nd,          Rad  X  .577 


"         "          "  revolving  around  any  point)  /  AS  _i_  „« 

between  its  two  ends.     Call  the  length  of  >  .       /  .       T  g  ., 
its  two  arms  A  and  a ^\/    3A  +  3°- 


It  is  plain  from  the  last  example,  that  any  body  may  have  any  number  of  radii  of  gyration ;  de- 
pending upon  the  position  of  the  cen  of  rev." 

Coal  used  by  engines  per  horse-power  per  hour.  Condensing  engines, 
!K  to  '^A  **;  non-condensing,  3  to  7  fcs ;  depending  on  the  quality  of  the  coal, 
perfection  of  engines,  &c. 

*  Perfectly  correct,  only  when  infinitely  thin.    When  the  thickness  is  appreciable,  first  measure 
the  rad  to  the  center  of  the  thickness  ;  then  mult  by  .817  for  a  practical  approximation. 
t  Here  rad  means  the  rad  of  the  circular  base  of  the  cone. 


496  MORTAR,   BRICKS,    ETC. 

MOETAR,  BRICKS,  &c. 


Art,  1.  Mortar.  The  proportion  of  1  measure  of  quicklime,  either  in  ir- 
regular lumps,  or  ground^*  and  5  measures  of  sand,  is  about  the  average  used  for 
common  mortar,  by  good  builders  in  our  principal  Atlantic  cities;  and  if  both 
materials  are  good,  and  well  mixed  (or  tempered)  with  clean  water,  the  mortar  is 
certainly  as  good  as  can  be  desired  for  such  ordinary  purposes  as  require  no  addi- 
tion of  hydraulic  cement.  The  bulk  of  the  mixed  mortar  will  usually  exceed  that 
of  the  dry  loose  sand  alone  about }/%  part. 

Quantity  required.  20  cub  ft,  or  16  struck  bushels  of  sand,  and  4  cub  ft,  or 

3. 2  struck  bushels  of  quicklime,  the  measures  slightly  shaken  iu  both  cases,  will  make  abt  2'2^  cub  ft  of 
mortar;  sufficient  to  lay  1000  bricks  oftho  ordinary  average  size  of  814  by  4  by  2  ins,  with  the  coarse 
mortar  joints  usual  in  interior  house-walls,  varying  say  from  %  to  %  inch.  With  such  joints,  1000 
such  bricks  will  make  2  cubic  yards  of  massive  work  ;  and  nearly  %  of  the  mass  will  be  mortar.  For 
outside  or  showing  joints,  where  a  whiter  and  neater-looking  mortar  is  required,  house-builders  in- 
crease the  proportion  of  lime  to  1  in  4,  or  1  in  3.  For  mortar  of  fine  screened  gravel,  for  cellar-walls 
of  stone  rubble,  or  coarse  brickwork,  1  measure  of  lime  to  6  or  8  of  gravel,  is  asual ;  and  the  mortar 
is  good.  In  average  rough  massive  rubble,  as  in  the  foregoing  brickwork,  about  %  of  the  mass  is 
mortar:  consequently  a  cubic  yard  will  require  about  as  much  as  500  such  bricks  ;  or  10  cubic  feet,  (8 
struck  bushels)  of  sand ;  and  2  cub  ft,  or  1.6  bushels  of  quicklime.  Superior,  well-scabbled  rubble, 
carefully  laid,  will  contain  but  about  TJ-  of  its  bulk  of  mortar ;  or  5}£  cub  ft  sand,  and  1.1  cub  ft  lime, 
per  cub  yard. 

For  public  engineering  works,  especially  in  massive  ones,  or  where  exposed  to  dampness,  an  addi- 
tion should  be  made  in  either  of  the  foregoing  mortars,  of  a  quantity  of  good  hyd 
cement,  equal  to  about  %  of  the  lime;  or  still  better,  %  of  the  lime  should  be 

omitted,  and  an  equal  measure  of  cement  be  substituted  for  it.  If  exposed  to  water  while 
quite  new,  use  little  or  no  lime  outside. 

With  bricks  of  8^  by  4  by  2  ins,  the  following  are  the  quantities  of  mor- 
tar ami  of  bricks  for  a  cubic  yard  of  massive  work. 


Thickness 
of  Joints. 

Proportion  of  Mortar 
in  the  whole  mass. 

No.  of  Bricks 
per  cub  yard. 

No.  of  Bricks 
per  cub  toot. 

574  

21.26 

1    "      • 

«        f 

522   

19.33 

:  ••  ?::. 

475   

,  17.60 

-  1  

..   433   .. 

..    16.04 

In  estimating  for  bricks  in  massive  work,  allow  2  or  3  per  ct  for  waste ; 

and  in  common  buildings,  5  per  ct.  or  more.  Much  of  the  waste  is  incurred  in  cutting  bricks  to  fit 
angles,  Ac.  In  Philadelphia  a  barrel  of  lump  lime  is  allowed  for  1000  bricks ;  or  for  2  perches  (25  cub 
ft  each)  of  rough  cellar-wall  rubble.  Somewhat  less  mortar  per  1000  is  contained  in  thin  walls,  than 
in  massive  engineering  structures  ;  because  the  former  have  proportionally  more  outside  face,  which 
does  not  require  to  be  covered  with  mortar;  but  thin  walls  involve  more  waste  while  building;  so  that 
both  require  about  the  same  quantity  of  materials  to  be  provided.  Careful  experiments  show  that 
mortar  becomes  harder,  and  more  adhesive  to  brick  or  stone,  if  the  proportion  of  lime  is  increased. 
Hence,  on  our  public  works  the  proportion  of  one  measure  of  quicklime  to  3  of  sand,  is  usually  spec- 
ified, but  probably  never  used. 

Lime  is  usually  sold  in  lump,  by  the  barrel,*  of  about  230  Ibs  net, 

or  250  fts  gross.    A  heaped  bushel  of  lump  lime  averages  about  75  fts.    Ground  quicklime, 

loose,  averages  about  70  fts  per  struck  bushel ;  and  3  bushels  loose  just  fill  a  common  Hour  barrel ;  but 
from  3.5  to  3.75  bushels,  or  245  to  280  fts  can  readily  be  compacted  into  a  barrel. 

General  remarks  on  mortar  and  lime.  On  too  great  a  pro- 
portion of  our  public  works,  the  common  lime  mortar  may  be  seen  to  be  rotten  and  useless,  where  it 
has  been  exposed  to  moisture  ;  which  will  be  carried  by  the  capillary  action  of  earth  to  several  feet 
above  the  natural  surface;  or  as  far  below  the  artificial  surface  of  embankments  deposited  behind 
abutments,  retaining-walls,  &c.  The  same  will  frequently  be  seen  in  the  soffits  of  arches  under  em- 
bankments. Common  lime  mortar,  thus  exposed  to  constant  moisture,  will  never  harden  properly. 
Even  when  very  old  and  hard,  it  absorbs  water  freely.  Cement  also  does  so,  but  hardens. 

Brickdust,  or  burnt  clay,  improves  common  mortar ;  and  makes  it  hydraulic. 
In  localities  where  sand  cannot  be  obtained,  burnt  clay,  ground,  may  be  substituted;  and  will  gen- 
erally give  a  better  mortar. 

Protection  of  quicklime  from  moisture,  even  that  of  the  air,  is 
absolutely  essential,  otherwise  it  undergoes  the  process  of  air-slacking,  or 

*  Price  of  quicklime  in  lump  in  Philadelphia,  about  $1.20  to  $1.30  per 

barrel.     By  the  bushel,  without  barrel,  about  35  cts.     Ground  lime,  about  25  cents  more  per  barrel. 

Sand,  sea,  per  cart  load  of  20  struck  bushels,  $2  to  $2.50.  Bar  sand  $1  to  $1.25 
(from  the  river  at  the  city).  In  188O. 


TAR,  BRICKS,  ETC.  497 

spontaneous  slacking Jby'which  it  becomes  reduced  to  powder  as  when  slacked  by  water  as  usual, 
but  without  heatiiyjK'and  with  but  little  swelling.  As  this  air  slacking  requires  from  a  few  months  to 
a  year  or  more.ji«pending  on  quality  and  exposure,  it  gives  the  lime  time  to  absorb  sufficient  carbonic 

acid  from  me  air  to  injure  or  destroy  its  efficacy.  But  quicklime  will  keep 
good  for  a  long-  time  if  first  ground,  and  then  well  packed  in  air-tight 
barrels.  The  grinding  also  breaks  down  refractory  particles  found  in  all  limes,  and  which  injure  the 
mortar  by  not  slacking  until  it  has  been  made  and  used.  For  the  same  reason  it  is  better  that  lime 
should  not  be  made  into  mortar  as  soon  as  it  is  slacked,  but  be  allowed  to  remain  slacked  for  a  day  or 
two  (or  even  several)  protected  from  rain,  sun,  and  dust. 

Lime  slacked  in  great  bulk  may  char  or  even  set  fire  to  wood. 
Lime  paste  and  mortar  will  keep  for  years,  and  improve,  if  well 

buried  in  the  earth.-    Also  for  months  if  merely  covered  In  heaps  under  shelter,  with  a  thick  layer  of 
sand.     The  paste  shrinks  and  cracks  in  drying  ;  but  the  sand  in  mortar  prevents  this. 
As  approximate  averages  varying  much  according  to  the  character  and 

degree  of  burning  of  the  limestone;  ana  to  the  fineness  or  coarseness  of  the  sand,  one  measure  of 
good  quicklime,  either  in  lump,  or  ground ;  if  wet  with  about  %  a  measure  of  water,  will  within  less 
than  an  hour,  slack  to  about  2  measures  of  dry  powder.  And  if  to  this  powder  there  be  added  about 
%  more  measures  of  water,  and  3  measures  of  dry  sand,  and  the  whole  thoroughly  mixed,  the  result 
will  be  about  3^  measures  of  mortar.  Or  the  same  slacked  dry  powder,  with  about  1  measure  of 
water,  and  5  measures  of  sand,  will  make  about  5%  measures  of  mortar.  In  both  cases  the  bulk  of 
the  mortar  will  be  about  %  part  greater  than  that  of  the  dry  sand  alone.  If  %  of  a  measure  of  water 
be  used  for  slackiug,  the  result,  instead  of  a  dry  powder,  will  be  about  ll/i  measures  of  stiff  paste ;  or 
with  1  whole  measure  of  water  for  slacking,  the  result  will  be  about  1%  measures  of  thin  paste,  of 
about  the  proper  consistence  for  mixing  with  the  sand.  Very  pure,  fat  limes,  slack  quickly,  and  make 
about  from  2  to  3  measures  of  powder ;  while  poor,  meagre  ones,  require  more  time,  and  swell  less. 
Slow  slacking,  and  small  swelling,  in  case  the  lime  has  been  properly  burnt,  are  not  in  general  bad 
properties  ;  but  on  the  contrary,  usually  indicate  that  it  is  to  some  extent  hydraulic.  In  this  case  it 
makes  a  better  mortar ;  especially  for  works  exposed  to  moisture,  or  to  the  weather.  Very  pure  limes 
are  the  worst  of  all  for  such  exposures;  or  are  bad  weather-limes  ;  and  in  important  works,  should 
never  be  used  without  cement. 

Shell  lime  appears  to  be  about  the  same  as  that  from  the  purest  limestones; 

but  that  from  chalk  is  still  more  inferior,  and  will  not  bear  more  than  about  1^  measures  of  sand; 
its  mortar  never  becomes  very  hard.  Madrepores  (commonly  called  coral)  appear  to  furnish  a  lime 
intermediate  between  those  of  chalk  and  limestone.  They  require  to  be  but  moderately  burnt. 

The  average  weight;  of  common  hardened  mortar  is  about  105  to  115  fts 

per  cub  ft. 

Grout  is  merely  common  mortar  made  so  thin  as  to  flow  almost  like  cream. 
It  is  intended  to  fill  interstices  left  in  the  mortar -joints  of  rough  masonry;  but  unless  it  contains  a 
large  amount  of  cement,  it  is  probably  entirely  worthless;  since  the  great  quantity  of  water  injures 
the  properties  of  lime;  and  moreover,  its  ingredients  separate  from  each  other;  the  sand  settling  be- 
low the  lime.  Besides  this,  it  will  never  harden  thoroughly  in  the  interior  of  thick  masses  of  ma- 
sonry ;  indeed,  the  same  may  probably  be  said  of  any  common  lime  mortar.  In  such  positions,  it  has 
been  found  to  be  perfectly  soft,  after  the  lapse  of  many  years. 

Both  the  sand  and  the  water  for  lime  mortar,  should  be  free  from  clay  and 

salt.  The  clay  may  be  removed  by  thorough  washing;  but  it  is  extremely  dif- 
ficult to  get  rid  of  the  salt  from  seashore  sand,  even  by  repeated  washings.  Enough  will  generally 
remain  to  keep  the  work  damp,  and  to  produce  efflorescences  of  nitre  on  the  surface;  whether  with 
lime,  or  with  cement  mortar.  Slacking  by  salt  water  gives  less  paste  than  fresh. 

Mortar  should  not  be  mixed  upon  the  surface  of  clayey  ground  ;  but  a  rough  board,  brick,  or  stone 
platform  should  be  interposed.  Pit  sand  sifted  from  decomposed  gneiss,  and  other  allied  rocks,  is  ex- 
cellent for  mortar;  its  sharp  angles  making  with  the  lime  a  more  coherent  mass  than  the  rounded 
grains  r.f  river  or  sea  sand.  Mortar  should  be  applied  wetter  in  hot  than  in  cold  weather;  especially 
in  brickwork  ;  otherwise  the  water  is  too  much  absorbed  by  the  masonry,  and  the  mortar  is  thereby 
injured. 

The  tenacity,  or  cohesive  strength,  that  is,  the  resistance  to  a  pull 

of  good  common  lime  mortar  of  the  usual  proportions  of  lime  and  sand,  and  6  months  old,  is  about 
?rom  15  to  30  fts  per  sq  inch  ;  or  .96  to  1.9  tons  per  sq  ft.  With  less  sand,  or  with  greater  age,  it  will 
be  stronger. 

The  crushing  strength  of  good  common  mortar  6  months  old  is  from  150 

to  300  ft>s  per  sq  inch,  or  9.7  to  19.3  tons  per  sq  foot. 

The  sliding  resistance,  or  that  which  common  mortar  opposes  to  any 

force  tending  to  make  one  course  of  masonry  slide  upon  another,  is  stated  by  Roudelet,  to  be  but  5  ft>s 
per  sq  inch  ;  or  about  Jtf  ton  per  sq  ft,  in  mortar  6  months  old. 

Transverse  strength  of  good  common  mortar  6  months  old.    A  bar  1 

inch  square  and  12  ins  clear  span,  breaks  with  a  center  load  of  4  to  8  Ebs. 

The  lime  in  mortar  decays  wood  rapidly,  especially  in  close, 

damp  situations.  Still  the  soaking  of  timber  for  a  week  or  two  in  a  solution  of  quicklime  in  water 
appears  to  act  as  a  preservative.  Iron*  so  completely  embedded  in  mortar  as  to  exclude  air  and 
moisture,  has  been  found  perfect  after  1400  years ;  but  if  the  mortar  admits  moisture  the  iron  decays. 
So,  probably,  with  other  metals. 

The  adhesion  to  common  bricks,  or  to  rough  nibble  at  any 

age  will  average  about  %  of  the  cohesive  strength  at  the  same  age;  or  say  12  to  24  fbsper  sq  inch,  or 
.75  to  1.5  ton  per  sq  ft  at  6  months  old.  If  care  be  taken  to  exclude  dust  entirely,  by  dipping  each 
brick  into  water  before  laying  it,  or  by  sprinkling  the  stone  by  a  hose,  &c,  the  adhesion  will  be  in- 
creased. On  the  other  hand,  much  dust  may  almost  prevent  any  adhesion  at  all.  The  precaution  of 
wetting  is  especially  necessary  in  very  hot  weather,  to  prevent  the  warm  bricks  or  stone  from  kill- 
ing the  mortar  by  the  rapid  absorption  and  evaporation  of  its  water.  The  adhesion  to  very 
smooth  hard  pressed  bricks,  or  to  smoothly  dressed  or  sawed  stone  is  considerably  less. 


498 


MORTAR,    BRICKS,    ETC. 


Art.  2.    Bricks,  size,  wt.  Ac.*    A  common  size  in  our  eastern  cities  is 

about  8.25  X  4  X  2  ins ;  which  is  equal  to  66  cub  ins  ;  or  26.2  bricks  to  a  cub  ft ;  or  707  bricks  to  a  cub 
yard.  For  the  number  required  with  mortar,  see  table,  p  496. 

In  ordering  a  large  number  a  minimum  limit  of  dimension  should  be  specified  in  order  to  prevent 
fraud.  A  brick  X  inch  le-s  each  way  than  the  above,  contains  but-52.5  cub  ins  ;  thus  requiring  full 
25  per  cent  more  bricks  to  do  the  same  work  ;  in  addition  to  25  per  ct  more  cost  for  laying,  which  is 
generally  paid  for  by  the  1000. 

The  weig-ht  of  a  grood  common  brick  of  8.25  X  4  X  2  ins,  will  aver- 
age about  4.5  tts ;  or  118  tts  per  cub  ft  —  3186  Ibs  or  1.42  tons  per  cub  yard  ;  or  2.01  tong  per  10OO. 

A  j;<MMl  pressed  brick  of  the  same  size  will  average  about  5  Ibs,  =  131  Ibs 
per  cub  ft  =  3537  Ibs  or  1.58  tons  per  cub  yd  ;  or  2.23  tons  per  1OOO. 

Immersed  in  water,  either  of  them  will  in  a  few  minutes  absorb  from 
*4  to  %  ft  of  water ;  the  last  being  about  ^  of  the  weight  of  a  hand  moulded  one ;  or  ^  of  its  own 
bulk.  Since  the  weight  of  hardened  mortar  averages  but  little  less  than  that  of  good  common  brick, 
we  may  for  ordinary  calculations  assume  the  weight  of  such  brickwork  at  1.4  tons  per  cub  yard :  1.3 
tons  per  perch  of  25 cub  ft;  or  116  tts  per  cub  ft;  or  for  machine-moulded,  at  1.56  tons  per  cub  yd; 
1.44  tons  per  perch ;  or  129  Ibs  per  cub  ft. 

Allowing  for  the  usual  waste  in  cutting  bricks  to  fit  corners,  jambs,  &c,  the  average  number  of 
8>4  X  4  X  2,  required  per  sq  foot  of  wall  is  as  follows  : 

Thickness  of  Wall.  No.  of  Bricks. 

S^ius,  or  1  brick  14 

12?i  "      orl»^"     21 

17      "      or  2      "     28 

21^"      or2^"     35 

25%"      or3      ••     42 

Laying  per  day*  a  bricklayer,  with  a  laborer  to  keep  him  supplied  with  materials,  will  in 
common  house  walls,  lay  on  an  average  about  1500  bricks  per  day  of  10  working  hours,  lu  tiie  neater 
outer  faces  of  back  buildings,  from  1000  to  1200  ;  in  good  ordinary  street  fronts,  800  to  1000 ;  or  of  the 
very  finest  lower  story  faces  used  in  street  fronts,  from  150  to  300,  depending  on  the  number  of  angles, 
&c.  In  plain  massive  engineering  work,  he  should  average  about  2000  per  day,  or  4  cub  yds;  und 
in  large  arches,  about  1500,  or  3  cub  yds.t 

Since  bricks  shrink  about  yj  part  of  each  dimension  in  drying  and  burning,  the  moulds  should  be 
about  y1,-  Part  Iar8er  every  way  than  the  burnt  brick  is  intended  to  be. 

Good  well-burnt  bricks  will  ring  when  two  are  struck  together. 

At  the  brick-yards  about  Philadelphia,  a  brick-moulder's  work  is  2333  bricks  per  day ;  or  14000  per 
week.  He  is  assisted  by  two  boys,  one  of  whom  supplies  the  prepared  clay,  moulding  sand,  and 
•water ;  while  the  other  carries  away  the  bricks  as  they  are  moulded.  A  fourth  person  arranges  them 
in  rows  for  drying.  About  %  of  a  cord,  or  96  cub  ft  of  wood,  is  allowed  per  1000  for  burning. 

Paving?  Witll  brick.  In  our  cities  this  is  done  over  a  6-inch  layer  of  gravel,  which 
should  be  free  from  clay,  and  well  consolidated.  With  bricks  of  8)4X4X2  ins,  with  joints  of  from  X 
to  14  inch  wide,  a  sq  yard  requires,  flatwise,  as  is  usual  in  streets,  38  bricks;  edgewise,  73;  endwise, 

'    '  iks  and  gravel,  will  in  10  hours  pave  about 
Lt-ii  done,  suiid  is  brushed  into  the  joints. 

Art.  3.  The  Crushing1  Strength  Of  bricks  of  course  varies  greatly.  A 
rather  soft  one  will  crush  under  from  450  to  600  Ibs  per  sq  inch  ;  or  about  30  to  40  tons  per  sq  ft ;  while 
a  first-rate  machine-pressed  one  will  require  about  200  to  400  tons  per  square  foot.  This  last  is 
about  the  crushing  limit  of  the  best  sandstone;  %  as  much  as  the  best  marbles  or  limestones  ;  and  % 
as  much  as  the  best  granites,  or  roofing  slates.  But  masses  of  brickwork  crush  under  much  smaller 
loads  than  single  bricks.  In  some  English  experiments,  small  cubical  masses  only  9  inches  on  each 
edge,  laid  in  cement,  crushed  under  27  to  40  tons  per  sq  ft.  Others,  with  piers  9  ins  square,  and  2  ft 
3  ins  high,  in  cement,  only  two  days  after  being  built,  required  44  to  62  tons  per  sq  ft  to  crush  them. 
Another,  of  pressed  brick,  in  best  Portland  cement,  is  said  to  have  withstood  202  tons  per  sq  ft;  and 
with  common  lime  mortar  only  y±  as  much. 

It  must,  however,  be  remembered,  that  cracking  and  splitting  usually  commence  under  about  one- 
half  the  crushing  loads.  To  be  safe,  the  load  should  not  exceed  %  or  fa  of  the  crushing  one ;  and 
•o  with  stone.  Moreover,  these  experiments  were  made  upon  low  masses ;  but  the  strength  decreases 
with  the* proportion  of  the  height  to  the  thickness. 

The  pressure  at  the  base  of  a  brick  shot-tower  in  Baltimore,  246  feet  high,  is  estimated  at  6>£  tons 

The  Peerless  Brick  CO,  office  208  South  Seventh  St,  Philada.  make  superb  smooth 
(not  shining)  bricks  of  various  shapes  and  colors  (as  white,  black,  gray,  butt,  brown,  red,  &c)  (or 
ornamental  architectural  purposes.  Their  standard  size  is  8%  X  4^  X  2%  =  82  cub  ins,  or  H  larger 
than  the  above  8%  X  4  X  2  ins.  For  either  plain  rectangular  or  voussoir  shapes  their  price  in  1880 
is  $4  per  hundred  if  red  ;  and  $5.50  if  of  other  colors.  For  simple  curved  mouldings  $4.50  to  $5.50  if 
red  ;  and  $6  to  $7  of  other  colors.  The  color  extends  throughout  the  body  of  the  brick.  With  a  few 
of  these  judiciously  distributed  among  common  bricks,  beautiful  architectural  eflects  may  be  pro 
duced,  both  indoors  and  out.  at  far  less  cost  than  in  stone. 

The  same  Co  will,  if  a  sufficient  order  is  given,  furnish  voussoir  bricks  for  specified  radii,  but  of  the 
quality  and  finish  of  ordinary  good  hard  brick,  at  from  25  to  50  per  ct  advance  on  prevailing  market 
rates  of  common  plain  ones. 

*  Prices  in  Philada  in  1873.  Bricks  alone;  Salmon,  or  soft,  $8  per  1000.  Hard  brick,  $10. 
Back  stretchers  (generally  used  for  the  facings  of  back  buildings,  &c,)  $16.  Paving  brick,  $15. 
Pressed,  (for  lower  stories'of  first-class  fronts,)  $30. 

t  Bricklaying ;  including  mortar  and  scaffolding :  averaging  an  entire  dwelling.  $8  per  1000.  Best 
pressed  bricks  in  first-class  fronts,  $15  to  $20.  In  188O  all  average  about  2O  per  ct  le«a. 


149.    An  average  workman,  with  a  laborer  to  supplv  the  bricks  and  gravel,  will  in  10  hours  pave  about 
2000  bricks ;  or  53  sq  yds  flat,  27  edgewise,  13  endwise.    When  ' 


,    ETC.  499 

per  sq  ft;  and  In  a  briok  chimne^r'at  Glasgow,  Scotland,  468  feet  high,  at  9  tons.  Professor  Rankine 
calculates  that  in  heavy  gale^Chis  is  increased  to  15  tons,  on  the  leeward  side.  The  walls  of  both  are 
of  course  much  thicker  ^MJottom  than  at  top.  With  walls  100  feet  high,  of  uniform  thickness,  the 
pressure  at  base  wouhi^Se  5.4  tons  per  sq  ft. 

With  our  present^mperfect  knowledge  ou  this  subject,  it  cannot  be  considered  safe  to  expose  even 
first-class  pressed  brickwork,  in  cement,  to  more  than  12  or  15  tons  per  sq  ft ;  or  good  hand-moulded, 
to  more  than-two-thirds  as  much. 

Tensile  strength  01  brick,  40  to  400  fbs  per  sq  inch ;  or  2.6  to  26  tons  per  sq  ft. 
The  English  ro«l  of  brickwork  is  306  cub  feet,  or  l\%  cub  yards;  and 

requires  about  4500  bricks  of  the  English  standard  size ;  with  about  75  cub  ft  of  mortar.  The  English 
hundred  of  lime,  is  a  cub  yd. 

Frozen  mortsir»  There  is  risk  in  using  common  mortar  in  cold  weather.  If  the  cold 
should  continue  lorig  euough  to  allow  the  frozen  mortar  to  set  well,  the  work  may  remain  safe  ,  but  if 
a  warm  day  should  occur  between  the  freezing  and  the  setting  of  the  mortar,  the  sun  shining  on  owe 
side  of  the  wall  may  melt  the  mortar  on  that  side,  while  that  on  the  other  side  may  remain  frozen 
hard.  In  that  case,  the  wall  will  be  apt  to  fall ;  or  if  it  does  not,  it  will  at  least  always  be  weak ;  for 
mortar  that  has  partially  set  while  frozen,  if  then  melted,  will  never  regain  its  strength.  By  the 
writer's  own  trials  hydraulic  cements  seemed  not  to  be  injured  by  freezing. 

Experiments  for  rendering  brick  masonry  impervious  to 

•Water.  Abstract  of  a  paper  read  before  the  American  Society  of  Civil  Engineers,  May  4,  1870, 
by  William  L.  Dearborn,  Civil  Engineer,  member  of  the  Society. 

The  face  walls  of  the  Back  Bays  of  the  Gate- houses  of  the  new  Croton  Eeservoir,  located  north 
of  Eighty-sixth  Street,  in  Central  Park,  were  built  of  the  best  quality  of  hard-burnt  brick;  laid  in 
mortar  composed  of  hydraulic  cement  of  New  York,  and  sand  mixed  in  the  proportion  of  one  measure 
of  cement  to  two  of  sand.  The  space  between  the  walls  is  4  ft ;  and  was  filled  with  concrete.  The  face 
•walls  were  laid  up  with  great  care,  and  every  precaution  was  taken  to  have  the  joints  well  filled  and 
/nsure  good  work.  They  are  12  ins  thick,  and  40  ft  high ;  and  the  Bays  when  full  generally  have  36  ft 
of  water  in  them. 

When  the  reservoir  was  first  filled,  and  the  water  was  let  into  the  Gate-houses,  it  was  found  to  filter 
through  these  walls  to  a  considerable  amount.  As  soon  as  this  was  discovered,  the  water  was  drawn 
•wt  of  the  Bays,  with  the  intention  of  attempting  to  remedy  or  prevent  this  infiltration.  After  care- 
fully considering  several  modes  of  accomplishing  the  object  desired,  I  came  to  the  conclusion  to  try 
"  Svlvester's  Process  for  Repelling  Moisture  from  External  Walls." 

The  process  consists  in  using  two  washes  or  solutions  for  covering  the  surface  of  brick  walls  :  one 
composed  of  Castile  soap  and  water  ;  and  one  of  alum  and  water.  The  proportions  are  :  three-quar- 
"«rs  of  a  pound  of  soap  to  one  gallon  of  water;  and  half  a  pound  of  alum  to  four  gallons  of  water; 
both  substances  to  be  perfectly  dissolved  in  the  water  before  being  used. 

The  walls  should  be  perfectly  clean  and  dry  ;  and  the  temperature  of  the  air  should  not  be  below  50 
degrees  Fahrenheit,  when  the'compositions  are  applied. 

The  first,  or  soap  wash,  should  be  laid  on  when  at  boiling  heat,  with  a  flat  brush,  taking  care  not 
to  form  a  froth  on  the  brickwork.  This  wash  should  remain  twenty-four  hours  ;  so  as  to  become  dry 
and  hard  before  the  second  or  alum  wash  is  applied ;  which  should  be  done  in  the  same  manner  as 
She  first.  The  temperature  of  this  wash  when  applied  may  be  60°  or  70° ;  and  it  should  also  remain 
twenty-four  hours  before  a  second  coat  of  the  soap  wash  is  put  on  ;  and  these  coats  are  to  be  repeated 
alternately  until  the  walls  are  made  impervious  to  water. 

The  alum  and  soap  thus  combined  form  an  Insoluble  compound,  filling  the  pores  of  the  masonry, 
and  entirely  preventing  the  water  from  penetrating  the  walls. 

Before  applying  these  compositions  to  the  walls  of  the  Bays,  some  experiments  were  made  to  test 
Che  absorption  of  water  by  bricks  under  pressure  after  being  covered  with  these  washes,  in  order  to 
determine  how  many  coats  the  wall  would  require  to  render  them  impervious  to  water. 

To  do  this,  a  strong  wooden  box  was  made,  put  together  with  screws,  large  enough  to  hold  2  bricks ; 
and  on  the  top  was  inserted  an  inch  pine  forty  feet  long. 

In  this  box  were  placed  two  bricks  after  being  made  perfectly  dry.  and  then  covered  with  a  coat  of 
each  of  the  washes,  as  before  directed,  and  weighed.  They  were  then  subjected  to  the  pressure  of  a 
column  of  water  40  ft  high  ;  and.  after  remaining  a  sufficient  length  of  time,  they  were  taken  out  and 
weighed  again,  to  ascertain  the  amount  of  water  they  had  absorbed. 

The  bricks  were  then  dried,  and  again  coated  with  the  washes  and  weighed,  and  subjected  to  press- 
ure as  before;  and  this  operation  was  repeated  until  the  bricks  were  found  not  to  absorb  any  water. 
Four  coatings  rendered  the  bricks  impenetrable  under  the  pressure  of  40  ft  head. 

The  mean  weight  of  the  bricks  (dry)  before  being  coated,  was  &%  Ibs;  the  mean  absorption  wai 
one-half  pound  of  water.  An  hydrometer  was  used  in  testing  the  solutions, 

As  this  experiment  was  made  in  the  fall  and  winter,  (1863,)  after  the  temporary  roofs  were  put  on 
to  the  Gate  house,  artificial  heat  had  to  be  resorted  to,  to  dry  the  walls  and  keep  the  air  at  a  proper 
temperature.  The  cost  was  10.06  cts  per  sq  ft.  As  soon  as  the  last  coat  had  become  hard,  the  water 
fras  let  into  the  Bays,  and  the  walls  were  found  to  be  perfectly  impervious  to  water ;  and  they  still  re- 
main  so  in  1870,  after  about  6^  years. 

BRICK  ARCH  (FOOTWAY  OP  HIGH  BRinoK).  The  brick  arch  of  the  footway  of  High  Bridge  is  the 
arc  of  a  circle  29  ft  6  in  radius  ;  and  is  12  in  thick  ;  the  width  on  top  is  17  ft;  and  the  length  covered 
was  1381  ft. 

The  first  two  courses  of  the  brick  of  the  arch  are  composed  of  the  best  hard-burnt  brick,  laid  edge- 
wise in  mortar  composed  of  one  part,  by  measure,  of  hydraulic  cement  of  New  York,  and  two  parts 
of  sand.  The  top  of  these  bricks,  and  the  inside  of  the  granite  coping  against  which  the  two  top 
courses  of  brick  rest,  was,  when  they  were  perfectly  dry,  covered  with  a  coat  of  asphalt  one-half  an 
inch  thick,  laid  on  when  the  asphalt  was  heated  to  a  temperature  of  from  360°  to  518°  Fahrenheit. 

On  top  of  this  was  laid  a  course  of  brick  flatwise,  dipped  in  asphalt,  and  laid  when  the  asphalt  was 
hot ;  and  the  joints  were  run  full  of  hot  asphalt. 

On  top  of  this  a  course  of  pressed  brick  was  laid  flatwise  in  hydraulic  cement  mortar,  forming  the 
paving  and  floor  of  the  bridge.  This  asphalt  was  the  Trinidad  variety  ;  and  was  mixed  with  10  per 
cent,  by  measure,  of  coal  tar;  and  25  per  cent  of  sand.  A  few  experiments  for  testing  the  strength 
of  this  asphalt,  when  used  to  cement  bricks  together,  were  made,  and  two  of  them  are  given  below. 

Six  bricks,  pressed  together  flatwise,  with  asphalt  joints,  were,  after  lying  six  mouths,  broken. 
The  distance  between  the  supports  was  12  ins  ;  breaking  weight,  900  tts  ;  area  of  single  joint,  28J$  sq 
ins.  The  asphalt  adhered  so  strongly  to  the  brick  as  to  tear  away  the  surface  in  many  places. 


500  CEMENT,   CONCRETE,    ETC. 

Two  bricks  pressed  together  end  to  end,  cemented  with  asphalt,  were,  after  lying  6  months,  broken. 

The  distance  between  the  supports  was  10  ins;  area  of  joint,  8>£  sq  ins;  breaking  weight,  150  tt>s. 

The  area  of  the  bridge  covered  with  asphalted  brick,  was  23065  sq  ft.  There  was  used  94200  Bs  of 
asphalt,  33  barrels  of  coal  tar,  10  cub  yds  of  sand,  93HOO  bricks. 

The  time  occupied  was  109  days  of  masons,  aud  148  days  of  laborers.  Two  masons  and  two  labor- 
ers will  melt  and  spread,  of  the' first  coat,  1650  sq  ft  per  day.  The  total  cost  of  this  coat  was  5.25 
cents  per  sq  ft,  exclusive  of  duty  on  asphalt.  There  were  three  grooves,  2  ins  wide  by  4  ins  deep, 
made  entirely  across  the  brick  arch,  and  immediately  under  the  first  coat  of  asphalt,  dividing  the 
arch  into  four  equal  parts.  These  grooves  were  filled  with  elastic  paint  cement. 

This  arrangement  was  intended  to  guard  against  the  evil  eflects  of  the  contraction  of  the  arch  in 
winter;  as  it  was  expected  to  yield  slightly  at  these  points,  and  at  no  other  point;  and  then  the 
elastic  cement  would  prevent  any  leakage  there. 

The  entire  experiment  has  proved  a  very  successful  one,  and  the  arch  has  remained  perfectly  tight. 

In  proposing  the  above  plan  for  working  the  asphalt  with  the  brickwork,  the  object  was  to  avoid 
depending  on  a  large  aontiuued  surface  of  asphalt,  as  is  usual  in  covering  arches,  which  very  fre- 
quently cracks  from  the  greater  contraction  of  the  asphalt  than  that  of  the  masonry  with  which  it  is 
in  contact ;  the  extent  of  the  asphalt  on  this  work  being  only  about  one-quarter  of  an  inch  to  each 
brick.  This  is  deemed  to  be  an  essential  element  in  the  success  of  the  impervious  covering." 

A  cheap  and  effective  process  for  preventing  the  percolation  of  water  through  the  arches  of  aque- 
ducts, and  even  of  bridges,  is  a  great  desideratum.  Many  expensive  trials  with  resinous  compounds 
have  proved  failures.  Hydraulic  cement  appears  to  merely  diminish  the  evil.  Much  of  the  trouble 
is  probably  due  to  cracks  produced  by  changes  of  temperature. 

Art.  4.  Hydraulic  Cements.*  Certain  limestones,  when  burnt,  will  not  slack 
with  water;  but  when  the  burnt  stone  is  finelv  ground,  and  made  into  a  paste,  it  possesses  the  pro- 
perty of  hardening  under  water;  and  is  therefore  called  hydraulic  cement.  So  long  as  the  propor- 
tion of  those  ingredients  which  impart  hydraulicity,  is  so  small  that  the  burnt  stone  will  slack  ;  bat 
still  make  a  paste  or  a  mortar,  which  will  harden  under  water,  it  is  called  hydraulic  lime.  This  does 
not  harden  so  promptly,  or  to  so  great  a  degree,  as  the  cements.  Hydraulic  limes  slack  more  slowly, 
and  swell  less,  in  proportion  to  their  hydraulicity ;  some  requiring  many  hours.  Artificial  hydraulic 
limes  and  cements,  of  excellent  quality,  may  be  made  by  mixing  lime  and  clay  thoroughly  together; 
then  moulding  the  mixture  into  blocks  like  bricks;  which  are  first  dried,  then  burnt,  aud  finely 
ground.  The  celebrated  artificial  English  Portland  cement,  is  made  by  grinding  together  in  water 
chalk  and  clay.  The  fine  particles  are  floated  away  to  other  vessels,  and  allowed  to  settle  as  a  paste ; 

which  is  then  collected,  moulded,  dried,  burnt,  and  ground.  Natural  Port- 
land is  that  made  from  limestone,  or  other  material  of  very  rare  occurrence, 

•which  combines  naturally  that  proportion  of  lime  and  clay  which  gives  the  above  artificial  Portlands 
their  pre-eminence.  This  alone  constitutes  its  difference  from  our  common  natural  hyd  cements. 

The  weig-ht  of  the  Rosendale,  and  other  American  cements, 

together  with  some  foreign  ones,  will  be  found  on  page  385.  Saylor's  Port* 
land  weighs  about  120  ft>s  per  struck  bushel. 

The  writer  found  by  10 years'  trial  that  if,  after  setting,  dampness  is  absolutely  excluded,  Cements 
preserve  iron,  lead,  zinc,  copper,  and  brass ;  and  that  Plaster  of  Paris  preserves  all  except 
iron,  which  it  rusts  somewhat  unless  galvanized.  Lime-mortar  probably  preserves  all  of  them, 
if  kept  free  from  damp. 

Protection  from  moisture,  even  that  of  the  air,  is  very  essential  for  the 

preservation  of  cements,  as  well  as  of  quicklime.  On  this  account  the  barrels  are  generally  lined 
with  stout  paper.  With  this  precaution,  aided  by  keeping  the  barrels  stored  in  a  dry  place,  raised 
above  the  ground,  the  cement,  although  it  may  require  more  time  to  set,  will  not  otherwise  very 
appreciably  deteriorate  for  six  months;  but  after  H  or  16  months,  Gillmore  says  it  is  unfit  for  use  m 
important  works.  But  in  lumps,  kept  dry,  it  will  remain  good  for  2 or  3 years;  and  may  be  ground 
as  required  for  use. 

Good  Portland  cement  is  stated  by  good  authority  rather  to  improve  by  free  exposure  to  the  air 
under  cover ;  but  whether  this  is  correct  or  not,  we  cannot  say. 

Restoration  by  rebarning-  may  be  effected.  If  the  injured  ground  ce- 
ment is  spread  in  a  thin  layer,  on  a  red-hot  iron  plate,  for  about  15  minutes,  its  good  qualities  will  be 
in  a  great  measure  restored.  The  time  should  be  ascertained  by  trial.  If  it  has  been  actually  wet, 
and  lumpy,  or  cemented  into  a  mass,  it  should  first  be  broken  into  small  pieces,  and  then  ground.  Or 
these  pieces  may  be  first  kiln-burnt  at  a  bright  red-heat  for  about  1>$  hours ;  and  then  ground. 

Art.  5.  For  roughcasting1,  or  stuccoing  the  outside  of  walls,  very  few 
hydraulic  cements  are  fit.  Mr.  Downing,  in  his  work  on  "  Country  Houses,"  excepts  that  from  Berlin, 
Connecticut,  as  the  only  one  within  his  extended  knowledge,  that  is  suitable.  Portland  cement  ia 
said  to  be  good  for  that  purpose.  A  wall  with  a  northern  exposure  in  Philada  was  coated  with  it  in 
i860 ;  and  appears  to  be  in  perfect  condition  in  1880. 

Quantity  required.  A  barrel  of  cement,  300  ft>s ;  and  2  barrels  of  sand,  (6 
bushels,  or  1%  cub  ft;)  mixed  with  about  %  a  barrel  of  water,  will  make  about  8  cub  ft  of  mortar, 

192  sq  ft  of  mortar-joint  X  inch  thick  =  21  #  sq  yarda. 

288''       «       ••          "      K     "         "      =  32       "       " 

384  ' "      X     "         "      -*2«  "       " 

768  "       "       »         "      H    "        "     =85*  •«      " 

*  Prices  of  hyd  cements  in  Philada,  1880,  by  the  large  importing  firm 

of  French,  Richards  &  Co,  corner  of  York  Avenue  and  Callowhill  St.  English  Portland,  $3.25  to 
$4  per  barrel  of  about  400  fts  gross;  according  to  quality  and  quantity.  Saylor's  Portland,  per 
barrel  of  400  fts  gross,  $3  to  $3.50.  Rosendale,  per  barrel  of  about  300  fts  net,  Si. '25  to  §1.50. 
Other  C.  8  cements,  per  barrel  of  800  fts  net,  $1.15  to  $1.50.  Ground  calcined  plaster  of 
Paris,  selected,  barrel  of  300  fts  net,  $2  to  $2.25.  Commercial,  barrel  of  about  250  tts  net,  $1.50 
to  $1.60.  In  1882  English  and  Saylor's  Portland  about  10  per  ct  less. 


CEJJHEOT,   CONCRETE,   ETC.  501 

Or,  to  lay  1  cubic  yard^tff-522  bricks  of  8^  by  4,  by  2  ins,  with  joints  %  inch  thick;  or  a  cubic  yard 
of  roughly  scrabbletf-fubble  stonework.  The  quantity  of  sand  may  be  increased,  however,  to  3  or  4 
measures  for  ordinary  work. 

JPointiiig  mortar.  Gen  Gillmore  recommends  "  1  part  by  weight  of  good 
cement  powder,  to  3  or  3%  parts  of  sand.  To  be  mixed  under  shelter,  and  in  quantities  of  only  2  or 
3  pints  at  a  time,  using  very  little  water,  so  that  the  mortar,  when  ready  for  use,  shall  appear  rather 
incoherent,  and  quite  deficient  in  plasticity.  The  joints  being  previously  scraped  out  to  a  depth  of 
at  least  %  an  inch,  the  mortar  is  put  in  by  the  trowel  ;  a  straight-edge  being  held  just  below  the  joint, 
if  straight,  as  an  auxiliary.  The  mortar  is  then  to  be  well  calked  into  the  joint  by  a  calking- 
iron  and  hammer;  then  more  mortar  is  put  in,  and  calked,  until  the  joint  is  full.  It  is  then  rubbed 
and  polished  under  as  great  pressure  as  the  mason  can  exert.  If  the  joints  are  very  One  they  should 
be  enlarged  by  a  stonecutter,  to  about  -A-  inch,  to  receive  the  pointing.  The  wall  should  be  well  wet 
before  the  pointing  is  put  in,  and  kept  in  such  condition  as  neither  to  give  water  to,  nor  take  it  from  the 
mortar.  In  hot  weather,  the  pointing  should  be  kept  sheltered  for  some  days  from  the  sun,  so  as  not 
to  dry  too  quickly."  Why  not  finish  joints  at  once,  without  subsequent  pointing  ?  Author. 

Art.  6.  Color  is  no  indication  of  strength  in  hyd  cements.  The  finer 
they  are  ground  the  better.  At  least  90  per  ct  should  pass  through  a 

sieve  of  50  meshes  per  lineal  inch,  of  Wire  No  35  Amer  wire  gauge  (.0056  inch  thick)  ;  or  2500  meshes 
per  sq  inch.  Weight  is  a  good  indication  when  equally  well  ground.  A  flat 
cake  of  good  cement  paste  placed  in  water  as  soon  as  it  admits  of  so  doing  safely,  and  left  in  it  for  a 

week,  should  show  no  cracks.  New  cement  is  not  as  good  as  when  a  few 
weeks  old.  The  term  Setting  does  not  imply  that  the  cement  has  hardened 

to  any  great  extent,  but  merely  that  it  has  ceased  to  be  pasty  and  has  become  brittle.  Quick  setting 
cements  may  do  this  sufficiently  to  allow  small  experimental  samples  to  be  lifted  and  handled  care- 
fully within  flve  to  thirty  minutes  ;  while  others  may  require  from  one  to  eight  or  more  hours. 

Slow  setting  does  not  indicate  inferiority,  for  many  of  the  very 

best  are  the  slowest  setting.  A  layer  of  very  quick  setting  cement  may  partially  set,  especially  in 
warm  weather,  before  the  masonry  is  properly  lowered  and  adjusted  upon  it,  and  any  disturbance 
after  setting  has  commenced  is  prejudicial.  Such  are  to  be  regarded  with  suspicion,  and  sub- 
mitted to  longer  tests  than  slow  ones.  Still,  quick  setting  ones  are  best  in  certain  cases,  as  when 
exposed  to  running  water,  &c.  They  may  be  rendered  slower  by  adding  a  bulk  of  lime  paste  equal  to  5 
or  15  per  ct  of  the  cement  paste,  without  weakening  them  seriously.  As  a  general  rule  cements 

set  and  harden  better  in  water  than  in  air,  especially  in  warm 

weather.  If,  however,  the  temp  for  the  first  few  days  does  not  exceed  55°  to  65°  Fah,  there  seems  to 
be  no  appreciable  difference  in  this  respect;  but  in  warm  air  cement  dries  instead  of  setting,  and  thus 
loses  most  of  its  strength.  In  hot  weather  every  precaution  should  be  used  against  this. 

The  time  reqd  to  attain  the  greatest  hardness  is  many  years,  but 

after  about  a  year  the  increase  is  usually  very  small  and  slow,  especially  with  neat  cement.  More- 
over, auy  subsequent  increase  is  a  matter  of  little  importance,  because  generally  by  that  time,  and 
often  much  sooner,  the  work  is  completed  and  exposed  to  its  maximum  strains.  §and  retards 
setting,  and  weakens  the  cement  paste.  But  although  with  sand  the  strength  of  the  mortar  may 
never  attain  to  that  of  the  neat  paste,  yet  it  increases  with  age  in  a  greater  proportion  ;  so  that  a 
neat  paste  which  at  the  end  of  a  year  would  be  but  twice  as  strong  as  in  7  days,  may  with  sand  yield 
a  mortar  which  at  the  end  of  a  year  will  be  8,  4,  or  5  times  as  strong  as  it  was  in  7  days.  Good  Port- 
lands neat  usually  have  at  the  end  of  a  year  from  1.5  to  2  times  their  strength  at  the  end  of  7  days  ; 
and  the  American  natural  cements,  Rosendale,  Louisville,  Cumberland,  &c,  from  2.5  to  3.5  times; 
but  inasmuch  as  Portlands  average  (roughly  speaking)  about  5  or  6  times  the  strength  of  the  others 
in  7  days,  they  still  average  about  2.5  to  3  times  as  strong  in  a  year  or  longer.  Cements  of  the  same 

class  differ  much  in  their  rapidity  of  hardening.    One  may  at  the  end  of 

a  month  gain  nearly  one-half,  and  another  not  more  than  one-sixth  of  its  increase  at  the  end  of  a 
ear,  at  which  time  both  may  have  about  the  same  strength.  Hence,  tests  for  1  week  or  1  month  are 
y  no  means  conclusive  as  to  their  final  comparative  merits. 
There  seems  to  be  a  period  occurring  from  a  few  weeks  to  several  months  after  having  been  laid,  at 
which  cement  and  its  mortars  for  a  short  time  not  only  cease  from  hardening,  but  actually  lose 
Strength.  They  then  recover,  and  the  hardening  goes  on  as  before.  It  has  been  suggested  that 
this  opinion  has  originated  in  some  oversight  of  the  experimenters,  but  the  writer  believes  it  to  be 
founded  on  fact.  In  his  expts  with  various  hvd  cements  of  the  consistence  of  mortar,  even  without 
sand,  the  writer  detected  no  change  of  bulk  in  setting. 

Art.  7.  Mr.  Wm.  W.  Ulaclay,  €.  E.  (see  his  very  instructive  paper  in 
Trans.  Am.  Soc.  C.  E.,  Dec,  1877),  found  that  in  the  testing  of  cements  the 

temperature  of  the  air  and  water  had  far  more  influence  than  bad  before  been  suspected.  Thus  neat 
Portland  moulded  in  air  at  30°  Fah,  and  kept  6  days  in  water  at  40°,  had  a  tensile  strength  of  but 
156  ft>s  per  sq  inch,  while  that  kept  in  water  of  70°  had  299  Ibs,  or  nearly  twice  as  much.  Other  bars 
moulded  in  air  at  60°,  after  6  days  in  water  of  40°,  broke  with  113  fbs  tensile  per  sq  inch,  while  those 
in  water  at  70°  required  254  Ibs,  or  about  2.25  times  as  much.  But  at  the  end  of  only  20  days  the 
strengths  of  these  last  were  as  212  to  336  fts,  or  as  1  to  1  .6  ;  the  weaker  one  having  in  that  time  gained 
rapidly  on  the  stronger.  As  longer  time  would  of  course  bring  them  still  nearer  to  an  equality,  the 
ultimate  effects  of  temperature  within  certain  limits  are  fortunately  not  so  important  in  actual  prac- 
tice as  the  first  expts  might  lead  us  to  infer.  Work  must  go  on  notwithstanding  changes  of  tempera- 
ture, but  we  must  take  care  that  our  mortar  shall  at  all  times  be  strong  enough  even  under  their  most 
injurious  influences.  Cements  in  open  air  are  certainly  more  or  less  injured  by  drying  instead  of 
setting,  as  the  temp  exceeds  about  65°  to  70°.  But  if  mixed  only  ia  small  quantities  at  a  time,  and 
quickly  laid  in  masonry  of  dampened  stone,  so  as  to  be  sheltered  from  the  air,  the  injury  is  much 
reduce'd.  The  sand  and  stone  should  both  be  damp,  not  wet,  in  hot  weather,  and  a  little  more  water 
may  be  used  in  the  cement  paste:  also  if  possible  not  only  the  mortar  while  being  mixed,  but  the 
masonry  also  should  then  be  shaded  Mr  Maclay  found  that  6  day  specimens  of  neat  Portland  broken 
direct  from  the  water  were  much  stronger  than  if  first  left  24  hours  to  dry  in  the  shade  at  tolerably 
high  temps.  But  the  reverse  occurs  with  such  U.  S.  natural  cements  as  Rosendale.  &c,  the  strength 
of  some  being  largely  increased  by  such  drying.  Experiments  in  Europe  with  Portlands  kept  3  months 


y 
b 


502 


CEMENT,    CONCRETE,   ETC. 


in  water,  seemed  to  show  the  weakest  period  for  such  to  be  at  2  days'  exposure,  when  the  strength  was 
but  half  as  great  as  when  first  taken  from  the  water.  But  on  the  4th  day  they  were  even  stronger  than 
at  first;  and  the  strength  then  increased  with  time  as  if  there  had  been  no  interruption. 

The  effects  of  colcl,  although  it  retards  the  setting,  do  not  appear  to  be 

serious  otherwise.  If  the  cement  mortar  even  freezes  almost  as  quickly  as  the  masonry  is  laid  with 
it,  it  does  not  seem  to  depreciate  appreciably.  The  writer  has  found  this  to  be  the  case  also  with  lime 
mortar,  even  when  a  few  hours  after  freezing  the  temp  became  so  high  as  to  soften  the  frozen  mortar 
again.  But  although  the  mortar  of  either  lime  or  cement  may  not  be  thereby  injured,  the  work,  espe- 
cially in  thin  brick  walls,  may  be  ruined  and  overthrown.  Thus  if  soon  after  the  mortar  through  th« 
entire  thickness  of  such  a  wall  be  frozen,  the  sun  shines  on  one  face  of  it  so  as  to  soften  the  mortar 
of  that  face,  while  the  mortar  behind  it  remains  hard,  it  is  plain  that  the  wall  will  be  liable  to  settle 
at  the  heated  face,  and  at  least  bend  outward  if  it  does  not  fall.  The  writer  has  observed  that  coat- 
ings of  cement  applied  to  the  backs  of  arches  on  the  approach  of  winter,  aud  left  unprotected,  were 

entirely  broken  up  and  worthless  on  resuming  work  the  next  spring.  The 
heating1  of  sand  and  cement  in  freezing  weather  seems  to  be  a  bad  prac- 
tice, especially  if  to  be  placed  in  cold  water.  But  for  use  out  of  water  Mr  Maclay  says  they  may  be 
heated  to  50°  or  60°.  Cold  water  for  mixing  is  probably  no  farther  inju- 
rious than  that  it  retards  the  setting.  All  cements  when  mixed  with  sand  to  a  proper  consistence  for 

mortar  will  fall  to  pieces  if  placed  in  water  before  setting  has  commenced. 
Portlands  do  so  even  without  sand ;  but  U.  S.  natural  cements  of  good  quality  do  not. 

Art.  8.    Strength  of  cements.    The  strength  as  before  stated  is  much 

affected  by  the  temp  of  the  air  and  water,  as  also  by  the  degree  of  force  with  which  the  cement  is 
pressed  into  the  moulds ;  by  the  extent  of  setting  before  being  put  into  the  water,  and  of  drying  when 
taken  out;  and  still  more  by  the  consideration  of  whether  or  not  it  sets  while  under  the  influence  of 
pressure,  which  increases  the  strength  materially.  On  this  account  cements  in  actual  masonry  may 
under  ordinary  circumstances  give  better  results  than  in-door  expts.  These  causes,  together  with  tbe 

degree  of  thoroughness  of  the  mixing-  or  gang- ing-,  the  proportion  of  water 

used,  and  other  considerations  may  easily  affect  the  results  100  per  ct,  or  even  much  more.  Hence, 
the  discrepancies  in  the  reports  of  different  experimenters. 

Rein.    Portlands  require  more  water  than  the  common  U.  S. 

cements,  and  shrink  less  in  mixing.  See  next  Art.  Also,  mortars  require  more  than  concrete,  espe- 
cially when  the  last  is  to  be  well  rammed,  in  which  case  it  should  be  merely 
moist,  so  aa  barely  to  cohere  when  pressed  into  a  ball  by  hand.  If  more  water  is  present,  the  consoli- 
dation by  ramming  is  proportionally  imperfect.  To  assure  himself  that 
the  quality  of  cement  furnished  is  equal  to  that  contracted  for,  the  engineer 

should  reserve  the  right  to  bore  with  a  long  auger  into  any  part  of  each  barrel,  and  to  reject  every 
barrel  of  which  the  sample  drawn  out  does  not  satisfy  the  stipulations.  On  works  using  large  quan- 
tities, there  should  be  one  person  specially  detailed  to  this  duty.  One  advantage  of  very  strong 
cements  is  their  economy,  even  at  a  higher  cost,  in  allowing  the  use  of  a  larger  proportion  of  the 
cheaper  ingredients,  sand,  gravel,  and  broken  stone.  Almost  any  common  U.  S.  cement,  if  of  good 
quality,  will  with  1.5  or  2  measures  of  sand  give  a  mortar  strong  enough  for  most  engineering  pur- 
poses :  but  a  good  Portland  will  give  one  equally  strong  with  3  or  4  meas  of  sand ;  and  will,  therefore, 
be  equally  cheap  at  twice  the  price;  beside  requiring  the  handling,  storing,  and  testing  of  only  half 
the  number  of  barrels. 

After  what  has  been  said  it  is  plain  that  great  latitude  must  be  allowed  in  attempting  to  prepare  a 
table  of  approximate  average  strengths.  The  writer  can  pretend  to  nothing  more  than  the  following, 
which  is  deduced  from  reliable  reports,  aided  by  a  few  experiments  of  his  own  on  transverse  strengths, 
a  summary  of  which  last  forms  the  last  column.  It  is  singular  that  most  experimenters  test  only  the 
tensile  strength,  the  coefficient  of  which  is  seldom  wanted  in  practice. 

If  one  measure  of  cement  slightly  shaken  be  mixed  to  a  paste  with  about  .35  meas  of  water  if  a 
common  U.  S.  cement,  or  about  .40  meas  if  Portland,  in  the  shade,  and  in  a  temp  of  from  60°  to  90°, 
this  paste  will  occupj^bout  .7  meas  if  common,  and  about  86  if  Portland,  when  well  pressed  into 
wooden  moulds  by  the  fingers  (protected  from  corrosion  by  gloves  of  rubber  or  buckskin).  If  then 
allowed  from  80  minutes  to  some  hours  (according  to  its  setting  properties)  to  set ;  then  removed  from 
the  moulds,  and  at  the  end  of  '24  hours  total,  placed  in  water  of  the  above  limits  of  temp  for  7  days, 
and  brokeu  at  once  when  taken  from  the  water,  the  samples  will  generally  exhibit  about  the  following 
strengths.  Those  for  compression  are  supposed  to  be  cubes  ;  and  those  for  transverse  strength  in  the 
table  were  beams  1  inch  square,  and  12  ins  clear  span,  loaded  at  the  center. 

Table  A.    Average  Strengths  of  neat  Cements  after  6  days  in 
water,  and  broken  directly  from  the  water. 


Tensile,  fts 

1"  X 

per  sqin. 

per  sqin. 

per  sq  ft. 

i"  x  12 

'.fts. 

Portlands,  artificial,  either  foreign,  or  the 
"  National  "  of  Kingston,  N.Y. 

200  to  350 

1400  to  2400 

90  to  154 

25  to 

45 

"         Saylor's  natural,  Coplay,  Penn.. 

170  to  370 
40  to    70 

1100  to  1700 
250  to    450 

71  to  109 
16  to    29 

26 
3  to 

7 

All  below  the  lowest  of  these  should  be  rejected  ;  the  average  of  the  table  may  be  considered  fair; 

and  all  above  the  highest  superior.     After  only  24  hours  in  water  the 

strength  of  the  common  ^U.  S.  cements  averages  about  half  that  for  6  days,  but  with  considerable 

variations  both  ways.    In  like  manner  at  the  end  of  a  year  neat  Portlands 

average  from  1.5  to  2  times  as  strong  as  in  6  days  ;  and  our  common  cements  from  2.5  to  3.5  times. 

The  London  board  of  works  require  that  Portlands  after  7  days  in  water 

shall  have  at  least  35  Ibs  transverse,  and  350  Ibs  tensile  strength.     Some  have  reqd  500  or  more  ten- 
sile to  break  them.    For  Portlands  the  writer  found  the  transverse 

strengths  of  several  well  known  English  brands  moulded  as  before  described,  to  be  26  to  40  Bbs  after  7 
days  in  water;  National  Portland  of  Kingston,  N.  Y.,  40  and  46;  Baylor's  Portland  (only  2  trials)  26 


f,    CONCRETE,    ETC. 


503 


fts.    ToepfTer,  G-^awitz,  «fc  Co,  of  Stettin,  Germany,  warrant  all  their 

Portland  (known  &aJA& "  Stera  "  brand)  equal  to  500  fts  tensile  after  7  days  in  water.  Some  of  it  has 
borne  760  fts.  ./ 

Mr.  J.  Ilerbert  Shedd,  as  Engineer  of  the  Water  Works  and  Sewers  of 

Providence-fTl.  I.,  rejected  all  Rosendale  which  when  mixed  to  a  stiff  paste,  and  allowed  30  min  in  air 
to  set,  and  then  put  into  water  for  only  24  hours,  broke  with  less  tension  than  70  fts.  At  first  he 
found some  that  broke  with  10  to  15  fts;  some  that  would  not  set  at  all  in  water;  and  but  little  that 
bore  30  fts.  Now  samples  frequently  bear  100  fts  or  more;  but  that  usually  sold  still  rarely  exceeds 

40  to  50,  and  frequently  scarce  half  as  much.    The  Sewer  Department  of 

St.  Louis.  Missouri,  requires  all  Louisville,  Kentucky,  cement  to  bear  at  least 
40  fts  tensile  after  24  hours  in  water.  Some  of  it  now  shows  as  high  as  100  or  more;  and  60  or  70 
would  have  heen  adopted  as  the  mininuin,  but  for  the  fear  that  it  would  have  encouraged  the  making 
of  too  quick  setting  cement.  Most  of  that  sold  will  probably  not  exceed  30  fts. 

Art.  9.    Cement  mortar  is  cement  mixed  with  water  and  sand  only. 

The  writer  found  that  for  making  cement  pastes  of  about  equal  consistency  and  fit  for  mortar  by 
themselves,  the  English  Portlands,  slightly  shaken  in  the  measure,  required  an  average  of  about  .4 
of  their  own  bulk  of  water;  and  the  D.  S.  common  cements  about  .35.  The  Portland  pastes  when 
thoroughly  mixed  and  slightly  pressed  by  hand  into  a  box  shrank  about  one-eighth  of  their  bulk  as 
dry  shaken  cement;  and  the  others  about  one-fourth  ;  or  in  other  words  the  common  U.  S.  cements 
shrink  about  twice  as  much  as  the  Portlands  ;  and  these  are  about  the  proper  proportions  to  assume 
in  estimating  the  quantity  of  cement  for  theoretically  filling  the  voids  in  sand. 

But  when  sand  is  added,  more  water  is  reqd.    It  is  impossible  to  lay 

down  rules  for  all  cases,  but  as  a  very  rough  average,  mortar  will  require  an  addition  equal  to  about 
.2  of  the  bulk  of  dry  sand ;  varying  of  course  with  the  weather,  &c.  Trial  on  the  work  in  hand  is 
better  than  rules. 

Any  addition  of  sand  weakens  cement,  especially  as  regards  ten- 
sion; as  it  does  also  lime  mortar.  But  economy  requires  its  use.  Sand  also  retards  the  setting,  so 
that  cement  which  by  itself  would  set  in  half  an  hour,  may  not  do  so  for  some  days  if  mixed  with  a 
large  proportion  of  sand.  This  weakening  effect  will  of  course  vary  with  different  cements,  and  with 

many  circumstances  inferable  from  Art.  7,  &c.    As  a  rough  average  the 

following  is  perhaps  not  far  from  the  truth  as  regards  either  tensile  or  transverse  strength  when  not 
rammed. 


Sand. 

0 

M 

i 

VA 

2 

3 

4 

5 

6 

7 

8 

Strength. 

1 

% 

y* 

.4 

1A 

.3 

% 

t 

Ye 

* 

Y* 

The  crushing-  strength  does  not  diminish  so  rapidly ;  but  for  each  pro- 
portion of  sand  we  may  take  the  strength  preceding  it  in  the  table,  as  an  approximation.  Moreover 
the  crushing  strength  with  sand  increases  with  age  much  more  rapidly  than  the  tensile  ;  and  the  more 
EO  the  greater  the  proportion  of  sand. 

As  a  general  rule  with  cements  of  good  quality  we  shall  have  mortars  fit  for  most  engineering  pur- 
poses if  we  do  not  exceed  from  1  to  1.5  measures  of  dry  sand  to  1  of  the  common  cements ;  or  from  2 
to  3  of  sand  to  I  of  Portland. 

The  shearing  strength  of  neat  cements  averages  about  one-fourth  of  the 

tensile. 

The  adhesion  of  cements  to  bricks  or  rough  rubble,  at  dif- 
ferent ages,  and  whether  neat  or  with  sand,  may  probably  be  taken  at  an  average  of  about  three- 
fourths  of  the  cohesive  or  tensile  strength  of  the  cement  or  mortar  at  the  same  age.  If  the  bricks  and 
stoae  are  moist  and  entirely  free  from  dust  when  laid,  the  adhesion  is  increased  ;  whereas  if  very  dry 
and  dusty,  especially  in  hot  weather,  it  may  be  reduced  almost  to  0.  The  adhesion  to  very  hard, 
smooth  bricks,  or  to  finely  dressed  or  sawed  masonry  is  less. 

The  voids  in  sand  of  pure  quartz  like  that  found  on  most  of  our  sea- 
shores, when  perfectly  dry  and  loose,  occupy  from  .303  of  the  mass  in  sand  weighing  1 15  fts  per  cub  ft, 
to  .515  in  that.weighing  80  fts.  Usually,  however,  such  dry  sand  weighs  say  from  105  fts  with  voids  of 
.364  :  to  95  fts,  with  voids  .424 ;  the  mean  being  100  fts,  with  voids  .394.*  But  the  wet  sand  in  mortar 
occupies  about  from  5  to  7  per  cent  less  space  than  when  dry  ;  the  shrinkage  averaging  say  6  per  ct 
or  ^  part ;  thus  making  the  voids  .323  of  the  105  ft  sand  when  wet ;  and  .387  of  the  95  ft  ;  the  mean 
of  which  is  .355.  But  to  allow  for  imperfect  mixing,  &c,  it  is  better  to  assume  the  voids  at  .4  of  the 

*  If  greater  accuracy  is  desired  pour  into  a  graduated  cylindrical 

measuring-glass  100  measures  of  dry  sand.     Pour  this  out,  and  fill  the  glass  up  to  60  measures  with 

water.    Into  this  sprinkle  slowly  the  same  100  measures  of  dry  sand.  These 

will  now  be  found  to  fill  the  glass  only  to  say  94  measures,  having  shrunk  say  6  perct;  while  the 
water  will  reach  to  say  121  measures  ;  of  which  121—94—27  measures  will  be  above  the  sand  :  leaving 
60—27  —  33  measures  filling  the  voids  in  94  measures  of  wet  sand;  showing  the  voids  in  the  wet  sand 

to  be  H  —  .351  of  the  wet  mass.    If  the  sand  is  poured  into  the  water 

hastily,  air  is  carried  in  with  it,  the  voids  will  not  be  filled,  and  the  result  will  be  quite  different. 

Since  a  cubic  foot  of  pure  quartz  weighs  165  Ibs,  it  follows  that 

if  we  weigh  a  cubic  foot  of  pure  dry  sand  either  loose  or  rammed,  then  as  165  is  to  the  wt  found,  so  is 
1  to  the  solid  part  of  the  sand.  And  if  this  solid  part  be  subtracted  from  1,  the  remainder  will  be  the 
voids,  as  below. 

Wt  in  Ibs  per  cub  ft  dry       80       85       90       95      100      105      110      115 
Proportion  of  solid  .485     .515     .546     .576     .606     .636     .667     .697 

Proportion  of  voids  .515     .485     .454     .424     .394     .364     .333     .303 


504  CEMENT,   CONCRETE,   ETC. 

dry  sand.  Moreover,  since  the  cements,  as  before  stated,  shrink  more  or  leas  when  mixed  with  water, 
and  worked  up  into  mortar,  it  would  be  as  well  to  assume  that  to  make  sufficient  paste  to  thoroughly 
fill  the  voids,  we  should  not  use  a  less  volume  of  dry  common  cement,  slightly  shaken,  than  half  the 

bulk  of  the  dry  sand ;  or  than  .45  of  the  bulk  if  Portland.  The  bulk  of  the 
mixed  mortar  will  then  be  about  equal  to  or  a  trifle  less  than  that  of  the  dry 

sand  alone. 

The  best  sand  is  that  with  grains  of  very  uneven  sizes,  and  sharp.  The 
more  uneven  the  sizes  the  smaller  are  the  voids,  and  the  heavier  is  the  sand.  It  should  be  well 
washed  if  it  contains  clay  or  mud,  for  these  are  very  injurious  to  mortar  or  concrete. 

Art.  1O.  Cement  concrete  or  Be  ton,  is  the  foregoing  cement  mortar 

mixed  with  gravel  or  broken  stone,  brick,  oyster  shells,  &c,  or  with  all  together.  In  concrete  as  in 
mortar,  it  is  advisable  on  the  score  of  strength  that  all  the  voids  be  filled  or  more  than  filled.  Those 
of  broken  stone  of  tolerably  uniform  size  and  shape  are  about  .5  of  the  mass  ;  with  more  irregularity 
of  size  and  shape  they  may  decrease  to  .4.  Those  of  gravel  vary  like  those  of  sand,  and  had  like  it 
better  be  taken  at  .5  when  estimating  the  dry  cement.  We  shall  then  have  as  follows.  . 

For  1  cub  yd  of  concrete  of  stone,  gravel,  and  sand,  without 
voids. 

1  cub  yd  broken  stone  with  .5  of  its  bulk  voids,  requiring     .5  cub  yd  gravel. 


0.5  cub  yd  gravel  "     .5  of 

0.25  cub  yd  sand  "    .5 


.25  cub  yd  sand. 

.1'25  (or  %)  cub  yd  dry  cement. 


It  Is  probable  that  mistakes  have  occurred  from  inadvertently  assuming  that  because  the 
voids  in  a  broken  mass,  constitute  a  certain  proportion  of  the  bulk  of  said  mass  ;  therefore,  the  original 
solid  has  swelled  in  only  that  same  proportion.  Thus,  if  a  solid  cubic  yard  of  stone  be  broken  into  small 
irregular  pieces,  which  have  among  themselves  about  the  same  proportions  of  large  and  small  cues,  as 
usually  occurs  in  quarrying,  or  in  railroad  rock-cuttings  ;  and  if  these  be  loosely  thrown  into  a  heap, 
the  .47  of  this  heap,  or  rather  less  than  half  of  it,  will  be  voids.  But  it  does  not  follow,  therefore,  that 
the  original  solid  cub  yd  has  swelled  only  .47,  or  nearly  one-half,  or  makes  only  1J£  cub  yds  of  broken 
stone  ;  although  many  young  engineers  would  probably  consider  this  a  very  full  allowance  ;  and  would 
suppose  that  they  were  quite  just  to  the  company,  if  they  counted  for  the  contractor  one  solid  yard 
of  excavation  for  every  1%  yds  of  fragments.  Now,  it  is  plain  that  if  .47  of  the  broken  heap  are  voids, 
the  remaining  .53  must  be  stone.  But  these  .53  constituted  the  original  solid  cubic  yard  :  and  they 
still  remain  equal  to  it  in  actual  solidity.  Hence  we  must  say  as  follows  :  If  .53  of  the  broken  mass 
occupies  oue  cub  yd  of  actual  space,  how  much  space  will  the  whole  mass  occupy  ;  or, 
Of  the  Cub  yd  Entire  Cub  yds 

broken  mass.       of  space.       broken  mass.  of  space. 


Hence,  we  see  that  a  solid  cub  yd  of  stone,  when  so  broken,  swells  to  1.9,  or  nearly  2  cub  yds  ;  and 
hence  a  proper  allowance  to  a  contractor,  would  be  1  cub  yd  solid,  for  every  1.9  cub  yds  of  pieces  ;  or 


the  yds  of  pfeces  must  be  divided  by  1.9  for  the  solid  yards. 
If  we  know  that  a  cubic  yard  of  any  stone,  break: 


,1  TC  ™w~  vu-u , —, ~.  jaks  to,  say  1.9  yds,  then  to  find  the  proportions  of 

voids,  and  solid,  in  the  broken  mass,  proceed  thus  :  The  solid  part  of  the  broken  mass  must  occupy  1 
cub  yd  of  space;  and  the  question  is  what  part  of  1.9  yds  does  this  1  yd  constitute.    The  answer  is 

—   53.  therefore  53  hundredths  of  the  broken  mass  is  solid;  and  of  course  the  remaining  47  bun- 

1.9 

dredths  are  voids. 

If  a  cubic  foot  solid  weighs  N  Ibs  ;  but  when  broken  up,  or  ground,  only  n  Ibs  per  cub  ft,  then  n 
divided  by  N,  will  be  the  proportion  of  solid  in  the  broken  mass. 

If  the  broken  stone  is  loosely  piled  up,  it  will  occupy  a  little  less  space,  say  about  1.8  cub  yds ;  in 
which  case  the  voids  will  be  .44 ;  and  the  solid,  .56  of  the  mass.  We  will  here  venture  to  express  our 
doubts  whether  hard  rock  when  blasted  and  made  into  embankment,  settles  to  less  than  1%  yds  for 
every  solid  yd.  Mr  Ellwood  Morris  gives  as  the  result  of  certain  embankment  of  hard  sandstone, 
made  under  his  supervision,  an  increase  of  bulk  of  y5^;  or  in  other  words,  that  1  cub  yd  of  rock  in 
place,  made  lj5^,  or  1.417  yds  of  embankment  This  corresponds  to  very  nearly  .7  solid;  and  .3 
voids ;  while  1%  yds  to  1  solid,  corresponds  to  .6  solid;  and  .4  voids.  The  rough  sides  of  rock  excava- 
tions, make  it  difficult  to  measure  them  with  accuracy;  and  we  cannot  but  suspect,  that  something 
of  this  kind  has  interfered  with  the  results  obtained  by  Mr  Morris.  He,  however,  may  be  right,  and 

By  some  careful  experiments  of  our  own,  an  ordinary  pure  sand  from  the  sea  shore,  perfectly  dry, 
and  loose,  weighed  97  Ibs  per  cub  ft;  and  its  voids  were  .41,  and  the  solid  .59 of  the  mass.  By  thorough 
shaking,  and  jarring,  it  could  be  settled  the  .1333  part,  (halfway  between  ^,  and  ^,)  and  no  more. 
It  then  weighed  112  Ibs  per  cub  ft;  and  its  voids  were  then  .32;  and  the  solid,  .68  of  the  mass. 

Another  pure  quartz  sand,  of  much  finer  grain,  perfectly  dry  and  loose,  weighed  but  88  fts  per  cub 
ft ;  the  voids  were  .466 ;  and  the  solid  .534  of  the  mass.  By  thorough  shaking  and  jarring  it  could 
be  reduced;  like  the  former,  onlv  the  .1333  part:  it  then  weighed  101.6  Ibs  per  cub  ft;  and  its  voids 
were  .384  ;  and  the  solid  .616.  Another,  consisting  of  the  finest  sifted  grains,  of  the  last,  weighed  82 
Ibs  per  cub  ft ;  so  that  its  voids,  and  solid,  each  were  very  nearly  .5  of  the  mass.  This  could  be  com- 
pacted about  %  part;  and  then  weighed  98J^  8>s  per  cub  ft. 

The  first,  or  coarsest  of  these  sands,  when  quite  moist,  but  not  wet,  perfectly  loose,  weighed  but  86 
Ibs  per  cub  ff  or  11  Ibs  less  than  when  dry.  It  could  be  rammed  in  thin  layers,  until  it  settled  one- 
fifth  part ;  and  then  weighed  107^  Ibs  per  cub  ft.  Voids  .348. t  solid  .652. 

The  second  sand,  similarlv  moist  and  loose,  weighed  but  69  Ibs  per  cub  ft ;  or  19  Ibs  less  than  when 
dry.  It  could  be  rammed  in  thin  layers,  until  it  settled  %  part ;  and  then  weighed  103^  Ibs  per  cub 

None  olf 'these  s°ands  when  dry,  and  loose,  if  poured  gently  into  water  to  a  depth  of  15  inches,  set- 
tled more  than  about  one-fifteenth  part;  the  coarsest  one,  considerably  less. 


•NCRETE,   ETC.  505 


Here  the  .125  cub  yd  of  drv>«6ment  constitutes  one-eighth  of  the  single  mass ;  or  one-fourteenth  of 
all  the  dry  ingredients  as^aleasured  separately. 

For  1  ciib  jpcTof  concrete  of  broken  stone  and  sand  without 
X^  voids. 

1  cub  yd  broken  stone,  with  .5  of  its  bulk  voids  requiring  |  .5  cub  yd  sand. 
.5  cub  yard  sand,  "    .5  "  "        "  "         |  .25  cub  yd  dry  cement. 

The  strength  of  concrete  is  affected  by  the  quality  of  the  broken  stone, 

as  well  as  by  that  of  the  cement,  the  degree  of  ramming,  &c.  Cubes  of  either  of  the  above  with  Port- 
land, as  well  as  one  composed  of  1  meas  of  good  Portland  to  5  of  sand  only,  well  made,  and  rammed, 
should  either  in  air  or  in  water  require  to  crush  them  at  different  ages,  not  less  than  about  as  follows. 

Agre  in  months 1  36  9         12 

Tons  per  sq  ft 15  40  65  85         100 

Under  favorable  conditions  of  materials,  workmanship  and  weather,  the  strengths  may  be  from  50 
to  100  per  ct  greater.    For  transverse  strength  as  beams  see  p  507. 
If  not  rammed  the  strength  will  average  about  one-third  part  less. 
With  common  U.  S.  cements,  if  of  good  quality  from  .2  to  .3  of  the 

strength  of  Portland  concrete  may  be  had. 

Slow  setting  cements  are  best  for  concrete,  especially  when  to  be 

rammed. 
It  may  not  be  amiss  to  state  here  that  when  masonry  is  backed  by 

concrete  the  two  are  liable  in  time  to  crack  apart  from  unequal  settlement, 
especially  if  the  ramming  has  not  been  thorough;  also  that  in  variable  climates 
cast  iron  cylinders  filled  with  concrete  are  frequently  split  horizon- 
tally by  unequal  expansion  and  contraction.  In  such  structures  it  is  safest  to  consider  the  cyls  as 
mere  moulds  for  the  concrete:  and  to  depend  upon  the  last  only  for  sustaining  the  load. 

The  concrete  for  the  New  York  City  docks  consists  of  1  measure 

of  either  English  or  Saylor's  Portland,  2  of  sand,  5  of  broken  stone  (hard  trap).  That  made  of  Eng- 
lish Portland,  after  drying  a  few  days,  and  then  being  immersed  6  weeks,  required  about  30  tons  per 

sq  ft  to  crush  it.  Saylor's  would  probably  require  the  same.  At  the  Missis- 
sippi Jetties,  (see  "South  Pass  Jetties"  by  Max  E.  Schmidt,  C.  E.,  Trans  Am 

Soc  C  E,  Aug  1879)  Saylor's  Portland  1 ;  sand  2.76;  gravel  1.46;  broken  stone  5. 

In  the  foundations  of  the  Washington  Monument  at  Washington,  D.C., 

(1880)  English  Portland  (J.  B.  White  &  Bros)  1 ;  sand  2  ;  gravel  3  ;  broken  stone  4 ;  and  according  to 
a  Govt.  Report,  has  a  crushing  strength  of  155  tons  per  sq  ft  when  7.5  months  old. 
At  Croton  Dam,  N.  Y.,  (1870;  Kosendale  1 ;  sand  2;  broken  stone  4.5.  Some 

at  the  same  work,  and  deposited  under  water,  had  6  meas  of  stone;  and  at  the  end  of  a  year  had  be- 
come so  hard  that  it  was  found  necessary  to  drill  and  blast  a  portion  that  had  to  be  removed. 

I, ime  with  cement  weakens  all  of  them,  but  General  Q.  A.  Gillmore,  our 

best  authority,  repeatedly  states  that  even  in  important  concrete  work  in  either  the  air  or  water,  (pro- 
vided the  water  does  not  come  into  contact  with  it  until  setting  takes  place),  from  .25  to  even  .5  of  the 
neat  cement  paste  of  the  U.  S.  common  cements  may  be  replaced  by  lime  paste  without  serious  dimi- 
nution of  either  strength  or  hydraulicity ;  and  with  decided  economy.  It  retards  the  setting  which 
is  often  of  great  advantage,  especially  with  quick  setting  cements  which  at  times  cannot  on  that  ac- 
count be  advantageously  used  without  some  lime. 

Moulded  blocks  of  Portland  concrete  of  even  50  tons  wt  can  generally  be 

handled  and  removed  to  their  places  in  from  1  to  2  weeks. 

Ramming  of  concrete,  when  properly  done,  consolidates  the  mass  about 

6  or  6  per  ct,  rendering  it  less  porous,  and  very  materially  stronger.  The  rammers  are  like  those 
used  in  street  paving,  of  wood,  about  4  ft  long,  6  to  8  ins  diam  at  foot  with  a  lifting  handle,  and  shod 
with  iron;  weight  about  35  ftts.  They  are  let  fall  6  or  8  ins.  The  men  using  them,  if  standing 
on  the  concrete,  should  wear  india-rubber  boots  to  preserve  their  feet  from  corrosion  by  the  cement. 

Ramming-  cannot  be  done  under  water,  except  partially,  when  the 

concrete  is  enclosed  in  bags.   A  rake  may,  however,  be  used  gently  for  levelling  concrete  under  water. 

Blake's  Stone  Crusher  (Co,  New  Haven,  Connecticut),  is  useful  for 

breaking  the  stone  more  cheaply  than  by  hand  on  a  large  work.  The  two  sizes  best  adapted  to  this 
purpose  cost  about  $900  and  $1300 ;  break  6  to  7  cub  yds  per  hour ;  and  require  steam-engines  of  about 
8  to  10  horse  power  to  run  them  properly.  According  to  Mr.  J.  J.  R.  Croes,  C.  E.  (see  "  Construction 
of  Croton  Dam,"  Trans.  Am.  Soc.  C.  E.,  Feb.  1875),  a  machine  will  require  about  as  follows:  1 
engine  man,  1  or  2  men  to  break  the  larger  stones  to  a  size  that  will  enter  the  machine,  1  driver  to 
horse-cart,  1  man  to  feed  the  stone  into  the  machine,  2  to  keep  him  supplied  with  stone,  1  at  the 
screen,  2  wheeling  away  the  broken  stone  to  the  stone-heap,  1  or  2  to  receive  it  at  the  heap.  Say  10 

or  12  men  in  all.  The  size  of  the  broken  stone  for  concrete  is  gen- 
erally specified  not  to  exceed  about  2  ins  on  any  edge ;  but  if  it  is  well  freed  from  dust  by  screening  or 
washing,  all  sizes  from  .5  to  4  ins  on  any  edge  may  be  used,  care  being  taken  that  the  other  ingredi- 
ents completely  fill  the  voids. 

The  common  (not  Portland)  cements,  when  used  as  mortar  for  brickwork,  often  disfigure  it,  especi- 
ally near  sea  coasts,  and  in  damp  climates,  by  white  efflorescences  which 

sometimes  spread  over  the  entire  exposed  face  of  the  work,  and  also  injure  the  bricks.  This  also 
occurs  in  stone  masonry,  but  to  a  much  less  extent,  and  is  confined  to  the  mortar  joints;  and  injures 
only  porous  stone.  It  is  usually  a  hydrous  carbonate  of  soda  or  of  potash  often  containing  other  salts. 
Gen'l  Gillmore  recommends  as  a  preventive  to  add  to  every  300  Its  (1  barrel)  of  the  cement  powder, 
100  fts  of  quicklime,  and  from  8  to  12  Tbs  of  any  cheap  animal  fat.  The  fat  to  be  well  incorporated 
with  the  quicklime  before  slacking  it  preparatory  to  adding  it  to  the  cement.  This  addition  will  re- 
tard the  setting,  and  somewhat  diminish  the  strength  of  the  cement.  It  is  also  said  by  others  that 
linseed  oil  at  the  rate  of  2  gallons  to  300  fts  of  dry  cement,  either  with  or  without  lime,  will  in  all 
exposures  prevent  efflorescence ,  but  like  the  fat  it  greatly  retards  setting,  and  weakens. 


506  CEMENTS,   CONCRETE,   ETC. 

Concrete  is  good  for  bringing;  up  an  uneven  foundation  to 
a  level  before  starting  the  masonry.  By  this  means  the  number  of  horizontal 
joints  in  the  masonry  is  equalized,  and  unequal  settlement  is  thereby  prevented. 

Concrete  may  readily  be  deposited  under  water  in  the  usual 

way  of  lowering  it,  soon  after  it  is  mixed,  in  a  V  shaped  box  of  wood  or  plate-iron,  with  a  lid  that 
may  be  closed  while  the  box  descends.  The  lid  however  is  often  omitted.  This  box  is  BO  arranged 
that  on  reaching  bottom  a  pin  may  be  drawn  out  by  a  string  reaching  to  the  surface,  thus  permitting 
one  of  the  sloping  sides  to  swing  open  below,  and  allow  the  concrete  to  fall  out.  The  box  is  then 
raised  to  be  refilled.  In  large  works  the  box  may  contain  a  cubic  yard  or  more,  and  should  be  sus- 
pended from  a  travelling  crane,  by  which  it  can  readily  be  brought  over  any  required  spot  in  the 
work.  The  concrete  may  if  necessary  be  genily  levelled  by  a  rake  soon  after  it  leaves  the  box.  Its 
consistency  and  strength  will  of  course  be  impaired  bv  falling  through  the  water  from  the  box  :  and 
moreover  it  cannot  be  rammed  under  water  without  still  greater  injury.  Still,  if  good  it  will  in  due 
time  become  sufficiently  strong  for  all  engineering  purposes.  Concrete  has  been  safely  deposited  in 
the  above  manner  in  depths  of  50  ft. 

The  Treniie,  sometimes  used  for  depositing  concrete  under  water,  is  a  box 

of  wood  or  of  plate  iron,  round  or  square,  and  open  at  top  and  bottom ;  and  of  a  length  suited  to  the 
depth  of  water.  It  may  be  about  18  ins  diam.  Its  top,  which  is  always  kept  above  water,  is  hopper- 
shaped,  for  receiving  the  concrete  more  readily.  It  is  moved  laterally  and  vertically  by  a  travelling 
crane  or  other  device  suited  to  the  case.  Its  lower  end  rests  on  the  river  bottom,  or  on  the  deposited 
concrete.  In  commencing  operations,  its  lower  end  resting  on  the  river  bottom,  it  is  first  entirely 
filled  with  concrete,  which  (to  prevent  its  being  washed  to  pieces  by  falling  through  the  water  in  the 
tremie)  is  lowered  in  a  cylindrical  tub,  with  a  bottom  somewhat  like  the  box  before  described,  which 
can  be  opened  when  it  arrives  at  its  proper  place.  After  being  filled  it  in  kept  so  by  throwing  fresh 
concrete  into  the  hopper  to  supply  the  place  of  that  which  gradually  falls  out  below, 'as  the  tremie  is 
lifted  a  little  to  allow  it  to  do  so.  The  wt  of  the  filled  tremie  compacts  the  concrete  as  it  is  deposited. 
A  tremie  had  better  widen  out  downwards,  to  allow  the  concrete  to  fall  ont  more  readily.  See  "  Gill- 
more  on  Cements." 
The  area  upon  which  it  is  deposited  must  previously  be  surrounded 

by  some  kind  of  enclosure,  to  prevent  the  concrete  from  spreading  beyond  its  proper  limits ;  and  to 
serve  as  a  mould  to  give  it  its  intended  shape.  This  enclosure  must  be  so  strong  that  its  sides  may 
not  be  bulged  outwards  by  the  weight  of  the  concrete.  It  will  usually  be  a  close  crib  of  timber  oV 
plate-iron  without  a  bottom  :  and  will  remain  after  the  work  is  done.  If  of  timber  it  may  require  an 
outer  row  of  cells,  to  be  filled  with  stone  or  gravel  for  sinking  it  into  place.  Care  must  be  taken  to 
prevent  the  escape  of  the  concrete  through  open  spaces  under  the  sides  of  the  crib  or  enclosure.  To 
this  end  the  crib  may  be  scribed  to  suit  the  inequalities  of  the  bottom  when  the  latter  cannot  readily 
be  levelled  off.  Or  iuside  sheet  piles  will  be  better  in  some  cases;  or  an  outer  or  inner  broad  flap  of 
tarpaulin  may  be  fastened  all  around  the  lower  edge  of  the  crib,  and  be  weighted  with  stone  or  gravel 
to  keep  it  in  place  on  the  bottom.  Broken  stone  or  gravel  or  even  earth  (the  last  two  where  there  is 
no  current)  heaped  up  outside  of  a  weak  crib  will  prevent  the  bulging  outwards  of  its  sides  by  the 
pressure  of  the  concrete.  After  the  concrete  has  been  carried  up  to  within  some  feet  of  low  water, 
and  levelled  off,  the  masonry  may  be  started  upon  it  by  means  of  a  caisson  (page  316) ;  or  by  men  in 
diving  dresses.  Or  if  the  concrete  reaches  very  nearly  to  low  water,  a  first  deep  course  of  stone  may 
be  laid  and  the  work  thus  brought  at  once  above  low  water  without  any  such  aids. 

The  concrete  should  extend  out  from  2  to  5  feet  (according  to 

the  case)  beyond  the  base  of  the  masonry.  All  soft  mild  should  be  removed 
before  depositing  concrete.  BjlJJS  partly  11  lied  With  Concrete,  and  merely  thrown 
into  the  water  may  be  useful  in  certain  cases.  If  the  texture  of  the  bags  is  slightly  open,  a  portion 
of  the  cement  will  ooze  out,  and  bind  the  whole  into  a  tolerably  compact  mass.  Such  bags,  by  the  aid 
of  divers,  may  be  employed  for  stopping  leaks,  underpinning,  and  various  other  purposes,  that  may 
suggest  themselves.  Such  bags  may  be  rammed  to  some  extent. 

Tarpaulin  may  be  spread  over  deep  seams  in  rock  to  prevent 

the  loss  of  concrete ;  and  in  some  cases,  to  prevent  it  from  being  washed  away  by  springs. 

There  is  much  room   for  judgment  in  the  various  applications  of 

concrete;  especially  under  water. 

Concrete  has  been  used  in  very  large  masses;  as  in  the  founda- 
tion of  a  graving  dock  at  Toulon,  France;  where  it  was  deposited  to  a  thickness  of  15  feet,  over  an 
area  of  400  ft  by  100  ft ;  forming,  as  it  were,  a  single  artificial  stone  of  that  size.  It  was  deposited 
under  water;  an  immense  mould  having  first  been  prepared  for  its  reception,  by  enclosing  the  area 
with  close  piling,  1  ned  inside  with  tarpaulin.  On  top  of  this  foundation  were  similarly  built,  likewise 
under  water,  the  sides  of  the  dock  ;  inside  of  great  boxes,  or  enclosures  of  timber,  conforming  to  the 
shape.  The  last  deposits  of  concrete  were  then  faced  with  masonry.  Walls  of  buildings  are  also  fre- 
quently built  of  cement  concrete  deposited  between  planks  as  a  mould  ;  and  which  are  moved  upward 
as  the  building  goes  on.  Flues  may  be  made  in  these  walls  by  ramming  concrete  around  a  tube,  which 

can  afterwards  be  lifted  out;  and  be  used  for  the  next  course  above.  The 
dome  of  the  Pantheon,  at  Rome,  142  ft  diam,  and  now  nearly  2000  years 
old,  is  of  concrete.  The  R.  R.  viaducts  Pont  Napoleon,  and  Pont 

d'Alma,  at  Paris,  have  arches  of  115  and  141  feet  span,  of  concrete. 

With  regard  to  the  mixing  of  concrete,  Gen  Gillrnore  gives  the 

method  pursued  and  described  by  Lieut  Wright.  The  gravel  and  pebbles  being  first  separated  by 
screening,  into  different  sizes,  "  the  concrete  was  prepared  by  spreading  out  the  gravel  on  a  platform 
of  rough  boards,  in  a  layer  from  8  to  12  ins  thick ;  the  smaller  pebbles  at  the  bottom,  and  the  larger 
ones  on  top.  The  mortar  was  then  spread  over  the  gravel  as  uniformly  as  possible.  The  materials 
were  theu  mixed  by  4  men  :  2  with  shovels,  and  2  with  hoes  ;  the  former  facing  each  other,  and  always 
working  from  the  outside  of  the  heap,  to  the  center.  They  then  went  back  to  the  outside,  and  re- 
peated this  operation,  until  the  whole  mass  was  turned.  The  men  with  hoes  worked  each  in  conjunc- 
tion with  a  shoveller,  and  were  required  to  rub  each  shovelful  well  into  the  mortar,  as  it  was  turned 
and  spread,  or  rather  scattered  on  the  platform  by  a  jerking  motion.  The  heap  was  turned  over  a 
second  time,  in  the  same  manner,  but  in  the  opposite  .direction  ;  and  the  ingredients  were  thus  thor- 
oughly incorporated ;  the  surface  of  every  pebble  being  well  covered  with  mortar.  Two  turnings 
usually  sufficed,  and  the  concrete  was  then  carried  to  the  foundation  in  which  it  was  to  be  used.  The 
•uccess  of  the  operation  depends,  however,  entirely  upon  the  proper  management  of  the  hoe  and 


CONCRETE,    ETC.  507 

shovel ;  and  althougte'fhig  may  be  easily  learned  by  the  laborer,  yet  he  seldom  acquires  it  without  the 
particular  attention  of  the  overseer."    It  is  bard  work. 

Or  simple  machinery  is  sometimes  employed  for  incorporat- 
ing the  ingredients  of  concrete,  when  large  quantities  are  required.  A  machine  that  has  been  much 
used  successfully  in  Germany,  consists  simply  of  a  cylinder  about  13  ft  long,  and  4  ft  diam,  open  at 
both  euds ;  and  lined  on  the  inside,  which  is  perfectly  smooth,  with  sheet  iron.  It  is  inclined  6  or  8 
degrees  with  the  horizon.  This  cylinder  is  made  to  revolve  15  or  20  times  per  min,  by  means  of  a 
simple  leather  strap  or  band  arouud  its  outside;  and  to  which  motion  is  given  by  a  locomotive,  which 
at  the  same  time  worked  a  heavy  mill  for  mixing  the  mortar.  This  simple  machine  easily  turns  out 
from  105  to  130  cub  yds  of  concrete  in  10  hours  ;  and  when  worked  in  connection  with  a  mortar  mill, 
at  a  trifling  expense." 

"  When  concrete  is  deposited  in  water,  especially  in  the  sea,  a  pulpy  gelatinous  fluid  exudes  from 
the  cement,  and  rises  to  the  surface.  This  causes  the  water  to  assume  a  milky  hue;  hence  the  term 
laltance,  which  Preach  engineers  apply  to  this  substance.  As  it  sets  very  imperfectly,  and 
with  some  varieties* of  cements  scarcely  at  all,  its  interposition  between  the  layers  of  concrete,  even 
in  moderate  quantities,  will  have  a  tendency  to  lessen,  more  or  less  sensibly,  the  continuity  and 
strength  of  the  mass.  It  is  usually  removed  from  the  enclosed  space  by  pumps.  Its  proportion  is 
greatly  diminished  by  reducing  the  area  of  concrete  exposed  to  the  water,  by  using  large,  boxes,  say 
from  1  to  \}4  cub  yds  capacity,  for  immersing  the  concrete." 

Weight  of  good  concrete  130  to  160  Ibs  per  cub  ft,  dry. 

Cost  of  concrete  $5  to  $9  per  cub  yard  if  roughly  deposited ;  and  $9  to  $15 
if  first  made  into  blocks;  depending  on  size,  cement,  locality,  wages,  &c. 

M.  F.  Colonel's  beton.  The  artificial  stone  which  bears  this  engineer's 
name  has  for  several  years  been  used  in  France  with  perfect  success,  not  only  for 
dwellings,  depots,  large  city  sewers,  &c,  but  for  the  piers  and  arches  of  bridges, 
light-houses,  &c.  Bridge  arches  of  116  ft  span,  and  of  low  rise,  have  been  built  of 
it.  It  is  composed  of  5  measures  of  sand,  1  of  sifted  dry-slaked  lime,  and  from  ^ 
to  %  measure  of  ground  Portland  cement.  Or  of  sand  6,  cement  1,  lime  %;  &c. 

These  are  first  well  mixed  together  dry,  and  then  placed  in  a  mixing-mill ;  at  the  same  time 
sprinkling  them  with  .3  to  A  measure  of  water,  so  as  to  moisten  them  slightly,  without  wetting 
them.  They  are  then  thoroughly  incorporated  by  mixing,  until  they  form  a  stiff  pasty  mass, 
slightly  coherent.  This  is  then  placed  in  a  mould,  in  successive  thin  layers,  each  of  which  is  well 
compacted  by  blows  of  a  16  B)  rammer.  The  top  of  each  layer  may  be  scored  or  cross-cut,  to  make 
the  next  one  unite  better  with  it.  Owing  to  the  small  proportion  of  water,  it  sets  soon;  and  may 
generally  be  taken  from  the  mould  in  from  a  few  hours  to  a  few  days,  depending  on  the  size  of  the 
block ;  and  left  to  harden.  River  sand  is  the  best,  inasmuch  as  it  requires  less  lime  and  cement 
than.pit  sand,  to  make  equally  good  stone.(?)  The  cement  should  be  a  rather  slow-setting  one;  and 
both  it  and  the  lime  should  be  screened,  to  exclude  lumps.  About  \%  bushels,  or  \%  cub  ft  of 
the  dry  materials,  make  1  cubic  foot  of  finished  stone,  weighing  about  140  Ibs;  resisting  100  to 
150  tons  per  square  foot  at  o  months  old.  250  to  400  in  2  years.  Arches  of  it  are  made  no  thicker 
than  brick  ones.  An  arch,  pier,  wall,  foundation,  &c,  may  be  built  of  it,  as  one  stone,  instead  of 
in  separate  blocks.  In  sewers  the  centers  may  be  struck  within  10  to  15  hours  after  the  arch  is 
finished  ;  and  the  water  may  be  admitted  within  a  week  or  less.  The  distinctive  features  of  Coignet's 
beton  are:  the  very  small  proportion  of  water;  the  thorough  incorporation  of  the  ingredients;  and 
the  consolidation  of  the  separate  layers  by  ramming.  It  is  difficult  for  a  person  who  has  never  seen 
the  process,  to  credit  the  rapidity,  facility,  and  economy  with  which  blocks  of  good  stone  can  be 
made  by  it.  Its  cost,  as  compared  with  perfectly  plain  dressed  granite,  does  not  exceed  one-half; 
while  for  ornamental  work  it  compares  even  far  more  favorably.  Hence  the  Coignet  beton,  or  artifi- 
cial stone,  is  nothing  more  than  good,  well-prepared  mortar,  mixed  with  very  little  water :  and  well 
rammed  into  moulds,  in  successive  layers.  A  mixture  of  1  measure  of  hydraulic  cement,  and  3 
measures  of  sand,  similarly  treated,  has  been  successfully  used  in  the  U/S.,  for  some  years,  in 
building*  of  all  kinds.  Ornamental  work  can  be  furnished  at  J4  the  price  of  stone ;  and  will  answer 
equally  well.  F'or  full  information,  see  Gillmore's  "  Coignet  Beton." 

Both  Ransomes,  (England,)  and  the  Sorel  (Boston,  Mass,)  artificial  stones 
are  too  expensive  for  general  engineering  or  architectural  purposes. 

Transverse  Strength  of  Concrete  Beams.* 

Average  results  of  24  beams,  10  ins  square,  made  of  good  Rosendale  and  English 
Portland  cements,  pit  sand  and  screened  pebbles,  few  exceeding  1  inch  diam.  The 
beams  were  buried  for  6  months,  in  a  pit  4  ft  deep,  in  gravelly  soil,  exposed  to  the 
rain,  snow,  &c.  A  first  set  of  beams  all  broke  on  being  taken  from  the  moulds  after 
7  or  8  days,  although  carefully  handled.  To  avoid  this,  the  bottom  of  the  pit  itself 
was  rammed  to  a  smooth,  hard  surface :  immediately  upon  which  a  new  set  was 
made  by  ramming  the  concrete  into  2  inch  planed  plank  moulds  without  bottoms. 
The  moulds  were  removed  after  24  hours,  and  when  all  were  done  the  earth  was 
filled  in  over  the  undisturbed  beams.  Very  little  of  the  soil  adhered  to  them.  Their 
wt  in  all  cases  when  tested  was  about  150  Ibs  per  cub  ft,  or  520  Ibs  wt  of  5  ft  clear 
span  of  beam;  one  half  of  which,  or  260  Ibs,  must  be  deducted  from  the  cen  breakg 
loads  of  the  5  ft  spans  below;  and  124  Ibs  from  the  2  ft  4.5  ins  ones,  as  explained 
p  183.  The  coefficient  or  Constant  C  is  the  cen  breakg  load  in  Ibs  for  a  beam 
1  inch  square,  and  1  ft  clear  span,  like  those  in  table  p  185 ;  and  like  them  is  found 
by  the  formula  at  top  of  p  184.  Its  use  is  shown  by  the  formula  at  foot  of  p  184. 

-v-  Both  these  useful  tables  (the  only  ones  we  know  of  on  the  subject)  were  kindly  furnished  us  by 
Eliot  O.  Clarke,  Esq..  Principal  Assistant  iu  charge  of  the  Improved  Sewerage  Works  of  Boston, 
Mass. ,  for  which  the  experiments  were  made. 


508 


CEMENT,  CONCRETE,  ETC. 


Proportions  of  mate- 
rials by  measure. 

Center  Breaking- 
load  in  Ibs,  including  half 
wt  of  beam. 

Constant  c. 

Cement. 

Sand. 

Pebbles. 

Span  2  ft  4.5  ins. 

Span  5  ft. 

Rosendale  1 

2 

5 

1782 

690 

3.7 

"           1 

3 

7 

all  broke  in  handling 

Portland  1 

3 

7 

3926 

1995 

9,8 

1 

4 

9 

3648 

8.1 

*'           1 

6 

11 

2822 

1190 

6.2 

Tensile  Strength  of  Cement  Mortars, 

of  medium  coarse  sea-beach  sand,  and  good  Rosendale,  and  English  Portland  ce- 
ments; being  averages  of  about  25000  experiments  in  the  years  1878  to  1882.  The 
area  of  breaking  section  was  2.25  sq  ins.  The  proportions  of  sand  and  cement  were 
by  measure.  The  mortar  was  rammed  into  the  moulds,  and  the  specimens  were 
immersed  in  water  as  soon  as  they  would  bear  handling,  and  so  remained  for  1  day, 
or  1  week,  or  for  1,  6,  or  12  months.  The  strengths  are  in  Ibs  per  sq  iiicn. 


Rosendale. 


Neat. 

Cement  1.       Sand  1. 

Cement  1.      Sand  1.5. 

ID. 

1W. 

1M. 

6M. 

1Y. 

1W. 

IMi 

6M. 

1  Y. 

1W. 

1M. 

6M. 

1Y. 

71 

92 

145 

282 

290 

56 

116 

180 

236 

41 

90 

135 

210 

Cement  1.     Sand  2. 

Cement  1.       Sand  3. 

Cement  1.       Sand  5. 

|  22 

49 

105  1  169 

12 

25 

65 

121 

10 

36 

80 

Portland. 

Neat. 

Cement  1.       Sand  1. 

Cement  1.     Sand  1.5. 

102 

303 

412  |  468 

494 

160 

225 

347 

387 

Cement  1.     Sand  2. 

Cement  1.       Sand  3. 

Cement  1.       Sand  5. 

126 

163    279 

ooo 
oZo 

95 

130 

198 

257 

55 

78 

116 

145 

'PLASTERING.  509 

PLASTERING. 


THE  plastering  of  the  inside  walls  of  buildings,  whether  done  on  laths,  bricks,  or 
stone,  generally  consists  of  three  separate  coats  of  mortar.  The  first  of  these  is  called 
by  workmen  the  rough  or  scratch  coat;  arid  consists  of  about  1  measure  of  quicklime, 
to  4  of  sand  ;  (which  latter  need  not  be  of  the  purest  kind;)  and  %  measure  of  bul- 
lock or  horse  hair;  the  last  of  which  is  for  making  the  mortar  more  cohesive,  and 
less  liable  to  split  off  in  spots.  This  coat  is  about  %  to  ^  inch  thick ;  is  put  on 
roughly ;  and  should  be  pressed  by  the  trowel  with  sufficient  force  to  enter  perfectly 
between  and  behind  the  laths;  which  for  facilitating  this  should  not  be  nailed 
nearer  together  than  ^  an  inch.  In  rude  buildings,  or  in  cellars,  Ac,  this  is  often 
the  only  coat  used.  When  this  first  coat  has  been  left  for  one  or  more  days,  accord- 
ing to  the  dry  ness  of  the  air,  to  dry  slightly,  it  is  roughly  scored,  or  scratched,  (hence 
its  name,)  with  a  pointed  stick,  or  a  lath,  nearly  through  its  thickness,  by  lines  run- 
ning diagonally  across  each  other,  and  about  2  to  4  ins  apart.  This  gives  a  better 
hold  to  the  second  coat,  which  might  otherwise  peel  off.  If  the  first  coat  has  be- 
come too  dry,  it  is  well  also  to  dampen  it  slightly  as  the  second  one  is  put  on. 

The  second  coat  is  put  on  about  %  to  %  inch  thick,  of  the  same  hair  mortar,  or 
coarse,  stuff.  Before  it  becomes  hard,  it  is  roughed  over  by  a  hickory  broom,  or 
some  substitute,  to  make  the  third  coat  adhere  to  it  better. 

The  third  coat,  about  %  inch  thick,  contains  no  hair;  and  forgiving  it  a  still 
whiter  and  neater  appearance,  more  lime  is  used,  say  1  of  lime,  to  2  of  sand ;  and 
the  purest  sand  is  used.  This  mortar  is  by  plasterers  called  stucco;  a  name 
also  applied  to  mortar  when  used  for  plastering  the  outsides  of  buildings.  Or  in- 
stead of  stucco,  the  third  coat  may  be,  and  usually  is,  of  hard  finish,  or  gauge  stuff; 
which  consists  of  1  measure  of  ground  plaster  of  Paris,  to  about  2  of  quicklime, 
without  sand.  Hard  finish  works  easier;  but  is  not  as  good  as  stucco,  for  walls  in- 
tended to  be  painted  in  oil.  The  plaster  of  Paris  is  for  hastening  the  hardening. 

Either  of  these  third  coats  is  smoothed  or  polished  to  a  greater  or  less  extent,  according  to  whether 
it  is  to  show,  or  to  be  papered,  painted,  &c.  The  polishing  tools  are  merely,  the  trowel ;  the  hand- 
lioat,  (a  kind  of  wooden  trowel ;)  and  the  water-brush,  (a  short-handled  brush  for  wetting  the  surface 
part  at  a  time  with  water,  in  order  to  polish  more  freely.)  For  finer  polishing,  a  float  made  of  cork 
is  used.  The  smooth  piece  of  board  about  10  to  12  ins  square,  with  a  handle  beneath,  on  which  the 
plasterer  holds  his  mortar  until  he  pUts  it  on  to  the  wall  with  his  trowel,  is  called  a  hawk. 

The  more  thoroughly  each  coat  is  gone  over  with  the  water-brush  and  trowel,  (which  process  is 
called  hand-floating,)  the  firmer  and  stronger  will  it  be.  Frequently  only  two  coats  of  plastering  are 
put  on  in  inferior  rooms  ;  or  where  great  neatness  of  appearance  is  not  needed.  The  first  is  of  hair 
mortar,  or  coarse  stuff;  this  is  scratched  with  the  broom,  and  then  covered  by  the  finishing  coat  of 
finer  mortar,  (stucco.)  If  this  last  is  nearly  all  lime,  or  with  but  very  little  sand,  to  make  it  work 
easier,  it  is  called  a  slipped  coat.  Without  any  sand  it  is  called  ftne  stuff.  Neither  is  as  good  as 
stucco,  if  the  wall  is  to  be  papered.  When  this  is  the  case,  the  third  coat  also  may  have  a  little  hair, 
to  give  it  more  strength  ;  but  this  is  not  absolutely  necessary. 

A  very  good  effect  may  be  produced  in  station-houses,  churches,  &c,  by  only  two  coats  of  plaster  in 
•which  fine  clean  screened  gravel  is  used  instead  of  sand.  When  lined  into  regular  courses,  it  resem- 
bles a  buff-colored  sandstone,  very  agreeable  to  the  eye. 

In  purchasing  plastering  hair,  care  must  be  taken  that  it  has  not  been  taken  from  salted  hides; 
inasmuch  as  the  salt  will  make  the  walls  damp.  For  the  same  cause  sea-shore  sand  should  not  be 
used.  It  is  almost  impossible  to  wash  it  entirely  free  from  salt. 

In  brick  walls  intended  to  be  plastered,  the  mortar  joints  should  be  left  very  rough,  to  let  the  plas- 
ter adhere.  If  it  is  put  on  smooth  walls,  without  first  raking  out  the  mortar  to  the  depth  of  nearly 
an  inch,  it  is  very  apt  to  fall  off;  especially  from  outside  walls ;  as  can  be  seen  daily  in  any  of  our 
cities.  As  this  raking  out  of  brick  joints  is  tedious  and  expensive,  it  would  generally  be  better  to 
use  paint  rather  than  plaster.  The  walls  should  also  be  washed  clean  from  all  dust;  and  should  be 
slightly  dampened  as  the  plaster  is  put  on. 

To  imitate  granite  on  outer  walls :  after  the  second  or  smooth  coat  of  plaster  is  dry,  it  receives  a 
coat  of  lime  wash,  slightly  tinted  by  a  little  umber,  or  ochre,  Ac.  After  this  is  dry,  in  case  it  appears 
too  dark,  or  too  light,  another  may  be  applied  with  more  or  less  of  the  coloring  matter  in  it.  Finally, 
a  wash  of  lime  and  mineral-black  is  sprinkled  on  from  a  flat  brush,  to  imitate  the  black  specks  of 
granite.  By  this  simple  means,  a  skilful  workman  can  produce  excellent  imitations.  The  horizontal 
and  vertical  joints  of  the  imitation  masonry,  may  be  ruled  in  by  a  small  brush,  using  the  same  black 
wash,  and  a  long  straight-edge. 

The  rough  surfaces  of  all  walls  are  more  or  less  warped,  or  out  of  line ;  and  it  is  not  possible  for 
the  plasterer  to  rectify  this  perfectly  by  eye,  as  may  be  seen  in  almost  every  house.  Even  in  what 
are  called  first-class  ones,  a  quick  eye  can  generally  detect  unsightly  undulations  of  the  plastered 
•urfaces. 

To  prevent  this,  the  process  of  screed  ing:  is  resorted  to.    Screeds  are  a  kind  of 

gauge  or  guide,  formed  by  applying  to  the  first  rough  coat,  when  partly  dried,  horizontal  strips  of  the 
plastering  mortar,  about  8  ins  wide,  and  from  2  to  4  ft  apart  all  around  the  room-  These  are  made  to 
project  from  the  first  coat,  out  to  the  intended  face  of  the  second  one;  anrtarwhile  soft  are  carefully 
made  perfectly  straight,  and  out  of  wind  with  each  other,  by  means  of  the  plumb-line,  straight-edge. 
&c.  When  they  become  dry,  the  second  coat  is  put  on,  filling  up  the  broad  horizontal  spaces  between 
them  ;  and  is  readily  brought  to  a  perfectly  flat  surface,  corresponding  with  that  of  the  screeds,  by 
means  of  long  straight-edges  extending  over  two  or  more  of  the  latter. 

A  day's  work  at  plastering:. 

A  plasterer,  aided  by  one  or  two  laborers  to  mix  his  mortar,  and  to  keep  his  hawk  supplied,  can 
average  from  100  to  200  sanare  yards  a  day,  of  first  coat ;  about  %  as  much  of  second ;  and  half  as 

33 


510 


SLATING. 


»msh  of  third,  which  requires  more  care.    The  amount  will  depend  upon  the  number  of  angles,  sir* 
of  rooms,  whether  on  ceilings  or  on  walls,  &c,  &c. 

Gen  Gillmorc's  estimate  of  cost  of  plastering-*  100  square  yards 
with  2  or  with  3  coats.    Common  labor  $1  per  day. 


Materials. 

Three  Coats. 
Hard  finished  work. 

Two  Coats.. 
Slipped  coat  finish. 

Quicklime 

4  casks. 

8  - 

2000 
4  bushels. 
7  loads.* 
2^  bushels. 
13  Ibs. 
4  days. 
3  days. 

$4.00 
.85 
.70 
4.00 
.80 
2.00 
.25 
.90 
7.00 
3.00 
2.00 

3Ji  casks. 

2000. 
3  bushels. 
6  loads. 

13  Ibs. 
3J^  days. 
2  days. 

$3.33 

4.00 
.60 
1.80 

.90 
6.1'2 
2.00 
1.20 

$19.95 

"           for  fine  stuff 

Plaster  of  Paris 

Laths 

Hair 

Common  Sand  

White  Sand 

Nails  

Mason's  labor  .   . 

Cartage  

Cost  of  100  square  vards  

$25.50 

This  amounts  to  25 V6  cts  per  sq  yd  for  3  coats ;  and  say  20  cts  for  2  coats,    gee  Art  5,  P  500. 

Plastering  laths  are  usually  of  split  white  or  yellow  pine,  in  lengths  of 
about  3  to  4  ft  long ;  and  hence  called  3.  or  4  ft  laths.  They  are  about  1^  ins  wide,  by  %  inch  thick. 
They  are  nailed  up  horizontally,  about  %  inch  apart.  The  upright  studs  of  partitions  are  spaced  a* 
such  distances  apart,  (generally  about  15  ins  from  center  to  center,)  that  the  ends  of  the  laths  may 
be  nailed  to  them.  Laths  are  sold  by  the  bundle  of  1000  each.  A  square  foot  of  surface  requires  1  & 
four  feet  laths  ;  or  1000  such  laths  will  cover  666  sq  ft.  Sawed  laths  may  be  had  to  order,  ot  any  re- 
quired length.  A  carpenter  can  nail  up  the  laths  for  from  40  to  60  »q  yds  of  plastering  in  a  day  of 
10  hours ;  depending  on  the  number  of  angles  in  the  rooms,  &c. 

*  A  load  (one-horse),  both  in  the  U.  S.  and  in  England,  usually  means  a  cub  yd;  but  many 
dealers  adopt  20  struck  bushels  —  25  cub  ft  —  fully  a  ton. 


SLATING. 


ROOFING  slates  are  usually  from  ^  to  *<£  inch  thick;  about -^  being  a  commo* 
average.  They  may  be  nailed  either  to  a  sheeting  of  rough  boards  (c,  #,  in  the  fig) 
from  %  to  1J4  inch  thick,  (which  should  be,  but  rarely  are,  tongued  and  grooved,) 


*  Average  prices  of  plastering  in  Philada,  1873,  in  cts  per  sq  yard. 
Three  coats,  including  laths,  scaffold  &c,  50  to  55  cts.  Two  coats  35  to  40.  Three  coats  on  brick  or 
stone  (no  laths  reqd,)  50  to  55.  Outside  plastering,  60;  or  if  to  imitate  marble,  75.  Simple  plaster 
cornices,  1  to  2  cts  per  inch  of  girth,  per  ft  run.  Plaster  center  flowers  for  parlors,  $5  to  $15  each, 
put  up.  The  plastering  of  a  20  ft  front,  3  story  dwelling,  with  large  3  story  back  buildings,  81000  to 
$1300.  Stipulate  expressly  to  pay  only  for  surfaces  actually  plastered :  and  thu.-  avoid  extras,  even 
if  vou  have  to  pay  a  few  cts  more  per  yard. 


SLATING.  511 

laid  horizontallyxfirt5m  rafter  to  rafter;  or  sloping,  from  purlin  to  purlin,  as  tho  case 
may  be ;  or  to^stout  laths  t  tt  about  2  to  3  ins  wide,  and  from  1  to  1^  thick,  nailed 
to  the  rafters  at  distances  apart  to  suit  the  gauge  of  the  slates.  Two  nails  are  used  to 
each  sb*€e ;  one  near  each  upper  corner.  They  may  be  either  of  copper,  (which  is  the 
most  durable,  but  most  expensive,)  of  zinc,  or  of  either  galvanized  or  tinned  iron. 
The  last  two  are  generally  used ;  or  in  inferior  work,  merely  plain  iron  ones,  pre- 
viously boiled  in  linseed  oil,  as  a  partial  preservative  from  rust.  Rust,  however, 
sometimes  weakens  them  so  much  that  they  break ;  and  the  slates  are  blown  off  in 
high  winds,  to  the  danger  of  passers  by.  Since  good  slate  endures  for  along  series 
of  years,  it  is  true  economy  to  use  nails  that  are  equally  durable.  In  iron  roofs,  the 
slates,  instead  .of  being  nailed  to  boards,  are  sometimes  tied  directly  to  the  iron 
purlins,  by  wire.  A  square  of  slating,  shingling,  &c,  is  100  sq  ft. 
In  laboratories,  chemical  factories,  Ac,  subject  to  acid  fumes,  it  is  difficult  to 

provide  a  metal  fastening  that  will  not  be  eaten  away.  In  such  cases  it  is  best  to  depend  chiefly  upon 
a  layer  of  mortar  between  the  slates.  This  will  harden  before  the  metal  fastenings  give  way  ;  and 
will  hold  the  slates  in  place,  while  new  fastenings  are  being  inserted. 

The  least  pitch  considered  advisable  for  a  roof,  to  prevent  rain  or  snow  from  being  driven 
through  the  interstices  between  the  slates,  is  about  26^° ;  or  1  vert  to  2  hor ;  which  corresponds  to  a 
rise  of  y±  the  span  in  a  common  double  pitched  roof.  Bat  even  at  steeper  pitches,  rain,  and  more 
particularly  snow,  will  be  forced  through  the  roof  by  violent  winds  ;  especially  if  laths  alone  be  used, 
or  even  boarding  alone.  To  avoid  this,  a  layer  of  mortar  about  y±  inch  thick,  may  be  spread  over 
the  touching  surfaces  of  the  slates  if  on  laths.  If  on  boards,  the  same  process  may  be  adopted  ;  or 
the.more  common  one  of  first  covering  the  boards  with  a  layer  of  what  is  called  slating  felt;  hut 
which  in  reality  is  merely  thick  brown  paper,  soaked  in  tar.  This  is  sold  in  long  continuous  rolls, 
28  ins  wide,  and  weighing  from  40  to  50  fts.  A  50  ft  roll  will  cover  about  300  sq  ft  of  roof.  With 
proper  precautions  against  the  admission  of  rain  and  snow,  a  pitch  as  flat  as  1  in  2^,  or  even  1  in 
8,  may  be  adopted. 

The  thickness  of  slate  on  a  roof  Is  double;  except  at  the  laps  is,  is,  &c,  where  it  is  triple.  The 
lap  is  measured  from  the  nail  hole  (under  t)  of  the  lower  slate,  to  the  lower  edge  or  tail,  s,  of  the 
•pper  one  ;  and  is  usually  about  3  ins.  In  order  that  the  showing  lower  edges  of  the  slates  shall, 
when  laid,  form  regular  straight  lines  along  the  roof,  the  nail  holes  are  made  at  equal  distances  from 
said  lower  edges  ;  so  that  any  irregularity  of  length  is  concealed  from  view  at  the  hidden  heads  of 
the  slates.  The  slater  estimates  the  length  of  his  slate  from  the  nail  hole  to  the  tail  •  discarding  the 
narrow  strip  between  the  nail  hole  and  the  head.  If  from  this  reduced  length  the  lap  be  deducted, 
then  one-half  of  the  remainder  will  be  the  gauge,  weathering,  or  margin,  of  the  slating;  or,  in  other 
words,  the  showing  or  exposed  width  of  the  courses  of  Blates.  The  gauge  in  ins  multiplied  by  the 
width  of  a  slate  in  ins,  gives  the  area  in  sq  ins  of  finished  roof  covered  by  a  single  slate ;  and  if  144 
(the  sq  ins  in  a  sq  footl  be  divided  by  this  area,  the  quotient  will  be  the  number  of  slates  required  per 
sq  ft  of  roof.  The  upper  side  of  a  shite  is  called  its  back;  the  lower  one,  its  bed. 

Slating,  like  shingling,  must  evidently  be  commenced  at  the  eaves,  and  extended  upward.  Since 
the  beds  of  the  slates  are  not  exactly  parallel  to  the  boarding,  and  consequently  do  not  rest  flat  upon 
It,  those  at  the  lower  edge  10  would  easily  be  broken.  To  prevent  this,  a  tilting  strip  (a  stout  wide 
lath,  with  its  upper  side  planed  a  little  bevelling,  to  suit  the  slope  of  the  slates)  is  first  nailed  around 
near  the  eaves,  for  the  tails  of  the  lowest  course  of  slates  to  rest  on.  This  is  shown  on  a  larger  scale 
atT. 

Slate  of  the  best  quality  has  a  glistening  semi-metallic  appearance,  somewhat  like  that  of  a  surface 
of  paper  rubbed  with  black-lead  pencil.  That  of  a  dull  earthy  aspect,  is  softer,  more  absorbent,  and 
consequently  more  liable  to  yield  to  atmospheric  influences,  rain,  frost,  &c.  Iron  pyrites  frequently 
occurs  in  slate;  and  since  it  always  decomposes  and  leaves  holes,  should  never  be  ad'mitted  on  a  roof. 
Of  two  qualities  of  slate,  that  which  absorbs  the  least  weight  of  water,  when  pieces  of  equal  size  are 
soaked  for  an  hour  or  two,  is  eenerallv  the  best;  being  least  liable  to  split  by  frost,  and  become 
weather-worn.  This  t«st  is  easily  applied. 

In  England  the  different  sizes  are  distinguished  by  absurd  names  of  no  meaning.  In  the 
United  States  they  are  called  6  by  12's  ;  16  by  24's,  &c,  according  to  their  measures  in  inches.  They 
may  be  cut  to  order,  of  almost  any  prescribed  dimensions,  or  shape.  Those  in  common  use  vary  from 
about  7  by  14,  to  12  by  18.  The  first  forms  about  5  to  6  inch  courses;  and  the  last  about  7  to  8  inch; 
depending  upon  how"  far  from  the  head  the  nail  holes  are  pierced.  The  farther  this  is,  the  firmer 
will  the  slating  be. 

Slate  roofs,  like  iron  ones,  heat  the  rooms  immediately  below  them  very  much.  This  is  somewhat 
diminished  when  the  slates  are  on  boards,  instead  of  laths;  and  still  niore  bv  a  coat  of  plaster  be- 
neath. They  are  also  liable  to  break  when  walked  on ;  less  so  when  bedded  in  mortar. 

Weight  of  slate  roofs.  Slate  weighs  about  175  fts  per  cub  foot;  therefore, 
a  sq  ft,  H  inch  thick,  weighs  about  1.8  Ibs;  y3^,  2.7  fts;  and  %  thick,  3.6  fts.  But  owing  to  the 
overlapping,  a  square  foot  of  roof  requires  about  2^  sq  ft  of  slate  of  ordinary  sizes ;  and  if  the  slate 
is  laid  on  boards  an  inch  thick,  tae  weight  per  sq  ft  of  roof  will  be  increased  about  2%  fts  ;  or  with 
\y4  inch  boards,  2.8  fts.  Laths  will  weigh  about  %  ft  per  sq  ft  of  roof. 
Hence, 

Appro*  Weight 
of  one  Rq  ft  of 
Slating,  in  Ibs. 

Slate  Hinch.  thick  on  laths 4.75 

"  "  on  1  inch  boards 6.75 


. 

"  3-16  "                 on  laths 7.00 

on  1  inch  boards 9.00 

"    "     "                onlVf"        "       9.55 

"    J4    "                 on  laths 9.25 

"     "     "                 on  1  inch  boards 11.25 

onlJ4"        " 11.80 

If  slating  felt  is  used,  add  ^  ft  ;  or  if  the  slates  are  bedded  in  }±  inch  of  mortar,  add  3  tbs. 


512 


SHINGLES. 


For  the  total  weight  borne  by  the  roof  trusses,  that  of  the  purlins  also  must  be  added.  •  This  wil) 
not  vary  much  from  the  limits  of  1^'  to  3  Ibs  per  sq  ft  in  roofs  of  moderate  span.  Add  tor  wind  and 
snow,  say  20  fts  per  sq  ft;  *  and  finally  add  the  weight  of  the  truss  itself. 

For  stopping*  the  joints  between  slates  (or  shingles,  &c)  and  chimneys, 
dormer  windows,  &c,  a  mixture  of  stirt'  white  lead  paint,  as  sold  by  the  keg,  with  sand  enough  to  pre- 
vent it  from  running,  is  very  good;  especially  if  protected  by  a  covering  of  strips  of  lead,  or  copper, 
tin,  &c.  nailed  to  the  mortar-joints  of  the  chimneys,  after  being  bent  m,  as  to  enter  said  joints  ;  which 
should  be  scraped  out  for  an  inch  in  depth,  and  afterward  refilled.  Mortar  protected  in  the  same 
way,  or  even  unprotected,  is  often  used  for  the  purpose  ;  but  is  not  equal  to  the  paint  and  sand.  Mor- 
tar a  few  days  old,  (to  allow  refractory  particles  cf  lime  to  alack,)  mixed  with  blacksmith's  cinder* 
and  molasses,  is  much  used  for  this  purpose;  and  becomes  very  hard,  and  effective. 


SHINGLES, 


WHITE  cedar  shingles  are  the  best  in  use ;  and  when  of  good  quality  will  last  40  or 
50  years  in  our  Northern  States.  They  are  usually  27  ins  long;  by  from  6  to  7  ins 
wide ;  about  ^  inch  thick  at  upper  end ;  and  about  %  at  lower  end  or  butt ;  and  are 
laid  in  courses  about  8%  ins  wide ;  so  that  not  quite  ^  of  a  shingle  is  exposed  to  the 
weather. 

They  ar«  usually  laid  in  three  thicknesses ;  except  for  an  inch  or  two  at  the  upper  ends,  where  there 
are  four.  Thev  are  nailed  to  sawed  shiugling-laths  of  oak  or  yellow  pine;  about  16  ft  long;  2^  ins 
wide,  and  1  inch  thick ;  placed  in  horizontal  rows  about  8%  ins  apart.  These  are  nailed  to  the  raft- 
ers, or  purlins ;  which,  for  laths  of  the  foregoing  size,  should  not  be  more  than  2  ft  apart  from  center 
to  center.  Two  nails  are  used  to  each  shingle,  near  its  upper  end.  They  should  not  be  of  less  size 
than  400  to  a  ft.  Those  of  wrought  iron  being  the  strongest,  are  the  best;  cut  ones  are  apt  to  break 
by  the  warping  of  the  shingles.  Two  pounds  of  such  nails  will  suffice  for  100  sq  ft  of  roof,  including 
w'aste.  An  average  shingle  7^  ins  wide,  in  8}4  inch  courses,  exposes  63%  sq  ins ;  making  '2%  shingles 
to  a  sq  ft  of  roof;  but  to  allow  for  waste,  and  narrow  shingles,  it  is  better  in  practice  to  allow  about  3 
shingles  to  a  sq  ft. 

Shingling,  like  slating,  must  plainly  be  begun  at  the  eaves :  and  extended  upward.  For  closing  the 
joints  between  the  shingles,  and  chimneys,  dormer  windows,  &c,  see  at  end  of  Slating. 

Cypress  and  white  pine  are  also  much  used  for  shingles,  being  much  cheaper,  hut  scarcely  half  as 
durable,  t  All  shingles  wear  quite  thin  in  time  by  rain  and  exposure.  In  warm  damp  climates  they 
all  decay  within  6  to  12  years. 


PAINTING. 

THE  principal  material  used  in  house-painting,  is  either  white  lead,  or  oxide  of 
Bine,  ground  in  raw  (unboiled)  linseed  oil,  by  a  mill,  to  the  consistency  of  a  thick 
paste.  In  this  condition,  it  is  sold  by  the  manufacturers  in  kegs  of  25,  50,  and  100 
ros.  To  prepare  it  for  actual  use,  merely  requires  the  addition  of  more  linseed  oil, 
say  3  or  4  pints  to  10  Ibs  of  the  keg  paint,  for  thinning  it  sufficiently  to  flow  readily 
under  the  brush. 

Good  painting  requires  4  or  5  coats ;  but  usually  only  4  are  used  in  principal  rooms ;  and  8  in  inferior 
ones.  Each  coat  must  be  allowed  to  dry  perfectly  before  the  next  one  is  put  on.  One  ft  of  the  keg 
paint  will,  after  being  thinned,  cover  about  2  sq  yds  of  first  coat;  3  yds  of  second;  and  4  yds  of  each 
subsequent  coat ;  or  1  sq  yd  of  3  coats  will  require  in  all,  1.08  fts  ;  of  4  coats,  1%  fts ;  of  5  coats,  1.58 
fts.  The  reason  why  the  first  coats  require  so  much  more  than  the  subsequent  ones,  is  that  the  bare 
surface  of  the  wood  absorbs  it  more. 

When,  as  is  usual,  raw  or  unboiled  oil  is  used  for  thinning,  dryers  must  be  added  to  it;  otherwise 
the  paint  might  require  several  weeks  to  harden;  whereas,  with  dryers,  from  1  to  3  days,  according 
to  the  weather,  suffice  for  each  coat  to  become  hard  enough  to  receive  the  next  one.  The  dryers  most 
commonly  used,  are  powdered  litharge,  in  the  proportion  of  one  heaped  teaspoonful ;  or  Japan  var- 
nish, 1  table-spoonful,  to  10  fts  of  the  keg  paint.  Either  sugar  of  lead,  or  sulphate  of  zinc,  may  also 
be  used  instead  of  litharge ;  and  in  the  same  proportion.  Although  both  litharge  and  Japan  varnish 
are  dark-colored,  yet  the  quantity  is  so  small  as  not  to  appreciably  affect  the  whiteness  of  the  paint. 
If  the  varnish  is  used  in  excess,  as  is  often  done  in  the  hurry  to  have  work  finished,  it  produces 
cracks  all  over  the  surface.  No  dryer  is  necessary  if  painters'  boiled  oil  be  used  for  thinning.  Mere 
boiling  will  not  cause  oil  to  harden  more  rapidly";  but  that  intended  for  painters,  has  litharge  added 
to  it  previously  to  boiling  ,  in  the  proportion  of  1*4  fts  to  each  10  gallons  of  raw  oil.  In  some  works 
written  for  the  use  of  house  painters,  it  is  asserted  that  boiling  renders  the  oil  too  thick  for  any  but 
coarse  outdoor  work.  But  this  is  entirely  a  mistake;  for  if  the  boiling  be  properly  done,  the  oil 
will  be  quite  thin  enough  tor  the  best  inside  work ;  and  will  moreover  be  clearer  than  while  raw ;  and 


*  Price  of  slate,  felt,  and  slating  in  Philada  in  1873,  is  from  12  to  14  cts  per  sq  ft,  according  to 
quality  of  slate;  kind  of  nails.  &c ;  but  exclusive  of  boarding.  With  copper  nails  add  2  cts  per  sq 
ft.  The  slate  from  Peach  Bottom,  York  Co,  Penna,  is  the  best  in  the  State.  It  commands  2  or  3  cts 
per  sq  ft  more  than  the  others.  A  roof  of  leaded  tin,  will  cost  about  the  same  as  one  of  slate;  and 
not.  much  more  than  half  as  much  as  good  cedar  shingles,  (in  Philada.)  Felt  about  4  cts  per  ft. 

f  Price  of  shingles  in  Philada.  in  1873 :  Best  cedar  (a*out  7  to  8  ins  wide,  by  27  ins  long,)  $50  to  $60. 
White  pine  or  cypress,  $40  to  $50.  Shingling  laths,  $3  to  $4  per  1000.  Cedar  shingles,  laths,  nails, 
and  shingling  complete,  30  cts  per  sq  ft  of  roof;  or  about  twice  aa  much  as  slate  or  leaded  tin  roofing, 

-"•"-""—•  '•->  -J-tij£rt' 


'PAINTING.  513 

Will  impart  to  the  palufttffsui  face  a  more  shilling  appearance.  The  heat  should  be  barely  sufficient 
to  produce  boilmglxor  about  t>UO°  Fan.  The  boiliug  should  continue  about  1^  hours;  the  oil  being 
thoroughly  stirnwai  short  intervals,  to  prevent  the  litharge  from  settling  at  the  bottom.  The  fire 
may  then  beutfiowed  to  subside;  when  the  operation  will  be  completed.  A  sedimeut  will  then  form 
at  the  bottsfu  ;  which  must  be  left  behind  when  the  oil  is  poured  off.  Although  no  dryer  is  necessary 
'  il,  still  a  little  litharge  may  be  added  when  great  expedition  demands  it.  Painters  rarely 
use  tuis  oil.  on  account  of  its  trifling  increase  of  cost. 

Another  substance  much  used  with  the  thinning  oil,  (except  for  the  first  coat,)  is  spirits  of  turpen- 
tine ;  called  "  turp"  by  the  workmen.  The  quantity  of  oil  may  be  diminished,  to  the  extent  of  the 
added  turp.  This  being  more  fluid  than  oil,  causes  the  paint  to  work  more  pleasantly  under  the  brush. 
It  moreover  diminishes  the  tendency  of  the  paint  to  become  yellow ;  especially  in  rooms  kept  closed 
for  some  time.  It  is  also  much  cheaper  than  oil.  It  should  not  be  used,  or  but  sparingly,  for  exposed 
outdoor  work;  inasmuch  as  its  tendency  is  to  impair  the  tirmuess  of  the  paint;  and  although  its 
effects  are  scarcely  appreciable  indoors,  they  are  quite  apparent  when  the  work  has  to  resist  the 
weather.  As  the  fashions  change  in  house-painting,  the  surface  is  at  times  required  to  present  a 
shining  or  glossy  finish :  at  other  times  a  dead  one  is  in  vogue.  Tbe  glossy  one  is  that  which  the 
paint  will  uuturaily  have,  provided  that  no  more  turp  than  oil  be  used  in  the  thinning.  The  dead 
finish  is  obtained 'by  using  no  oil,  but  turp  alone,  for  the  last  coat;  which  in  that  case  is  called  a 
flatting  coat.  Although  turp  is  not  properly  a  dryer,  still,  as  it  evaporates  quickly,  it  facilitates  the 
hardening  of  the  paint. 

In  outdoor  work  it  is  usually  advisable  to  use  more  dryer  than  inside,  so  that  the  paint  may  sooner 
become  hard  enough  not  to  be  injured  by  dust  or  rain.  Otherwise  less  would  be  better. 

When,  instead  of  a  white  finish,  one  of  seme  other  color  is  required,  the  coloring  ingredient  is 
mixed  with  the  white  paint  to  be  used  in  the  last  coat  only  ;  although  two  coloring  coats  are  some- 
times found  to  be  necessary  before  a  satisfactory  effect  is  produced.  The  coloring  ingredients  may  be 
indigo,  lampblack,  terra  sienna,  umber,  ochre,  chrome  yellow,  Venetian  red,  red  lead,  &c,  &c;  which 
ara  ground  in  oil,  ready  for  sale,  by  the  manufacturers  of  the  white-lead  and  zinc  paints.  They  are 
simply  well  stirred  iuto  the  white  paint. 

All  surfaces  to  be  painted,  should  first  be  thoroughly  dry,  and  free  from  dust.  If  on  wood,  all 
plane-marks,  and  other  slight  irregularities,  should  first  be  smoothed  off  by  sand-paper,  when  the 
neatest  finish  is  required.  Also,  all  heads  of  nails  must  be  punched  to  about  %  in«h  below  the  sur- 
face. To  prevent  knots  from  showing  through  the  finished  work,  (as  those  in  white  or  yellow  pine 
would  do,  on  account  of  the  contained  turpentine,)  thev  must  first  be  killed,  as  it  is  termed.  A  usual 
and  effective  way  of  doing  this,  is  by  covering  them  with  two  coats  of  shellac  varnish ;  which,  when 
dry,  should  be  smoothed  by  sand-paper.  Another  mode,  not  quite  so  certain,  is  by  one  or  two  coats 
of  white  lead  mixed  with  thin  glue- water,  or  size,  as  it  is  called. 

After  these  preparations,  the  first,  or  priming  coat,  is  put  on  ;  in  which  there  should  be  no  turp; 
because  it  would  sink  at  once  into  the  bare  wood,  leaving  the  white  lead  behind  it,  in  a  nearly  dry 
friable  condition.  After  this  the  nail  holes,  cracks,  &c,  must  be  filled  with  common  glaziers'  putty, 
made  of  whiting  (fine  clean  washed  chalk)  and  raw  linseed  oil ;  boiled  oil  will  not  answer ;  the  putty 
would  be  friable.  The  puny  would  be  apt  to  fall  out,  if  put  in  before  priming,  because  the  wood 
would  absorb  the  oil,  and  the  putty  would  then  shrink.  After  the  first  coat  is  perfectly  dry,  the 
second  one  is  put  on  ;  and  for  it  about  1  measure  of  turp  may  be  mixed  with  3  measures  of  the  thin- 
ning oil.  In  the  third,  and  any  subsequent  coats,  equal  measures  of  turp  and  oil,  may  be  used  for 
thinning,  if  the  work  is  required  to  dry  with  a  gloss ;  but  if  it  is  to  finish  dead,  the  last  coat  must 
be  a  flatting  one;  or  one  in  which  the  "thinning  oil  is  entirely  omitted,  and  turp  alone  substituted 
for  it. 

Painters  generally  clean  their  brushes  by  merely  pressing  out  most  of  the  paint  with  a  knife ;  and 
then  keep  them  in  water  until  further  use.  If  to  be  put  away  for  some  time,  they  may  be  thoroughly 
cleaned  by  turp;  or  by  soap  and  water.  To  prevent  a  hard  skin  from  forming  on  the  top  of  their 
paint  when  not  used  for  some  days,  they  pour  on  a  little  oil. 

The  best  paints  for  preserving?  iron  exposed  to  the  weather, 

appear  to  be  pulverized  oxides  of  iron,  such  as  yellow  and  red  iron  ochres;  or  brown  hematite  iron 
ores  finely  ground;  and  simply  mixed  with  linseed  oil.  and  a  dryer.  White  lead  applied  directly  to 
the  iron,  requires  incessant  renewal :  and  indeed  probably  exerts  a  corrosive  effect.  It  may.  how- 
ever, be  applied  over  the  more  durable  colors,  when  appearance  requires  it.  Red  lead  is  said  to  be 
very  durable,  when  pure.  An  instance  is  recorded  of  pump-rods,  in  a  well  200  ft  deep,  near  London, 
which,  having  first  been  thus  painted,  were  in  use  for  45  years;  and  at  the  expiration  of  that  time, 
their  weight  was  found  to  be  precisely  the  same  as  when  new;  thus  showing  that  rust  had  not 
affected  them. 

When  the  size  of  the  exposed  iron  admits  of  it,  its  freedom  from  rust  may  be  very  much  promoted 
by  first  heating  it  thoroughly;  and  then  dipping  it  into,  or  washing  it  well  with,  hot  linseed  oil; 
which  will  then  penetrate  into  the  interior  of  the  iron.  For  tinned  iron  exposed  to  the  weather,  on 
roofs,  rain  pipes,  Ac,  Spanish  brown  is  a  very  durable  color.  The  tin  is  frequently  found  perfectly 
bright  and  protected,  when  this  color  has  been  used,  after  an  exposure  of  40  or  50  years.  White 
paint  washes  off  in  a  few  years  by  rain. 

Plastered  walls  should  if  possible  be  all  wed  to  dry  for  at  least  a  year,  before  being  painted  in  oil ; 
otherwise  the  paint  will  be  liable  to  blister.  They  may,  if  preferred,  be  frescoed  (water-colors, 
mixed  with  size)  to  the  desired  tint  during  the  interval. 

The  painting  of  unseasoned  wood  hastens  its  decay.  If  the  surface  to  be  painted  is  greasy,  the 
grease  must  first  be  removed  by  water  in  which  is  dissolved  some  lime. 

Washes  for  outside  work.  Downing,  in  his  work  on  country  houses, 
recommends  the  following:  For  wood-work;  in  a  tight  bushel,  slack  half  a  bushel  of  fresh  lime,  by 
pouring  over  it  boiling  water  sufficient  to  cover  it  4  or  5  ins  deep;  stirring  it  until  slacked.  Add  2 
tts  of  sulphate  of  zinc  (white  vitriol)  dissolved  in  water.  Add  water  enough  to  bring  all  to  the  con- 
sistence of  thick  whitewash.  Apply  with  a  whitewash  brush.  This  wash  is  white;  but  it  may  b« 
colored  by  adding  powdered  ochre  Indian  rod,  umber,  &c.  If  lampblack  is  added  to  water  colors,  U 


*  Average  cost  of  Painting*  in  Philada,  1873,  including  scaffold  &c,  per 

square  yard.  Four  coats  in  plain  colors  40  cts ;  3  coats,  35.  Graining  in  imitation  of  oak,  walnut 
&c,  90.  White  lead  ground  in  oil,  in  kees  15  cts  per  Ib.  The  cost  of  painting  and  glazing  a  20  ft  front, 
3  story  dwelling,  with  large  3  story  back  buildings,  $600  to  $700.  A  church  of  60  by  80  ft,  with  base- 
meat  story,  and  galleries,  $2500  to  $3000.  Avoid  extras ;  or  stipulate  for  them  in  advance. 


514 


GLASS,  AND   GLAZING. 


should  first  be  thoroughly  dissolved  in  alcohol.    The  sulphate  of  zin 
iu  a  tew  weeks. 


lauses  the  wash  to  become  hard 


For  brick,  masonry,  or  rougrh-cast.    Slack  ^  a  bushel  of  lime  as 

before;  then  flli  the  barret  %  full  of  water,  and  add  a  bushel  of  hydraulic-  cement.  Add  3  fts  of  sul- 
phate of  ziuc.  previously  dissolved  in  water.  The  whole  should  be  of  the  thickness  of  paint;  and 
may  be  put  on  with  a  whitewash  brush.  The  wash  is  improved  by  stirring  iu  a  peck  of  white  sand, 
just  before  using  it.  It  may  be  colored,  if  desired,  like  the  preceding. 

He  also  gives  the  following  cheap  oil-paint  for  outside  work  on  wood,  brick,  stone,  &c  ;  and  says  it 
becomes  far  harder  and  more  durable  than  common  paint:  One  measure  of  ground  fresh  quicklime; 
add  the  same  quantity  of  fine  white  sand,  or  fine  coal  ashes:  and  twice  as  much  fresh  wood  ashes; 
all  the  foregoing  to  be  passed  through  a  tine  sieve.  Mix  well  together  dry.  Mix  with  as  much  raw 
linseed  oil  as  will  make  the  mixture  as  thin  as  paint.  Apply  with  a  painter's  brush.  It  may  be  col- 
ored like  the  foregoing,  taking  care  to  mix  the  colors  well  with  oil  before  adding  them.  It  is  best  t« 

Also,  another,  said  to  stand  15  to  '20  years  :  50  fts  best  white  lead  :  10  quarts  raw  linseed  oil :  %  fl> 
dryer;  50  ftis  finely  sifted  sharp  clean  sand;  2  tt>s  raw  umber.  Add  very  little,  say  %  pint  of  tur- 
pentine. Apply  with  a  large  brush. 

Coment  for  stopping*  joints,  such  as  around  chimneys,  &c,  &c.  White 
lead  ground  in  oil,  as  sold  by  the  keg;" mixed  with  enough  pure  sand  to  make  a  stiff  paste  that  will 
not  run.  It  grows  hard  by  exposure,  and  resists  heat,  cold,  and  water.  Pieces  of  stone  may  be 
strongly  cemented  together  by  it,  allowing  a  few  mouths  for  proper  hardening. 

Whitewash  lor  inside  work.,  according  to  Mr.  Downing,  "is  made  more 
fixed  atid  permanent,  by  adding  2  quarts  of  thin  size  to  a  pailful  of  the  wash,  just  before  using. 
The  best  size  for  this  purpose  is  made  of  shreds  of  glove  leather;  but  any  clean  size  of  good  quality 
will  answer."  as  thin  glue-water.  We  will  add,  that  the  common  practice  of  mixing  salt  with  white- 
wash, should  not  be  permitted.  Paper  pasted  on  a  wall  which  has  previously  been  covered  with  salt 
whitewash,  is  very  apt  to  become  wet.  and  loose,  and  to  fall  off  during  damp  weather.  The  white- 
wash shnuld  he  scraped  off.  and  the  wall  or  partition  covered  with  a  coat  or  two  of  thin  size,  to  pro- 
tect the  paper  from  the  effect  of  the  salt  that  may  still  adhere  to  the  plaster. 


GLASS,  AND  GLAZING. 


WINDOW  glass  is  sold  by  the  box ;  and  whatever  may  be  the  size  of  the  panes,  a 
box  contains  as  nearly  50  sq  feet  of  glass  as  the  dimensions  of  the  panes  will  admit 
of  In  the  following  table,  those  numbers  which  have  no  +  after  them,  denote  that 
they  amount  to  precisely  50  sq  ft;  while  in  the  others  a  part  of  a  pane  would  have 
to  be  added  to  make  up  the  60  ft. 

Panes  of  any  size  may  be  made  to  order  by  the  manufacturers.  The  sizes  given  In  the  following 
table,  as  well  as  many  others,  are  generally  to  be  had  ready  made.  Ordinary  window  glass  of  all  the 
sizes  iu  the  table,  is  about  one-sixteenth  of  an  inch  thick  ;  and  this  is  the  thickness  supposed  to  be 
intended  when  a  greater  one  is  not  specified.  Double-thick  glass  is  nearly  ^  inch  ;  and  its  price  is 
50  per  ct  more  than  the  single  thick.  It  is  of  course  much  stronger  than'the  single.  ** 

The  panes  are  confined  to  the  sash  by  glaziers'  putty,  made  of  whiting  (powdered  chalk)  and  raw 
linseed  oil ;  and  by  small  triangular  pieces  of  thin  tin.  about  %  inch  on  a  side,  which  uphold  the 
glass  while  the  putty  is  being  put  on  ;  and  are  allowed  to  remain  afterward,  as  a  protection  while  the 
putty  continues  soft. 

TABLE  OF  XFMBERS  OF  PAXES  IX  A  BOX. 


Size  in  ins. 

Panes 
to  a  'box. 

Size  in  ins. 

Panes 
to  a  box. 

Size  in  ins. 

Panes 
to  a  box. 

Size  in  ins. 

Pane 
to  a  bo 

6X8 
6  X  10 
6  X   12 

150 
120 
100 

11  X  12 
11   X  14 
11   X  16 

54  4 
46  - 

40  - 

14  X  22 
14  X  24 
14  X  26 

23  + 
21  + 
19  + 

18  X  28 
18  X  30 
18  X  33 

14  4- 

13  4 
12  4 

7X8 

128  + 

11  X  18 

36  - 

14  X  28 

18  + 

18  X  36 

11  4 

7  X  10 
7  X  12 
7  X   14 

102  + 
85  + 
73  + 

11   X  20 
11   X  22 
12  X  14 

32  - 
29  - 
42  - 

15  X  16 
15  X  18 
15  X  20 

30 
26  + 
24 

20  X  24 
20  X  26 
20  X  28 

15 
134 
12  4 

8  X   10 

90 

12  X  16 

37  - 

15  X  22 

21  + 

20  X  30 

12 

8  X   12 

75 

12  X  18 

33  - 

15  X  24 

20 

20  X  33 

10  4; 

8  X   H 

8  X   16 

644- 
53  + 

12  X  20 
12  X  22 

30 
27  - 

15  X  26 

15  X  28 

18  + 
17  + 

20  X   36 
20  X  40 

10 
9 

9  X   10 

80 

12   X   24 

25  - 

15  X  30 

16 

22  X  26 

12  - 

9  X   12 

(H  + 

13   X   14 

39  - 

16  X  18 

25 

22  X  30 

10  - 

9  X   U 

57  + 

13  X  16 

3*  - 

16  X  20 

22  + 

22  X  36 

9  - 

9  <  n 

50 

13  X  18 

30  - 

16  X   22 

20  + 

22  X   40 

8  - 

9  X  18 

4t  + 

:j   X   20 

27  - 

16  X   24 

18  + 

22  X  44 

7  - 

T)  X  12 

60 

:»  X  22 

25  - 

1H  X   28 

16  + 

24  X  27 

n  - 

li)  X  It 

51  + 

3  X  24 

23  - 

16  X   32 

14  + 

24  X  30 

10 

1')  X  15 

48 

:',  X  26 

21  - 

18  X   20 

20 

2t  X  34 

8  - 

li)  X   18 

45 

4  X  16 

32  - 

18  X  22 

18  + 

24  X  38 

7  - 

10  X  13 

40 

4  X   18 

28  - 

18  X   24 

16  + 

24  X  42     ;       7  - 

1J  X  20 

36 

4  X  20 

25  - 

18  X   26 

15  + 

24  X  43 

6  - 

WATER. 


515 


The  best  qualities  ofXmerican  glass  made  in  the  vicinity  of  Philadelphia,  Boston,  Plttsbnrg,  Ac, 
ar«  for  most  purelyjrfse/wZ  purposes,  as  good  as  those  from  foreign  countries:  but  when  the  highest 
degree  of  &euu<//Xrequired,  as  in  the  lower  front  windows  of  first-class  dwellings,  fancy  stores,  &c, 
polished  plate-glass  of  England,  France,  or  Germany,  must  be  used;  although  the  price  for  moderate 
sized  panes  is  from  5  to  8  times  as  great  as  that  of  the  best  quality  single- thick  American,  as  given 
in  the  following  table.*  Its  perfectly  smooth  surface,  free  from  distorted  reflections,  also  makes  it  the 
best  for  covering  pictures;  still,  if  carefully  selected  American  panes  be  used  for  this  purpose,  few 
except  critics  in  glass  will  detect  the  difference.  The  polishing  of  glass  plates  is  not  done  in  the 
United  States  as  yet. 

A  thick  glass  is  made  expressly  for  flooring,  up  to  1*4  ins  thick, 

and  up  to  40  ins  by  5  ft  dimensions.  Also,  for  skylights,  from  y±  to  ^  inch  thick.  This  cim  be  fur- 
Dished  to  order  of  any  size  up  to  40  ins  by  8  or  10  ft.  Any  of  these  can  also  be  had  ground.  Grind- 
ing prevents  the  entrance  of  the  full  glare  of  the  sun  ;  and,  moreover,  diffuses  the  light  over  a  much 
greater  width  of  space  below. 

Strength  of  glass.  Tensile  2500  to  9000  8>s  per  sq  inch.  Boston  rods  by 
author,  3500  to  5200.  Crushing  strength,  6000  to  10000  fts  per  sq  inch.  Transversely,  (by  the 
writer's  trials,)  Millville,  N.  J.r  flooring  glass,  1  inch  square,  and  1  foot  between  the  end  supports, 
breaks  under  a  center  load  of  about  170  fts  ;  consequent.y,  it  is  considerably  stronger  than  granite, 
except  as  regards  crushing;  in  which  the  two  are  about  equal. 

REMARK.  Windo.w  and  other  glass  which  contains  an  excess  of  potash  or  of  soda  is  very  liable  to 
become  dull  ia  time,  owing  to  the  decomposition  of  those  ingredients  by  atmospheric  Influences. 


WATEE, 


PURE  water,  as  boiled  and  distilled,  is  composed  of  the  two  gases,  hydrogen  and 
oxygen ;  in  the  proportions  of  2  measures  hyd,  to  1  of  ox;  or  1  weight  of  hyd,  to  8 
of  ox.  Ordinarily,  however,  it  contains  several  foreign  ingredients,  as  carbonic,  and 
other  acids;  and  soluble  mineral,  or  organic  substances.  "When  it  contains  much 
lime,  it  is  said  to  be  hard  ;  and  will  not  make  a  good  lather  with  soap.  The  air  in 
its  ordinary  state  contains  about  4  grains  of  water  per  cub  ft. 

The  average  pressure  of  the  air  at  sea  level,  will  balance  a  column  of  water  of  34  ft  In  vert  height, 
or  about  30  ins  of  mercury.  At  its  boiling  point  of  212°  Fan,  its  bulk  Is  about  ^3  greater  than  at  70°. 

Us  weight  per  cub  ft  is  taken  at  62%  fts,  or  1000  ounces  avoir;  bat  62% 

fts  would  be  nearer  the  truth,  as  per  table  below.  It  is  about  815  times  heavier  than  air,  when 
both  are  at  the  temp  of  62°  ;  and  the  barom  at  30  ins.  With  barom  at  30  ins  the  wt  of  perfectly  pure 
water  is  as  follows.  At  max  density  62.425  fts. 


Lbs  per 
Cub  Ft. 


Temp  in  Deg 
of  Fah. 

32^ 6-2.417 

40° 62.423 

50° 62  409 

60° 62.367 


Temp  in  Deg 
of  Fah. 


Lbs  per 
Cub  Ft. 

70° 62.302 

80° 62.218 

90® 62.119 

212° 59.7 


*The  prices  for  American  single-thick   glass,  per  box,  in 

small  orders  are  (in  Philadelphia,  in  1873,)  approximately  as  below.    Double-thick  50  per  ct  more. 
Liberal  discounts  are  made  on  heavy  orders.    For  ground  glass  add  about  $2.50  per  box. 


Size  in  Inches. 

1st  Quality. 

2d  Quality. 

3d  Quality. 

4th  Qnality. 

From 

to 

6  by    8 

8  by  10 

$  5.00 

$  4.50 

$4.25 

$  4.00 

8  by  11 

10  bv  15 

5.25 

4.75 

4.40 

4.20 

11  by  14 

12  bv  18 

6.00 

5.50 

5.00 

4.65 

14  by  16 

16  bv  24 

6.25 

6.75 

5.25 

4.90 

18  by  22 

18  by  30 

7.50 

6.75 

5.75 

5.25 

20  by  30 

24  by  30 

9.25 

8.25 

6.50 

6.00 

24  by  31 

24  by  36 

10.00 

8.75 

7.00 

6.25 

28  by  46 

30  by  48 

11.50 

10.50 

8.50 

30  bv  50 

82  bv  52 

12.25 

11.25 

9.00 

34  by  58 

34  by  60 

15.00 

14.00 

11.00 

The  charge  by  glaziers  for  putting  the  glass  into  new  windows,  including  putty,  tins,  and  two  coats 
of  paint  to  the  sash,  (one  of  which  is  a  priming  coat,)  is  (1873)  equal  to  the  co'st  of  the  glass  or  the 
above  prices.  See  footnote,  p  513. 

For  reglazing  old  sash,  removing  the  broken  panes,  the  charge  is  about  twice  as  great.  In  small 
quantities,  the  following  are  also  approximate  prices  for  American  glass  :  Large  plates  of  U  inch 
thick,  75  cts  to  $1  per  sq  ft.  One  inch  thick.  Si  .40  to  $1.80;  if  either  is  ground.  10  to  15  cts  addi- 
tional per  sq  ft.  Ribbed  glass,  ^  inch  thick,  50 cts  ;  Klnch,60ctspersqft.  Stained  glass,  single  thick- 
ness, (^  inch,)  or  figured  white  enamelled  glass,  (single  thickness,)  60  to  75  cts  per  sq  ft.  Superior 
thicker  strong  figured  glass,  first  ground,  and  the  transparent  figures  then  formed  by  polishing  awar 
portions  of  the  ground  surface,  $1.25  to  $2.00  per  sq  ft. 

Mufted  glass  is  an  inferior  article  of  fanciful  colored  patterns,  attached  by  some  Imperfect  process 
which  allows  them  to  peel  off  after  a  year  or  two  of  exposure  to  the  weather. 


516 


WATER. 


The  weight  of  water  affords  an  easy  way  to  find  the  cubic  contents  of  a  vessel.  First  weigh  the  ves- 
sel by  itself;  and  then  full  of  water.  The  diff  will  be  the  weight  of  the  water ;  and  this  divided  by 
62.3  or  by  the  number  in  the  table  opp  the  temp  of  water,  will  be  the  contents  in  cub  ft. 

To  obtain  the  size  of  commercial  measures  by  means  of  the 
weight  of  water. 

At  the  common  temperature  of  from  70°  to  75°  Fan,  a  cub  foot  of  fresh  water  weighs  very  approxi- 
mately 62L1  B)d  avoir.  A  cubic  half  foot,  (6  ius  on  each  edge,)  7.78125  fi>s.  A  cub  quarter  foot,  (3  ina 
on  each  edge,)  .97268  B>.  A  cub  yard,  1680.75  Ibs;  or  .75034  ton.  A  cub  half  yd,  (18  ins  on  each  edge,) 
•A:).094  fos:  or  .0938  ton.  A  cub'iuch,  .036024  ft  ;  or  .576384 ounce;  or  9.2222  drams  ;  or  252. 170  grains. 
An  inch  square,  and  one  foot  long,  .432292  B>.  Also  1  ft>  =•  27.75903  cub  ins,  or  a  cube  of  3.028  ins  on  au 
edge.  An  ounce,  1.735  cub  ins  ;  a  ton,  35.984  cub  ft,  all  near  enough  for  common  use. 

Original. 


Liquid  Measures. 

r.  s.  cm  

U    S    Pint                           

Lbs  Avoir, 
of  "Water. 
.26005* 
1.0402 

Liquid  and  I>ry.    Lbs  Avoir. 

of  Water. 
British  Imp  Gill  31211* 
"        "     Pint  ,  1.248:iR 

V   S    Quart                                   

2.0804 
8.3216 
262.1310 

1  2104 

"        "     Quart... 

2  49715 

U.  S.  Gallon  8  Ibs  5^  oz  
U.  S.  Wine  Barrel,  31^  Gall  

I>ry  Measures. 

U   S   Pint. 

"        "      Gallon.. 

9  9SS6 

"        "     Peck 

19  9772 

"        "     Bushel.. 

.79  9088 

*  4.9942  ;  or  ve 

French 

Centilitre  

•y  nearly  5  ounces. 

Measures. 

021981 

U    S    Quart 

.     2.4208 

.     9.6834 

U.  S.  Gallon  

U   S   Peck 

19  3668 

Decilitre  

2198J 

*  Or  4  ounces  ;  2  drams  ;  15.6625  grs. 

Litre  

2.1981 

Decalitre,  or  Ceutist 

ere  21.9808 
2198  0786 

t  Or  5.6271  drai 
J  3.5169  ounces. 

ns  ;  or  153.866  gra. 

Its  max  density  is  when  its  temp  is  a  little  more  than  39°  Fah ;  or  about  7° 

warmer  than  the  freezing  point.  By  best  authorities  39.2°.  From  about  39°  it  expands  either  by 
cold,  or  by  heat.  When  the  temp  of  32°  reduces  it  to  Ice,  its  wt  is  but  about  57.2  Ibs  per  cub  ft : 
aud  its  sp  gr  about  .9175,  according  to  the  latest  determination  by  L.  Dutour.  Hence,  as  ice,  it  haa 
expanded  J^-  part  of  its  original  bulk  as  water ;  and  the  sudden  expansive  force  exerted  at  the  mo- 
ment of  freezing,  is  sufficiently  great  to  split  iron  water-pipes  ;  being  probably  not  less  than  30000  Ibs 
per  sq  inch.  Instances  have  occurred  of  its  splitting  cast  tubular  posts  of  iron  bridges,  aud  of  ordi- 
nary buildings,  when  full  of  rain  water  from  exposure.  It  also  loosens  and  throws  down  masses  of 
rock,  through  the  joints  of  which  rain  or  spring  water  has  found  its  way.  Retaining-  walls  also  are  some- 
times  overthrown,  or  at  least  bulged,  by  the  freezing  of  water  which  has  settled  between  their  backs 
and  the  earth  filling  which  they  sustain  ;  and  walls  which  are  not  founded  at  a  sufficient  depth,  are 
often  lifted  upward  by  the  same  process. 

It  is  said  that  in  a  glass  tube  }\  inch  in  diarn,  water  will  not  freeze  until  the  temp  is  reduced  to 
23°;  and  in  tubes  of  less  than  -^Q- inch,  to  3°  or  4°.  Neither  will  it  freeze  until  considerably  colder 
than  32°  in  rapid  running  streams.  Anchor  ice,  sometimes  found  at  depths  as 
great  as  25  ft.  consists  of  an  aggregation  of  small  crystals  or  needles  of  ice  frozen  at  the  surface  of 
rapid  open  water ;  and  probably  carried  below  by  the  force  of  the  stream.  It  does  not  form  under 
frozen  water. 

Since  ice  floats  in  -water;  and  a  floating  body  displaces  a  wt  of  the  liquid 

equal  to  its  own  wt,  it  follows  that  a  cub  ft  of  floating  ice  weighing  57.2  fts,  must  displace  57.2  R>-*  of 
water.  But  57.2  Ibs  of  water,  one  ft  square,  is  11  ius  deep;  therefore,  floating  ice  of  a  cubical  or  par- 
allelopipedal  shape,  will  have  y^-  of  its  volume  under  water;  and  only  y1^  above;  and  a  sq  ft  of  ice 
of  any  thickness,  will  require  a  wt  equal  to  y*y  of  its  own  wt  to  sink  it  to~the  surf  of  the  water.  In 
practice,  however,  this  must  be  regarded  merely  as  a  close  approximation,  since  the  wt  of  ice  is  some- 
what affected  by  enclosed  air- bubbles. 

Pure  water  is  usually  assumed  to  boil  at  212°  F,  in  the  open  air,  at  the  level  of 
the  sea ;  the  barom  being  at  30  ins ;  and  at  about  1°  less  for  every  520  ft  above  sea 


at  from  212°  to  220°;  and  it  is  stated  that  if  all  air  be  previously  extracted,  it 
requires  275°. 

It  evaporates  at  all  temps;  dissolves  more  substances  than  any  other 
agent ;  and  has  a  greater  capacity  for  heat  than  any  other  known  substance. 

It  is  compressed  at  the  rate  of  about  the  ^T^?TT  part,  (or  about  T^  of  an 
inch  in  ISj1^  ft,)  by  each  atmosphere  or  pressure  of  15  Ibs  per  sq  inch.  Wheu 
the  pres  is  removed,  its  elasticity  restores  its  original  bulk. 

Effect  oil  metals.  The  lime  contained  in  many  waters,  forms  deposits  in 
metallic  water-pipes ,  and  in  channels  of  earthenware,  or  of  masonry ;  especially  if  the  current  b« 


WATER.  517 

•low.  Some  other^tffTstances  do  the  same;  obstructing  the  flow  of  the  water  to  such  an  extent,  that 
it  is  always  expedient  to  us'e  pipes  of  diameters  larger  than  would  otherwise  be  necessary.  See  Hy- 
draulics, Ar><Z7.  The  lime  also  forms  very  hard  incrustations  at  the  bottoms  of  boilers;  very  much 
iinpairin^their  efficiency  ;  and  rendering  them  more  liable  to  burst.  It  is,  therefore,  unfit  tor  loco- 
motives. We  have  seen  it  stated  that  the  Southwestern  11  R  Co,  England,  prevent  this  lime  deposit, 
along  their  limestone  sections,  by  dissolving  1  ounce  of  sal-ammoniac  to  feO  gallons  of  water. 

The  salt  of  sea  water  forms  similar  deposits  in  boilers;  as  also  does  mud,  and  other  impurities. 

Water,  either  when  very  pure,  as  rain  water;  or  when  it  contains  carbonic  acid,  (as  most  water 
does,)  produces  carbonate  of  lead  in  lead  pipes;  and  as  tnis  is  an  active  poison,  such  pipes  should 
not  be  used  for  such  waters.  Tinned  lead  pipes  may  be  substituted  for  them.  If,  however,  sulphate  of 
lime  also  be  present,  as  is  very  frequently  the  case,  this  eflect  is  not  always  produced;  and  several 
other  substances  usually  found  in  spring  and  river  water,  also  diminish  it  to  a  greater  or  less  degree. 
Fresh  water  corrodes  wrought  iron  more  rapidly  than  cast ;  but  the  reverse  appears  to  be  the  case 
with  sea  water;  although  it  also  affects  wrought  iron  very  quickly ;  so  that  thick  flakes  may  be  de- 
tached from  it  with  ease.  The  corrosion  of  iron  or  steel  by  sea  water  increases  with  the  carbon. 
Cast-iron  cannons  from  a  vessel  which  had  been  sunk  in  the  fresh  water  of  the  Delaware  River  for 
more  than  40  years,  were  perfectly  free  from  rust.  Gen  Pasley,  who  had  examined  the  metals  found 
in  the  ships  Royal  George,  and  Edgar,  the  first  of  which  had  remained  sunk  in  the  sea  for  62  years, 
and  the  last  for  133  years,  "stated  that  the  cast  iron  had  generally  become  quite  soft;  and  in  some 
cases  resembled  plumbago.  Some  of  the  shot  when  exposed  to  the  air  became  hot ;  and  burst  into 
many  pieces.  The  wrought  iron  was  not  so  much  injured,  except  when  in  contact  Kith  copper,  or 
brass  gun-metal.  Neither  of  these  last  were  much  affected,  except  when  in  contact  with  iron.  Some 
of  the  wrought  iron  was  reworked  by  a  blacksmith,  and  pronounced  superior  to  modern  iron."  "  Mr. 
Cottam  stated  that  some  of  the  guns  had  been  carefully  removed  in  their  soft  state,  to  the  Tower  of 
London;  and  in  time  (within  4  years)  resumed  their  original  hardness.  Brass  cannons  from  the 
Mary  Rose,  which  had  been  sunk  in  the  sea  for  292  years,  were  considerably  honeycombed  in  spots 
only;  (perhaps  where  iron  had  been  in  contact  with'them.)  The  old  cannons,  of  wrought-iron  bars 
hooped  together,  were  corroded  about  y±  inch  deep ;  but  had  probably  been  protected  by  mud.  The 
oast-iron  shot  became  redhot  on  exposure  to  the  air;  and  fell  to  pieces  like  dry  clay  1  " 

"  Unprotected  parts  of  cast-iron  sluice- valves,  on  the  sea  gates  of  the  Calidonian  canal,  were  con- 
verted into  a  soft  plumbaginous  substance,  to  a  depth  of  %  of  an  inch,  within  4  years:  but  where 
they  had  been  coated  with  common  Swedish  tar.  they  were  entirely  uninjured.  This  softening  effect 
on  cast  iron  appears  to  be  as  rapid  even  when  the  water  is  but  slightly  brackish  ;  and  that  only  at  inter- 
vals. It  also  takes  place  on  oast  iron  imbedded  in  salt  earth.  Some  water  pipes  thus  laid  near  the 
Liverpool  docks,  at  the  expiration  of  20  years  were  soft  enough  to  be  cut  by  a  knife;  while  the  same 
kind,  on  higher  ground  beyond  the  influence  of  the  sea  water,  were  as  good  as  new  at  the  end  of  50 
years." 

Observation  has.  however,  shown  that  the  rapidity  of  this  action  depends  much  on  the  quality  of 
the  iron  ;  that  which  is  dark-colored,  and  contains  much  carbon  mechanically  combined  with  it,  cor- 
rodes most  rapidly ;  while  hard  white,  or  light-gray  castings  remain  secure  for  a  long  time.  Some 
oast-iron  sea-piles  of  this  character,  showed  no  deterioration  in  40  years.  See  note,  p  324. 

Contact  with  brass  or  copper  is  said  to  induce  a  galvanic  action  which 
greatly  hastens  decay  in  either  fresh  or  salt  water.  Some  muskets  were  recovered  from  a  wreck  which 
had  been  submerged  in  sea  water  for  70  years  near  New  York.  The  brass  parts  were  in  perfect  con- 
dition ;  but  the  iron  parts  had  entirely  disappeared.  Galvanizing  (coating  with  zinc;  acts  as  a  pre- 
servative to  the  iron,  but  at  the  expense  of  the  zinc,  which  soon  disappears.  The  iron  then  corrodes. 
If  iron  be  well  heated,  and  then  coated  with  hot  coal-tar,  it  will  resist  the  action  of  either  salt  or  fresh 
water  for  many  years.  It  is  very  important  that  the  tar  be  perfectly  purified.  Such  a  coating,  or  one 
of  paint,  will  not  prevent  barnacles  and  other  shells  from  attaching  themselves  to  the  iron.  Asphal- 
turn,  if  pure,  answers  as  well  as  coal  tar.  Copper  and  bronze  are  very  little  affected  by  sea  water. 

No  galvanic  action  has  been  detected  where  brass  ferules  are  Inserted  into  the  water  pipes  in  Philada. 

The  most  prejudicial  exposure  for  iron,  as  well  as  for  wood,  is  that 

to  alternate  wet  and  dry.  At  some  dangerous  spots  in  Long  Island  Sound,  it  has  been  the  practice  to 
drive  round  bars  of  rolled  iron  about  4  inches  diara,  for  supporting  signals.  These  wear  away  most 
rapidly  between  high  and  low  water;  at  the  rate  of  about  an  inch  in  depth  in  20  years:  in  which 
time  the  4-inch  bar  becomes  reduced  to  a  2  inch  one,  along  that  portion  of  it.  Under  fresh  water 
especially,  or  under  ground,  a  thin  coating  of  such  tar,  so  applied,  will  protect  iron,  such  as  water- 
pipes,  >tc,  for  a  long  time.  The  sulphuric  acid  contained  in  the  water  from  coal  mines  corrodes  iron 
pipes  rapidly. 

In  the  fresh  water  of  canals,  iron  boats  have  continued  in  service  from 

20  to  40  years. 

Wood  remains  sound  for  centuries  under  either  fresh  or  salt  water,  if  not  exposed  to  be  worn  away 
by  the  action  of  currents  ;  or  to  be  destroyed  by  marine  insects. 

Sea  water  differs  a  little  in  weight,  at  diff  places ;  but  at  the  same  place  it  is  appreciably  the  same 
at  all  depths;  and  may  be  generally  assumed  at  about  64  Ibs;  or  1%  per  cub  ft  more  than  fresh.  The 
additional  \%  fts,  or  -$\-%  part  of  its  entire  weight,  is  chiefly  common  salt.  Sea  water  freezes  at 
27°  Fah ;  the  ice  is  fresh.' 

A  teaspoonful  of  powdered  alum,  well  stirred  into  a  bucket  of  dirty  water,  will  generally  purify  it 
sufficiently  within  a  few  hours  to  be  drinkable.  If  a  hole  3  or  4  ft  deep  be  dug  in  the  sand  of  the  sea- 
shore, the  infiltrating  water  will  usually  be  sufficiently  fresh  for  washing  with  soap ;  or  even  for 
drinking.  It  is  also  stated  that  water  may  be  preserved  sweet  for  many  years  by  placing  in  the  con- 
taining vessel  1  ounce  of  black  oxide  of  manganese  for  each  gallon  of  water. 

It  Is  said  that  water  kept  ill  zsinc  tanks;  or  flowing  through  iron 
tubes  galvanized  inside,  rapidly  becomes  poisoned  by  soluble  salts  of  zinc  formed  thereby  ;  and  it  is 
recommended  to  coat  zinc  surfaces  with  asphalt  varnish  to  prevent  this.  Yft.  in  the  city  of  Hartford, 
Conn,  service  pipes  of  iron  galvanized  inside  and  out,  were  adopted  in  1855.  fit  the  recommendation 
of  the  water  commissioners;  and  have  been  in  use  ever  since.  They  are  likewise  need  in  Philadel- 
phia and  other  cities  to  a  considerable  extent.  In  manv  hotels  and  other  buildincs  in  Boston,  the 
"  Seamless  Drawn  Brass  Tube  "  of  the  American  Tubes  Works  at  Boston,  has  for  many  years  been  in 
use  for  service  pipe;  and  has  given  great  satisfaction.  It  is  stated  that  the  softest  water  may  be  kept 
In  brass  vessels  for  years  without  any  deleterious  result. 

The  action  of  lead  upon  some  waters  (even  pure  ones)  is  highly  poisonous.    The  subject,  however, 


518 


RAIN. 


is  a  complicated  one.  An  Injurious  ingredient  may  be  attended  by  another  which  neutralizes  its 
action.  Organic  mutter,  whether  vegetable  or  animal,  is  injurious.  Garbouic  acid,  when  uotiu  excess, 
is  harmless,  bee  near  bottom  of  page  379. 

Ice  may  be  so  impure  that  its  water  is  dangerous  to  drink. 


RAIN. 


The  quantity  that  falls  annually  in  any  one  place,  varies 

greatly  from  year  to  year  ;  the  extremes  being  frequently  greater  than  2  to  1.  In  making  calculations 
for  collecting  water  in  reservoirs,  whether  for  feeding  canals,  or  for  supplying  cities,  we  cannot  safely 
assume  more  than  the  minimum  fall  observed  for  many  years  ;  or  rather,  somewhat  less.  And  from 
even  this  must  be  deducted  the  amount  (a  quite  considerable  one)  lost  by  evaporation  and  leakage 
after  it  has  been  collected.  The  following  table  shows  in  some  cases,  the  average  annual  falls  j  and 
in  others,  the  least  and  the  greatest  ones  observed  at  several  places ;  including  snow  -water.  It  ia 
highly  probable  that  most  of  the  results  are  merely  approximate.  See  Evaporation,  p.  521. 


Inches 
per  au. 

Augusta,  Georgia 23 

Albany,  N.York 31  to  51 

Arkansas 41 

Bath,  Maine 30  to  50 


Inches 
per  an. 
Port  Laramie,  Nebraska  ...............      20 

Fort  Worth,  Texas  ....................      41 

FortMcIntosh,  " 


,        .................... 

Fort  Dallas,  Oregon  ................... 


30  to  39 

34  to  45 

35 


, 

Key  West,  Flor 
Lebanon,  Pe 
Michigan 
Monterey,  Cal  ..............  .  ..........      12^ 

Marietta,  Ohio  ..............  .  .........  35  to  54 

New  Orleans,  Louisiana  ...............      51 

New  Fane,  Vermont  ..................  36  to  74  1 

New  England  ...............  average.  .      47 

Natchez,  Miss  .........................  37  to  58 

New  York  State  ............  average..      36J4 

Ohio  .......................        "       ..      36 

Philadelphia,  Penna  ..................  23  to  59 

av  for  32  years,  to  1870.  .  .       45.8  * 
Pennsylvania  ..............  average..      41 

Savannah,  Georgia  ...................  30  to  60 

Stow,  Mass  ...........................  33  to  49 

St.Louis,  Mo  ..........................      42 

Washington,  D.  C  ................  .  ----      41 

West  Chester,  Penna  ..................  39  to  54 

Williamstown,  Mass  ...................  26  to  39 


Baltimore,  Md 40 

Boston,  Mass 25  to  46 

Charleston,  S.  C 40  to  76  ! 

Canada 36 

Carlisle,  Penna 34 

Detroit,  Michigan 30 

Frankford,  Peuna 33  to  54 

Fort  Gaston,  California,  in  9  months.     129 

Fort  Yuma,  Cal 3}£ 

Port  (not  Fort)  Orford,  Oregon 69 

Fort  Pike,  Louisiana 72 

Fort  Pierce,  E.  Florida 63 

Fort  Conrad.  New  Mexico 6% 

Fort  Kent,  Maine 36»4 

Fort  Preble,     "     45^ 

Fort  Constitution,  N.  Hamp 35^ 

Fort  Adams,  Rhode  Island 52>$ 

Fort  Hamilton,  N.  York  Harbor 43% 

Fort  Niagara,  N.Y 3H{ 

Fort  Monroe,  Virg 51 

Fort  Kearney,  Nebraska 28 

The  greatest  fall  recorded  in  one  day  in  Philada,  was  6  ins, 

•which  fell  iu  2  hours,  in  July,  1842.  The  greatest  in  any  month,  was  12  ins,  in  the  same  month.  It 
has  not  reached  9  ins  per  month,  more  than  5  or  6  times,  in  25  years.  During  a  tremendous  rain  at 
Norristown,  Ponna,  in  1865,  the  writer  saw  evidence  that  at  least  9  ins  fell  within  5  hours.  At  Ge- 
noa, Italy,  on  one  occasion,  32  ins  fell  in  24  hours  ;  at  Geneva.  Switzerland,  H  ins  in  3  hours  ;  at  Mar- 
seilles, France,  13  ins  iu  14  hours;  in  Chicago,  Sept.  1878,  .97  inch  in  7  minutes. 

Near  London,  England,  the  mean  total  fall  for  many  years  is  23  ins.  On 
one  occasion,  6  ins  fell  in  1%  hour!  In  the  mountain  districts  of  the  English  lakes,  the  fail  is  enor- 
mous ;  reaching  in  some  years  to  180,  or  240  ins ;  or  from  15  to  20  ft !  while,  in  the  adjacent  neighbor- 
hood, it  is  but  40  to  60  ins.  At  Liverpool,  the  average  is  34  ins;  at  Edinburg,  30;  Glasgow,  22;  Ire- 
1'iiirt.  36;  Madras,  47;  Calcutta,  60;  maximum  for  16  years,  82;  Delhi,  21;  Gibraltar,  30;  Adelaide, 
Australia,  23  :  West  Indies.  36  to  9(i ;  Rome,  39.  On  the  Khassya  hills  north  of  Calcutta,  5CO  ins,  or 
41  ft  8  in«.  have  fallen  in  the  6  rainy  mouths  I  la  other  mountainous  districts  of  India,  annual  falls 
of  10  to  20  ft  are  common. 

It  requires  a  quite  heavy  rain,  for  24  hours,  to  yield  the  depth  of  an  inch;  still,  inasmuch  as  at 
rare  iutervals  falls  of  as  much  as  from  1  to  3,  or  even  6  ins  per  hour  occur,  these  latter  depths 
must  be  considered  in  planning  sewers,  culverts,  etc.  See  Remark,  p  566. 

As  a  general  rule,  more  rain  falls  in  warm  than  in  cold 
countries;  and  more  in  elevated  regions  than  in  low  ones.  Local  peculiarities, 
however,  sometimes  reverse  this;  and  also  cause  great  diffs  in  the  amounts  in  places  quite  near  each 
other;  as  in  the  English  lake  districts  just  alluded  to.  It  is  sometimes  difficult  to  account  for  these 
variations.  In  some  lagoons  in  New  Granada,  South  America,  the  writer  has  known  three  or  lour 
heavy  rains  to  occur  weekly  for  some  months,  during  which  not  a  drop  fell  on  hills  about  1000  feet 
hk'h.  \\irhin  10  miles'  distance,  and  within  full  sight.  At  another  locality,  almost  a  dead-level  plain, 
fullv  %  <>f  the  rains  that  fell  for  2  years,  at  a  spot  2  miles  from  his  residence,  occurred  in  the  morning ; 
•while  those  which  fell  about  3  miles  from  it,  in  an  opposite  direction,  were  in  the  afternoon. 

"  The  returns  of  several  rain  gauges  in  the  Longdendale  district,  England,  for  1847,  gave  the  rain- 
falls at  diff  altitudes  above  the  sea,  as  follows:  " 


At  1750  ft  altitude  ...........  56.5  ins. 

1800          "          ...........  62.1   " 


At  500  ft  altitude  ..............  4fi.fi  ins. 

809          "         ..............  50.5    " 

1700          "         ..............  52.1    " 

"  The  annual  average  fall  at  Edinhurp,  200  ft  above  the  sea.  In  three  successive  years,  -was  30  ins. 
In  the  Pent-land  hills,  a  few  miles  south,  and  700  ft  above  sea,  37.4  ins  ;  and  at  Carlops,  similarly  situ- 
ated  near  the  last,  but  900  ft  above  sea,  49.2  ins." 

*  In  1869.  during  which  occurred  the  greatest  drought  known  in  Philada  for  at  least  50  years,  ii 

•was  4S.84  ius. 


-AIR.  519 


There  ai*T»robably  but  few  places  in  the  United  States, 

•where  an  annual  fall  of  2  feet  may  not  be  safely  relied  on;  and  since,  as  an  ordinary  average,  cue- 
half  of  it  majrbe  collected  into  reservoirs,  we  should  have  for  a  square  mile  of  drainage,  one  foot  deep, 
2787S4GO  cift  ft  ;  equal  tt»  7<w79  cub  ft  per  day.  Allowing  4  cub  ft,  or  30  gallons  per  day  for  each  per- 
son; and  making  no.  deduction  for  evaporation  and  filtration,  this  would  supply  a  population  of  100^)5 
persons  ;  or  a  square  of  33'4  ft  on  a  side,  would  in  like  manner  suffice  for  one  person.  From  -^-t-  to 
y8^-  of  all  the  water  annually  resulting  from  rain  and  snow,  passes  off  into  the  neighboring  rivulets; 
and  thence  into  tlie  larger  streams  and  rivers  ;  or  may  be  collected  into  reservoirs.  Under  ordinary 
circumstances  of  locality,  about  M  may  usually  be  thus  secured.  The  difference  is  owing  chiefly  to 
the  distance  which  the  water  may  have  to  run  ;  the  rntes  of  absorption  of  various  soil'  ;  the  rate  of 
descent  of  the  sides  of  the  valleys  leading  to  the  streams;  the  season  of  the  year,  &c,  &c. 

All  inch  of  rain  amounts  to  363O  cub  ft:  or  27155  U.S.  gals;  or  101.3 

tons  per  acre  ;  or  to  2323200  cub  ft  ;  or  17378743  U.  S.  gals  ;  or  64821  tons  per  sq  mile  at  62^  fos. 

The  most  destructive  rains  are  usually  those  which  fall  upon  snow,  under  which  the  ground  is  frozen 
BO  as  not  to  absorb  water. 


SNOW, 


Trials  at  difT  times  by  the  writer,  showed  the  wt  of  freshly  fallen 

enow  to  vary  from  about  5  to  12  Bis  per  cub  ft;  apparently  depending  chiefly  upon  the  degree  of  hu- 
midity of  the  air  through  which  it  had  passed.  On  oue  occasion  when  mingled  snow  and  hail  had 
fallen  to  the  depth  of  6  ins,  he  fouud  its  wt  to  be  31  E>s  per  cub  ft.  It  was  very  dry  and  incoherent. 
A  cub  ft  of  heavy  snow  may,  by  a  gentle  sprinkling  of  water,  be  converted  into  ano'ut  half  a  cub  foot 
of  slush,  weighing  20  Ibs ;  'which  will  not  slide  or  run  off  from  a  shingled  roof  sloping  30°,  if  the 
weather  is  cold.  A  cub  block  of  snow  saturated  with  water  until  it  weighed  45  H>s  per  cub  fr,  just  sl'd 
on  a  rough  board  inclined  at  45°;  on  a  smoothly  planed  one  at  30°  j  and  on  slate  at  18°;  all  approxi- 
mate. A  prism  of  snow,  saturated  to  52  E>s  per  cub  ft;  one  inch  square,  and  4  ins  high,  here  a  wt  of  7 
Ibs  ;  which  at  first  compressed  it  about  }£  part  of  its  length.  European  engineers  consider  6  fts  per 
sq  ft  of  roof,  to  be  sufficient  allowance  for  the  wt  of  snow  ;  and  8  Ibs  for  the  pressure  of  wind ;  total, 
14  Ibs.  The  writer  thinks  that  in  the  U.  S.  the  allowance  for  snow  should  not  be  taken  at  las  than 
12  EH  ;  or  t'.)e  total  for  snow  and  wind,  at  20  Bs.  There  is  no  dancer  that  snow  on  a  roof  will  become 
saturated  to  the  extent  just  alluded  to;  because  a  rain  that  would  supply  the  necessary  quantitv  of 
water,  would  also  by  its  violence  wash  away  the  snow;  but  we  entertain  no  doubt  whatever  that  the 
united  pressures  from  snow  and  wind,  in  o'ur  Northern  States,  do  actually  at,  times  reach,  and  even 
surpass,  20  Ibs  per  pq  ft  of  roof.  See  Table  4,  p  301,  of  Trusses.  The  limit  of  pprpetnnl  snow  at,  the 
equator  is  nt  the  height  of  about  1GOOO  ft,  or  *ij  3  miles  above  sea-lfvel;  in  lat  45°  north  or  south,  it 
is  about  half  that  height;  while  near  the  poles  it  is  about  at  sea  level. 


AIE.-ATMOSPHEEE, 

The  atmosphere  is  known  to  extend  to  at  least  45  miles 
above  the  earth.  Its  composition  is  about  .79  measures  of  nitrogen  gas,  and 

.21  of  oxygen  gas ;  or  about  .77  nit,  .23  ox,  by  weight.  It  generally  contains,  however,  a  trace  of 
•water;  carbonic  acid,  aud  carburetted  hydrogen  gases;  and  still  less  ammonia. 

Vhen  the  barom  is  at  30  ins,  aud  the  temperature  60°  Fah,  air  weighs  about  ^  ^  part  as  much  as 
•water;  or  535  grains-  1.224  commercial  ounces  —  .0765  commercial  ft,  per  cub  ft.  Or  13.072  cub  ft 
weigh  1  ft  ;  uutJ  a  cub  yd,  2.0(>6  B>s.  Or  a  cube  of  30.82  ft  on  each  edge,  1  ton.  When  colder  it  weighs 
more  per  cub  ft,  and  vice  versa,  at  the  rate  of  about  a  grain  per  deg  of  Fah.  The  average  weight  of 
the  entire  atmospheric  column,  (at  least  45  miles  high,)  at  sea  level,  is  14%  fts  avoir  per  sq  inch  ;  or 
2124  fts  per  sq  f t ;  =  weight  of  a  column  of  water  34  ft.  or  of  mercury  30  inches  nigjj.  ,  ins  u 
what  is  usually  called  the  "  pressure  of  the  air."  At  %  mile  above  sea  level  it  is  but  14.02  fts  per 
sq  inch;  at  %  mile  13.33;  at  %  mile,  12.66;  at  1  mile,  12.02;  &tl%  mile,  11.42;  at  1^  mile  10  88- 
aud  at  2  miles,  9.80  Bxs.  Therefore,  a  pump  in  a  high  region,  will  not  lift  water  to  as  g'reat  a  height 
as  in  a  low  one.  The  pres  of  air,  like  that  of  water,  is,  at  any  given  point,  equal  in  all  directions 

Tt  is  often  stated  that  the  temperature  of  the  atmosphere  lowers  or  becomes  colder,  at  the  rate  of  1° 
Fah  for  each  300  ft  of  ascent  above  the  enrth's  surface  ;  but  this  is  liable  to  many  exceptions,  and  varies 
much  with  local  causes.  Actual  observation  in  balloons  seems  to  snow  that  up  to  the  first  1000  ft,  about 
200  ft  to  1°,  is  nearer  the  truth  ;  at  2000  ft,  250 ;  at  4000  ft,  300 ;  and  at  a  mile.  350. 

In  breathing,  a  grown  person  at  rest  requires  from  .25  to  .35  of  a  cub  ft  of 

air  per  minute;  which  when  breathed  vitiates  from  3^  to  5  cub  ft.  When  walking  or  hard  at  work, 
he  breathes  and  vitiates  two  or  three  times  as  much.  About  5  cub  ft  of  fresh  JMI-  per  i>ei>ou  per 
minute  is  reqd  for  the  perfect  ventilation  of  rooms  in  winter;  8  in  summer.  Hospitals  40  to  80. 

Beneath  the  general  level  of  the  surface  of  the  earth  in  temperate  re- 

5 ions,  a  tolerably  uniform  temp  of  about  50°  to  60°  Fah  exists  at  the  depth  of  about  50  to  60  ft ;  and 
acreases  about  1°  for  each  additional  50  to  60  ft;  all  subject,  however,  to  considerable  deviations 
from  n.any  lociil  causes.  In  the  Rose  Bridge  colliery.  Kngland,  at  the  depth  of  2424  ft.  the  tempera 
ture  of  the  coal  is  93.4°  Fah;  and  at  the  bottom  of  a' boring  4lfi9  ft  deep,  near  Berlin,  the  temp  is  119° 

The  air  is  a  very  slow  oomlnetor  of  heat;  hence  hollow  walls  serve 

to  retain  the  heat  in  dwellings;  besides  keeping  them  dry.  It  rushes  into  a  vacuum  nenr  sea  level 
with  a  vel  of  about  1157ft  por  sec:  or  JS^  miles  per  minute;  or  about  as  fast  as  sound  ordinarily 
travels  through  quiet  air.  See  Sound,  p  173. 

Ulie  all  other  elastic  fluids,  it  expands  equally  with  equal 
increases  of  temperatnre.  Every  increase  of  5°  Fah,  expands  tho  luilk 
of  any  of  them  slieht.lv  more  than  1  per  cent  of  that  which  it  has  at  0°  Fah  ;  or  500°  about  doubles  its 
bulk  at  zero.  The  bulk  of  any  of  them  diminishes  inversely  in  proportion  to  the  total  pressure  to 


520 


WIND. 


which  it  is  subjected.    Thus,  if  we  have  a  cylinder  open  at  top,  aud  1  ft  deep,  full  of  air  at  it«  natural 

Srea  of  about  15  tt»s  per  sq  inch ;  if  by  means  of  a  piston  we  apply  an  additional  pres  of  15  fts  per  aq 
ich,  making  30  fts  in  all,  or  twice  as  much  as  the  nut  pres,  theu  the  uir  will  be  compressed  into  tt 
ins  of  depth  of  the  cylinder,  or  cue-half  of  what  it  occupied  before.  Or  it  we  apply  45  fts  additional, 
making  60  fts  ia  all,  or  4  times  the  natural  pres,  then  the  air  will  be  compressed  into  y^  of  the  depth 
of  the  cylinder.  Experiment  shows  that  this  holds  good  with  air  at  least  up  to  pressures  of  about  750 
Ibs  per  sq  inch,  or  50  times  its  uat  pres;  the  air  in  this  case  occupying  the  -^  part  of  its  natural 
bulk.  In  like  manner  the  bulk  will  increase  as  the  total  pres  is  diminished  ;  so  that  if  we  remove  our 
additional  45  fts  per  sq  inch,  the  air  in  the  cylinder  will  regain  its  original  bulk,  aud  will  precisely  fill 
the  cylinder.  Substances  which  follow  these  laws,  are  said  to  be  perfectly  tUstic.  Under  a  pres  of 
about  5><|  tons  per  sq  inch,  air  would  become  as  dense,  or  would  weigh  as  much  per  cub  ft,  as  water. 
Since  the  air  at  the  surface  of  the  earth  is  pressed  14%  fts  per  sq  inch  by  the  atmosphere  above  it, 
and  since  this  is  equal  to  the  wt  of  a  column  of  water  1  inch  sq,  and  34  ft  high,  it  follows  that  at 
depths  of  34,  68,  102  ft,  &c.  below  water,  air  will  be  compressed  into  %,  %,  y±.  &c,  of  its  bulk  at  the 
surf;  because  at  those  depths  it  is  exposed  to  pressures  equal  to  2,  3,  4,  &c,  times  14%  fts  per  sq  inch, 
inasmuch  as  the  pres  of  the  atmosphere  on  the  surf,  is  in  each  case  to  be  added  to  that  of  the  water. 
The  pres  of  the  water  alone  at  those  depths,  would  be  but  1,  2,  3,  &c,  times  14%  Ibs  per  sq  inch. 

Tile  greatest  heat  of  the  air  in  the  sun  probably  never  exceeds  145° 

Pah  ;  nor  the  greatest  cold  — 74°  at  night.  About  130°  above,  aud  40°  below  zero,  are  the  extremes 
in  the  U.  S.  east  of  the  Mississippi :  and  65°  below  in  the  N.  W. ;  all  at  common  ground  level.  It  ia 
stated,  however,  that — 81°  has  been  observed  in  N.E.Siberia:  and  -j-  101°  Fah  in  the  shade  in 
Paris;  and  +  153°  in  the  suu  at  Greenwich  Observatory,  both  in  July,  1881. 

Ill  a  diving-bell,  men,  after  some  experience,  can  readily  work  for  several 

hours  at  a  depth  of  51  feet;  or  under  a  pressure  of  2^  atmospheres  ;  or  37^  Ibs  per  sq  inch.  But  at 
90  ft  deep ;  or  under  3.64  atmospheres  ;  or  nearly  55  fts  per  sq  inch,  they  can  work  for  but  about  an 
hour,  without  serious  suffering  from  paralysis  ;  or  even  danger  of  death.  'Still  at  the  St.  Louis  bridge 
some  work  was  done  at  a  depth  of  110 %  ft;  pres  63.7  fts  per  sq  inch. 

The  dew  point  is  that  temp  f varying)  at  which  the  air  deposits  its  vapor. 


WIND. 


by  Smeaton,  is'prepared.     On  Mt.  Washington,  N.  H.,  180  miles  per  hour  has  been  observed. 


Vel.  in  Miles 
per  Hour. 

Vel.  in  Ft. 
per  Sec. 

Pres.  in  Lbs. 
per  Sq.  Ft. 

Remarks. 

1 

1.467 
2  933 

.005 
.020 

Hardly  perceptible. 



3 

4'.400 

.045 

' 

ct>  ^_^/ 

4 

5.867 

.080 

5 

7.33 

.125 

•n  .•"."."".- 

nil 

10 

14.67 

.5 

lt/\l  ^ 

18.33 

.781 

Fresh  breeze. 

o 

15 

22. 

1.125 

20 

25 

29.33 
36.67 

2. 

3.125 

Brisk  wind. 

The  pres  against 
a     semicylindrical 

30 

44. 

4.5 

Strong  wind. 

surface  a  c 

b  nom 

40 

58.67 

8. 

High  wind. 

is  about  ha 

If  that 

50 

73.33 

12.5 

Storm. 

against     th 

e      fiat 

60 

88. 

18. 

Violent  storm. 

aurf  abnm 

80 

117.3 

32. 

Hurricane. 

100 

146.7 

50. 

Violent  hurricane,  uprooting  large  trees. 

'   Tredgold  recommends  to  allow  4O  Ibs  per  sq  ft  of  roof  for  the 

pres  of  wind  against  it;  but  as  roofs  are  constructed  with  a  slope,  and  consequently  do  not  receive 
the  full  force  of  the  wind,  this  is  plainly  too  much.*  Moreover,  only  one-half  of  a  roof  is  usually  ex- 
posed, even  thus  partially,  to  the  wind.  Probably  the  force  in  such  cases  varies  approximately  as  the 
sines  of  the  angles  of  slopes.  According  to  observations  in  Liverpool,  in  1860,  a  wind  of  38  miles  per 
hour,  produced  a  pres  of  14  Ibs  per  sq  ft  against  an  object  perp  to  it;  and  on«  of  70  miles  per  hour, 
(the  severest  i?ale  on  record  at  that  city.)  42  fts  per  sq  foot.  These  would  make  the  pres  per  sq  ft, 


requ 


treat  as 


given  in  Smeaton's  table.     We  should  ourselves  give  the  preference  to  the  Livprpool  ol 

ft.  "  It  is  stated  that  as  high  as  55  fts  has  been  observed  at  Glasgow.     High  winds  often  lift  roofs. 

The  gauge  at  Oirard  College.  Philada,  broke  under  a  strain  of  42 

fts  per  sq  ft:  a  tornado  passing  at  the  moment,  within  y±  mile. 

By  inversion  of  Smeaton's  rule,  if  the  force  in  fts  per  sq  ft,  be  mult  by  200.  the  sq  rt  of  the  prod 
will  give  the  vel  in  miles  per  hour.     Smeaton's  rule  is  used  by  the  U.  S   Signal  Service. 

*  The  writer  thinks  8  fts  per  sq  foot  of  ordinary  double-sloping  roofs,  or  16  ft«  for  thed-rooft,  Bnffl- 
•tent  allowance  for  pres  of  wind.     Sett  Table  4,  p  301 ;  also  Snow,  p  519. 


HYDROSTATICS.  521 

EVAPORATION,  FILTRATION,  AND  LEAKAGE. 

The  amount  of  evaporation  from  surfaces  of  water  exposed  to 

the  natural  effects  of  the  open  air,  is  of  course  greater  in  summer  thau  in  winter;  although  it  is  quite 
perceptible  in  even  tbe  coldest  weather.  It  is  greater  iu  shallow  water  thaii  in  deep,  inasmuch  as  the 
bottom  also  becomes  heated  by  the  sun.  It  is  greater  in  running,  than  in  standing  water;  on  much 
the  same  principle  that  it  is  greater  during  winds  than  calms.  It  is  probable  that  the  average  daily 
loss  from  a  reservoir  of  moderate  depth,  from  evaporation  alone,  throughout  the  3  warmer  moLths 
of  the  year,  (June,  July,  August,)  rarely  exceeds  about  y^W  inch,  in  any  part  of  the  United  States.  Or 
y1-  inch  during  the  9  colder  months  ;  except  in  the  Southern  States.  These  two  averages  would  give 
a  daily  one  of  .15  inch  ;  or  a  total  annual  loss  of  55  ins,  or  4  ft  7  ins.  It  probably  is  3.5  to  4  ft. 

By  some  trials  by  the  writer,  in  the  tropics,  ponds  of  pure  water 

8  ft  deep,  in  a  stiff  retentive  clay,  and  fully  exposed  to  a  very  hot  sun  all  day,  lost  during  the  dry  sea- 
son, precisely  2  ins  in  16  days  ;  or  %  inch  per  day  ;  while  the  evaporation  from  a  glass  tumbler  WHS 
y^  inch  per  day.  The  air  in  that  region  is  highly  charged  with  moisture;  and  the  dews  are  heavy. 
Every  day  during  the  trial  the  thermometer  reached  from  115°  to  125°  in  the  sun. 

The  total  annual  evaporation  in  several  parts  of  England  and  Scotland  is  stated  to  average  from  22 
to  38  ins  ;  at  Paris,  34  ;  Boston,  Mass,  32  ;  many  places  in  the  U.  S.,  30  to  36  ins.  This  last  would  give 
a  daily  average  of  y1^  inch  for  the  whole  year.  Such  statements,  however,  are  of  very  little  value, 
unless  accompanied  by  memoranda  of  the  circumstances  of  the  case;  such  as  the  depth,  exposure, 
size  and  nature  of  the  vessel,  pond.  &c.  which  contains  the  water.  &c.  Sometimes  the  total  annual 
evaporation  from  a  district  of  country  exceeds  the  rain  fall  ;  and  vice  versa. 

On  canals,  reservoirs,  &c,  it  is  usual  to  combine  the  loss  by  evaporation, 

with  that  by  filtration.  The  last  is  that  which  soaks  into  the  earth;  and  of  which  some  portion 
passes  entirely  through  the  banks,  (when  in  embankt;)  and  if  in  very  small  quantity,  may  be  dried 
up  by  the  sun  and  air  as  fast  as  it  reaches  the  outside;  so  as  not  to  exhibit  itself  as  water;  but  if  in 
greater  quantity,  it  becomes  apparent,  as  leakage. 

E.  H.  Cxi  11.  €  E,  states  the  average  evaporation  and  filtra- 
tion on  the  Sanely  and  Beaver  canal.  Ohio,  (38  ft  wide  at  water  sur- 

face; 26  ft  at  bottom  ;  and  4  ft  deep,)  to  be  but  13  cub  ft  per  mile  per  minute,  in  a  dry  season.  Here 
the  exposed  water  surf  in  one  mile  is  200640  sq  ft;  and  in  order,  with  this  surf,  to  loj«e  13  cub  ft  per 
min,  or  18720  cub  ft  per  day  of  24  hours,  the  quantity  lost  must  be  ^^^  -  0933  ft  -  l  ^  inch  in 


depth  per  day.  Moreover,  one  mile  of  the  canal  contains  675840  cub  ft  ;  therefore,  the  number  of  days 
reqd  for  the  combined  evaporation  and  filtration  to  amount  to  as  much  as  all  the  water  in  the  canal,  is 

*LZJ?  ?JL?_  —  36  davs.     Observations  In  warm  weather  on  a  22-mile  reach  of  the  Chenango  canal,  N 
18720 

York,  (40;  28;  and  4  ft,)  gave  65^  cub  ft  per  mile  per  min  ;  or  5  times  as  much  as  in  the  preceding 
case.  This  rate  would  empty  the  canal  in  about  8  days.  Besides  this  there  was  an  excessive  leakage 
at  the  gates  of  a  lock,  (of  only  5^  ft  lift.)  of  479  cub  ft  per  min.  22  cub  ft  per  mile  per  min  ;  and  at 
aqueducts*and  waste-  weirs,  others  amounting  to  19  cub  ft  per  mile  per  min.  The  leakage  at  other 
locks  with  lifts  of  8  ft,  or  less,  did  not  exceed  about  350  cub  ft  per  miu.  at  each.  On  otjier  canals,  it 
has  been  found  to  be  from  50,  to  500  ft  per  min.  On  the  Chesapeake  and  Ohio  canal,  (where  50,  32, 
and  6  ft,)  Mr.  Fisk,  C  E.  estimated  the  loss  bv  evap  and  filtration  in  2  weeks  of  warm  wenther,  to  be 

3  uai  to  all  the  water  in  the  canal.    Professor  Kaitkiiio  assumes  2  ins  per 
ay,  for  leakage  of  canal  bed,  and  evaporation,  on  English 

gates,  on  the  original  Erie  canal,  (40,  28.  and  4  ft,;  at  100  cub  ft  per  m'ile  per  min  :  or  144000  cub  ft 
per  day.  The  water  surf  in  a  mile  is  211200  sq  ft  ;  therefore,  the  daily  loss  would  be  equal  to  a  depth  of 

211     0  0  ~  >682  ft>  ~  8Hy  ^  ln8'     SeC  Cnd  °f  RalU>  P  519' 

On  the  Delaware  division  of  the  Pennsylvania  canals,  when 

the  supply  is  temporarily  shut  off  from  any  long  reach,  the  water  falls  from  4  to  8  ins  per  day.  The 
filtration  will  of  course  be  much  greater  on  embankts,  than  in  cuts.  In  some  of  our  canals,  the  depth 
at  high  embankts  becomes  quite  considerable;  the  earth,  from  motives  of  economy,  not  being  filled  in 
level  under  the  bottom  of  the  canal;  hut  merely  left  to  form  its  own  natural  slopes.  At  one  spot  at 
least,  on  the  Ches  and  Ohio  canal,  where  one  side  is  a  natural  face  of  vertical  rock,  this  depth  is  40 
ft.  Such  depths  increase  the  leakage  very  greatly  ;  especially  when,  as  is  frequently  the  case  the  em- 
bankts are  not  puddled  ;  and  the  practice  is  not  to  be  commended,  for  other  reasons  also. 

The  total  average  loss  from  reservoirs  of  moderate  depths, 

In  case  the  earthen  dams  be  constructed  with  proper  care,  and  well  settled  by  time,  will  not  exceed 
about  from  ^  to  1  inch  per  day  ;  but  in  new  ones,  it  will  usually  be  considerably  greater. 

The  loss  from  ditches,  or  channels  of  small   area,  i*  much 

jrreatpr  than  that  from  navigable  canals;  so  that  long  canal  feeders  usually  deliver  but  a  small  pro- 
portion of  the  water  which  enters  them  at  their  heads. 


HYDKOSTATICS. 


Art.  1.  Hydrostatics  treats  of  the  pressure  of  quiet  water ; 
and  other  liquids.  The  pros  of  liquids  against  any  point  of  any  surf  upon 
which  they  act,  whether  said  surf  be  curved  or  plain,  is  always  at  right  angles  to  that 


522 


HYDROSTATICS. 


Soint.  At  any  given  depth,  the  pres  of  water  is  equal  in  every  direction ;  and  is  in 
i  i  ect  proportion  to  the  v?rt  depth  below  the  surf.  In  all  cases  whatever,  the  total  pres 
of  quiet  water  against,  and  perp  to  any  surf,  is  equal  to  the  wt  of  a  uniform  column 
of  water,  the  area  of  whose  cross-section  parallel  to  its  base,  is  everywhere  equal  to  the 
area  of  the  surf  pressed ;  and  whose  height  is  equal  to  the  vurt  depth  of  the  cen  of 
grav  of  the  surf  pressed,  beluw  the  hor  surf  of  the  water.  This  fact  is  one  of  those 
important  ones  of  frequent  application,  which  the  young  student  should  impress 
firmly  upon  his  memory.  The  wt  of  a  cub  ft  of  fresh  water  is  usually  assumed  to 
be  ti:i^  ft>s  avoir;  which  is  sufficiently  correct  for  ordinary  engineering  purposes;, 
although  62^  is  nearer  the  truth  for  ordinary  temperatures  of  about  70°  i'ah.  Hence, 

To  find  the  total  pres  of  quiet  water  against,  and  perp  to 
any  surf  whatever,  as  a  dam,  embkt,  lock-gate,  <£c;  or  the  bottom,  side,  or  top 
of  any  containing  vessel,  water-pipe,  etc,  whether  said  surf  be  vert,  hor,  or  inclined  at 
any  angle  whatever ;  or  whether  it  be  flat,  or  curved;  or  whether  it  reach  to  the.  surf  of 
the  water,  or  be  entirely  below  it : 

Rule.  Mult  together  the  area,  in  sq  ft,  of  the  surf  pressed  ;  the  vert  depth  in  ft 

of  its  cen  of  grav  below  the  surf  of  the  water;  and  the  constant  number  62.5.     The  prod  will  be  the 
r«qd  pres  in  pounds. -X- 

Ex.  1.  The  wall  A,  Fig  1,  is  50  ft  long ;  and  the  depth,  no,  of  water  pressing  against  its  vert  back  is 
uniformly  10  ft.  What  pres  does  the  wall  sustain? 

The  area  of  surf  pressed  is  50  X  10  =  500  sq  ft.  And  the  vert  depth  of  its  cen  of  grav  below  the  surf 
of  the  water  is  5  ft ;  hence, 

500  X  5  X  62.5  =  156250  pounds,  or  about  70  tons,  the  pres  reqd. 

The  pres  in  this  case  being  perp  to  a 
vert  surf,  is  horizontal;  tending  either 
to  overturn  the  wall;  or  to  make  it 
slide  on  its  base.  The  center  of  press- 
ure is  at  c ;  or  ^  of  the  vert  depth, 
from  the  bottom. 

Ex.  2.  As  in  the  foregoing  case, 
the  wall  B,  Fig  1M,  is  50  ft  long;  and 
the  vert  depth  of  water  is  10  ft ;  but  it 
presses  against  the  sloping  side  of  the 
wall ;  n  o  being  15  ft.  What  is  the 
total  pres,  or  the  pres  perp  to  no;  or 
in  the  direction  of  the  arrow? 

Here  the  area  of  surf  pressed  is  50  X  15  =  750  sq  ft.  And  the  vert  depth  of  its  ceu  of  grav  below 
the  surf  of  the  water  is  5  ft,  as  before  ;  hence, 

750  X  5  X  62.5  —  234375  pounds,  or  about  105  tons,  the  total  pres  reqd. 
The  cen  of  pres  is  before,  is  at  c,  X  the  depth,  from  the  bottom.  , 

In  such  cases,  the  total  pres  perp  lo  n  o,  may  be  considered  as  resolved  into  two  pressures  ;  one  of 
them  acting  hor,  either  to  overthrow  the  wall,  or  to  make  it  slide ;  and  the  other  acting  vert  to  hold  it 
in  its  place.  And  if  the  sloped  line  n  o  be  taken  at  any  scale  to  represent  the  total  pres,  then  will  the 
vert  line  m  o,  measured  by  the  same  scale,  represent  the  hor  pres  ;  and  the  hor  line  m  n,  the  vert 
one.  See  Art  34,  Force  in  Rigid  Bodies.  Therefore,  so  long  as  the  vert  depth  of  water  remains  the 
same,  the  hor  pres  remains  the  same,  no  matter  what  may  be  the  slope  of  no;  bill  the  vert,  as  well 
as  the  total  pres,  will  increase  with  n  o.  See  Art  4.  In  Fig  2,  the  pres  tends  to  lift  the  wall. 

RKM.  1.  This  total  pres  of  the  water  is  of  course  distributed  over  the  entire  depth  of  the  wetted 
part  of  the  back  of  the  wall ;  being  least  at  top,  and  gradually  increasing  toward  the  bottom  ;  but  so 
far  as  regards  the  united  action  of  every  portion  of  it,  in  tending  to  overthrow  the  wall,  considered  as 
a  single  mass  of  masonry,  incapable  of  being  bent  or  broken,  it  may  all  be  assumed  to  be  applied  at 
c-  dist  from  the  bottom  of  the  water,  ^  of  its  vert  depth  ;  or,  which  amounts  to  the  same  thing,  at 
^  of  the  sloping  dist  o  n,  Figs  1^  and  2.  See  Art  57,  of  Force  in  Rigid  Bodies,  p  482. 

REM.  2.  It  follows,  from  the  foregoing  rule,  that  the  amount 
of  pres  of  water  against  any  surf  is  entirely  independent  of 
the  quantity  of  the  water,  so  long  as  the  area  pressed,  and  the  vert 
depth  of  its  cen  of  grav  below  the  level  surf  remain  unchanged.  The  wall  A  or  B  would  sustain  as 
great  a  pres  from  a  layer  of  water  only  an  inc-h  thick  behind  it.  as  if  the  water  had  extended  back 
for  miles.  From  this  cause,  retaining-walls  of  mortar  masonry  carelessly  backed,  have  been  bulged, 
and  cracked,  by  the  infiltration  of  rain  behind  them ;  while  walls  of  dry  masonry  would  have  per- 
mitted the  water  to  escape  through  the  open  joints  ;  and  would  therefore  have  stood  safely. 

Also  in  vessels  a,  fc.  Fig  2^,  of  any  size  or  shape  whatever,  if  they  contain 
the  same  vert  depth  of  water  fand  have  equal  bases  o  o,  pressed  by  said  depths 
of  water,  the  pressures  on  the  bases  will  all  be  equal,  without  any  regard  to 
I      the  quantity  of  water.     Or,  if  we  have  two  water-pipes  of  the  same  diam,  both 
V  )      full  of  water,  one  standing  vertically.  10  feet  long ;  and  the  other  20  miles  long, 

J  O\  /       and  laid  at  an  inclination  of  ^  ft  per  mile,  so  as  to  make  its  vert  depth  of  water 

*-- - ~        also  10  ft,  then  the  pressures  at  the  bases  of  the  two  pipes  will  be  the  same. 

This  pres  of  water,  independently  of  its  quantity,  is  called  the  hydrrfstatic  par- 
adox.    In  the  vessel  a,  the  pres  on  the  base  is  much  greater  than  the  wtof  the 
water;  but  in  o,  it  is  less. 
RKM.  3.    Since  the  pres  of  water  against  any  point  of  a  surf,  is  at  right  angles  to  that  point,  it  fol- 


*  This  is  strictlv  true  as  regards  the  pres  of  the  water  alone, ;  and  this  is  usually  all  that  is  required. 
But  it  must  be  borne  in  mind  that  the  surf  of  the  water  is  itself  pressed  by  the  air;  to  the  average 
extent  (near  the  level  of  the  sea)  of  about  14.7  fts  per  sq  inch  ;  or  2117  Tbs.  or  nearly  1  ton  per  sq  foot. 
Therefore,  to  find  the  true  total  pres,  we  should  mnlt  the  area  in  sq  ft  of  the  surf  pressed  by  the  water. 
fcy  2117  fts  ;  and  add  the  prod  to  the  water-pres  given  by  the  rule.  But  in  ordinary  engineering  cases, 


HYDROSTATICS. 


523 


IOTTS  that  props  pp,  for  strengthening  such  structures  as  the  sloping  dam  D,  Pig  3,  should  be  placed 
at  right  angles  to  them  i»*order  to  oppose  the  greatest  possible  resistance  to  the  pres.  Other  consid- 
erations may  at  tiuieXpr  event  our  doiug  so;  thus  the  outer  prop,  p.  if  BO  placed,  would  be  in  danger 
of  beiug  bro'ken  byfce,  or  logs  tumbling  over  the  dam ;  and  therefore,  had  better  be  more  nearly 
vertical.  / 

RKM.  4,  It  follows,  from  the  foregoing  rule,  that  in  a  cubical  vessel,  filled  with  water,  the  pres  on 
the  base  is  equal  to  the  weight  of  the  water;  that  on  each  of  the  four  sides,  to  half  the  weight  of  the 
water;  and  that  on  the  bottom  and  the  4  sides  together,  to  3  times  the  wt  of  the  water.  In  a  conical 
vessel,  forming  an  entire  cone,  the  pres  on  its  hor  base  is  equal  to  3  times  the  wt  of  the  water ;  and  so 
likewise  in  a  pyramidal  vessel;  for  in  both  cases  the  wt  of  the  water  is  but  %  that  of  a  uniform  column 
of  water  of  the  same  height.  In  a  full  spherical  vessel,  the  total  pres  against  its  entire  interior  surf, 
is  also  equal  to  3  times  the  wt  of  the  water,  as  in  a  cubical  one. 

Since  the  pres  increases  with  the  depth,  the  props  in  the  dam,  Fig  3, 
should  be  closest  together  near  the  bottom  ;  also  the  hoops  of  a  tank. 

The  following;  Table  gives  the  pres  to  the  nearest 
E>  per  sq  ft  at  diff  vert  depths  ;  and  also  the  total  pi  es  against  a  plane  one     ^ 
foot  wide  extending  vert  from  the  surface  to  those  depths.     The  first  in- 
creases as  the  depths ;  the  last  as  the  squares  of  the  depths. 

For  the  pres  in  Ibs  per  sq  inch  at  any  given  depth,  mult  the  depth  in  ft 
by  .434.  For  tbs  per  sq  ft,  mult  by  62.5.  For  tons  per  so  ft,  mult  by  .0279.  For 
the  depth  in  1't  at  which  any  given  pres  exists,  divide  the  ft>s  per  sq  inch  by 

.434;  or  the  fts  per  sq  ft  by  62.5:  or  the  tons  per  sq  ft  by  .0279. 


D 
in 
Ft. 

Per 

a 

Tot 
P. 

D 
Ft. 

Per 

a 

Tot 
P. 

I) 
in 
Ft. 

Per 

K 

Tot 
P. 

D 
in 
Ft. 

Per 
F?. 

Tot 
P. 

D 
in 
Ft. 

Per 

il 

Tot 
P. 

1 

62. 

31 

11 

687. 

3781 

21 

1312. 

13781 

31 

1937. 

30031 

41 

2562. 

52531 

2 

125. 

125 

12 

750. 

4500 

22 

1375. 

15125 

32 

2000. 

32000 

42 

•2625. 

55125 

3 

187. 

281 

13 

812. 

5281 

23 

1437. 

16531 

33 

2062. 

34031 

43 

2687. 

57781 

4 

250. 

500 

14 

875. 

6125 

24 

1500. 

18000 

34 

2125. 

36125 

44 

2750. 

60500 

5 

312. 

781 

15 

937. 

7031 

25 

1562. 

19531 

35 

2187. 

38281 

45 

2812. 

63281 

6 

375. 

1125 

16 

1000. 

8000 

•26 

1625. 

21125 

36 

2250. 

40500 

46 

2875. 

66125 

7 

437. 

1531 

17 

1062. 

9031 

27 

1687. 

22781 

37 

2312. 

42781 

47 

2937. 

69031 

8 

500 

2000 

18 

1125. 

10125 

28 

1750. 

24500 

38 

2375. 

45125 

48 

3000. 

72000 

9 

562. 

2531 

19 

1187. 

11281 

29 

1812. 

26281 

39 

2437. 

47531 

49 

3062. 

75031 

10 

625. 

3125 

20 

1250. 

12500 

30 

1875. 

28125 

40 

2500. 

50000 

50 

3125. 

78125 

Thus  we  see  that  at  the  depth  of  36  ft,  the  pres  of  water  against  a  single  sq  ft  of  surf,  whether  hor, 
rert,  or  oblique,  is  fully  1  ton  ;  requiring  great  precaution  to  prevent  leakage,  or  breaking.  At  72  ft, 
it  would  be  2  tons.  &c.  A  pres  of  62^  fts  per  sq  ft  gives  a  pres  of  .434  fts  per  sq  inch. 

Further;  let  a  6,  Fig  3>^,  be  a  tube  of  36  ft  vert  height ;  full  of  water ;  with  a  bore  so 
small  that  the  tube  would  contain  say  only  one  pound  of  water;  and  let  this  tube  open  at 
its  lower  end  into  a  vessel  also  full  of  water;  tlie  top  and  bottom  of  which  are  8  ft  apart. 
Then  the  1  Tb  of  water  in  the  tube,  will  cause  each  sq  ft  of  the  top  of  the  vessel,  (which 
is  36  ft  below  the  surf  of  the  water  in  the  tube)  to  be  pressed  upward  with  a  force  of  2250 
fts,  as  per  table.  Each  sq  ft  of  the  bottom  of  the  vessel  (which  is  44  ft  below  the-  surf 
of  the  water  in  the  tube)  will  be  pressed  downward  with  a  force  of  2750  Ibs ;  and  any  par- 
ticular sq  ft  of  the  sides  of  the  vessel,  will  be  pressed  hor  outward,  with  the  force  given 
in  the  table,  opposite  to  the  depth  of  the  cen  of  grav  of  said  sq  ft  below  the  same  water 
surf  of  the  top  of  the  tube,  whatever  said  depth  may  happen  to  be.  Or,  suppose,  first 
only  th3  ?ower  vessel  to  be  filled  with  water,  and  its  inner  surf  to  be  sustaining  the  pres 
arising  therefrom  ;  if  we  then  flil  the  36  ft  tube  with  its  1  Ib  of  water,  this  1  ft  will  create 
an  additional  pres  of  2250  tbs  against  every  sq  ft  of  said  inner  surf;  so  that  if  each  of  the  6  sides  of 
the  vessel  be  8ft  square;  or  contain  in  all  384  sq  ft  of  inner  surf,  this  1  ft  of  water  will  produce  addi- 
tional pres  of  864000  Ibs.  or  full  385  tons,  against  them.  If  we  then  press  upon  the  top  of  the  water 
with  our  thumb  to  the  extent  of  1  ft,  we  shall  thereby  redouble  this  enormous  pres.  This  fact,  how- 
ever, belongs  to  Art.  7,  p  526. 

Art.  2.    Surfaces   pressed    on    both  sides;    and   immersed. 

When  two  bodies  of  water  of  diff  depths,  press  against  two  oppo- 
site sides  of  a  plane  which  is  completely  immersed,  whether  vert  or 
eloping;  as,  for  instance,  against  the  two  sides  a  6,  no,  Fig  4;  or 
the  two  sides  d  e.  c  r,  then,  the  total  pres  against  i  b,  i  e,  a  b,  n  o,  or 
c  r,  &c,  may  still  be  found  by  the  foregoing  rule,  in  Art  1  :  but  the 
XXCESS  of  pres  against  the  part  a  b,  or  d  e,  of  the  immersed  plane, 
beyond  the  counier-pres  against  the  opposite  part  n  o,  or  c  r,  will 
be  equal  to  the  wt  of  a  column  of  water  whose  section  is  equal 
to  the  area  of  the  part  a  b,  or  d  e.  (as  the  case  may  be;)  and 
whose  vert  height  is  equal  to  m  n.  or  xp.  the  vert  diff  of  level  of  the 
two  bodies  of  water.  Consequently,  this  excess  of  outward  pres  is 
found  by  mult  together,  the  area  of  a  6  or  d  e,  in  sq  ft;  the  vert 
height  m  n  or  xp,  in  ft ;  and  the  constant  62.5  fts  wt  of  a  cub  ft  of 
water.  Thus,  if  a  6  is  10  ft  high,  and  20  ft  long :  and  the  vert  height 

mn,  12  ft;  then  the  excess  of  pres  against  a  6,  over  that  against  no,  will  be  10  X  20  X 12  X  62.5=150000 
fts.  The  excess  will  be  greater  on  d  e,  than  on  a  b,  although  both  are  exposed  to  the  same  vert  depths 
mn,  xp  ;  because  the  area  of  d  e  is  greater  than  that  of  a  b.  Moreover,  this  excess  of  outward  pres 
is  equally  distributed  over  the  entire  area  of  a  b  or  d  e ;  being  no  greater  at  b  and  e,  than  at  a  or  d; 
in  other  words,  every  sq  ft  of  area  of  a  6  or  deis  pressed  outward  at  right  angles  to  its  surf,  by  an 
excess  of  force  equal  to  the  wt  of  a  column  of  water  1  ft  sq  ;  and  of  a  height  equal  to  m  n,  or  xp. 


this  pres  of  the  air  may.  and  should  be  omitted  ;  because  it  is  counterbalanced  by  an  equal  pres  of  air 
against  the  opposite  side,  face,  or  surf  of  the  pressed  body.  It  becomes  necessary,  therefore,  to  take 
it  into  consideration  only  when  the  opposite  face  of  the  body  is  not  exposed  to  a  counterbalancing 
atmospheric  pressure;  as  when  there  is  a  vacuum  on  that  side. 


524 


HYDROSTATICS. 


JfilK 


This  will  be  understood  by  means  of  Pig  5.  which  may  represent  five 

Slauks,  1,  2,  3,  4,  and  5,  forming  a  dam,  and  seen  endwise;  each  one  1  ft 
i  depth,  and  say  20  ft  long  hor ;  making  the  area  of  each  surf  pressed, 
equal  to  20  sq  ft.  The  pres  in  tts  against  each  separate  20  sq  ft  of  area, 
calculated  by  the  rule  in  Art  1,  is  shown  in  the  tig.  Now,  the  outward 
pres  against  the  upper  immersed  '20  ft  area,  or  that  of  plank  3,  is  2125  ttis  ; 
while  the  counter-pres  against  it  trom  the  other  side  is  625  tts  ;  making 
the  excess  of  outward  pres  equal  to  3125  —  625  =  2500  B>s.  Again,  at  the 
lowest  plank,  number  5,  the  outward  pres  exceeds  the  inward  one  by 
5625  —  3125  =r  2500  tts,  the  same  as  in  the  upper  one.  And  so  of  any  other 
equal  area  of  surf,  at  any  depth  whatever  ;  the  excess  depending  upon  the 
vert  height  of  m  /t,  will  be  equally  distributed  over  a  b.  it  only  remains 
to  show  that  the  total  excess  of  outward  pres  against  a  b,  is  equal  in 
amount  to  the  wt  of  a  uniform  column  of  water  with  a  base  equal  in  area 
to  a  6.  and  with  a  height  equal  to  m  n.  Thus,  we  have  seen  that  in  the 
instance  before  us,  the  excess  amounts  to  3  times  2500  tt>3,  or  to  7500  tt>s. 
Now,  the  wt  of  the  column  of  water  will  be  60  (or  area  of  a  fc)  X  wt  n  (or 
2  ft)  X  62.5  B)s  —  7500  Bt>s ;  or  the  same  as  the  excess  pres  on  a  6. 

The  excess  of  pres  against  the  entire  side  s  b,  over  that  against  n  o,  is 
evidently  the  diff  between  those  two  pressures  calculated  respectively  by  the  rule  in  Art  1. 

Art.  3.  Surfaces  of  equal  widths,  commencing-  at  the  level 
of  the  water,  but  extending  to  diff  depths,  measured  vert; 
and  having1  the  same  inclination  to  the  surf  of  the  water; 
sustain  total  pressures  proportional  to  the  squares  of  those 
depths. 

In  Fig  6,  let  the  two  vert  sides,  a  no  t,  and  b  m  c  s,  of  a  vessel, 

4,  5,  &c,  times  greater  than  the  depth  n  o,  the  pres  against  the  surf 
b  mcs,  will  be  4,  9,  16.  25,  &c,  times  greater  than  that  against  ano  t. 
This  will  be  seen  by  referring  to  the  pressures  figured  on  the  left  side 
of  Fig  5,  where,  as  stated  in  Art  2,  the  surf  of  plank  1,  exposed  to 
the  pr«s  on  the  left  side,  is  20  sq  ft ;  that  of  planks  1  and  2,  40  sq  ft ; 
that  of  planks  1,  2,  and  3,  60  sq  ft,  &c.  All  these  surfs  commence  at 
the  level  of  the  water;  and  all  of  them  being  vert,  are  of  course  at 
the  same  inclination  with  the  water  surf;  but  their  depths  are  re- 
spectively 1,  2,  and  3  ft.  The  pres  against  the  surf  of  1,  is  625  tts; 
that  against  the  surf  of  1,  2,  is  625 -f- 1875  —  2500 ;  and  that  against 
the  surf  of  1,  2.  3,  is  625  -j- 1875+  31 25  =  5625.  But  2500  is  four  times 
625;  and  5625  is  nine  times  625.  And  the  pres  against  the  entire 
surf  s  &,  (which  is  5  times  as  deep  as  plank  1,)  is  25  times  as  great  as  that  against  plank  1 ;  or 
625  X  25  — 15625  fts  -  the  sum  of  all  the  pressures  marked  on  the  left  side  of  Fig  5. 

This  follows,  from  the  Rule  in  Art  1 ;  for  twice  the  area  of  surf,  mult  by  twice  the  vert  depth  of  the 
oen  of  grav  below  the  surf,  must  give  4  times  the  pres :  three  times  the  area,  by  three  times  the  depth, 
must  give  9  times  the  pres,  &c.  See  third  columns  of  table,  p  523. 

It  follows,  also,  that  at  any  particular  point,  or  against  any  given  area  placed  at  various  depths,  the 
pres  will  increase  simply  as  the  vert  depth  :  thus,  if  there  be  three  areas,  each  one  sq  ft,  placed  in 
the  same  positions,  but  with  their  centers  of  grav  respectively  8,  16,  and  24  ft  below  the  surf,  the  pres 
against  them  will  he  respectively  as  8,  16,  and  24;  or  as},  2,  and  3.  See  se  -on  '  crlumns  table,  p  523. 

Art.  4.  The  pressure  of  quiet  water,  in  any  one  given  di- 
rection, against  any  given  surf,  whether  vert,  hor,  inclined,  flat,  or  curved,  is  equal 
to  the  wt  of  a  uniform  column  of  water,  the  area  of  whose  section,  parallel  to  its  base,  is  everywhere 
equal  to  the  area  of  the  projection*  of  the  pressed  surf  taken  perp  to  the  given  direction;  and  the 
height  of  the  column  equal  to  the  vert  depth  of  the  cen  of  grav  of  the  pressed  surf  below  the  upper 
surf  of  the  water.  Hence  the 

RULE.  To  find  the  pres  in  Ibs,  mult  together  the  area 

in  sq  ft  of  the  projection  taken  at  right  angles  to  the  given  direction;  the 
vert  depth  in  ft  of  the  cen  of  grav  of  the  pressed  surf  below  the  upper  surf 
of  the  water ;  and  the  constant  62.5  B>s  wt  of  a  cub  ft  of  water. 

Ex.  Let  m  cs  n,  Fig  7,  be  an  inclined  surf,  sustaining  the  pres  of  water 
which  is  level  with  its  top  m  c.  Then  the  total  pres  against  me  s  n,  and  at 
right  angles  to  it,  as  found  by  the  rule  in  Art  1,  is  an  illustration  of  the  pres- 
ent rule ;  because  the  projection  of  mean,  taken  at  right  angles  to  the  given 
direction,  or  parallel  tomcsn,  is  in  fact  me  s  n  itself,  or  equal  to  it.  Hence 
the  rule  in  Art  1  is  merely  a  simple  modification  of  the  present  one,  appli- 
cable to  the  case  of  total  pres  against  any  surf. 

But  if  it  be  reqd  to  find  only  the  vert  or  downward  pres 
against  m  c  s  n,  in  pounds,  mult  together  the  area  of  the  hor  projection  aocm 
in  sq  ft;  the  vert  depth  in  ft  of  the  cen  of  grav  of  m  c  s  n  below  the  surf;  and  62.5,  Or  if  only  the 
hor  pres  against  m  c  s  n  be  sought,  mult  together  the  area  of  the  vert  projection  a  o  s  n;  the  vert 
depth  of  the  cen  of  grav  of  me  an;  and  62.5. 

In  Fig  8  also,  the  total  pres  against  efg  h  is  found  by  rule  in  Art  1 :  while 
the  hor  and  vert  pressures  against  it  are  found  as  in  Fig  7,  by  using  the  projec- 
tions efk  i,  and  k  i  g  h.  In  Fig  7  the  vert  pres  is  downward;  while  in  Fig  8 
it  is  upward ;  but  this  circumstance  in  no  respect  affects  the  rule. 

RKM.  I.  It  will  be  observed  in  both  figs,  that  the  vert  projections,  aosn,  efki, 
will  remain  the  same,  no  matter  what  may  be  the  inclination  of  the  pressed 
purfs  me  s  n,  and  efg  h;  the  degree  of  inclination  therefore  does  not  influence 
the  hor  pres,  but  only  the  total,  and  the  vert  ones. 


a 


*  See  Projection,  in  our  glossary. 


fDROSTATICS. 


525 


IW10 


;  with  the  same  depth :  and  62.5 


Again,  let  Fig  9pej5resent  a  conical  vessel  full  of  water; 

•i£s  base  6  c,  2  ft  d^eem;  its  vert  height  a  n,  3  ft ;  then  the  circumf  of  the  base  will  be 
li.2832  ft;  the  ap«aof  the  base  3.1416  sq  ft;  the  length  of  its  slant  side  a  6  or  a  c,  3.16 
(I;  the  areaf'of  its  curved  slanting  sides  will  be  6'2832  X  3-16  —  y  9a  sq  ft;  and  the 

vert  depth  of  the  cen  of  grt 
height  a  n  from  the  apex  a,  < 

Here,  to  tiud  the  total  pres  against  the  base,  we  have  by  rule  in  Art  1,  3.1416  X  3 
X  62.5  —  589.05  tt>s.  For  the  total  pres  against  the  slant  sides,  by  the  same  rule, 
9.93  X  2  X  62.5  —  1241.25  tt>s.  For  the  vert  pres  upward  against  the  entire  area  of  the 
ilaut  sides,  we  have  given  the  area  of  the  base  (which  is  here  the  hor  projection  of 
the  slant  sides)  ~  3.H16;  and  the  vert  depth  of  theceu  of  gray  of  the  slant  sides,  2  ft.  Therefore, 
3.1416  X  2  X  62.5  =:  35*2.7  fts,  the  upward  vert  pres. 

Finally,  for  the  hor  pres  in  any  given  direction  against  the  slant  sides  of  one  half  of  the  cone,  we 
have  the  vert  projection  of  that  half,  represented  by  the  triangle  ale,  with  its  base  2  ft,  and  its  perp 
height  3  ft ;  and  consequently ,  with  an  area  of  8  sq  ft.  The  depth  of  its  ceu  of  grav  is  2  ft :  therefore, 
3  X  2  X  62.5  =  375  fts,  the'reqd  hor  pres.* 

In  Fig  10,  widen  represents  a  vessel  full  of  water,  the  total  pres 
against  the  semi-cylindrical  surf  avemdk.  and  perp  to  it,  must  be 
also  hor.  because  the  surf  is  vert;  but  inasmuch  as  the  surf  is  curved, 
this  total  pres,  as  found  by  rule  in  Art  1,  acts  against  it  in  many  di- 
rections, wuich  might  be  represented  by  an  infinite  number  of  radii 
drawn  from  o  as  a  center.  But  let  it  be  reqd  to  find  the  hor  pres  in 
ft>s,  in  one  direction  only,  say  parallel  to  o  .e,  or  perp  to  a  d;  which 
would  be  the  force  tending  to  tear  the  curved  surf  away  from  th«  flat 
sides  a  6  «  v,  and  desk,  by  producing  fractures  along  the  lines  a  v 
and  d  k  ;  or  which  would  tend  to  burst  a  pipe  or  other  cylinder.  Ju 
this  case,  mult  together  the  area  of  the  vert  projection  a  d  k  v  in  sq 
ft;  the  depth  of  the  cen  of  grav  of  the  curved  surf  in  ft;  (which,  in 
the  semi-cyliuder  would  be  half  of  e  m,  or  of  o  i;)  and  62.5.  Since 
the  resulting  pres  is  resisted  equally  by  the  strength  of  the  vessel 
along  the  two  lines  a  v  and  d  k.  it  is  "plain  that  each  single  thickness 
along  those  lines  need  only  be  sufficient  to  resist  safely  one  half  of  it; 
and  so  in  the  case  of  pipes,  or  other  cylinders,  such  as  hooped  cisterns 
or  tanks.  See  Art  16,  p  531. 

Should  the  pres  against  only  one  half  of  the  curved  surf,  as  edmk 
be  sought,  and  in  a  direction  parallel  to  o  d,  tending  to  produce  frac- 
tures along  the  lines  e  m,  and  d  k,  then  use  the  vert  projection  oen  ' 
as  before. 

It  follows,  that  if  the  face  of  a  metallic  piston  be  made  concave  or  convex,  no  more  pres  will  be  reqd 
to  force  the  piston  through  any  dist,  than  if  it  were  fiat;  lor  the  pres  against  the  face  of  the  piston, 
in  the  direction  in  which  it  moves,  must  be  measured  by  the  area  of  a  projection  of  that  face,  taken 
at  right  angles  to  said  direction  ;  and  the  area  of  said  projection  will  be  the  same  in  all  three  cases. 

HEM.  2.    If  a  bridge  pier,  or  other  construction, 

Fi;;-  1O  V£.  be  founded  on  sand  or  gravel,  or  on  any  kind  of 

foundation  through  which  water  may  find  its  way  underneath,  even  in  a  very  thin 
sheet,  then  the  upward  pres  of  the  water  will  take'effect  upon  the  pier ;  and  win  tend 
to  lift  it,  with  a  force  equal  to  the  wt,  of  the  water  displaced  by  the  pier ;  (see  Arts  17 
and  18 ;)  or  in  other  words,  the  effective  wt  of  the  submerged  portion  of  the  pier,  will 
be  reduced  62^  fts  per  cub  ft;  or  nearly  the  half  of  the  ordinary  wt  of  masonry. 

But  if  the  foundation  be  on  rock,  covered  with  a  layer 
of  cement  to  prevent  the  infiltration  of  water  beneath  the  masonry,  no  such  effect 
will  be  produced;  but  on  the  contrary,  the  vert  pres  downward,  afforded  by  the  bat- 
tering sides  of  the  pier,  and  bv  its  offsets,  will  tend  to  hold  it  down,  and  thus  increase  its  stability  ; 
which,  in  quiet  water,  will  then  actually  be  greater  than  on  land. 

Art.  5.  To  divide  a  rectangular  surf, 
whether  vert  as  ft  b  c  d,  or  inclined  as 
m  nop,  Fig  11,  whose  top  a  b  or  m  n  is 
level  with  the  surf  of  the  water,  by  a 
hor  line  .r  2,  such  that  the  total  pres 
against  the  part  above  said  hor  line, 
shall  equal  that  against  the  part  be- 
low it. 

RULE.  Mult  one  half  of  the  length  of  6  c,  or  m  p,  as  the  case 
may  be,  by  the  constant  number  1.4142;  the  prod  will  be  b  2, 

Ex.  '  Let  ft  c=  12  ft.  Then  6  X  1.4142  =  8.4852  ft ;  or  6  2. 
Let  TO  p  —  16  ft.  Then  8  X  1.4142  =  11.3136  ft,  or  m  x. 

REM.  The  line  x  2,  thus  found,  must  not  be  confounded  with 
the  cen  of  pres,  which  is  entirely  diff.  See  Art  8. 

Art.  6.  In  a  rectangular  surf,  whether  vert  as  a  b  c  df  or  in- 
clined as  m  n  op,  Fig  11,  whose  top  a  b  or  m  n  coincides  with 
the  surf  of  the  water,  to  find  any  number  of  points,  as  1,2,  Ac. 
through  which  if  hor  lines,  as  1  .<?,  2.r,  «fce,  be  drawn,  they  will 
divide  the  given  siirf  into  smaller  rectangles,  all  of  wrhich 
shall  sustain  equal  pressures. 

RULE.  First  fix  on  the  number  of  small  rectangles  reqd.  Then  for  point  1  from  the  top,  mult  the 
number  1 ,  by  this  number  of  rectangles.  Take  the  sq  rt  of  the  prod.  Mult  this  sq  rt  by  the  entire  length 


*  In  a  spher«  filled  with  a  fluid  the  total  inside  pres 

34 


=  3  times  wt  of  fluid. 


526 


HYDROSTATICS. 


b  e  or  m  p,  as  the  case  may  be.  Div  the  prod  by  the  number  of  rectangles.  The  quot  will  be  the  dlst 
b  1.  or  n  1,  as  the  case  may  be. 

For  the  dist  6  2,  or  n2,  proceed  in  precisely  the  same  way;  only  instead  of  the  number  1,  use  the 
number  2  to  be  mult  by  the  number  of  rectangles:  and  so  use  successively  the  numbers  3,  4.  5,  &c, 
if  it  be  reqd  to  find  that  number  tif  points. 

Ex.  Let  b  c  =  10  ft ;  and  let  it  be  reqd  to  find  2  points,  1  and  2,  for  dividing  the  rectangular  surf 
abed  into  3  rectangular  parts,  which  shall  sustain  equal  pressures.  Here  we  have  for  point  1, 

lX3=r3.     The  sqvt  of  3=1.732.     And  1.732  X  10  (or  b  c)  =  17.32.     And  ...1_7ji?__  =5.773ft  =  61, 

3  rectangles 
For  point  2,  we  have 

2X3  =  6.   The  sqrt  of  6  =  2.449.   And  2.449  X  10  (or  6  c)  =  24.49.    And *—— =  8.163  ft  =  b  2. 

3  rectangles 
And  so  for  any  number  of  poiuta. 

REM.  1.  This  rule  will  be  found  useful  in  spacing  the  cross- 
bars of  lock-gates;  the  hoops  around  cylindrical  cisterns; 
and  the  props  to  a  structure,  like  Fig  3,  p  523. 

REM.  2.  For  dividing  any  surf,  as  o  b  c  d,  Fig  12.  which  is  not 
rectangular,  in  the  same  manner, 

with  an  accuracy  sufficient  for  most  practical  purposes,  per- 
haps the  following  method  is  as  convenient  as  any. 

RULE.  First  div  the  surf,  as  in  Fig  12,  into  several  small 
hor  parts,  equal  or  not,  at  pleasure.  Then  by  Rule  in  Art  1, 
find  the  pres  on  each  part  separately,  as  is  supposed  to  be 
done  in  the  numbers  on  the  left  hand  of  the  fig.  The  sum  of 
these  (in  this  case  15510)  is  the  total  pres  against  the  entire 
surf  o  b  c  d.  Now  suppose  we  wish  to  div  this  surf  in  4  parts 
bearing  equal  pres;  first  div  15510  by  4  =  3878.  Then  begin- 
ning at  the  top,  add  together  a  number  of  the  separate 
pressures  sufficient  to  amount  to  3878 ;  by  this  means  find 
point  1.  Then  proceed  with  the  additio'n  until  the  sum 
amounts  to  twice  3878,  or  7756,  which  will  indicate  point  2; 
and  in  the  same  manner  find  point  3.  by  adding  up  to  three 
times  3878,  or  11634.  Then  the  hor  dotted  lines  ruled  through 
points  1,  2,  and  3,  will  give  the  reqd  divisions  approximately. 
In  this  manner  the  hoops  of  conical,  and  other  shaped  ves- 
sels, may  be  spaced  nearly  enough  for  practical  purposes. 

Art.  7.  The  transmission  of  pressure  through  water.  Wa- 
ter, in  common  with  other  fluids,  possesses  the  important 
property  of  transmitting  pres  equally  in  all  directions.  Thus, 

suppose  the  vessel,  Fig  13,  to  be  entirely  closed,  and  filled  with  water; 
and  suppose  the  transverse  area  of  T,C,  D,  and  E,  to  be  each  equal  to  one 
sq  inch.  Then,  if  by  means  of  a  piston,  or  otherwise,  a  pres  of  1  ft.  1 
ton,  or  any  other  amount,  be  applied  to  the  one  sq  inch  of  area  of  T,  C, 
D.  or  E,  every  sq  inch  of  the  inner  surf  of  the  vessel,  and  of  the  pipe  a, 
will  instantly  receive,  at  right  angles  to  itself,  an  equal  pres  of  1  ft,  or 
1  ton,  &c:  in  addition  to  the  pres  which  it  before  sustained  from  the 
water  itself ;  and  this  will  occur  if  the  vessel  consist  of  parts  even  miles 
asunder ;  as,  for  instance,  if  T  were  miles  distant  from  E :  and  united 
to  it  by  a  long  series  of  tubes.  If  the  vessel  were  a  strong  steam  boiler 
full  of  water,  a  single  pres  of  a  few  hundred  pounds  at  T,  C,  &c,  would 
burst  it.  See  also  flg  3J^,  p  523. 

The  hydrostatic  press  acts  on  this  prin- 
ciple.    Any  body,  within  the  vessel,  would  also  receive 
an  equal  additional  pres  on  each  sq  inch  of  its  surf. 

If  the  top  of  T  be  open,  the  air  will  press  upon  the  sq  inch  of  the  exposed  surf  of  water  to  the  extent 
of  nearly  15  fts ;  and  the  same  degree  of  pres  will  also  be  transmitted  to  every  sq  inch  of  the  interior 
surf  of  the  vessel,  and  its  connecting  tubes ;  but  no  danger  of  bursting  will  result  from  this  atmo- 
spheric pres,  because  the  air  also  presses  every  sq  inch  of  the  outside  of  the  vessel  to  the  same  extent. 

Air,  and  other  gaseous  fluids,  transmit  pres  equally  in  all 
directions,  like  liquids;  but  not  as  rapidly. 

Art.  8.     The  center  of  pressure.    Let  Fig  14 

represent  a  vessel  full  of  water,  and  suppose  the  side  P  to  be  perfectly 
loose,  so  as  to  be  thrown  outward  by  the  slightest  pres  of  the  water  from 
within.     Now,  there  is  but  one  single  point,  P,  in  every  surf  so  pressed, 
— .     no  matter  what  its  shape  may  be,  to  which  if  we  apply  a  force  equal  to 
\     the  pres  of  the  water,  and  in'a  direction  opposite  to  said  pres,  the  side  P 

(will  be  thereby  prevented  from  yielding.      Such  point  is  called  the  cen- 
ter of  pressure.     It  must  not  "be  understood  by  thin  that  the  actual 
amount  of  pres  of  the  water  against  that  part  of  the  surface  which  is 
<     above  the  hor  dotted  line  passing  through  P,  is  equal  to  that  of  the  water 
v     below  said  line ;  but  that  the  products  of  the  several  pressures  above  it, 
mult  by  their  several  leverages,  or  dists  from  P,  are  equal  to  the  products 
of  the  pressures  below,  mult  by  their  leverages ;  or,  in  other  words,  that 
the  moments  around  the  point  P,  of  the  pressures  above  the  line,  are 

-|-^.  equal  to  the  moments  of  those  below  it ;  so  that  if  a  hor  iron  rod  b  b  were 

JP~l(f  IT-  passed  entirely  through  the  side  P,  at  the  same  level  as  the  dotted  line, 

S  '    •  so  as  to  serve  as  a  hinge,  or  pivot  for  the  side  P  to  turn  on,  as  shown  in 

the  fig,  the  equal  moment*  above  and  below  the  bar  would  prevent  the  aide  from  turning.    P.  482. 


Nb 


ROSTATICS. 


527 


Art.  9.     To  fiiitf  'the  cen  of  pres  of  a  quiet 
fluid,  againjtfa  plane  surface.    Fig  15. 

1.  The  ceri lofpres  of  a  quiet  fluid,  against  one  si.Je  of  any  plane  rec- 
tangular surface,  (that  is,  a  plane  surf  of  uniform  width  throughout  its 
depth,)  wtfetber  vert  as  eo,  or  incliued  as  c  a,  (or  inclined  in  the  opposite 
direction  ;)  and  whose  top  c,  or  e,  coincides  with  the  hor  water  surf;  is 
distant  vert  below  the  water  surf,  two-thirds  of  the  vert  depth,  sx,  from 
uaid  water  surf  to  the  bottom  of  the  plane;  as  at  n,  and  i.     Inasmuch  as 
a  hor  line  at  %  of  the  depth  of  ax,  intersects  both  ca  and  eo  at  %  of  their 
lengths  respectively,  we  might  say  at  once  that  the  center  of  pres  against 
a  rectangular  plane  is  at  %  of  its' length  below  the  water  surface. 

Throughout  Art  9  any  measure,  as  yard,  foot,  or  inch 
Ac,  may  be  used. 

2.  But  if  the  hor  top  a,  or  o,  Fig  16,  of  the  rectangular  plane  ag,  or 
o  A,  be  covered  to  some  depth  with  water,  then  the  vert  depth  sm,  of  the 
cen  of  pres  d,  or  e,  below  the  surf  of  the  water,  will  be  equal  to 

fcube  of  sc  —  cube  of  sw 
of 
square  of  a  c  —  square  of  s  w 

•where  sc  Is  the  vert  depth  of  the  bottom,  and  sw  the  vert  depth  of  the 
top,  of  the  pressed  surf,  below  the  water  surf.  Or,  in  words  :  From  the 
cube  of  sc,  take  the  cube  of  sw,  and  call  the  rem  a.  Then,  from  the 
square  of  «  c,  take  the  square  of  sw;  and  call  the  rem  6.  Div  o  by  b, 
and  take  two-thirds  of  the  quot  for  s  m. 


3.  When  a  plane  surf  of  any  shape  whatever,  whether 
rectangular,  triangular,  or  circular,  &c ;  whether  vert  as 
op,  Fig  17,  or  inclined  as  mn,  is  entirely  immersed,  so  as  to 
be  pressed  over  the  entire  area  of  both  sides  •  but  by  diff 
depths  of  water  on  its  two  sides  ;  then  the  cen  of  pres  coin- 
cides with  the  cen  of  grav  of  the  pressed  surf. 

In  the  3  foregoing  figures  the  supposed  surfaces  are  shown 
edgewise,  so  that  their  widths  do  not  appear. 


4.  In  any  triangular  plane  surf,  whether  right-angled,  or 
otherwise,  as  a  fee.  Fig  18;  whether  vert,  or  inclined;  the  base 
a  b  of  which  coincides  with  the  hor  surf  of  the  water  ;  the  cen 
of  pres  o,  will  be  in  the  center  of  the  line  c  r,  which  bisects  the 
base  a  b. 

5.  But  if  the  triangle,  as  a  s  c,  vert,  or  Inclined,  have  Its 
apex,  a,  at  the  surf  of  the  water ;  and  its  base  *  c,  hor ;  then  the 
cen  of  pres  x.  will  al«o  be  in  the  line  am  which  bisects  the  base; 
but  ax  will  be  %  of  am. 


6.    If  any  plane  triangle  a  b  c,  Fig  19,  base  up,  and  hor ;  have  its  base 

ab  covered  to  some  depth  nd,  with  water;  then  the  cen  of  pres  o,  will 

be  in  the  line  cs  which  bisects  the  base ;  and  no  will  be  equal  to 

7»»a  +  ('2mx  X  ma)  -f  3mq2 

(m»  +  2ma)  X  2. 


7.  The  center  of  pres  against  any 
plane  rectangular  surface,  Fig  20, 
whether  vert  as  m  n,  or  inclined  as 
po,  or  wx;  having  its  top  coinciding 
with  the  surf  of  the  water  ;  and 
pressed  by  diff  depths  of  water  on 
its  opposite  sides,  as  shown  in  the 
fig  ;  will  be  vert  below  the  upper 
water  surf,  a  dist  equal  to 

so  of  vert  \ 


Fig  16 


area  °f  turf 
n,  orpo,  or  wx 


half  of  *. 
naijof& 


(     area  of  surf 
Vcn,  or  «o,  ofsx 


haifof  r 
««</<>/  r 


528 


HYDROStATICS. 


8.    To  find  the  cen  of  pres  against  either  a  circular,  or  an  elliptic  surf,  pressed  on  one  side  only* 
whether  vert,  or  inclined  ;  and  having  its  top  either  coinciding  with  the  surf  of  the  water,  or  below  i( , 
Call  the  vert  depth  of  the  cen  of  pres  below  the  water  surf,  h. 
The  vert  (or  inclined,  as  the  case  may  be)  se/wi-diam  of  the  surf,  r. 
The  vert  dist  of  the  cen  of  the  pressed  surf,  below  the  water  surf,  d. 

Then,  h  — f-  d.    In  a  vert  circle  with  top  at  surf,  h  =  1%  rad. 

Art.  1O.    Walls  for  resisting-  the  pres  of  quiet  water.    A  study 

of  what  we  have  said  on  retaining- walls  for  earth,  will  be  of 
service  in  this  connection.  It  is  of  course  assumed  that  the 
water  does  not  find  its  way  under  the  wall ;  and  that  the  wall 
cannot  slide.  In  making  calculations  for  walls  to  resist  the  pres 

be  but  one  foot  in  Ir.nyth;  (not  height,  or  thickness;)  for  then 
the  number  of  cub  ft  contained  in  it,  is  equal  to  that  of  the  s^  ft 
of  area  of  its  cross-section,  or  profile;  so  that  these  sq  ft,  when 
mult  by  the  wt  of  a  cub  ft  of  the  masonry,  give  the  wt  of  tiie 
wall.  In  ordinary  cases,  it  is  well  for  safety  to  assume  that 
the  water  extends  down  to  the  very  bottom  line  of  the  wall. 

Now.  by  Art  1,  the  total  pres  of  quiet  water,  against  the  rec- 
tilineal back  of  a  wall,  whether  vert  or  sloping,  is  found  in 
tts,  by  mult  together  the  area  in  sq  ft  of  the  part  actually 
pressed,  Cor  in  contact  with  the  water;)  half  the  vert  depth  of 
the  water,  in  ft,  (being  the  vert  depth  of  the  cen  of  grav  of  a 
rectilineal  back,  below  the  surf;)  and  the  constant  62.5  tt»s; 
and  this  total  pres  is  always  perp  to  the  pressed  area. 
When  the  lack  of  the  watt  is  vert,  as  in  Fig  20^.  this  pres  p  is  of  course  less  than  when  it  is  bat- 
tered; and  is  also  hor  ;  and  it  tends  to  overthrow  the  wall,  by  making  it  revolve  around  its  outer 
toe,  or  edge  t.    Tbe  cen  of  pres  is  at  c ;  c«  being  %  the  vert  depth  on;  in  other  words,  the  entire 
pres  of  the  water,  so  far  as  regards  overthrowing  the  wall  as  one  mass,  (see  Art  1,  of  Force  in  Rigid 
Bodies,)  may  be  considered  as  concentrated  at  the  point  c :  where  it  acts  with  an  overthrowing  lever- 
age tl,  (see  Art*  41,  49,  50,  Force  in  Rigid  Bodies.)     The  pres  in  Ibs,  mult  by  this  leverage  in  feet, 
gives  the  moment  in  ft-fts  of  the  overturning  force ;  (see  Art  49,  Force 
in  Rigid  Bodies.)    The  wall,  on  the  other  hand,  resists  in  a  vert  di- 
rection g  a,  with  a  moment  equal  to  its  wt,  (supposed  to  be  concen- 
trated at  its  cen  of  grav  g,)  mult  by  the  hor  dist  at,  which  consti- 
tutes the  leverage  of  the  wt  with  respect  to  the  point  t  as  a  fulcrum. 
If  the  moment  of  the  water  is  greater  thnn  that  of  the  wall,  the  lat- 

Fl  ter  will  be  overthrown  ;  but  if  less,  it  will  stand. 

*-»^_      /      9  q  REM.  I.     Art  49  of  Force  in  Rigid  Bodies,  will  sufficiently  explain 

the  subjects  of  moments  and  leverage;  and  make  it  evident  that  the 
same  prinjiple  applies  also  to  sloping  backs,  as  in  Fig  21.     Here  the 
overturning  moment  of  the  water  is  equal  to  its  calculated  pres  p  X 
its  leverage  tl;  while  the  moment  of  stability  of  the  wall  is  equal  to 
its  wt  X  its  leverage  at.     By  aid  of  a  drawing  to  a  scale,  we  may  on 
this  principle  ascertain  whether  any  proposed  wall  will  stand.     For 
we  have  only  to  calculate  the  pres  p;  then  apply  it  at  c.  and  at  right 
angles  to  the  hack;  prolong  it  ~to  1;  'measure'?  2  by  the  same  scale. 
Then  calculate  the  wt  of  wall ;  find  its  cen  of  grav  "g :  draw  g  a  vert, 
and  measure  the  leverage  a  t.    We  then  have  the  data  for  calculating  the  two  moments.    For  finding 
the  cen  of  grav,  see  Cen  of  Grav,  Trapezoid,  p  442. 
RIM.  2.    If  the  water,  instead  of  being  quiet,  Is  liable  to  waves,  the  wall  should  be  made  thicker. 

Art.  11.  To  find  the  thickness,  a  c,  of  a  vert 
wall.  Vis  22«  sustaining  quiet  wa*er  level  with 
its  top,  and  as  deep  as  the  wall  is  high;  so  as 
to  resist  being:  overturned. 

RULE.  Divthe  number  1,  orl1^.  2,  3,  or  Ac,  (according  as  the resistance  of 
the  wall  is  reqd  to  be  either  just  equal  to,  or  1  ^.  2,  3.  or  Ac,  times  as  great 
as  the  overturning  notion  of  the  water.)  by  3  times  the  sp  grav  of  the  mate- 
rial of  which  the  wall  is  built.  Take  the  sq  rt  of  the  quot.  Mult  this  sq  rt 
bv  the  vert  depth  of  the  water  in  ft.  The  prod  will  be  the  reqd  thickness, 

The  sp  gr  of  a  dressed  granite  wall  may  be  taken  at  2.5 ;  of  dressed  sand- 
stone, 2.2;  common  mortar  rubble.  2:  brickwork.  1.8. 

T^  j*  O  O  Ex.    What  must  he  the  thickness  of  a  vert  wall,  built  of  mortar  rubble  of 

In  1  0   &  Lj        a  so  gr  of  2  ;  and  reqd  to  present  a  resistance  equal  to  1 .5  time*  the  pres  of 
\i?  the  water ;  the  depth  of  water,  or  height  of  wall  a  n,  each  being  20  ft?  Here, 

1.5-r6  =  .25;  and  the  sq  rt  of  .25=  .5;  and  .5X  20  =  10  ft  =  ac. 

Or  mult  the  ht  by  the  proper  decimal  below. 


Sp.  Gr. 

Lbs  per 
Cub  Ft. 

Resist  =  1.5  pres. 

Resist  =  2  pres. 

Resist  =  3  pres. 

Dressed  Granite... 
Dressed  Sandstone 
Mortar  Rubble  
Brickwork  

2.5 

t 

156 
137 
125 
112 

.447 
.477 
.500 
.527 

.516 
.550 
.578 
.609 

.633 
.674 
.707 
.746 

To  change  a  vert  wall  into  a  battered  one.  see  Art  8.  p  389. 

Art.  12.    To  find  the  thickness,  w  n,  at  the  base  of  a  rigrht- 

angled  triangular  wall  wmn,  Fig  23.  sustaining  at  its  vert  back  wm,  the  pressure  of  qui«t  water 
1—..1  •arith  it.  ton    «•./*  «>  r)»»n  i\«  the  wall  ii  hiffh.  in  a«  la  r«sier  baimr  overturned. 


fDKOSTATICS. 


529 


RULE.     DiT  the  nupafcer  1, 1  V$,  or  Ac,  (according  as  the  resistance  of  the  wall 
it  reqd  to  be  jus^etfual  to,  or  1&,  2,  or  &c,  times  as  great  as  the  overturning  ~" 
action  of  thejirftter,)  by  twice  the  sp  gr  of  the  material  of  which  the  wall  is 
built.     TaJe€  the  sq  rt  of  the  quot.     Mult  this  sq  rt  by  the  depth  of  water,  or 
height  of  wail  in  ft.     The  prod  will  be  the  required  thickness,  m  n,  in  feet. 

Or,  (original ;)  mult  the  thickness,  m  o,  of  a  vert  wall  by  1.225. 

Ex.  As  before ;  wall  of  rubble,  of  sp  gr  of  2  ;  resistance  to  be  1.5  times  the 
pros  of  the  water,  depth  20  ft.  Reqd  the  thickness,  mn.  Here, 

1.5-r  4=r.375.     The  sq  rt  of  .375- .6124 ;  aud  .6124  X  20-12.25  ft  ~m  n. 

Or   mull   the  tit  by  tlie   proper  decimal  be-  _ 
low. 

REM.  The  pressure  against  a  wall  sustaining  water  is  not  increased  bv  a  de- 
posit of  earth  on  the  same  side  at  its  natural  slope,  if  the  earth  is  imne'rvious 
to  water  and  in  sufficient  quantity  to  prevent  the  water  from  reaching  the  wall. 


Sp.  Or. 

Lbs. 

Resist  =1.  5  pres. 

Resist  —  2  pres. 

Resist  =  3  pres. 

Dressed  Granite... 

2.5 

156 

.548 

.633 

.775 

Dressed  Sandstone. 

2.2 

137 

.584 

.675 

826 

Mortar  Rubble  

2. 

125 

.613 

.707 

.866 

Brickwork.   . 

1  8 

112 

646 

746 

Notwithstanding  their  greater  thickness  at  base,  such  triangular  walls  contain,  as  seen  by  the  fig, 
not  much  more  than  half  the  quantity  of  masonry  reqd  for  vert  ones  of  equal  strength.  This  is  owing 
to  the  fact  that  their  cen  of  grav  is  thrown  farther  back  ;  thus  increasing  the  leverage  by  which  the 
wt  of  the  wall  resists  overthrow. 

Art.  13.  To  find  the  thickness,  a  b,  at 
the  base  of  a  wall.  Fig  24,  with  a  vert 
back,  ft  af  and  with  a  face,  it  ft,  having 
any  given  batter;  to  sustain  the  pres 
of  quiet  water  level  with  its  top.  and  \,.r 
as  deep  as  the  wall  is  high;  so  as  to 
resist  being  overturned. 

RULE.  Square  the  vert  depth  of  the  water,  or  wall,  in  feet. 
Mult  this  square  by  1.  or  by  \%,  2.  3,  or.  Ac.  (according  as  the 
resistance  of  the  wall  is  reqd  to  be  just  equal  to.  or  1%.  2,  3,  or, 
Ac.  times  as  great  as  the  overturning  action  of  the  water.)  Call 
the  prod  a.  Square  the  entire  amount  of  batter,  h  n,  in  feet. 
Mult  this  square  by  the  sp  gr  of  the  masonry  of  which  the  wall 

the  sum  by  3  times  the  sp  pr  of  the  masonry.    Take  the  sq  rt  of 
the  quot    This  sq  rt  will  be  the  reqd  base,  a  b. 

Ex.  Rubble  wall  20  ft  hi^h,  sp  gr  2  ;  ba-ter  of  face  4  inches  to  a  ft ;  or  total  batter  hn,  =  6.666  ft. 
What  must  be  the  thickness  a  b,  at  base,  that  the  resistance  of  the  wall  shall  be  1.5  times  the  pres 
of  the  water? 

Here,  the  square  of  20  —  400;  and  400  X  1.5  —  600,  or  prod  a.     Again,  the  square  of  6.666  =  44.44; 

and  44.44  X  2  =  88.88,  or  prod  b.    Now,  600  -4-  88.88  =  688.88 ;  aud  6^-  =  114.81 ;    aud   )/lU.bl  =: 

I 

10.715,  —  ab,  the  base  reqd. 
The  following  table  is  drawn  up  from  this  rule  : 

Or  mult  the  lit  sa  by  the  proper  decimal  below. 


Dressed  Granite.. 
Dressed  Sandstone 
Mortar  Rubble  
Brickwork  

£3 
& 

25 

2.2 
1 

1.8 

Wt.of 
a  cub. 
ft.  of 
Wall. 

Lbs. 

156 
137 
125 
112 

The  Resist  of  the  Wall  to  be  equal 
o  1>£  times  the  pres  of  the  Water. 

The  Resist  of  the  Wall  to  be  equal 
to  twice  the  pres  of  the  Water. 

Batter 
1  in  to 
a  foot. 

.449 
.480 
.502 
.530 

Batter 
2  ins.  to 
a  foot. 

.458 
.488 
.510 
.539 

Batter 
4  ins.  to 
a  foot. 

Batter 
6  ins.  to 
a  foot. 

Batter 
1  in.  to 
a  foot. 

Batter 
2  ins.  to 
a  foot. 

Batter 
4  ins.  to 
a  foot. 

Batter 
6  ins.  to 
a  foot. 

.593 
.622 
.646 
.674 

.487 
.515 
.536 

.5(52 

.532 

.558 
.578 
.602 

.519 
.552 
.571 
.610 

.526 
.560 
.586 
.618 

.551 
.583 
.609 
.640 

The  greater  the  batter,  and  consequently  the  greater  the  base,  the  less  masonry  is  reqd  to  secure 
the  same  strength. 

Art.  14.  The  following:  table  (although  not  scrupulously 
correct)  shows  how  greatly  the  safety  of  a  wail  sustaining 
water,  is  aifected  by  its  form;  the  quantity  of  masonry  re- 
maining the  same.  It  will  be  observed  that  the  vert  wall  is  the  least  safe 
of  all  in  the  table.  The  overturning  tendency  of  the  water  is  here  taken  as  1. 

Iii  France  are  dam  walls  from  4O  to  7O  ft  high,  with  bases  of -fjj 

the  height;  the  face  battering  y4^  of  the  height;  the  back,  against  which  the  water  presses,  being 
vert.  These  correspond  to  No.  12  of  the  following  table,  for  which  the  resist  is  2.6.  But  if  the  water 
pressed  against  the  battered  side,  the  resist  would  be  4.9 :  as  at  No.  6,  of  the  table  ;  but  on  this  point 
see  the  next  Art,  inasmuch  as  theory  and  practice  differ  here. 


530 


HYDROSTATICS. 


All  these  walls  contain  precisely  the  same 
quantity  of  masonry.    The  masonry  is  supposed 
to  be  mortar  rubble,  weighing  125  fbs  per  cubic  foot  ;  or  twice  as  much 

Base  in 
parts  of 
height. 

Appro* 
resist  ot 
wall. 

safety  also  will  be  greater  or  less,  in  precisely  the  same  proportion. 

1 

•  Vertical  wall  

5 

1  5 

2 

Face  vertical  ;  back  batters  one-tenth  height  

55 

1  8 

3 

"           "             "           "         onc-flfth         "       

6 

2  2 

4 

625 

2  6 

5 

•«          «            •'          •«        one-third       "       

667 

8  5 

• 

'«          •'           "          "        four-tenths    "      

7 

4  9 

7 

"          "            «'          "        one-half         "       

75 

14  0 

8 

Back  vertical  ;  face  batters  one-tenth  height  

.55 

1.8 

9 
10 

"          "            "          "        one-fourth     " 

.6 

625 

2.1 
2  2 

H 

••          "            *•          "        one-  third       "       

667 

2  4 

12 

"          "            "          "        four-tenths   "       

.7 

2.6 

13 
14 

Back  and  face  each  batter  one-  tenth  height  

'e 

2.9 
2  2 

15 

"          "           "          •<        one-fifth        "        

7 

3  4 

16 

75 

4  6 

17 

"           "             "           "         one  third       " 

833 

9  0 

18 

"            "              "            "         four  tenths    " 

| 

36  0) 

rbx 


When  the  base  of  a  triangular  wall,  of  sp  grav  2,  is  less  than  %  the  height,  the  wall  is  theoretically 
safest  when  the  water  presses  the  vert  side ;  but  if  the  base  is  greater  than  }£  the  height,  it  is  safest 
with  the  water  on  the  battered  side  ;  but  for  the  practical  view,  see  Art.  15. 

Art.  15.  Our  statement  that  walls  are  stronger  with  the  water 
pressing?  ag'aiiist  the  sloping*  back  rather  than  against  the  vert  one, 
holds  good  so  long  as  both  the  wall  and  the  foundation  may  be  considered  unyielding,  as  has  been 
the  assumption  of  writers  until  within  a  short  period.  But  of  late  years  the  erection  of  many  enor- 
mous reservoir  walls,  some  exceeding  160  ft  iu  ht,  (the  chief  danger  in  which  arises  from  their  own 
wt.)  has  induced  European  scientists,  Rankine  among  others,  to  investigate  the  subject  very  thor- 
oughly. 

They  agree  that  the  water  should  press  against  the  vert  back,  because  the  pressure  is  then  less  ; 
and  the  resultaat  falls  farther  back  from  the  toe,  thus  distributing  the  pressure  more  equally  not 
only  through  the  masonry,  but  also  along  the  foundation,  thereby  lessening  the  liability  of  either  to 
yield.  These  practical  considerations,  they  maintain,  far  outweigh 
that  of  the  higher  theoretical  safety  when  the  water  presses  on  the 
slant  back.  To  make  this  clearer,  Fig  25,  drawn  carefully  to  scale, 
represents  a  dam  wall  at  Poona.  Hindoostan,  de- 
signed by  Mr.  Fife,  C.  E.  of  England.  It  is  built  of  heavy  mortar 
rubble  of  150  fts  per  cub  ft.  Its  total  vert  ht  is  100  ft ;  thickness  at 
base  60  ft  9  ins  ;  and  at  top  13  ft  9  ins.  The  front  slopes  42  ft.  and  the 
back  5  ft,  in  100.  Its  cen  of  grav  is  at  G ;  and  1  ft  in  length  of  the 
wall  weighs  249.4  tons.  Its  foundation  is  7  ft  deep ;  but  we  shall  here 
assume  the  water  to  press  against  its  entire  back  xv.  This  pres 
would  be  139.6  tons.  On  ru  it  would  be  151.4  tons.  Through  G  draw 
a  vert  line  G  s ;  and  from  c,  where  the  direction  of  the  pres  P  of  the 
water  strikes  G  s,  lay  off  en  by  scale  to  represent  the  139.6  tons  water 
pres  against  x  v ;  and  c  t  the  249.4  tons  wt  of  1  ft  length  of  wall.  Com 
plete  the  parallelogram  of  forces  cnmt,  and  draw  its  diagonal  c  m. 
Then  cm  represents  the  resultant  of  all  the  pressures  upon  the  base 
u  v,  and  at  I  it  cuts  the  base  20  ft  back  from  the  toe  u.  Doing  the  same 
with  the  151.4  tons  pressure  .p  against  the  back  r  u,  we  get  the  result- 
ant o  y,  which  cuts  the  base  at  i,  only  12.7  ft  back  from  the  toe  v ;  or 
7.3  feet  less  than  I  is  from  u.  The  points  I  and  tare  called 
centers  of  resistance  of  the  base;  or  centers 
of  pres  upon  the  base.  See  Rem  1,  page  492.  Now  if 
the  back  xv  were  truly  vert,  the  theoretical  safety  against  overturning 
around  u  as  a  fulcrum  the  writer  finds  would  be  2  ;  with  the  actual  xv, 
battered  5  ft,  it  is  2.2;  and  with  the  water  pressing  against  ru  it 
would  be  3,  around  the  other  toe  v;  or  50  per  ct  greater  than  with  the 
rert  back  ;  or  36  per  ct  greater  than  with  the  actual  back  xv.  If  we  treat  a  portion  of  the  wall  (as  r 
xhf)  as  if  it  were  an  entire  wall,  with  the  water  first  against  one  back,  and  then  against  the  other, 
we  shall  obtain  the  centers  of  resistance  d  and  z  in  the  bed-joint/ A.  If  in  the  same  manner  we  as- 
•ume  three  or  four  such  walls,  each  having  the  top  rx,  we  may  find  the  centers  rof  resistance  at  other 
bed-joints.  Through  these  centers  draw  the  slightly  curved  dotted  lines  b,  d,  etc,  I :  and  6,  z,  etc,  t; 

called  lines  of  resistance,  or  lines  of  pressure;*  which  at  any  bed- 
joint  in  the  ht  of  the  wall  show  the  point  at  which  the  pres  upon  it  may  be  assumed  to  be  concen- 
trated. It  does  not  thow  in  what  direction  the  pres  comes  to  that  point.  Said  direction  is  that  of 
the  resultant  that  cuts  at  that  point.  In  Fig  25,  it  is  seen  that  when  the  water  presses  the  back  xv, 
the  line  bdl  of  pres  falls  farther  within  the  wall  than  when  the  water  is  against  ru,  in  which  case 
6  z  i  would  be  the  line.  Hence  both  the  wall  and  its  foundation  are  less  liable  to  derangement  by  frac- 
ture or  crushing  when  the  water  presses  against  xv;  and  the  earth  foundation  u  v  is  then  more 
evenly  loaded,  and  hence  less  liable  to  yield  unequally  so  as  to  cause  cracks  in  the  wall ;  and 

•*  Moseley  unfortunately  applies  this  last  term  to  another  line  seldom  if  ever  used  by  engineers 
»nd  'or  which  "  Hue  of  resultants  "  would  probably  have  answered  as  well. 


STATICS.  531 

on  this  account  x  v  is  made  theJjSck  of  the  wall,  notwithstanding  that  the  theoretical  safety  against 
overturning  would  be  36  pejxa  greater  if  the  water  pressed  against  r  u.  According  to  Rankine,  the 
dist  e  I  ,  or  K  i,  from  theartuer  e  of  the  base  u  v,  to  the  point  where  the  resultant  c  m  or  o  y  (as  the  case 
may  be)  cuts  tbe  ba^e-fshould  not  exceed  .25  of  the  base  u  v,  unless  the  foundation  is  unusually  firm  ; 
and  on  soft  foundations  it  should  cut  at  or  near  tbe  center  e.  In  tbePoona  wall  it  is  .173  of  tbe  base. 
He  says,  howler,  that  tbe  common  practice  among  British  engineers  is  to  make  si  or  ei  from  .3  to 
.375,  or  from  three-tenths  to  three-eighths  o  the  base.  The  first  would  bring  tbe  resultant  as  near. 
and  tbe  last  considerably  nearer  to  the  toe  than  i  in  the  Poona  dam  would  be  to  the  toe  v  if  the  water 
pressed  against  ru. 

If  from  the  end  m  or  y  of  the  resultant,  we  draw  m  2  or  y  a  horizontal,  then  c  2  or  oa  (as  the  case 
may  be)  will  measure  the  entire  vert  pres  on  the  base  u  v  ;  and  m  2,  or  y  a  will  measure  the  hor  pres 
against  the  back  of  the  wall,  which  tends  to  make  the  courses  of  masonry  slide  on  each  other  as  well 
as  to  make  tbe  whole  wall  slide  on  its  foundation.  Having  this  hor  pres*,  and  knowing  that  the  fric- 
tion of  masonry  on  masonry  is  about  .6  of  the  wt,  or  of  the  vert  pres  upon  it,  it  is  easy  to  ascertain  if 
the  wall  is  safe  in  this  respect  or  not.  Also,  there  can  be  no  sliding  at  any  bed-joint  at  which  the 
resultant  makes  an  angle  not  greater  than  about  30°  or  32°  with  a  line  at  right  angles  to  said  joint. 
See  Art  63,  p  486;  and  table,  page  599.  Such  sliding  never  occurs  in  walls  of  ordinary  forms.  Good 
mortar  well  set  aids  against  sliding  ;  but  it  is  better  not  to  rely  upon  it.  Entire  walls  have  slidden 
on  slippery  foundations.  See  Rem  2,  p  331  :  Rem  2,  p  339;  and  Art  9,  p  340,  for  preventives. 

When  the  resultant  cuts  at  the  center  e  of  the  base,  the 

pres  is  distributed  equally  over  the  entire  base;  and  its  amount  per  sq  ft  of  base  will  be  = 
to  «   ft  '  which  is  the  m«a«   pres.     But  when  as  at  I  or  i  it  cuts 


areMMM  to  «q  ft  ' 

nearer  to  a  toe,  the  pres  will  be  greatest  at  that  toe.  and  will  diminish  regularly  to  the  other  end  of 

the  base.    When  el  or  ei  (as  the  case  may  be)  is  not  greater  than  one-sixth  of  uv,  then  the  max  pres 

persq  ft  at  the  nearest  toe  is  =  mean  presx(l  +  (6  -  —  —  )j.    When  el  or  ei  is  more  thaa 


/  2  v 

I3(.S  — eloref)   \ 
*  u«        ' 


one-sixth  of  u  v,  then  max  pres  per  sq  ft  at  nearest  toe  —  mean   pres  X 

The  max  pres  allowed  should  evidently  not  exceed  the  aafe  strength  per  sq  ft  of  the  masonry  or  soil. 

To  tind  the  least  pres  or  that  at  the  toe  farthest  from  the  resultant, 

subtract  the  mean  pres  from  the  max  pres ;  mult  the  remainder  by  2  ;  take  the  prod  from  the  max 
pres.  It  may  happen  that  the  prod  is  greater  than  the  max  pres ;  and  when  this  is  the  case  it  shows 

that  there  is  tension  at  the   farthest  toe;  that  is,  that  the  wall  at 

that  toe  does  not  pres  at  all  upon  the  base,  but  has  a  tendency  to  rise,  or  to  pull  away  from  it.  Thin 
may  he  exemplified  by  H,  Fig  25,  thus,  draw  a  hor  line  ai  for  the  base  :  at  its  center  c  draw  the  vert 
c  v,  equal  to  the  mean  vert  pres  ;  and  at  i  draw  it  for  the  max  pres.  Also  draw  tvn;  then  an  give* 
the  least  vert  pres  on  the  base,  being  that  at  a.  And  any  vert  line  drawn  from  t  n  to  any  point  on 
a  i  will  give  the  vert  pres  at  said  point.  But  if  tbe  mean  pres  should  be  as  small  as  c  e,  then  when 
we  draw  teo  we  find  that  a  portion  g  o  of  it  falls  below  the  base  a  i  ;  showing  that  at  g  pres  ceases, 
while  tension  begins,  and  increases  to  ao  at  a.  In  order  that  there  shall 
be  no  tension,  the  resultant  cm  or  oy  must  not  cut  the  base  uv  nearer  to  a 
toe  than  one-third  of  u  v,  which  is  the  dist  in  the  Poona  wall. 

Rankine  states  that  in  order  that  the  masonry  shall  be  secure 
against  crushing?,  the  dist  e  /,  or  e  i  from  the  resultant  to  the  center  e  of  the 

/          Twice  the  total  vert  pres  on  the  base  uv          \ 
base  must  not  exceed  .5  -  /  3  time8  the  safe  crushing^nguTof  masonry^  )  .    The  vert  Pre« 

sq  ft  X  area  of  base  uv  in  sq  ft.  ' 

and  the  safe  strength  both  to  be  in  the  same  terms,  tons  or  Jbs. 

First-class  rubble  in  cement  mortar,  or  good  cement  concrete,  should  be  safe  with  8  tons  pres  per 
sq  ft,  which  limit  will  rarely  be  reached  ;  and  sound  earth  or  gravel  foundations  sunk  to  a  depth 
sufficient  to  protect  them  from  frost,  rain,  sliding,  <fec,  should  be  safe  with  from  2  to  4  tons  per  sq  ft. 

The  foundation  must  be  secure  against  not  only  the  wt  of  wall,  but  against  its  resultant  with  the 
water  pres. 

Art.  16.    In  California  is  this  dam  of  a  mining  reservoir,  built  of 

rough  stone  without  mortar,  founded  on  rock.     Ht,  70  ft ;  base,  50  :  top,  6;  water-slope,  30  ft;  outer. 

slope,  14.     To  prevent  leaking  the  water-slope  is  only  covered  with  3  inch  plank 

bolted  horizontally  to  12  by  12  inch  strings,  built  into  the  stone-work.     All  laid 

with  some  care  by  hand,  except  a  core  of  about  one-fifth  of  the  mass,  which  was 

roughly  thrown  in.    Cost  about  $3  per  cub  yd.    It  has  been  in  use  since  1860.    The 

fig  is  drawn  to  scale. 

Rem.  If  a  dam  is  compactly  backed  with  earth 

at  its  natural  slope,  and  in  sufficient  quantity  to  prevent  the  water  from  reaching 
the  dura,  the  pressure  against  the  dam  will  not  be  increased. 

Art.  17.    To  find  the  thickness  of  a  cylinder  to 

resist  safely  the  pressure  of  water,  steam,  &c,  against  its  interior.    If  riveted, 
see  next  page.     Tbe  rules  of  Reuleaux  and  Lame  which  are  the  best,  amount  to 

this.    Rule.  Divide  the  ultimate  cohesion  of  the  material  in  ft>s  per  sq  inch,  by 

the  number  denoting  the  reqd  degree  of  safety.  The  quotient  is  the  aafe  cohesion.  Divide  the  given 
interior  pressure  in  B>s  per  sq  inch  by  this  safe  cohesion.  (Indeed  this  interior  pres  might  first  be 
reduced  15  fts  per  sq  inch,  on  account  of  the  outer  pres  of  the  air,  which  assists  the  pipe;  but  it  is 
so  small  that  this  is  rarely  done.)  Call  the  quot  m.  To  half  of  m  add  1.  Mult  the  sum  by  m.  Mult 
the  prod  by  the  inner  rad  in  ins. 

Rent.  1.    It  is  often  known  beforehand  (as  in  large  steam  boilers,)  that  the 

safe  thickness  will  he  less  than  one-thirtieth  of  the  rad;  and  in  that  case  we  may  find  the  thickness 
by  merely  mult  TO  by  inner  rad  in  ins. 

Example.    How  thick  should  be  the  cast  iron  cyl  of  a  hyd  press  of  14  ins 

bore,  or  7  ins  inner  rad,  to  resist  with  a  safety  of  4  an  internal  pres  of  2000  tt»s  per  sq  inch;  taking 
the  ult  cohesion  of  the  iron  at  18000  tt»s  per  sq  inch  ? 


532 


HYDROSTATICS. 


.2222  ;  to  which  adding  1  we  have  1.2222.  And  finally  1.2222  X  .4*44  X  7  =:  8.8  ins,  the  reqd  thickness. 
But  to  allow  for  irregular  casting,  air-bubbles,  &c,  make  it  .5  in  more,  or  4.3  ins. 

Rom.  2.  Want  of  uniformity  in  the  cooling:  of  thick  castings  makes 

them  proportionally  weaker  thau  thin  ones,  so  that  in  order  to  reduce  thickness  in  important  cases 
we  should  use  only  best  iron  remelted  3  or  4  times,  by  which  means  an  ult  cohesion  of  about  30000 

tbs  per  sq  inch  may  be  secured.  But  even  with  this  precaution  no  rule  will 
»Pl>ly  safely  in  practice  to  cast  cylinders  whose  thickness  exceeds  either 

about  8  to  10  ins,  or  the  inner  rad  however  small. 

Under  a  pres  of  8000  ft>s  per  sq  inch,  water  will   ooze  through  cast 
iron  8  or  1O  ins  thick ;  and  under  but  250  Jbs  per  sq  inch,  through  .5  inch. 
Table  of  thicknesses  of  single-riveted  wrought  iron  pipes, 

tanks,  standpipes.  Ac,  by  the  above  rule,  to  bear  with  a  safety  of  6  a  quiet  pressure  of  1000  ft  head 
of  water,  or  434  Ibs  \-e:  sq  inch  ;  the  ult  cob.  of  fair  quality  plate  iron  being  taken  at  480uO  ft>s  per  sq 
inch,  or  at  8000  fts  for  a  safety  of  6  ;  which  is  farther  reduced  to  8000  X  .56  —  4480  ft>s,  to  allow  tor 

weakening  by  rivet  holes;  for  single-riveted  cyls  have  but  about  .56  of  the 
strength  of  the  solid  sheet;  and  double- riveted  ones  about  .7.  With  the 

above  pres  and  other  data,  the  rule  here  leads  to  thickness  —  .1016  X  iuuer  rad  in  ins. 

For  a  similar  table  for  tanks,  see  p  484;  and  for  cast  iron  and 

lead  pipes,  loot  of  this,  and  top  of  next  page.  (Original.) 


Di. 
Ins. 

Ths. 
Ins. 

Di. 
Ins. 

Ths. 
Ins. 

Di. 

Ths. 

Di. 

Ths. 

Di. 

Ths. 

Di. 

Di. 

Ths. 

.5 
1.0 
1.5 
2.0 
3.0 
4.0 

.025 
.051 
.076 
.102 
.152 
.203 

5 

8 
8 
10 
12 
14 

.254 
.305 
.406 
.508 
.609 
.711 

16 
18 
20 

22 
24 
27 

.813 
.914 
1  016 
1.117 
1.219 
1.371 

30 
33 
36 
42 
48 
54 

1.52 
1  68 
1.83 
2.13 
2.44 
2.74 

60 
66 
72 
8i 
96 
108 

3.05 
3.35 
366 
4.27 
4.88 
5.49 

120 
132 
144 

19'2 
240 

288 

10 
11 
12 
16 
20 
24 

6.09 
6.70 
7.31 
9.75 
12.19 
14.63 

For  a  less  liead  or  pressure,  or  for  any  safety  less  than  6,  it  is  safe  and 

near  enough  in  practice,  to  reduce  the  thickness  of  wrought  iron  cyls  in  the  same  proportion  as  said 
head,  pres,  or  safety  is  less  than-the  tabular  one. 

l>oiible-riveted  cylinders.  Fairbairn  says,  are  about  1.25  times  as  strong 
as  single-riveted.  Hence  they  may  be  one-fifth  part  thinner.  Lap-welded 
ones  are  nearly  1.8  times  as  strong  as  single-riveted;  and  hence  may  be- only 

.56  as  thick. 

Many  continuous  miles  of  double- riveted  pipes  in  California  have 

beeu  in  use  for  years  with  safetys  of  but  2  to  2.6.    In  one  case  the  head  is  1720  ft,  with  a  pres  of  746  fts 
per  sq  inch ;  diam  11.5  ins ;  thickness,  .34  inch  ;  safety,  2.6  by  rule  p  531  for  such  iron  as  in  our  table, 

Cast  iron  city  water  pipes  require  a  somewhat  greater  thickness  than 

that  given  by  the  rule,  p  531,  to  enable  those  of  small  bore  to  endure  the  necessary  handling,  and  to 

Srovide  against  irregular  casting,  and  the  air- bubbles  or  voids  to  which  all  castings  are  more  or  less 
able.     In  the  writer's  opinion  experience  has  shown  that  if  we  employ  one-eighth  of  the  ultimate  co- 
hesion of  the  iron  in  using  the  rule  p  531,  and  then  add  .3  inch  to  every 
resulting  thickness,  weshall  obtain  satisfactory  practical  thicknesses.    In 
preparing  the  following  table  the  ult  con  of  the  cast  iron  is  taken  at 

18000  fbs  per  sq  inch  ;  and  in  the  table  .3  inch  is  added  to  each  result  of  the  rule. 

Table  of  Practical  Thicknesses  for  Cast  Iron  Water-pipes. 


Heads  of  Water,  in  Feet. 

1 

50    |     75 

100 

125 

150 

200 

250 

300 

400 

500 

600 

700    |    800      1000 

a 

Pressures  in  Pounds,  per  sq  inch. 

0 

21.7 

32.6 

43.4 

54.3 

65.1 

86.8 

109 

130 

174 

217    i 

260 

304    1    347 

434 

Thickness  of  Pipe  in  Inches.       Original. 

2 

.31 

.32 

.32 

.32 

.33 

.34 

.35 

,36 

.38 

.40 

.42 

.45   '      .47   ;      .51 

4 

.32 

.33 

.34 

.35 

.36 

.38 

.40 

.42 

.46 

.50 

.54 

.59  '      .63   j      .72 

6 

.33 

.35 

.36 

.37 

.39 

.42 

.45 

.48 

.54 

.60 

.67 

.73         .80         .94 

9 

.34 

.37 

.39 

.41 

.43 

.48 

.52 

.57 

.66 

.76 

.85 

.95 

1.05  !    1.25 

12 

.36 

.39 

.42 

.45 

.48 

.54 

.60 

.66 

.78 

.91 

1.03 

1.17 

1.30  ,    1.57 

15 

.37 

.41 

.45 

.48 

.52 

.60 

.67 

.75 

.90 

1.06 

1  22 

1.38 

1  55 

1.89 

18 

.39 

.43 

.48 

.52 

.56 

.65 

.75 

.84 

1.02 

1.21 

1.40 

1  60 

1.80 

2.20 

24 

.42 

.48 

.53 

.59 

.65 

.77 

.90 

1.01 

1.26 

1.51 

1.77 

2.03 

2.29 

2.84 

30 

.45 

.52 

.59 

.67 

.74 

.89 

1.04 

1.19 

1.51 

1.82 

2.13 

2.46 

2.79 

3.47 

36 

.47 

.56 

.65 

.74 

.83 

1.01 

1.19 

1.37 

1.75 

2.12 

2.50 

2.90 

3.29 

4.11 

48 

.53 

.65 

.77 

.89 

1.00 

1.24 

1.49 

1.73 

2.23 

2.73 

3.23 

3.76 

4.29 

5.38 

60 

.59 

.74 

.88 

1.03 

1.18 

1.48 

1.79 

2.08 

2.71 

3  33 

3.97 

4.63 

528 

6.65 

72 

.65 

.83 

1.00 

1.18 

1.36 

1.72 

2.09 

2.14 

3.19 

3.94 

4.70 

5.49 

6.28 

7.91 

84 

.71 

.91 

1.12 

1.33 

1.53 

1.95 

2.38 

2.80 

3.67 

4.55 

5.43 

6.36 

7.28 

9.18 

96 

.77 

1.00 

1.24 

1.47 

1.71 

2.19 

2.68 

3.15 

4.16 

5.15 

6.17 

7.22 

8.27 

10.5 

•OSTATICS. 


533 


Table  of  thicfcrtess  of  lend 


I  pipe   to  bear  internal  pressures  with  a 

•afety  of  6;  taking  UWlmimate  cohesion  of  lead  at  1400  Ibs  per  sq  inch.     By  rule  on  p  531. 

Item.  Although  these  thicknesses  are  safe  againstquiet  pressures.they  might  not 

resist  shockaprfused  by  too  sudden  closing  of  stop-cocks  against  running  water.  See  Service  pipes,  p  377. 


1 

1 

£ 

1 

I 

H 

100 

Hea 

200 

da  in 
300 

Feet. 
400 

500 

1 
1 

a 

1 
IK 
l^ 
IX 

2 

100 

Hea 
200 

dsin 
800 

Feet. 
400 

500 

Pr 
43.4 

esin] 
'86.8 

bs  per  sq  ir 

130    1    174 

ch. 
217 

Pr 

43.4 

esinl 

86.8 

bspe 
130 

p  sq  in 
174 

oh. 

217 

T 

.026 
.038 
.051 
.064 
.076 
.089 

hickn 
.055 
.083 

.111 

.138 
.166 
.193 

ess  in 
.089 
.134 
.179 

.223 
.268 
.313 

Inchc 

.128 
.192 
.256 
.320 
.383 
.447 

58. 

.171 

.256 
.341 
.427 
.512 
.597 

1 

.102 
.127 
.153 
.178 
.204 

hickn 

.221 
.276 
.332 
.387 
.442 

ess  in 

.357 
.447 
.536 
.626 

.714 

Inch* 
.511 
.639 
.767 
.895 
1.02 

Mi 

.682 
.853 
1.02 
1.20 
1.36 

Rein.  The  valves  or  stop-grates  of  water-pipes  must  be  closed 
slowly,  and  this  precaution  increases  with  their  diams.  Otherwise  the  sud- 
den arresting  of  the  momentum  of  the  running  water  will  create  a  great  pressure  against  the  pipes 
in  all  directions,  and  throughout  their  entire  length  behind  the  gate,  even  if  it  be  many  miles ;  thus 
endangering  their  bursting  at  any  point.  Hence  stop-gates  are  shut  by  screws,  (p  573)  which  pre- 
vent any  very  sudden  closing ;  but  in  large  diams  even  the  screws  must  be  worked  very  slowly  to 
avoid  bursting. 

Art.  17.    The  buoyancy  of  liquids.    When  a  body  is  placed  in  a 

liquid,  whether  it  float  or  sink,  it  evidently  displaces  a  bulk  of  the  liquid  equal  to  the  bulk  of  the 
immersed  portion  of  the  body  ;  and  the  body  in  both  cases,  and  at  any  depth,  aud  in  any  position 
whatever,  is  buoyed  up  by  the  liquid  with  a  force  equal  to  the  wt  of 
the  liquid  so  displaced.  Thus,  if  we  immerse  entirely  in  water  a  piece 
of  cork  c,  c,  Fig  26,  or  any  body  of  less  sp  gr  than  water,  the  cork  will 
by  its  wt,  or  force  of  gravity,  tend  to  descend  still  deeper;  but  the 
upward  buoyant  force  of  the  water,  being  greater  than  the  downward 
force  of  gravity  of  the  cork,  will  compel  the  latter  to  rise  with  a 
force  equal  to  theditf  between  the  two.  In  this  case,  the  cork  receives 
a  total  downward  pres  equal  to  the  wt  of  the  vert  column  of  water 
above  it,  shown  by  the  vert  Hues  in  vessel  1 ;  and  a  total  upward 
pres  equal  to  the  wt  of  the  column  shown  in  vessel  2.  The  diff  be- 
tween these  two  columns  is  evidently  (from  the  figs)  equal  to  the 
bulk  of  the  cork  itself;  therefore  the  diff  between  their  wts  or 
pressures,  (or,  in  other  words,  the  buoyancy  of  the  water,)  is  equal 
to  the  wt  or  pres  of  the  water  which  would  have  occupied  the  place 
of  the  cork :  or,  in  other  words,  of  the  water  which  is  displaced  by 
the  cork.  This  diff,  or  buoyancy,  will  plainly  be  the  same  at  any 
depth  whatever  of  entire  immersion.  Now  the  cork,  if  left  to  itself,  will  continue  to  rise  until  a  por- 
tion of  it  reaches  above  the  surf,  as  in  vessel  3  ;  so  that  the  downward  pressing  column  ceases  to 
exist;  aud  the  cork  is  then  pressed  downward  only  by  its  own  wt.  But  as  it  now  remains  station- 
ary, we  know  (from  the  fact  that  when  two  opposite  forces  keep  a  body  at  rest,  they  must  be  equal  to 
one  another)  that  the  upward  pres  of  the  water  must  be  equal  to  the  wt  of  the  cork.  But  the  upward 
pres  of  the  water  arises  only  from  the  shaded  column  shown  in  vessel  3;  and  this  column  is  tas  in 
the  case  of  total  immersion)  equal  to  the  bulk  of  water  displaced.  Therefore,  in  all  cases,  the  buoy- 
ancy is  equal  to  the  wt  of  water  displaced  ;  and  when  the  body  floats  on  the  surf,  the  buoyancy,  or 
the  wt  of  water  displaced,  is  also  equal  to  the  wt  of  the  body  itself. 

If  the  immersed  body  <%  c,  be  of  iron,  or  any  other  substance  spe- 
cifically heavier  than  water,  the  diff  between  the  upward  and  downward  pres  wilt  of  course  remain 
the  same ;  or  equal  to  the  wt  of  water  displaced.  But  the  wt  of  the  body  is  now  greater  than  that 
of  the  water  which  it  displaces;  or,  in  other  words,  the  downward  force  of  gravity  of  the  body  is 
greater  than  the  upward  buovant  force  of  the  displaced  water;  and  therefore  the  body  descends,  or 
sinks,  with  a  force  equal  to  the  diff  between  the  two.  Thus,  if  the  body  be  a  cub  ft  of  cast  iron, 
weighing  450  Ibs,  while  a  cub  ft  of  fresh  water  weighs  62J^  Ibs,  the  iron  will  descend  with  an  effective 
force  of  only  450  —  62^  =  387.5  fts. 

If  the  immersed  body  has  the  same  sp  j?r  as  the  fluid,  it  will 

neither  rise  nor  sink  ;  but  will  remain  wherever  it  is  placed  ;  because  then  the  wt  of  the  body,  and 

the  buoyancy  of  the  water,  are  equal.    For  floating  bodies, see  p  635. 

The  air  also  buoys  bodies  upward  to  an  extent  equal  to  the 
wt  of  air  displaced;  therefore,  although  a  pound  of  iron,  and  a  pound  of 

feathers,  weighed  in  the  air,  will  balance  each  other  yet  in  the  exhausted  bell-glass  of  an  air-pump 
the  feathers  will  outweigh  the  iron,  by  as  much  as  the  bulk  of  air  which  they  displaced  outweighs 
the  bulk  of  air  displaced  by  the  iron. 

A  balloon  rises  in  the  air  on  the  same  principle  that  corlt 
rises  in  water.  Its  ascending  force  is  equal  to  the  diff  between  its  wt  when 
full  of  ea<<.  and  the  wt  of  the  bulk  of  air  which  it  displaces.  The  balloon  does  not  actually  tend  to 
rise,  but  to  descend;  but  the  air  being,  bulk  for  bulk,  heavier  than  the  balloon,  pushes  the  latter 
upward  with  more  force  than  the  gravity,  or  the  wt  of  the  balloon,  exerts  to  bring  it  down.  So  also 
warm  smoke  has  no  tendency  in  itself  to  rise.  It  is  pushed  up  by  the  heavier  cold  air.  No  substance 
tends  to  rise:  but  all  tend  downward  teward  the  center  of  th«  earth. 


534 


HYDRAULICS. 


m 


Art.  18.  A  body  lighter  than  water,  if  placed  at 
tbe  bottom  of  a  vessel  containing:  water,  will  not 
rise  unless  the  water  can  get  under  it,  to  buoy  it, 
or  press  it  upward,  as  the  air  presses  a  balloon  or 
smoke  upward.  Thus,  if  one  side  of  a  block  of  light  wood, 
perfectly  flat  and  smooth,  be  placed  upon  the  similarly  flat  and  smooth  bottom  of  a 
vessel,  and  held  there  until  the  vessel  is  Oiled  with  water,  the  downward  pres  will 
keep  it  in  its  place,  until  water  insinuates  itself  beneath  through  the  pores  of  the 
wood.  But  if  the  wood  be  smoothly  varnished,  to  exclude  water  from  its  pores,  it 
will  remain  at  the  bottom. 


Fig  28 


On  the  other  hand,  a  piece  of  metal  may  be  pre- 
vented from  sinking:  in  water,  by  subjecting  it  to  a  suffi- 
cient upward  pres  only,  while  the  downward  pres  is  excluded.     Thus,  if  the  bottom 
TV      ft-*      °^  an  °Pen  glass  tube,  t,  Fig  27,  and  a  plate  of  iron  m,  be  made  smooth  enough  to  be 
|5  m  £,  /      water-tight  when  placed  as  in  the  fig ;  and  if  in  this  position  they  be  placed  in  a 
,J  vessel  of  water  to  a  depth  greater  than  about  8  times  the  thickness  of  the  iron.  th» 

upward  pres  of  the  water  will  hold  the  iron  in  its  place,  and  prevent  its  sinking,- 
because  it  is  pressed  upward  by  a  column  of  water  heavier  than  both  the  column  of  air,  and  its  own 
weight,  which  press  it  downward.  On  this  principle  iron  ships  float. 

HEM.  1.    A  retaining-- wall,  as  in  Fig:  28, 

founded  on  piles,  may  be  strong  enough  to  re- 
sist the  pres  of  the  earth  e  behind  it,  in  case  water  does  not  find 
its  way  underneath;  and  yet  may  be  overthrown  if  it  does;  or 
even  if  the  earth  «  «  around  the  heads  of  the  piles  becomes  satu- 
rated with  water  so  as  to  form  a  fluid  mud.  In  either  case,  the 
upward  pres  of  the  water  pgainst  the  bottom  of  the  wall  will  vir- 
tually reduce  the  wt  of  all  such  parts  as  are  below  the  water  surf, 
to  the  extent  of  62>£  H>s  per  cub  ft;  or  nearly  one-half  of  the  or- 
dinary wt  of  rubble  masonry  in  mortar. 

RKM.  2.  Although  the  piles  under  a  wall,  as  in  Fig  28,  may  be 
abundantly  sufficient  to  sustain  the  wt  of  the  wall ;  and  the  wall 
equally  strong  in  itself  to  resist  the  pres  of  the  backing  e;  yet  if 
the  soil  as  around  tbe  piles  be  soft,  both  they  and  the  wall  may  be  pushed  outward,  and  the  latter 
overthrown  by  the  pres  of  the  backing  e.  From  this  cause  the  wing-walls  of  bridges,  when  built 
on  piles  in  very  soft  soil,  are  frequently  bulged  outward  and  disfigured.  In  such  cases,  the  piling, 
and  the  wooden  platform  on  top  of  it,  should  extend  over  the  whole  space  between  the  walls;  or  else 
some  other  remedy  be  applied. 

Art.  19.  Draught  of  vessels.  Sincea  floating  body  displaces  a  wt  of liquid 
equal  to  the  wt  of  the  body,  we  may  determine  the  wt  of  a  vessel  and  its  cargo,  by  ascertaining  how 
many  cub  ft  of  water  they  displace.  The  cub  ft,  mult  by  62V£,  will  give  the  reqd  wt  in  fl>s.  Suppose, 
for  instance,  a  flat-boat,  with  vert  sides,  60  ft  long.  15  ft  wide,  and  drawing  unloaded  6  ins,  or  .5  of 
a  ft.  In  this  case  it  displaces  60  X  15  X  .5  =  450  cub  ft  of  water  ;  which  weighs  450  X  62J^  =  28125 
Tbs ;  which  consequently  is  the  wt  of  the  boat  also.  If  the  cargo  then  be  put  in.  and  found  to  sink 
the  boat  2  ft  more,  we  have  for  the  wt  of  water  displaced  by  the  cargo  alone,  60  X  15  X  '^  X  62>^  = 
112500  Its  ;  which  is  also  the  wt  of  the  cargo.  So  also,  knowing  beforehand  the  wt  of  the  boat  and 
cargo,  and  the  dimensions  of  the  boat,  we  can  find  what  the  draught  will  be.  Thus,  if  the  wt  as  before 

140625 

be  140625  fts.  and  the  boat  60  X  15,  we  have  60  X  15  X  62^  =  56250;  and =  2  5ft  tbe  required 

56250 

draught.    In  vessels  of  more  complex  shapes,  aa  in  ordinary  sailing  vessels,  the  calculation  of  the 
amount  of  displacement  becomes  more  tedious ;  but  the  principle  remains  the  same. 

Art.  2O.  Compressibility  of  liquids.  Liquids  are  not  entirely  in- 
compressible ;  but  for  most  engineering  purposes  they  may  be  so  considered.  The  bulk  of  water  is 
diminished  but  about  one-thousandth  part  by  a  pres  of  324  fts  per  sq  inch,  or  22  atmospheres  ;  vary, 
ing  very  slightly  with  its  temperature.  It  is  perfectly  elastic  ;  regaining  its  original  bulk  when  the 
pres  is  removed. 


HYDRAULICS, 


A  rt.  1.   Hydraulics  treats  of  the  flow  or  motion  of  water  through 

pipes,  aqueducts,  rivers,  and  other  channels;  also-  through  orifices  or  openings  of  various  kinds :  of 
machinery  for  raising  water ;  as  well  as  that  in  which  water  furnishes  the  moving  power.  The  science 
of  hydraulics,  in  many  of  its  departments,  is  but  imperfectly  understood ;  therefore,  some  of  the  rules 
given  on  the  subject  are  to  be  regarded  merely  as  furnishing  close  approximations  to  the  truth. 

On  the  flow  of  water  through  pipes.    See  Caution,  p  566. 

Inasmuch  as  the  experiments  on  which  the  following  rules  are  based,  were  made  with  pipes  care- 
fully laid  in  straight  lines ;  and  perfectly  free  from  all  obstructions  to  the  flow  of  the  water,  some 
allowance  must  in  practice  be  made  for  this  circumstance.  Workmen  do  not  lay  long  lines  of  pipes 
in  perfectly  straight  lines ;  it  is  almost  impossible  to  avoid  very  numerous,  although  slight  devia- 
tions, both  vert  and  hor ;  the  soil  itself,  in  which  the  pipes  are  imbedded,  especially  when  in  embkt, 
will  settle  unequally  ;  especially  in  streets  liable  to  heavy  traffic,  which  not  only  frequently  deranges, 
but  occasional! v  breaks  water  pipes  whose  tops  are  3  or  4  ft  below  the  surf.  The  material  used  for 
calking  the  joints,  may  be  carelessly  left  projecting  into  the  interior,  and  thus  cause  obstructions;  the 
water  is  frequently  muddy,  or  is  impregnated  with  certain  salts,  or  gases,  which  form  deposits,  or 
incrustations,  wh  ch  materially  impede,  the  flow.  See  Art  27.  Moreover,  the  pipes  themselves  are  not 
cast  perfectly  straight,  or  smooth,  or  of  uniform  diam ;  and  irregular  swellings,  by  producing  eddies, 


HYDRAULICS. 


535 


retard  the  flow  as  wirtfas  contractions ;  and  accumulations  of  air  do  the  same.  Under  the  most  favor- 
able circuu»staric*«r  therefore,  it  is  expedient  to  make  the  diams  of  pipes,  even  lor  temporary  pur- 
poses, sufficiency  large  to  discharge  at  leant  20  per  ct  more  than  the  quantity  actually  needed ;  and 
if  there  is  qnseasion  to  anticipate  deposit,  or  incrustation,  a  still  larger  allowance  should  be  made  in 
permanent  pipes,  especially  iu  those  of  small  diam  ;  because  in  them,  the  same  thickness  of  incrusta- 
tion occupies  a  greater  comparative  portion  of  the  area.  Perhaps  it  would  be  best  to  allow  an  equal 
increase,  of  say  from  J^  to  1%  inch,  to  each  diam,  whether  great  or  small ;  inasmuch  as  the  thickness 
of  incrustation  will  be  the  same  for  all  diams,  or  nearly  so.  The  cost  of  pipes  does  not  increase  as 
rapidly  as  their  discharging  capacities  ;  thus,  if  the  diam  be  Increased  only  -fa  part,  the  disch  will 
be  increased  about  25  per  cent;  if  %  part,  nearly  50  per  cent,'  if  J^  part,  the  disch  will  be  doubled. 
See  Table  2.  Within  these  limits,  the  increase  of  thickness  for  the  larger  diams,  and  the  increased 
expense  of  laying,  will  add  but  little  to  the  cost;  vhich  will  therefore  augment  only  a  little  more 
rapidly  than  the  diahis. 
The  increased  diam  involves  no  waste  of  water ;  since  the  disch  may  be  regulated  by  stopcocks. 


LEVEL 


eL-li 


The  term  HEAD  or  TOTAI*  HEAD  of  water,  as  applied  to  the  flowage  of 

water  through  canals,  pipes,  or  openings  in  reservoirs,  &c,  means  the  vert  dist  i  v  or  p  o,  Fig  1,  from 
the  level  surf,  mi,  of  the  water  in  the  reservoir,  or  source  of  supply,  to  the  center  (or  more  properly  to 
the  cen  of  grav)  o,  of  the  orifice  (whether  the  end  of  a  pipe,  r  o,  t  o,  v  o,zo.  lo;  or  any  other  kind  of 
opening)  through  which  the  disch  takes  place  freely,  into  the  air ;  or  the  vert  dist  a  u,  or  /.a,  from 
the  same  surf,  m  i,  to  the  level  surf,  g  u,  of  the  water  in  the  lower  reservoir;  when  the  disch  takes 
place  under  water.  Thus,  in  the  case  of  disch  into  the  air,  the  vert  dist  i  v  orpo,  is  the  total  head 
for  either  of  the  pipes  ro,  t  o,  v  o,  zo,  or  I  o;  and  i  k  is  the  head  for  the  orifice,  k,  in  the  side  of  the 
reservoir.  And  for  disch  under  water,  au.  or  / g,  is  the  head  for  either  the  pipe  j,  or  the  opening  n; 
without  any  regard  whatever  to  their  depths  below  the  surf  of  the  lower  water;  which,  according  to 
the  older  authorities,  do  not  at  all  afiect  their  disch.  Weisbach,  however,  a  more  recent  experimenter, 
says  the  disch  will  be  about  -fa  part  less  under  water,  than  into  the  air. 

In  the  case  of  circular  pipes  or  openings ;  or  of  rectangular,  or  many  other  shaped  openings,  the 
center,  and  the  cen  of  grav  coincide:  but  iu  triangular  or  trapezoidal  ones,  they  do  not;  hence  in  all 
cases  of  disch  into  air  the  head  must  be  measured  totheceu  of  grav  of  the  disch'opening ;  as  it  is  this 
head  alone  that  causes  the  flow.  The  pres  which  it  imparts  to  the  water,  overcomes  the  friction  op- 
posed by  thu  sides  and  bottom  of  the  channel,  and  thus  enables  the  water  to  advance.  After  water 
has  descended  along  any  channel,  to  any  given  dist,  that  head  by  which  it  was  driven  through  that 
dist,  is  sometimes  called  lost  head;  or  the  water  is  said  to  have  lost  so  much  head  in  that  dist. 

A  portion  of  a  pipe  mav  have  a  head  greater  than  the  total  head  of  the  entire  pipe.  Thus  the 
point  6  in  the  pipe  lo,  has  a  head  6 1 ;  while  the  entire  pipe  has  only  the  head  p  o. 

Both  in  theory  and  in  practice  it  is  immaterial  as  regards 
the  vel,  and  tSie  quantity  of  water  discharged,  whether  the 
pipe  is  inclined  downward,  as  ro.  Fig*  1;  or  lior.  as  r  o:  or  in- 
clined upward,  as  lo\  provided  the  total  head  po,  and  also 
the  length  of  the  pipe,  remain  nn  changed.  If  one  pipe  is  longer 

than  another,  its  sides  will  evidently  present  more  friction  against  the  water,  and  thus  diminish  the 
vel  and  the  quantity  of  disch.  The  inclined  pipes,  r  o,  I  o,  being  of  course  a  little  longer  than  the 
nor  one  uo,  will  therefore  each  disch  a  trifle  less  water:  but  if  the  hor  one  were  extended  slightly 
beyond  o,  so  as  to  give  it  the  same  length  as  the  others,  then  each  of  the  three  would  disch  the  same 
quantity  in  the  same  time. 

Rem.  1.  It  is  evidently  necessary  that  the  upper  or  entry  endr,  v  or  Z  of  the  pipe 

be  so  far  at  least  below  the  surface  m  f  of  the  water  as  to  enable  the  water  first  to  overcome  the  resistance 
which  the  sharp  edges  of  the  end  of  the  pipe  oppose  to  its  entrance  ;  and  then  to  force  the  water  into  the 
pipe  with  the  same  vel  it  i*  intended  to  have  through  it.  The  total  head  p  o  must  be  considered  as  di- 
vided into  three  parts,  of  which  either  p  w  or  i  »  may  represent  the  two  devoted  to  the  two  above  dis- 
tinct duties  ;  while  the  third  part  or  the  remainder  w  o  or  s  v  overcomes  the  resistanoe  of  friction, 

&e,  inside  of  the  pipe,  and  is  hence  called  the  friction  head,  or  the  resist- 
ance head ;  while  the  first  two  parts  are  called  the  entry  head,  and  the 
velocity  head.  The  vel  head  is  the  same  as  the  theoretical  head  in  Table  10, 

p  552;  and  experiment  shows  that  with  the  usual  sharp  edged  entry,  the  entry  head  is  (near  enough 
for  practice)  half  as  great.  If  the  entry  is  shaped  like  Pig  7,  p  554,  the  entry  head  disappears  almost 
entirely.  This,  however,  has  hut  little  effect  on  the  vel  or  discharge,  except  in  pipes  shorter  than 
1000  diamg.  It  becomes  more  apparent  as  they  shorten.  The  friction  head  may  be  all  above,  or  all 


536 


HYDRAULICS. 


below  the  entrance  into  the  pipe,  or  part  above  aud  part  below,  without  affecting  the  vel  or  disahare* 
of  the  water. 

Since  the  friction,  in  pipes  of  the  same  diam,  increases  in  amount  in  the  same  proportion  as  their 
lengths,  the  water  when  it  first  enters  the  pipe  encounters  but  little  friction,  and  has  great  vel ;  but 
this  gradually  decreases  as  the  advancing  water  encounters  the  friction  along  increased  lengths  of 
the  pipe;  and  finally  becomes  slowest  when  the  water  fills  the  whole  length,  and  begins  to  flow  from 
the  disch  end  o.  The  vel  then  becomes  uniform  along  the  pipe  so  long  as  the  entry  and  vel  heads  (is) 
are  sufficient  to  allow  the  water  of  the  reservoir  to  enter  the  pipe  with  that  name  veL  If  even  much 
more  th;in  these  two  heads  is  left  above  the  entrance  into  the  pipe,  the  effect  of  the  surplus  is  not  at 
all  diminished  thereby  as  regards  overcoming  the  friction  along  the  inside  of  the  pipe;  and  con- 
sequently the  vel  of  'the  disch  will  undergo  no  change.  Thus,  if  i  g  he  sufficient  head  for  a  pipe  laid 
direct  from  «  to  o,  and  if  the  pipe  be  afterward  changed  to  the  position  I  o,  the  vel  and  disch  will  be 
the  same  in  both  positions.  Therefore  nothing  more  is  necessary  than  to  be  certain  that  the  depth 
below  the  surface  m  i  of  the  reservoir  shall  not  be  less  than  t  *.  Theoretically,  «  is  at  the  center  (or 
rather,  at  the  cen  of  grav)  of  the  entry  opening  of  the  pipe,  or  aperture ;  and  the  head  i  t.  above  it. 
is  equal  to  J.5  times  that  found  in  Table  10.  opposite  to  the  vel  in  the  pipe.  Thus,  if  we  have  found 
by  calculation  (by  Rule  1.  following)  that  the  total  head  p  o  will  produce  along  the  pipe  a  vel  of  6  ft 
per  sec,  we  find  in  Table  10,  opposite  the  vel  of  6  ft.  the  vel  head  .56  ft.  Therefore  f  «  —  .56  4-  .28  = 
.84  ft  is  the  least  dist  that  the  center  of  the  end  of  the  pipe  must  be  placed  below  the  surf  m  i.  But 
the  end  of  the  pipe  should  in  practice  always  be  entirely  below  water;  otherwise  air  and  floating  im- 
purities will  be  drawn  into  it,  and  cause  obstructions ;  therefore>  if  the  pipe  is  large,  say  in  this  case 
3  feet  in  diam,  it  is  evident  that  the  center  of  its  upper  end  cannot  be  placed  lees  than  \%  ft  under 
water;  or  nearly  3  times  as  deep  as  the  true  theoretical  vel  head.  Moreover,  the  water  surface  of 
reservoirs  is  always  liable  to  considerable  changes  of  height;  so  that  the  end  of  the  pipe  must  be 
placed  at  such  a  depth  that  the  water  can  flow  into  it  at  the  lowest  stages.  As  before  stated,  this  will 
be  attended  by  no  diminution  of  disch.  The  above  vel  of  6  ft  per  sec,  is  full  4  miles  an  hour,  and  is 
one  seldom  reached  in  water  pipes  in  practice  ;  hence  we  see  that  in  ordinary  cases  the  vel  aud  entry 
heads  together  need  in  theory  rarely  exceed  .56  -f-  -28  =  .84  of  a  foot,  which  is  usually  but  a  small 
part  of  the  total  head,  p  o. 

A  straight  line  a  o,  drawn  from  the  above  S  to  the  cen  of  grav  o  of  the  disch  end,  is  called  the 
tllllic  grade-lilt©  of  any  pipe  so,  r  o,  t  o,  vo,  z  o,  or  I  o,  commencing  at  the 
ir,  and  ending  at  o;  and  any  such  pipe  may  be  bent  into  easy  curves,  (having  radii  not  less 
than  5  diaras  of  the  pipe.)  as  t  o,  zo;  and  still  deliver  as  much  water  a*  a  straight  one  of  the  same 
actuil  length,  provided  the  tops  of  its  highest  bends  are  kept  below  this  hydraulic  grade-line;  and 
provided  that  arrangements  be  made  for  the  escape  of  any  air  that  may  accumulate  at  the  tops  of  the 
bends.  See  Art  5,  and  Fig  42. 

If  the  hydraulic  grade-line  so  be  divided  into  any  number  of  equal  parts,  as  «  c,  ex,  kc;  and  if 
the  actual" length  of  any  pipe  terminating  at  o,  be  afso  divided  into  the  same  number  of  equal  parts, 
then  will  b  c  be  the  friction  head  of  the  first  division  of  the  pipe ;  zx  that  for  the  lir.st  two  divisions, 
Ac.  If  the  pipe  commences  vertically  under  s,  like  all  those  in  Fig  1  :  and  is  straight,  like  r  o,  vo, 
or  lo,  the  equal  divisions  of  the  pipe  will  be  vert  under  those  of  the  line  so.  But  if  the  pipe  be 
curved  either  hor,  or  vert,  like  to  and  z  o,  this  of  course  will  not  be  the  case. 

REM.  2.  A  great  diff  exists  between  the  condition  of  a  hor 
pipe,  and  one  inclining-  downward  from  the  reservoir,  in 
case  it  should  become  necessary  to  prolong*  them  in  their 
original  directions.  Thus,  if  we  extend  the  length  of  the  hor  pipe  r  o  be- 
yond o,  it  is  plain  that  we  do  not  thereby  increase  the  total  head  of  water;  for  the  vert  dist  from  the 
surf  of  the  reservoir,  to  the  di*ch  end  of  the  pipe,  will  still  remain  equal  to  po.  Consequently  the 
additional  friction  along  the  sides  of  the  extended  pipe,  will  cause  the  water  to  flow  more  slowly  than 
before;  or,  in  other  words,  the  disch  will  be  less.  Now  suppose  an  inclined  pipe  so,  atfirtinp  from 
the  exact,  vel  head  s.  It  is  plain  that  every  equal  dist,  s  c,  ex,  &c.  along  this  pipe,  has  its  equal  fric- 
tion head  be,  d x,  &c;  and  however  far  the  pipe  may  be  extended  beyond  o.  with  the  same  degree  of 
inclination,  its  friction  head  will  be  extended  in  the  same  proportion;  and  consequently  the  friction 
along  the  additional  length  will  be  thereby  counteracted,  so  that  no  chance  will  take  place  in  the  vel 
of  the  flow;  nor,  consequently1,  in  the  quantity  discharged.  The  vel  head  will  remain  the  same  as 
before,  for  it  still  has  merely  to  supply  the  same  vel  of  water  as  at  first.  Such  a  pipe  is  said  to  be 

in  train. 

This  will  not  be  the  case  to  the  same  extent  if  the 
entry  end  of  the  pipe,  as  a.  Fig  1%,  be  placed  be- 
low  a;  for  it  is  evident  that  if  ao  be  the  original 
length  of  the  pipe,  and  wo  its  friction  hejvd ;  then 
if  we  double  this  length,  by  extending  it  to  n,  we 
do  not  thereby  double  the  friction  head ;  for  the 
new  friction  head  m  n  is  not  equal  to  twice  wo.  and 
the  disch  will  of  course  be  slower.  Still,  if  s  a  is 
(as  will  often  be  the  case  in  practice)  very  small  in 
proportion  to  the  friction  head  wo  of  the  original 
pipe,  then  mn  will  be  so  nearly  twice  w  o,  that  the 
diminution  of  di«ch  from  extending  the  pipe  to 
any  reqd  dist,  will  be  but  slight. 

REM.  3.  Of  the  outward,  or  bursting*  pressure,  of  water  in 
pipes.  When  any  pipe,  as,  for  instance,  any  of  those  in  Fig  1,  is  full  of  water  at 
rest,  this  pres  is  greater  than  when  the  water  is  flowing  through  it;  and  is  that  due  to  the  total  head 
above  the  point  pressed.  Thus,  at  the  point  4  on  the  pipe  r  o,  it  is  that  due  to  the  head  4,  1  :  at  the 
point  fi  in  the  pipe  I  o,  that  due  to  the  head  6,  1  :  at  the  point  o  in  any  of  the  pipes,  that  due  to  o  p, 
Ac.  Therefore  it  may  always  be  readilv  calculated  in  Ibs.  by  the  Rule  p  522,  of  Hydrostatics; 
namely,  mult  together  the  area  in  sq  ft,  of  the  portion  pressed ;  the  total  head,  in  ft.  above  its  cen  of 
grav  ;  'and  the  constant  number  62.5.  If  the  discharge  end  of  the  pine  be  partially  opened,  the  water 
will  move  slowly,  and  the  prea  will  become  less,  and  when  the  water  ii  flowing  freely  through  any 


Fi 


HYDRAULICS. 


537 


full  pipe  of  iMrttorm  diam,  the  ends  of  which  are  entirely  open,  it  becomes  still  more  reduced;  and 
ff  the  plp^has  its  entry  end  precisely  at  the  vel  head  s,  Fig  1,  aud  itself  lying  upon  the  hydraulic 
grade-line  s  o.  (a  case  which  never  occurs  iu  practice,)  there  will  be  uo  bursting  pres  ;  and  the  pipe 
will  experience  uo  pres  of  any  kiud.  except  that  of  the  weight  of  the  ruuuiug  water  upon  its  lower 
side.  But  if  any  part  of  the  ruuuiug  pipe,  or  all  of  it,  be  below  the  hyd  grade-line,  as  iu  all  the  pipes 
in  Fig  I.  then  every  point  of  such  pipe  will  be  subject  to  a  bursting  pres  due  to  a  head  equal  to  the 
vert  dist  of  said  point,  below  the  yradv-line ;  aud  which  consequently  may  be  calculated  in  Ibs,  in 

Precisely  the  same  way  as  in  the  preceding  case  of  water  at  rest.  Thus  the  point  4,  in  the  pipe  ro, 
'i*  I,  will  be  pressed  outward  by  the  head  4,  3  ;  the  point  5,  in  the  pipe  to,  by  the  head  5,  3;  the 
point  6,  iu  lo,  by  6.  3;  aud  the  point  7  will  be  pressed  outward  by  the  head  7,  8;  &c.  If  at  any 
point  iu  any  pipe  thus  below  the  hydraulic  grade-line,  aud  discharging  freely,  au  open  vert  commu- 
nicating pipe  be  inserted,  the  flowing  water  will  rise  in  it  to  the  level  of  said  grade-line.  Thus,  it  will 
rise  iu  a  pipe  from  6  to  3,  from  5  to  3,  from  4  to  3,  from  7  to  8,  &c.  Advantage  may  be  taken  of  this,  to 
supply  water,  or  establish  fountains,  at  intervals  along  a  great  line  of  pipes.  Small  vert  pipes  of  this 
kiud,  called  piezometers,  (pressure  measurers,)  either  made  of  glass,  or  else 
furuished  with  a  floating  index,  are  sometimes  used  for  detecting  the  position  of  accidental  obstruc- 
tions iu  a  line  of  pipes.  If  the  water  in  these  is  found  at  any  time  to  fall  below  its  proper  level,  it 
shows  that  the  pres  upon  it  has  become  diminished  by  some  obstruction  in  the  interval  between  it 


and  the  outlet,  or  disc 
becomes  a  maximum  ;  and  the  water 


some  obstruction,  or  partial  closing  of  the  pipe,  between  the  pie: 
end.    If  the  outlet  be  entirely  closed,  the  pres  on  the  piezometer  1 

will  rise  in  it  to  a  level  with  the  surf  mi  of  the  water  in  the  reservoir.  By  having  several  piezome- 
ters, the  point  at  which  an  obstruction  has  taken  place  can  be  approximately  ascertained;  aud  thus 
much  labor  saved  in  searching  for  it. 

If  the  lower  end  o.  Fig  1.  of  any  of  the  pipes,  instead  of  discharging  freely  into  the  air,  discharges 
under  water  of  which  t.h  represents  the  surf,  then  the  hydraulic  grade-line  must  be  drawn  from  s  to 
e,  instead  of  to  o  ;  and  p  e  becomes  the  head,  instead  of  p  o. 

RKM.  4     When  the  pipe  rises  above  the  hydraulic  grade-line 

in  any  part,  an  entire  change  of  condition  takes  place  throughout.  In  Fig  1^>,  the  pipe  agno  rises 
above  the  grade-line  so;  which,  however,  can  no  longer  be  properly  so  called;  for  the  pipe  must  now 
be  considered  as  divided  into  two  sections,  agn,  and  nyo;  each  having  its  own  grade-line,  as  t  n, 
no.  In  a  long  undulating  line, 

B 

m  "\ 

"T-- 

_  •  *-          • — -- i  n 

.._..!  n 


it  may  thus  become  necessary 
to  consider  many  separate  sec- 
tions, with  their  respective 
grade-lines.  In  our  fig,  the  sec- 
tion agn  has  the  friction  head 
TIT;  only  ;  and  the  shorter  section 
ny  o  has  the  greater  friction 
head  t  o ;  consequently  the  water 
would  move  more  slowly  in  agn 
than  in  nyo;  and  would  re- 
quire a  greater  diam  than  it,  in 
order  to  carrv  the  same  quan- 
tity of  water."  Or,  if  the  diam 

tire  pipe,  then  the  quantity  of 

water  delivered  at  o.  will  be  that 

due  to  the  small  head  n  b  only.    It  will  then  flow  from  a  to  nulling  that  portion  of  the  pipe;  but 

from  n  to  o  the  pipe  will  not  be  full,  but  will  carry  otf  the  water  as  iu  an  iron  gutter. 

The  bursting  pres  at  any  point,  as  g,  of  the  section  agn,  will  be  measured  by  its  corresponding  vert 
line,  as  g  c ;  and  if  the  sections  have  different  diams,  (proportioned  to  their  vels,)  the  pres  at  any  point, 
as  ?/,  in  section  nyo,  will  be  measured  by  its  corresponding  vert,  as  ?/ x.  But  if  both  sections  be  of 
the  same  diam,  then,  since  section  nyo  becomes  virtually  an  open  gutter,  it  can  experience  no  burst- 
ing pres. 

If  a  pipe  be  closed  suddenly,  the  arrested  momentum  of  the  flowing  water  will  exert  a  great  pres, 
sufficient  in  most  cases  to  burst  with  ease  any  ordinary  street  pipe.  These  are  therefore  closed  very 
slowly  by  valves  moved  by  screws;  see  Figs  35  to  36 j£.  Leaden  service-pipes  in  dwellings  are  fre- 
quently burst  by  closing  the  stopcock  too  quickly. 

RKM.  5.    Resistance  to  flow  in  pipes  of  diff  materials.    It  was 

formerly  supposed  that  the  material  of  which  the  pipe  was  made,  exerted  no  influence  on  the  flow, 
provided  the  insideswere  equally  smooth;  but  later  observations  show  that  this  is  not  the  case.  Weis- 
bach  states  that  in  wooden  pipes  of  from  2V£  to  4>£  ins  diam.  he  found  the  frictional  resistance  to  be 
as  much  as  ]%  times  as  great  as  in  equally  smooth  cast-iron  ones.  If  this  be  correct,  we  infer  that 
the  friction  heads  only,  in  table  p  544,  Ac,  should  be  multiplied  by  1.75  to  adapt  them  to  wooden 
pipes.  Moreover  it  is  said  that  Darcy  found  the  friction  in  corroded  iron  pipes  to  be  twice  as  great  as 
in  new  smooth  ones.  If  so  the  friction  heads  in  the  table  should  be  multiplied  by  2  for  such  pipes. 
Darcy  found  that  the  usual  formulas  (those  here  given)  agreed  sufficiently  closely  with  the  ac- 
tual results  with  perfectly  clean  smooth  cant-iron  pipes  ;  except  that  they  made  the  disch  rather  too 
large  in  small  pipes  ;  and  rather  too  small  in  lare;e  ones ;  hot  that  when  the  insides  of  the  pipes  were 
smoothly  covered  with  pitch,  the  disch  was  increased  about  %  part;  and  became  about  equal  to  that 
through  tubes  made  of  glass.  The  vel  hea.d/or  any  given  vel,  of  course,  remains  the  same  for  all  ma- 
terials and  diams  of  pipe  ;  only  \\\e  friction  head  will  vary.  See  Remark  2,  Art  4. 

The  later  researches  of  Ganguillet  and  lint  tor  on  this  subject  are  very  im- 
portant; see  p  651. 

Art.  2.  To  find  the  velocity,  and  the  quantity  discharged 
through  a  straight,  smooth,  cylindrical  cast-iron  pipe,  r  o, 
v  of  or  I  0^  Fig  1,  whose  length  is  not  shorter  than  4  times  its 
diam;  knowing  its  total  head  po;  its  length;  and  its  diam, 
or  bore. 


538 


HYDRAULICS. 


Rule.  Mult  thediam  in  feet,  by  the  total  head  in  feet.    Call  the  prod  a.    Add 


Approx  vel  in  ft  per  sec 

Total  length  in  ft +  54  diams  in  ft. 

For  any  head  not  less  than  at  the  rate  of  4  ft  per  mile,*  multiply 
theapprox  vel  thus  found  by  the  number  corresponding  to  the  diam  in  ft  in  the 
table  below.  If  the  pipe  is  m  good  order,  this  last  vel  will  probably  be  within  5 
to  10  per  ct  of  the  truth,  inasmuch  as  it  corresponds  with  Kutter's. 


Diam 
in  Ft. 

No. 

Diam 
in  Ft. 

No. 

Diam 
in  Ft. 

No. 

Diam 

in  Ft, 

No. 

.1 

.48 

.6 

.87 

1.5 

1.10 

4 

1.37 

.2 

.63 

.7 

.91 

2. 

1.18 

5 

1.42 

.3 

.71 

.8 

.95 

2.5 

1.24 

6 

1.46 

.4 

.77 

.9 

.98 

3. 

1.30 

7 

1.50 

.5 

.82 

1.0 

1.00 

35 

1.34 

10 

1.60 

Then  to  find  the  discharge  in  cub  ft  per  see,  mult  the  vel  last 
found,  by  the  area  of  circular  transverse  section  of  the  pipe  in  sq  ft.  Said  area 
may  be  taken  from  Table  3,  p  541. 

If  the  disch  end  of  the  pipe  is  under  water,  the  total  head  is  the 
vert  dist  between  the  surfs  of  the  water  of  the  two  reservoirs. 

Ex.  A  straight  pipe  a  mile,  or  5280  ft  long ;  with  a  diam  of  1  foot,  has  a  total 
fall  of  12  ft,  measured  from  the  water  surf  in  the  reservoir,  to  the  cen  of  grav  of 
its  lower  end  or  opening.  With  what  vel  will  the  water  flow  through  it ;  and  how 
much  will  be  dischd  per  sec? 

Here  the  diam  in  ft,  mult  by  the  total  head  in  ft  =  1  X  12  =  12  =  a.  Again, 
the  length  in  ft  is  5280 ;  and  54  times  the  diam  in  ft  is  54 ;  and  these  two  added 


5334 


=  .00225.     The  sq  rt  of 


together  =  5334.    And  the  prod  a,  div  by  5334,  = 

.00225  is  .04743.  And  .01743  X  constant  48  =  2.27  ft  per  sec  approx  vel.  The  num- 
ber in  the  above  table,  for  1  ft  diam  is  also  1 ;  therefore,  2.27  X  1  =  2.27  ft  per  sec 
the  reqd  vel. 

Dischargee.  The  area  of  cross  section  of  a  pipe  1  ft  diam  is  .7854  of  a  sq  ft. 
Hence,  2.27  vel  X  -7854  area  =  1.782  cub  ft  per  sec  disch. 

Item.  1.  Table  Hfo.  1,  calculated  by  this  rule,  shows  the  vels  and  dischgs 
through  a  pipe,  one  mile  long,  and  1  ft  diam,  under  different  heads.  But  they  will 
be  very  nearly  the  same  for  any  greater  lengths;  and  also  quite  approximate  for 
shorter  ones  not  less  than  1000  or  even  500  diams  long,  provided  that  in  all  cases  they 
have  the  same  RATE  O/HKAD  ;  that  is,  if  the  given  pipe  of  1  it  diam,  is  2  or  3  miles 
long,  it  must  have  2  or  3  times  as  much  head  as  the  pipe  in  the  table  which  is  1 
mile  long;  or  if  the  given  pipe  of  1  ft  diam,  is  %,  %,  or  %,  &c,  of  a  mile  long,  it 
must  have  but  y±,  ^,  ^  as  much  total  head  as  the  1  mile  one  in  the  table,  in  order 
to  have  very  nearly  the  same  vel  and  disch. 

Special  Note.  The  above  rule  and  formula  for  finding  the  first  or  approx 
vel,  are  modifications  by  Poncelet  of  the  original  by  Eytelwein  ;  and  were  until 
within  a  few  years,  quite  generally  accepted  by  engineers  as  correctly  applicable 
by  themselves  to  all  diams.  without  the  use  of  such  a  Table  as  the 
above.  But  later  experiments  have  proved  that  such  was  not  the  case,  and 
therefore  the  writer  has,  by  the  aid  of  Kutter's  formula,  added  the  above  table  of 
corrections.  See  "  €antion,"  p  566 ;  and  Matter's  formula,  p  650. 

Rem.  2.  When  we  speak  of  the  vel  of  water  in  a  pipe,  river,  &c,  we 
allude  to  the  mean  vel  of  the  entire  cross  section  of  the  water.  As  in  a  river  the 
vel  is  usually  greater  half  way  across  it,  and  at  the  surface,  than  it  is  at  the  bot- 
tom and  sides,  so  in  a  pipe  the  vel  is  actually  greater  at  the  center  of  its  cross  sec- 
tion than  at  its  circumf.  The  mean  vel  referred  to  in  our  rules  is  an  assumed  uni- 
form one  which  would  give  the  same  discharge  that  the  actual  ununiform  one  does. 

*  About  .9  of  an  Inch  per  1OO  ft. 


fDRAULIOS. 


539 


TABLE  1.  Oftlie  actual  velocities  and  discharges  through 
a  pipe,  1  ft  fii  diams  1  mile,  or  528O  diams  in  length;  and 
of  cast  iron  ;  smooth,  and  straight. 

Head  is  the  vert  dist  from  the  surf  of  the  water  in  the  reservoir,  to  the  ce'n 
of  grav  of  the  lower  end  of  the  pipe,  when  the  disch  is  into  the  air;  or  to  the  level  surface  of  the 
lower  reservoir,  when  the  disch  is  under  water. 


Head 
in  Ft. 

per 
100  Ft. 

Head 
in  Ft. 

per 
Mile. 

Velocity 
in  Ft. 
per 
Second. 

Discharge 
in 
Cub.  Ft. 
per 
Second. 

Discharge 
in 
Cub.  Ft. 
per 
24  Hours. 

Head 
in  Ft. 
per 
100  Ft. 

Head 
in  Ft. 
per 
Mile. 

Velocity 
ii  Ft. 
per 
Second. 

Discharge 
in 
Cub.  Ft. 
per 
Second. 

Discharge 
in 
Cub.  Ft. 
per 
24  Hours. 

.0019 

.1 

.208 

.1633 

14114 

1.515 

80. 

5.85 

4.602 

397613 

.0038 

.2 

.293 

.2301 

19880 

1.704 

90. 

6.23 

4.900 

423435 

.0057 

.3 

.359 

.2819 

24360 

1.894 

100. 

6.56 

5.144 

444312 

.0076 

.4 

.415 

.3267 

28229 

2.083 

110. 

6.87 

5.395 

466128 

.0095 

.5 

.464 

.3638 

31435 

2.272 

120. 

7.18 

5.639 

487209 

.0114 

.6 

.508 

.b989 

34464 

2.462 

130. 

7.47 

5.866 

506822 

.0132 

.7 

.549 

.4311 

37217 

2.652 

140. 

7.76 

6.094 

526521 

.0151 

.8 

.585 

.4602 

39760 

2.841 

150. 

8.05 

6.322 

546048 

.0170 

.9 

.623 

.4901 

42343 

3.030 

160. 

8.30 

6.534 

564576 

.0189 

1. 

.656 

.5144 

44431 

3.219 

170. 

8.55 

6.715 

580176 

.0237 

.25 

.735 

.5753 

49701 

3.408 

180. 

8.80 

6.903 

596418 

.0284 

.5 

.805 

.6322 

54604 

3.596 

190. 

9.04 

7.100 

613440 

.0331 

.75 

.871 

.6832 

59011 

3.788 

200. 

9.28 

7.276 

628704 

.0379 

2. 

.928 

.7276 

62870 

4.261 

225. 

9.84 

7.696 

664848 

.0426 

.25 

.984 

.7696 

66484 

4.735 

250. 

10.4 

8.168 

705728 

.0473 

.5 

1.04 

.8168 

70572 

5.208 

275. 

10.8 

8.482 

732844 

.0521 

.75 

1.08 

.8482 

73284 

5.682 

300. 

11.3 

8.914 

769824 

.0568 

3. 

1.13 

.8914 

76982 

6.629 

350. 

12.3 

9.621 

831168 

.0758 

4. 

1.31 

1.028 

88862 

7.576 

400. 

13.1 

10.28 

888624 

.0947 

5. 

1.47 

1.150 

99403 

8.532 

450. 

13.9 

10.91 

943056 

.1136 

6. 

1.61 

1.264 

109209 

9.47 

500. 

4. 

11.50 

994032 

.1325 

7. 

1.74 

1.366 

118022 

10.41 

550. 

5. 

12.09 

1044576 

,1514 

8. 

1.86 

1.455 

125740 

11.36 

600. 

6. 

12.64 

1092096 

-1703 

9. 

1.96 

1.539 

132969 

12.30 

650. 

6. 

13.11 

1132704 

4894 

to. 

2.08 

1.633 

141145 

13.25 

700. 

7. 

13.66 

1180224 

.2273 

12. 

2.27 

1.782 

153964 

14.20 

750. 

18.0 

14.13 

1220832 

.2652 

14. 

2.45 

1.924 

166233 

15.15 

800. 

18.6 

14.55 

1257408 

.3030 

16. 

2.62 

2.057 

177724 

16.09 

850. 

19.1 

15.00 

1296000 

.3409 

18. 

2.78 

2.183 

188611 

17.04          900. 

19.6 

15.39 

1329696 

•3788 

20. 

2.93 

2-301 

198806 

17.99          950. 

20.3 

15.94 

1377216 

.4735 

25. 

3.28 

2.572 

222156 

18.94 

1000. 

20.8 

16.33 

1411456 

.5682 

30. 

3.59 

2.819 

243604 

22.73 

1200. 

22.7 

17.82 

1539648 

.6629 

35. 

3.88 

3.047 

263260 

26.52 

1400. 

24.5 

19.24 

1662M6 

.7576 

40. 

4.15 

3.267 

282288 

30.30     j    1600. 

26.2 

20.57 

1777248 

.8523 

45. 

4.40 

3.451 

298209 

34.08     :    1800. 

27.8 

21.83 

1886112 

.9470 

50. 

4.64 

3.638 

314352 

37.87     1    2000. 

29.3 

23.01 

1988064 

1.136 

60. 

5.08 

3.989 

344649 

47.35     i    2500. 

32.8 

25.72 

2221560 

1.326 

70. 

5.49 

4.311 

372470 

56.81         3000. 

35.9 

28.19 

2436040 

To  reduce  cub  ft  to  U.  S.  gallons,  mult  by  7.48.  Since,  therefore,  8  cub  ft  are  equal  to  60  gals,  (very 
nearly,)  if  we  divide  the  cub  ft  per  24  hours,  by  8,  we  get  the  number  of  persons  that  may  be 
daily  supplied  with  60  gals  each,  by  a  pipe  constantly  running  full,  and  at  the  vel  given  in  the  third 
col.  This  condition  does  not  exist  in  city  water-pipes;  the  water  in  them  being  comparatively  stag- 
nant. Therefore,  the  results  of  the  rule  and  table  do  not  at  all  apply  to  them.  See  Art  33,  p  580. 

KKM.    If  the  pipe,  instead  of  being  straight,  has  easy  carves. 

(say  with  radii  not  less  than  5  diams  of  the  pipe,)  either  hor  or  vert,  the  disch  will  not  be  materifllly 
diminished,  so  long  as  the  total  heads,  and  total  actual  lengths  of  pipes  remain  the  same;  provided 

be  made  for  the  e«cape  of  air  accumulating  at  the  tops  of  the  curves.    See  Fig  42,  p  579. 

Notwithstanding  what  is  said  about  bends  on  pages  549,  550,  we 

advise  to  make  the  radius  as  much  more  than  5  diams  as  can  conveniently  be  done. 

To  find  either  the  area  of  pipe,  opening,  or  channel-way  ; 
or  the  mean  vel;  or  the  quantity  discharged,  when  the  other  two 

are  given.    This  applies  to  openings  in  the  sides  of  vessels,  to  rivers,  and  to  all  other  channels  aa 
well  as  to  pipes. 

Disch  in  cub  ft  Disch  in  cub  ft 

Area  in  _       per  sec°Dd  Mean  vel  _      per  secoud 

S(*feet"  mean  vel  in  in  ft  per  sec  &rfia  ,n 

feet  per  sec.  sq  feet. 

Disch  in  cub  ft  _  area  in  x    mean  vel  in 
per  second  sq  feet        ft  per  second. 

Or  all  the  terms  may  be  in  inches  instead  of  feet ;  and  minutes  or  hoars  instead  of  second*. 


540 


HYDRAULICS. 


TABLE  2.    (Original.) 

Of  the  rliani  of  long  pipe  reqcl  to  deliver  either  more  or  lea* 
water  than  that  of  1  ft  diam  in  Table  1,  under  the  same  rate  of  incli- 
nation, or  of  head  in  ft  per  mile.  To  Hud  the  disch  (but  uot  the  vel)  through  another  pipe,  not  less 
than  about  1000.  or  at  least  500,  of  its  own  diams  in  length ;  first  take  out  the  disch  through  the  1  ft 
one,  from  Table  1.  Div  the  reqd  disch  by  this  tabular  one.  Look  for  the  quot  in  the  third  column 
of  Table  '2,  of  proportions  of  disch  ;  and  opposite  to  it,  in  columns  1  and  2,  will  be  found  the  reqd 
diarn.  See  Rem  2,  p  543.  From  this  table  we  see  that  a  13-inch  pipe  will  deliver  nearly  1%  time* 
as  much  as  a  1  ft  one ;  a  14-inch  one,  nearly  1%  times;  a  15-inch  oue,  nearly  1%  times;  and  a  16- 
inch  one,  fully  twice  as  much  as  the  1  ft  one;  &c,  of  the  same  length  and  bead. 

This  use  of  this  table  is  not  sufficiently  correct  for  pipes 
less  than  about  1OOO  (or  at  furthest  5OO)  diams  long-;  therefore, 
for  shorter  ones,  use  preceding  Rule,  p  538.  For  more  on  finding  diams  required  for  a  giveu  dis- 
charge, see  Art  4.  See  Caution,  p  566. 


Diam. 

Diam. 

Proportion  of 
Disch.  to  that 

Diam. 

Diam. 

Proportion  of 
Disch.  to  that 

Diam. 

Diam. 

Proportion  of 
Disch.  to  that 

of  long 

of  long 

through  a  1  ft. 

of  long 

of  long 

through  a  1  ft. 

of  long 

of  long 

through  a  1  ft. 

Pipe,  In 
Inches. 

Pipe,  in 
Feet. 

lung  Pipe, 
with  the  same 

Pipe,  in 
Inches. 

Pipe,  in 
Feet. 

lona  Pipe, 
with  the  same 

Pipe,  in 
Inches. 

Pipe,  in 
Feet. 

long  Pipe, 
with  the  same 

head  per  mile. 

head  p-.-r  mile. 

head  per  mile. 

1 

.0833 

.0020 

10^ 

.8750 

.7157 

28 

2.333 

8.319 

13* 

.1250 

.0055 

11 

.9167 

.8044 

30 

2.5 

9.822 

2 

.1667 

.0113 

.9583 

.8987 

30J4 

2.521 

10.0 

.2083 

.0198 

12 

32 

2.667 

11.6 

3 

.2500 

.0310 

ISM 

1.'042 

!l06 

34 

2.833 

13.5 

3J* 

.2917 

.0458 

13 

1.083 

.221 

36 

3. 

15.5 

4 

.3333 

.0643 

14 

1.167 

.470 

38 

3.167 

17.8 

4/* 

.3750 

.0857 

15 

1.250 

.746 

40 

3.333 

20.2 

5 

.4167 

.1119 

16 

1.333 

2.053 

42 

3.5 

22.9 

.4583 

.1422 

17 

1.417 

2.388 

44 

3.667 

25.7 

6 

.5 

.1767 

18 

1.5 

2.754 

48 

4. 

32.0 

6Va 

.5417 

.2159 

19 

1.583 

3.153 

54 

4.5 

42.0 

7 

.5815 

.2600 

20 

1.667 

3.585 

60 

5. 

54.9 

.6250 

.3090 

21 

1.75 

4.051 

66 

5.5 

69.8 

8 

.66S7 

.3631 

22 

1.833 

4.551 

72 

6. 

85.8 

.7083 

.4220 

23 

1.917 

5.084 

78 

6.5 

104.8 

9 

.75 

.4871 

24 

2. 

5.649 

84 

7. 

126.1 

.7917 

.5575 

2.052 

6.000 

96 

176. 

10 

.8333 

.6337 

26 

2.167 

6.912 

120 

10'. 

302. 

Examples  of  the  use  of  Tables  1.  and  2.    Having  a  head  from  a 

reservoir  to  a  certain  point  of  delivery,  of  20  ft,  in  a  dist  of  1860  ft,  and  wishing  to  receive  6  cub  ft  of 
water  per  sec;  what  must  be  tiie  diam  of  a  pipe  to  accomplish  this? 

In  the  first  place,  we  find  that  a  fall  of  20  ft  in  1860,  is  equal  to  a  fall  of  1.075  ft  in  100  ft.    Then  we 
see  by  Table  1,  that  with  a  fall  of  1.075  ft  in  100,  a  long  pipe  of  1  ft  diam  yields  about  3.8  cub  ft  per  sec. 

But  we  want  „--  =  1.58  times  as  much  as  the  1  ft  pipe  can  deliver ;  and  by  Table  2,  we  see  that  the 

pipe,  to  do  this,  under  the  same  rate  of  head,  must  be  about  14}^  ins  in  diam.    In  practice,  we  should 
adopt  at  least  15  ins.     Near  enough,  we  may  say  that  double  the  diam  gives  5%  times  the  disch. 

TABLE  2^.    Weight  of  water  (at  62V£  Ibs  per  cub  foot)  con- 
tained in  one  foot  length  of  pipes  of  different  bores.   (Original.) 


Bore. 

Ins. 

Water. 
Lbs. 

Bore.  |  Water. 
Ins.   I     Lbs. 

Bore. 

Water. 
Lbs. 

Bore. 

Water. 
Lbs. 

Bore. 

Water. 
Lbs. 

Bore. 

Ins. 

Water. 

Lbs. 

K 

.00531 

2. 

1.3581 

3% 

5.0980 

T% 

19.098 

13V$ 

61.877 

22 

164.33 

K 

.02122 

% 

1.5331 

4. 

5.4323 

% 

20.392 

14. 

66.545 

23 

179.60 

.04775 

14 

1.7188 

6.1325 

8. 

21.729 

71.384 

24 

195.56 

H 

.08488 

5 

1.9150 

y* 

6.8750 

y± 

23.109 

15. 

76.392 

25 

212.20 

.13263 

3 

2.1220 

H 

7.6601 

y* 

24.530 

J* 

81.568 

26 

229.51 

% 

.19098 

M 

2.3395 

5. 

8.4880 

%    \    25.993 

16. 

86.916 

27 

247.51 

.25994 

X 

2.5676 

\/ 

9.3580 

9.          27.501 

y^ 

92.434 

28 

266.18 

I. 

.33952 

y» 

2.8063 

X 

10.270 

H       30.641 

17. 

98.121 

29 

285.53 

If 

.42969 

3 

3.0557 

11.225 

10.          33.952 

H 

103.97 

30 

305.57 

y± 

.53050 

%      3.3156 

6. 

12.223 

^       37-432 

18. 

110.00 

31        326.27 

y 

.64190 

3.5862 

13.262 

11. 

41.082 

VJ      116.20 

3'J         347.66 

% 

.76392 

iU 

3.8673 

ix 

14.345 

y* 

44.901 

19.      |  122.56 

33 

369.74 

RZ 

.89654 

1Z 

4.1591 

8/ 

15.469 

1? 

48.891 

129.10 

34 

392.48 

% 

1.0398 

RX 

4.4615 

7. 

16.636 

H 

53.04!) 

20. 

135.81 

35 

415.90 

%     1.1936 

% 

4.774f> 

17.846 

13. 

57.379 

21. 

149.73 

36 

440.00 

And  in  larger  pipes,  as  the  squares  of  their  bores.    Thus  a  pipe  of  40  or 

60  ins  bore,  will  contain  4  times  ag  much  as  one  of  20  or  30  ins  bore ;  and  one  of  -j&r,  %  as  much  at 
one  of  %  inch.     At  62J/4  Ibs  per  cub  ft,  a  sq  inch  of  water  1  ft  high  weighs  .432292  of  a  fl). 


HYDRAULICS. 


541 


TABLE 


reas  and  Contents  of  Pipes ;  and  square  roots 
of  IHams.     (Original  )  Correct. 


Diarn. 

iu 
lus. 

s 

Diam. 

in 
Feet. 

Area  in 
sq  ft,  also 
cub  ft, 
iu  1  foot 
length  of 
Pipe. 

Sq.  rt. 
of 
Diam. 
iu  Ft. 

Diam. 
iu 
lus. 

Diam. 
iu 
Feet. 

Area  in 

sq  ft,  also 
cub  ft, 
iu  1  foot 
length  of 
Pipe. 

Sq.  rt. 
of 
Diam. 
iu  Ft. 

Diam. 

lus. 

Diam. 
iu 
Feet. 

Area  in 
sq  ft,  also 
cub  ft, 
in  1  foot 
length  of 
Pipe. 

Sq.  rt. 
of 
Diam. 
in  Ft. 

U 

.0208 

,0003 

.145 

4. 

.3333 

.0873 

.579 

15. 

1.250 

1.227 

1.118 

516 

.0260 

.0005 

.161 

% 

.3438 

.0928 

.588 

24 

1.271 

1.268 

1.127 

H 

.0313 

.0008 

.177 

li 

.3542 

.0985 

.590 

/^ 

1.292 

1.310 

1.136 

716 

.0305 

.0010 

.191 

y» 

.3646 

.1040 

.604 

K 

1.313 

1  353 

1.146 

}i 

.OUT 

.0014 

.2C4 

x 

.3750 

.1104 

.612 

16. 

1.333 

1.396 

1.155 

916 

.0469 

.0017 

.217 

% 

.3854 

.1167 

.621 

M 

1.354 

1.440 

1.163 

y» 

.0521 

.0021 

.228 

H 

.3958 

.1231 

.629 

« 

1.375 

1.485 

1.172 

1116 

.0573 

.0026 

.239 

y» 

.4063 

.1296 

.637 

H 

1.396 

1.530 

1.181 

% 

.0025 

.0031 

.250 

5 

.4167 

.1363 

.645 

17. 

1.417 

1.&74 

1.180 

1816 

.0677 

.0036 

.2GO 

K 

.4271 

.1433 

.653 

i/ 

1.437 

1.623 

1.199 

H 

.0729 

.0042 

.270 

X 

.4375 

.1503 

.6CO 

% 

1.458 

1.670 

1.207 

15-16 

.0781 

.0048 

.280 

% 

.4479 

.1576 

jm 

X 

1.479 

1.718 

1.216 

1. 

.0803 

.0055 

,28i) 

X 

.4583 

.1650 

.677 

18. 

1.5 

1.767 

1.224 

1-16 

.0885 

.0002 

.297 

N 

.4688 

.1725 

.(JS5 

y* 

1.542 

1.867 

1.241 

M 

.C938 

.0009 

.305 

ai 

.4792 

.1803 

.(il)3 

19. 

1.583 

1  969 

1.258 

316 

.o;)90 

.0077 

.314 

y» 

.4896 

.1878 

.700 

M 

1.625 

2.074 

1.274 

X 

.1042 

.0085 

.322 

6. 

.5 

.n:64 

.707 

20. 

1  667 

2.18'! 

1.291 

516 

.1094 

.009* 

.330 

ft 

.5208 

.2131 

.722 

1A 

1.708 

2  292 

1.307 

% 

.1146 

.0103 

.338 

S 

.5417 

.2304 

.736 

21. 

1.750 

2.405 

1.323 

7-16 

.1198 

.0113 

JU4 

H 

.5625 

.2485 

.750 

y* 

1.791 

2.521 

1.339 

K 

.1250 

.0123 

.35  1 

7. 

.5833 

.2673 

.764 

22. 

1.833 

2.640 

1.354 

9-16 

.1302 

.0133 

.3<jl 

i^ 

.6042 

.2867 

.777 

H 

1.875 

2.761 

1.369 

M 

.1354 

.0144 

.363 

/4 

.6250 

.3068 

.791 

23. 

1.917 

2.885 

1.384 

lfl6 

.1106 

.0155 

.375 

H 

.6458 

.3276 

.803 

M 

1.958 

3.012 

I.b99 

U 

1458 

.0167 

.382 

8. 

.6007 

.3491 

.817 

24. 

2.COO 

3.142 

1.414 

13-16 

.1510 

.0179 

.389 

a 

.6875 

.3712 

.8-9 

25. 

2.083 

8.409 

1.443 

H 

.1563 

.0192 

.31'5 

^ 

.7083 

.3941 

.841 

2i. 

2.166 

3.(i87 

1.472 

15-16 

.1615 

.0205 

.402 

h 

.7292 

.4176 

.854 

27. 

2.250 

3.!'76 

1.500 

2. 

.1067 

.0218 

.408 

D. 

.75 

.4418 

.866 

28. 

2  333 

4.276 

1.528 

1-16 

.1719 

.0232 

.414 

H 

.7708 

.466T 

.879 

29. 

2416 

4.587 

1.555 

U 

.1771 

.0246 

.420 

y 

.7917 

.492-2 

.890 

30. 

2500 

4.Ji09 

1.581 

8-16 

.1823 

.0200 

.427 

:H 

.8125 

.5185 

.902 

31. 

2.584 

5.24J 

1.607 

5*16 

.1875 
.1927 

.0276 
.0291 

.433 
.440 

10. 

M 

.8333 
.8542 

.5454 
.5730 

.91;; 

.924 

32. 
33. 

2.666 
2.750 

5585 
5.940 

1.633 
1.658 

% 

.1979 

.0308 

.415 

1A 

.8750 

.6013 

.935 

34. 

2.834 

6.305 

1.683 

7-16 

.2031 

.0324 

.451 

H 

.8958 

.6303 

.946 

35. 

'2  916 

6681 

1.708 

* 

.2083 

.0341 

.457 

li. 

.9167 

.6600 

.957 

36. 

3000 

7.069 

1.732 

916 

.2135 

.0358 

.462 

i^ 

.9375 

.6903 

.%K 

:;8. 

3.166 

7.876 

1.779 

K 

.2188 

.0375 

Ml 

H 

.9583 

.7213 

.979 

40. 

3.333 

8.727 

1.825 

11-16 

.2240 

.0394 

.473 

H 

.9792 

.7530 

.9:i(> 

42. 

3.500 

£.621 

1.871 

« 

.2292 

.0412 

.478 

12. 

.7854    ' 

1.000 

4». 

3.666 

10.56 

1.914 

13-16 

.2344 

.§432 

.484 

v\ 

1  .021 

.8184 

1  010 

48. 

4.000 

12.57 

2.000 

1*16 

.23!>6 
.2448 

.0451 
.0471 

.489 
.495 

j 

1.042 
1  063 

.8522 
.8866 

1.020 
1.031 

60! 

4.500 
5.000 

15.90 
1963 

2.121 
2.236 

3. 

.2500 

.0491 

.500 

13. 

1.083 

.9218 

1.041 

66. 

5.500 

23.76 

2.345 

U 

.2604 

.0532 

.510 

H 

.104 

.9576 

1.051 

72. 

6.000 

28.27 

2.449 

.2708 

.0576 

.520 

M 

.125 

.9940 

1.060 

78. 

6.500 

33.18 

2.550 

y\ 

.2813 

.0621 

.530 

K 

.146 

1.031 

1.070 

84. 

7.000 

38.48 

2.646 

$4 

.2917 

.0668 

.540 

14. 

.167 

1.069 

1.080 

90. 

7.500 

44.18 

2.739 

K 

.3021 

.0716 

.550 

y*. 

.187 

1.108 

1.090 

96. 

8.000 

60.27 

2.828 

K 

.3125 

.0767 

.560 

* 

.208 

1.147 

1.099 

3 

* 

.32-29 

.0819 

.570 

H 

1.229 

1.187 

1.110 

For  con  tents  in  gallons,  see  p  46. 


35 


542  HYDRAULICS. 

Art.  3.  To  find  the  total  head  in  feet,  that  must  be  given  to  a  straight. 

smooth.  cylindrical  cast-iron  pipe,  not  less  than  4  diams  long,  to  enable  it  to  disch  agiveu  read  suau- 
tity  per  sec  ;  knowing  its  diam  in  feet.     See  Rem  2,  p  543.     Also,  Rem  p  539. 

Rur,K.  Square  the  given  disch  in  cub  ft  per  sec.  Add  together  the  total  length  of  the  pipe  in  feet, 
and  54  times  its  diam  in  ft.  Mult  the  sum  by  the  square  of  the  disch  just  found.  Call  the  prod  p. 
Next,  div  the  diam  in  ft  by  the  dec  .235.  Take,  from  Table  5,  the  fifth  power  of  the  quot.  Divp  by 
this  5th  power.  The  quot  will  be  the  reqd  head  in  ft. 

See  Caution,  p  566.  /Disch  in  cu&y         /   Length  hi  feet  \ 

Total  ftea<f_A    ft  per  sec     )     X  V+  54  diams  in  ft  ) 
Or,  as  a  formula,          infect      ~  (Diam,  in  ft  ^> 

J        !23d~  '• 

Ex.  A  straight,  clean,  cast  iron  pipe  6  ins,  or  .5  ft  diam.  and  20  ft  long,  is  reqd  tn  disch  3.066  cub 
ft  per  sec.  What  total  head  must  it  have?  Here  the  square  of  the  disch,  or  3.0662,  is  9.4.  The 
length  in  ft,  added  to  54  diam  in  ft  —  20  +  27  —  47  ;  and  9.4  X  47  =  441.8,  or  p.  Next,  the  diam  in 

ft,  div  by  .235,  or  —  =  2.128.    The  5th  power  of  2.128,  by  Table  5,  is  43.64.*    Hence  the  total  head 


REM.    The  following1  rule  is  more  simple;  but  is  applicable  only 

when  the  pipe  is  so  long  that  the  54  diams  may  be  omitted  from  the  calculation,  without  affecting  the 
result  to  a  practical  degree  ;  in  which  case  writers  call  it  a  long*  pipe.  If,  however,  we  take  it  as 
low  as  1000  diams,  the  resulting  head  will  be  but  about  6  per  ct,  or  y1^  part  too  small  ;  at  2000  diams,  -Ar 
too  small,  <fec. 

RULE  2.  Having  the  diam  of  the  pipe  in  ft,  find  its  area  in  sq  ft.  by  Tnble  3.  Div  the  reqd  disch 
in  cub  ft  per  sec,  by  this  area.  The  quot  will  be  the  vel  in  ft  per  sec.  Then 

Square  of  vel  v  „  , 

Head  in  ft  per  mile,  _   jn  ft  per  8ec  A  *••* 
iu  long  pipes 

Diam  in  feet. 

Ex.  How  much  head  in  feet  per  mile,  must  be  given  to  a  long  pipe  1  ft  diam,  to  enable  it  to  disch 
1.782  cub  ft  per  sec?  Here  the  area  of  the  pipe,  by  Table  8,  is  .7854  sq  ft.  And  —  —  ~  2.269  ft  per 
•ec  vel.  And  the  square  of  2.269  =  5.1484  ;  hence, 


Head  in     _  5.1484  X  2.8  _ 
ft  per  mile  ~       ;—  - 


-••  :    {     :  :  "**•  :  it.  head. 


Table  1,  which  is  for  a  pipe  5280  diams  long,  gives  the  head  12  ft  per  mile  ;  instead  af  11.84,  which 
is  near  enough. 

Art.  4.  To  find  the  diam  of  a  long,  smooth,  straight,  cylindrical  cast-iron 
pipe,  reqd  to  deliver  a  given  quantity  per  see:  knowing  its  length,  and  total  head.  No  simple  direct 
rule  can  be  given  for  the  diam  of  short  pipes.  For  those  longer  than  about  1000  diams,  (also  quit* 
approximate  even  for  500.)  we  may  use  the  following.  See  Rem  after  Table  1,  p  539. 

Square  the  disch  in  cub  ft  per  sec.  Mult  this  square  by  the  length  iu  ft.  Div  the  prod  by  the  head 
in  ft.  From  Table  5,  p  546.  take  the  5th  root  of  the  quot.  Mult  this  5th  root  by  the  dec  .236.  The 
prod  will  be  the  diam  in  feet.  Or  by  formula. 

See  Caution,  p  566.  5  /Square  of  discharge        length 

Diam  in  ft       —    4     /      in  cub  ft  per  tec      *    tn  ft 
of  a  long  pipe    ~    ^/     i  HeriHnJi 

Ex.  ^Vhat  diara  must  be  given  to  a  pipe  half  a  mile,  or  2640  ft  long  ;  with  a  total  head  of  6  ft  ;  in 
roder  to  furnish  1.782  cub  ft  per  sec  T 
First,  the  sq  of  the  disch,  or  1.7822  =  3.1755.     Hence  we  have 


Diam  _  /  - 

""-  V 


Or  for  the  diam  of  a  long  pipe,  see  Tables  1  and  2,  pp  539,  540. 


*  This  5th  power  43.64  was  calculated,  instead  of  being  taken  from  the  table  ;  in  order  to  illustrate 
the  example  more  accurately.  For  practical  purposes,  however,  they  can  be  taken  from  the  table 
near  enough  by  eye. 


HYDRAULICS. 


543 


REM.  1.  For  the  <lifnfi  of  a  short  pipe;  that  is,  of  one  less  than  about 
2000  diams  In  length,  the  la>CTule  and  formula  give  the  diams  of  such  pipes  too  small.  To  obtain 
correct  results,  in  suchca^es,  mult  the  diam  as  found  by  the  rule  or  formula,  by  the  corresponding 
multiplier  in  the  foU0wing  table. 

Table  Qfniultipliers  for  the  <liams  of  short  pipes  as  given 
by  the  preceding?  rule  or  formula,  iu  Art  4. 


See  Caution,  p  566.    Original. 


Length 
of  pipe 
in  diams. 

Mult'r. 

Length 
of  pipe 
in  diams. 

Mult'r. 

Length 
of  pipe 
in  diams. 

Mulfr. 

1000 
500 
300 
250 
200 
150 
125 

1.01 
1.02 
1.03 
1.04 
1.05 
1.06 
1.07 

100 
75 
50 
35 
25 
20 
15 

1.09 
1.11 
1.16 
1.20 
1.26 
1.30 
1.35 

12 
10 
8 
7 
6 
5 
4 

1.40 
1.45 
1.50 
1.54 
1.58 
1.63 
1.70 

For  tubes  shorter  than  4  diams,  see  Art  8,  p  553. 

REM.  2.     Recent  experimenters  state  that  the  old  formulas  in  use, 

although  generally  sufficiently  exact  for  ordinary  practice,  are  to  some  extent  defective.  Thus,  Darcy 
states  that  tor  small  diams. "the  disch  are  somewhat  too  great;  apd  for  large  diams,  too  small ;  and 
Weiibach  asserts  that  for  vels  less  than  1%  ft  per  sec,  (full  I  mile^per  hour.)  the  heads  given  by  our 
formulas  are  too  small ;  and  for  higher  vels,  too  great.  In  other  words,  these  writer*  agree  that  the 
elder  authorities  hitherto  followed,  have  not  correctly  valued  the/riction  heads.  Weisbach's  experi- 
ments included  much  greater  vels  than  those  of  bis  predecessors  ;  and  his  extraordinary  attainments 
in  science  entitle  his  results  to  great  confidence.  The  following  is  his  rule  for  finding  the  friction 
head  iu  anv  case.  For  the  total  head  we  have  only  to  add  to  this  the  vel  head  taken  from  the  next 
Table,  or  from  Table  10,  p  552  opposite  the  given  vel,  and  the  entry  head,  (equal  to  half  the  vel  head  ; 
see  Rem  1.  p  535,)  the  two  together  rarely  amounting  to  a  foot.  His  rule  applies  to  any  lengths  ex- 
ceeding  4  diams. 

Weisbach's  rule.    Square  the  reqd  vel  in  feet  per  sec.    Div  this  square  by 

64.4,  calling  the  quot  a.  Next,  take  the  sq  rt  of  the  vel.  Div  the  dec  .01716  by  this  sq  rt.  To  the  quot 
add  the  dec  .0144;  calling  the  sum  6.  Div  the  total  length  of  the  pipe  iu  feet,  by  its  bore  or  inner 


diam,  in  feet.    Call  the  quot  c.    Then, 


f  Caution,  p  566. 


Or  by  formula, 


The  friction  head  _ 
in  feet 


=  a  X  6  X  c. 


Frictio 


(.01716       V  Length  Vel2  in 

.0144  -j-  —   I  v  in  feet  v    ft  per  sec 

^/vel  in  ft   /  x  --DuHT         — 64l~ 
v       n«r  SPP    '  ot.t. 


per  sec 


in  feet 

By  this  formula,  or  by  the  next  table,  a  pipe  1  ft  diam.  and  a  mile  long,  with  a  total  head  of  56  ft, 
will  disch  265  cub  ft  per  min,  with  a  vel  of  5.6  ft  per  sec  ;  by  the  rule  in  Art  2,  or  by  Table  1,  the  disch 
will  be  230  cub  ft  per  min  ;  and  the  vel  4.9  ft  per  sec  ;  and  the  total  head  reqd  for  a  vel  and  disch  equal 
to  Weisbach's,  would  be  73  ft,  instead  of  his  56  ft. 

On  the  other  hand*  many  measurements  by  competent  engineers  seem  to  show  that  the  old 
formulas  give  all  the  accuracy  required  in  common  practice.  Mr  Simpson,  Prest  of  the  Soc  Civ 
Eng.  London,  states  that  he  has  gauged  the  flow  from  many  pipes  from  1  to  10  miles  long,  and  from 
12  to  30  ins  diam,  and  found  them  to  agree  very  exactly  with  the  formula 

See  Caution,  p  566. 

Vel  in  ft  per  sec  == 


/T5 

=  504     /  _ 

V       lengt; 


am  X  head 


;th  -|-  50  diams. 

all  in  feet;  which  gives  somewhat  more  than  Poncelet's,  at  top  of  page  538.  Poncelet's  is  the  one 
used  by  Fairbairn  in  his  Treatise  on  Mills.  Perfect  accuracy  is  not  attainable  in  such  matters  with 
our  present  imperfect  knowledge. 

The  following  useful  table  is  taken  from  Weale's  well- 
known  Engineer's  Pocket  Book. 

TABTjE  4%.    Of  the  vel,  and  discharge  of  water  through  straight,  smooth, 

cylindrical  cast-iron  pipes;  with  the  friction  head  required  for  each  100  feet  in  length  ;  and  also  the 
velocity  head.    (For  these  terms,  gee  Remark  1,  p  535.)    Calculated  by  means  of  Weisbach's  formula, 
by  James  Thomson,  A  M  ;  and  George  Fuller,  C  E,  Belfast,  Iretand.    The  vel  head  remains  the  same 
for  any  length  of  pipe  ;  being  dependent  only  on  the  velocity  of  the  water  in  the  pipe. 
The  entry  head  is  equal  to  half  the  vel  head ;  see  Rem  1,  p  535. 


544 


HYDRAULICS. 


TABLE 


See  Caution,  p  566. 


Vel.  in 

Feet 
per  Sec. 

Vel- 
bead  ii 
Feet. 

Diam.  in  Inches. 

3 

^ 

4 

^ 

5 

Fr  head 
in  Feet. 

Cub  ft 
per  Miu 

Frhead 
in  Feet. 

Cub  ft 
per  Miu 

Frhead 
in  Feet. 

Cub  ft 
per  Min 

Fr  head 
in  Feet. 

Cub  ft 
per  Min 

Frhead 
in  Feet. 

Cub  ft 
per  Min 

2.0 

.062 

.659 

5.89 

.565 

8.02 

.494 

10.4 

.439 

13.2 

.395  !   16.3 

2.2 

.075 

.780 

6.48 

.669 

8.82 

.585 

11.5 

.520 

14.6 

.46S 

18.0 

2.4 

.090 

.911 

7.07 

.781 

9.62 

.683 

12.5 

.607 

15.9 

.547 

19.6 

2.6 

.105 

1.05 

7.65 

.901 

10.4 

.788 

13.6 

.701 

17.2 

.631 

21.3 

2.8 

.122 

1.20 

8.24 

1.03 

11.2 

.900 

14.6 

.800 

18.5 

.720 

22.9 

3.0 

.140 

1.35 

8.83 

1.16 

12.0 

1.02 

15.7 

.905 

19.8 

.815 

24.5 

3.2 

.160 

1.52 

9.42 

1.31 

12.8 

1.14 

16.7 

1.02 

21.2 

.915 

26.2 

3.4 

.180 

1.70 

10.0 

1.46 

13.6 

1.27 

17.8 

1.13 

22.5 

1.02 

27.8 

3.6 

.202 

1.89 

10.6 

1.62 

14.4 

1.41 

18.8 

1.26 

23.8 

1.13 

29.4 

3.8 

.225 

2.08 

11.2 

1.78 

15.2 

1.56 

19.9 

1.39 

25.2 

1.25 

31.0 

4.0 

.250 

2.28 

11.8 

1.96 

16.0 

1.71 

20.9 

1.52 

26.5 

1.37 

32.7 

4.2 

.275 

2.49 

12.3 

2.14 

16.8 

1.87 

22.0 

1.66 

27.8 

1.50 

34.3 

4.4 

.302 

2.71 

12.9 

2.33 

176 

2.03 

23.0 

1.81 

29.1 

1.63 

36.0 

4.6 

.330 

2.94 

13.5 

2.52 

18.4 

2.21 

24.0 

1.96 

304 

1.76 

37.6 

4.8 

.360 

3.18 

14.1 

2.72 

19.2 

2.38 

25.1 

2.12 

31.8 

1.91 

39.2 

5.0 

.390 

3.43 

14.7 

2.94 

20.0 

?.57 

26.2 

2.28 

33.1 

2.05 

40.9 

5.2 

.422 

368 

15.3 

3.15 

20.8 

2.76 

27.2 

2.45 

34.4 

2.21 

42.5 

5.4 

.455 

3.94 

15.9 

3.38 

21.6 

2.96 

28.2 

2.63 

35.8 

2.37 

44.2 

5.6 

.490 

4.22 

16.5 

3.61 

22.4 

3.16 

29.3 

2.81 

37.1 

253 

45.8 

5.8 

.525 

4.50 

17.1 

3.85 

23.2 

3.37 

30.3 

3.00 

38.4 

2.70 

47.4 

6.0 

.562 

478 

17.7 

4.10 

24.0 

3.59 

31.4 

3.19 

39.7 

2.87 

49.1 

6.2 

.600 

5.08 

18.2 

4.36 

24.8 

3.81 

32.4 

3.39 

41.0 

3.05 

50.7 

6.4 

.640 

5.39 

18.8 

4.62 

25.6 

4.04 

33.5 

3.59 

42.4 

3.23 

52.3 

6.6 

.680 

5.70 

19.4 

4.89 

26.4 

4.28 

34.5 

3.80 

43.7 

3.42 

54.0 

6.8 

.722 

6.02 

20.0 

5.16 

27.3 

4.52 

35.6 

4.01 

45.0 

361 

55.6 

7.0 

.765 

6.35 

20.6 

5.45 

28.0 

4.77 

36.6 

4.24 

464 

3.81 

57.2 

1 

1 

Vel.  in 
Feet 
per  Sec. 

Vel- 
head  in 
Feet. 

Diam.  in  Inches. 

5 

7 

* 

9 

10 

Fr  head 
in  Feet. 

Cub  ft 
per  Min 

Fr  head 
in  Feet. 

Cub  ft 

per  Min 

Fr  head 
in  Feet. 

Cub  ft 
per  Min 

Frhead 
in  Feet. 

Cub  ft 
per  Min 

Frhead;  Cub  ft 
in  Feet,  per  Min 

20 

.062 

.329 

235 

.282 

320 

.247 

41.9 

.220 

53.0 

.198 

65.4 

2.2 

.075 

.390 

25.9 

.334 

35.3 

.293 

46.1 

.260 

58.3 

.234 

72.0 

2.4 

.090 

.456 

28.2 

.390 

38.5 

.342 

50.2 

.304 

63.6 

.273 

78.5 

2.6 

.105 

.526 

30.6 

.450 

41.7 

.394 

54.4 

.350 

68.9 

.315 

85.1 

2.8 

.122 

.600 

32.9 

.514 

449 

.450 

58.6 

.400 

74.2 

.360 

91  .« 

3.0 

.140 

.679 

3o.3 

.582 

48.1 

.509 

62.8 

.453 

79.5 

.407 

98.2 

3.2 

.160 

.763 

37.7 

.654 

51.3 

.572 

67.0 

.508 

84.8 

.458 

105 

3.4 

.180 

.851 

400 

.729 

54.5 

.638 

71.2 

.567 

90.1 

.510 

111 

3.6 

.202 

.943 

42.4 

.808 

57.7 

.707 

75.4 

.629 

95.4 

.566 

118 

3.8 

.225 

1.04 

44.7 

.832 

60.9 

.780 

79.6 

.693 

101 

.624 

124 

4.0 

.250 

1.14 

47.1 

.979 

64.1 

.856 

83.7 

.761 

106 

.685 

131 

4.2 

.275 

1.25 

495 

1.07 

67.3 

.935 

87.9 

.832 

111 

.748 

137 

4.4 

.302 

1.35 

51.8 

1.10 

70.5 

1.02 

92.1 

.905 

116 

.814 

144 

4.6 

.330 

1.47 

54.1 

1.26 

73.7 

1.10 

96.3 

.981 

122 

.883 

150 

4.8 

.360 

1.59 

56.5 

1.36 

76.9 

1.19 

100 

1.06 

127 

.954 

157 

5.0 

.390 

1.71 

589 

1.47 

80.2 

1.28 

105 

1.14 

132 

1.03 

163 

5.2 

.422 

184 

61.2 

1.58 

8:$.3 

1.38 

109 

1.23 

138 

1.10 

170 

5.4 

.455 

1.97 

63.6 

1.69 

86.6 

1.48 

113 

1.31 

143 

1.18 

177 

5.6 

.490 

2.11 

65.9 

1.81 

89.8 

1.58 

117 

1.40 

148 

1.26 

183 

5.8 

.525 

2.25 

68.3 

1.93 

93.0 

1.68 

121 

150 

154 

1.35 

190 

6.0 

.562 

2.39 

70.7 

205 

96.2 

1.79 

125 

1.59 

159 

1.43 

196 

6.2 

.600 

2.54 

73.0 

2.18 

99.4 

1.90 

130 

1.69 

164 

1.52 

203 

6.4 

.640 

269 

75.4 

2.31 

102 

2.02 

134 

1.79 

169 

1.61 

209 

6.6 

.680 

2.85 

77-7 

2.44 

106 

2.14 

138 

1.90 

175 

1.71 

216 

6.8 

.722 

3.01 

80.1 

2.58 

109 

2.26 

142 

2.01 

180 

1.81 

222 

7.0 

.765 

3.18 

82.4 

2.72 

112 

2.38 

146 

2.12 

185 

1.90 

229 

YDRAULICS. 


545 


TABLE  4%  —  (Continued.) 


Diam.  in  Inches. 

X 

Vel.  in 
Feet 

Vel- 
head  in 

11 

12 

13 

14 

15 

per  Sec. 

Feet. 

Frhead 

Cub  ft 

Frhead 

Cub  ft 

Fr  head 

Cub  ft 

Fr  head 

Cub  ft 

Fr  head 

Cub  ft 

in  Feet. 

per  Mir 

in  Feet. 

perMin 

in  Feet 

perMin 

in  Feet 

per  Min 

n  Feet. 

perMin 

2.0 

.062 

.180 

79.2 

.165 

94.2 

.152 

no 

.141 

128 

.132 

147 

2.2 

.075 

:2is 

87.1 

.195 

103 

.180 

121 

.167 

141 

.156 

162 

2.4 

.090 

.248 

95.0 

.228 

113 

.210 

133 

.195 

154 

.182 

176 

2.6 

.105 

.287 

103 

.263 

122 

.242 

144 

.225 

167 

.210 

191 

2.8 

.122 

.327 

111 

.300 

132 

.277 

156 

.257 

179 

.240 

206 

3.0 

.140 

.370 

119 

.339 

141 

.313 

166 

.291 

192 

.271 

221 

3.2 

.160 

.416 

127 

.381 

151 

.352 

177 

.327 

205 

.305 

235 

3.4 

.180 

.464 

134 

.425 

160 

.393 

188 

.365 

218 

.340 

250 

3.6 

.202 

.514 

142 

.472 

]69 

.435 

199 

.404 

231 

.377 

265 

3.8 

.225 

.567 

150 

.520 

179 

.480 

210 

.446 

243 

.416 

280 

4.0 

.250 

.623 

158 

.571 

188 

.527 

221 

.489 

256 

.457 

294 

4.2 

.275 

.680 

166 

.624 

198 

.576 

•232 

.534 

269 

.499 

309 

4.4 

.302 

.740 

174 

.679 

207 

.626 

243 

.582 

282 

.543 

324 

4.6 

.330 

.803 

182 

.736 

217 

.679 

254 

.631 

295 

.589 

339 

4.8 

.360 

.867 

190 

.795 

226 

.734 

265 

.682 

308 

.636 

353 

5.0 

.390 

.935 

198 

.857 

235 

.791 

276 

.734 

321 

.685 

368 

5.2 

.422 

1.00 

206 

.920 

245 

.850 

287 

.789 

333 

.736 

383 

5.4 

.455 

1.07 

214 

.986 

254 

.910 

298 

.845 

846 

.789 

397 

5.6 

.490 

1.15 

222 

1.05 

261 

.973 

309 

.903 

359 

.843 

412 

6.8 

.525 

1.22 

229 

1.12 

273 

1.04 

321 

.964 

372 

.899 

427 

6.0 

.562 

1.30 

237 

1.19 

283 

1.10 

332 

1.02 

385 

.957 

442 

6.2 

.600 

1.38 

245 

1.27 

292 

1.17 

343 

1.09 

397 

1.01 

456 

6.4 

.f)40 

1.47 

253 

1.35 

301 

1.24 

354 

1.15 

410 

1.08 

471 

6.6 

.680 

1.55 

261 

1.42 

311 

1.31 

365 

1.22 

423 

1.14 

486 

6.8 

.722 

1.64 

269 

1.50 

320 

1.39 

376 

1.29 

436 

1.20 

500 

7.0 

.765 

1.73 

277 

1.59 

330 

1.46 

387 

1.36 

449 

1.27 

515 

Diam.  in  Inches. 

Vel.  in 

Vel- 

16 

17 

18 

19 

20 

Feet 

head  in 

perSeo. 

Feet. 

Frhead 

Cub  ft 

Fr  head 

Cub  ft 

Frhead 

Cub  ft 

Fr  head 

Cub  ft 

Fr  head 

Cub  ft 

n  Feet. 

perMin 

in  Feet. 

perMin 

in  Feet. 

perMin 

in  Feet. 

perMin 

in  Feet. 

perMin 

2.0 

.062 

.123 

167 

.116 

189 

.110 

212 

.104 

236 

.099 

262 

2.2 

.075 

.146 

184 

.138 

208 

.130 

233 

.123 

260 

.117 

288 

2.4 

.090 

.171 

201 

.161 

227 

.152 

254 

.144 

283 

.137 

314 

2.6 

.105 

.197 

218 

.185 

246 

.175 

275 

.166 

307 

.158 

340 

2.8 

.122 

.225 

234 

.212 

265 

.200 

297 

.189 

331 

.180 

366 

3.0 

.140- 

.255 

251 

.240 

284. 

.226 

318 

.214 

354 

.204 

393 

3.2 

.160 

.286 

268 

.269 

302 

.254 

339 

.241 

378 

.229 

419 

3.4 

.180 

.319 

284 

.300 

321 

.283 

360 

.269 

401 

.255 

445 

3.6 

.202 

.354 

301 

.333 

340 

.314 

382 

.298 

425 

.283 

471 

3.8 

.225 

.390 

318 

.367 

359 

.347 

403 

.328 

449 

.312 

497 

4.0 

.250 

.428 

335 

.403 

378 

.380 

424 

.360 

472 

.342 

623 

4.2 

.275 

.468 

352 

.440 

397 

.416 

445 

.394 

496 

.374 

650 

4.4 

.302 

.509 

368 

.479 

416 

.452 

466 

.429 

519 

.407 

576 

4.6 

.330 

.552 

385 

.519 

435 

.490 

488 

.465 

543 

.441 

602 

4.8 

.360 

.596 

402 

.561 

454 

.530 

509 

.502  i  567 

.477 

628 

5.0 

.390 

.642 

419 

.605 

473 

.571 

530 

.541 

590 

.514 

654 

5.2 

.422 

.690  1  435 

.650 

492 

.614 

551 

.581 

614 

.562 

680 

5.4 

.455 

.740  1  452 

.696 

511 

.657 

572 

.623 

638 

.592 

707 

5.6 

.490 

.791     469 

.744 

529 

.703 

594 

.666 

661 

.632 

733 

5.8 

.525 

.843 

486 

.793 

548 

.749 

615 

.710 

685 

.674 

759 

6.0 

.562 

.897 

502 

.844 

567 

.798 

636 

.755 

709 

.718 

785 

6.2 

.600 

.953 

519 

.897 

586 

.847 

657 

.802 

732 

.762 

811 

6.4 

.640 

1.01 

536 

.951 

605 

.898 

678 

.851 

756 

.808 

838 

6.6 

.680 

1.07 

553 

1.01 

624 

.950 

700 

.900 

780 

.855 

864 

6.8 

.722 

1.13 

569 

1.06 

643 

1.00 

721 

.951 

803 

.904 

89G 

7.0 

.765 

1.19 

586 

1.12 

662 

1.06 

742 

1.00 

827 

.953 

916 

546 


HYDKAtJLIOS. 


TABLE  4J/£.  —  (Continued.) 


Vel,  in 
Feet 
perSec. 

Vel- 
head  in 
Feet. 

Diam.  iu  Inches. 

22 

24 

26 

28 

30 

Fr  head 
in  Feet, 

Cub  ft 
perMiu 

Fr  head 
in  Feet. 

Cub  ft 
per  MID 

Fr  head 
in  Feet. 

Cub  ft 
per  Miu 

Fr  head 
in  Feet. 

Cub  ft 
perMiu 

Kr  head 
m  Feet. 

Cub  ft 
per  Min 

2.0 

.062 

.090 

316 

.082 

377 

.076 

442 

.070 

513 

.066 

589 

2.2 

.075 

.106 

348 

.097 

414 

.090 

486 

.083 

564 

.078 

648 

2.4 

.090 

.124 

380 

.114 

452 

.105 

531 

.097 

616 

.091 

707 

2.6 

.105 

.143 

412 

.131 

490 

.121 

575 

.112 

667 

.105 

766 

2.8 

.122 

.164 

443 

.150 

528 

.138 

619 

.128 

718 

.120 

824 

3.0 

.140 

.185 

475 

.170 

565 

.157 

663 

.145 

770 

.136 

883 

3.2 

.160 

.208 

507 

.191 

603 

.176 

708 

.163 

821 

.152 

942 

3.4 

.180 

.232 

538 

.213 

641 

.196 

752 

.182 

872 

.170 

1001 

3.6 

.202 

.257 

570 

.236 

678 

.218 

796 

.202 

923 

.189 

1060 

3.8 

.225 

.284 

601 

.260 

716 

.240 

840 

.223 

974 

.208 

1119 

4.0 

.250 

.311 

633 

.285 

754 

.263 

885 

.244 

1026 

.228 

1178 

4.2 

.275 

.340 

665 

.312 

791 

.288 

923 

.267 

1077 

.249 

1237 

4.4 

.302 

.370 

697 

.339 

829 

.313 

973 

.290 

1129 

.271 

1296 

4.6 

.330 

.401 

728 

.368 

867 

.339 

1017 

.315 

1180 

.294 

1355 

4.8 

.360 

.434 

760 

.397 

905 

.367 

1062 

.341 

1231 

.318 

1414 

5.0 

.390 

.467 

792 

.428 

942 

.395 

1106 

.367 

1283 

.343 

1472 

5.2 

.422 

.502 

823 

.460 

980 

.425 

1150 

.394 

1334 

.368 

1531 

5.4 

.455 

.538 

855 

.493 

1018 

.455 

1194 

.423 

13S5 

.394 

1590 

5.6 

.490 

.575 

887 

.527 

1055 

.486 

1233 

.452 

1437 

.422 

1649 

5.8 

.525 

.613 

918 

.562 

1093 

.519 

1283 

.482 

1488 

.450 

1708 

6.0 

.562 

.652 

950 

.598 

1131 

.552 

1327 

.513 

1539 

.478 

1767 

6.2 

.600 

.693 

982 

.635 

1168 

.586 

1371 

.544 

1590 

.508 

1826 

6.4 

.610 

.735 

1013 

.673 

1206 

.622 

1416 

.577 

164^! 

.539 

1885 

6.6 

.680 

.778 

1045 

.713 

1244 

.658 

1460 

.611 

1693 

.570 

1943 

6.8 

.722 

.821 

1077 

.753 

1282 

.635 

1504 

.645 

1744 

.602 

2003 

7.0 

.765 

.867 

1109 

.794 

1319 

.733 

1548 

.681 

1796 

.635 

2061 

TABLE  3.    Of  fifth  roots  and  fifth  powers. 

For  most  practical  purposes  of  the  preceding  rules,  we  may  take  from  this  table,  the  fifth  root  of  th« 
power  nearest  to  that  whose  root  we  seek;  when  the  exact  o'ne  is  not  in  the  table. 


Power. 

No.  or 

Root. 

Power. 

No.  or 
Root. 

Power. 

No.  or 
Root. 

Power. 

Root. 

Power. 

Root. 

Power. 

No.  or 

Root. 

.0000100 

! 

.000142 

.170 

.004219 

.335 

.077760 

.60 

.695688 

.93 

8.11368 

1.52 

.000164 

.175 

.004544 

.340 

.084460 

.61 

.733904 

.94 

8.66171 

1.54 

.0000110 

.102 

.000189 

.180 

.00*888 

.345 

.091613 

.62 

.773781 

.95 

9.23896 

1.56 

.000217 

.185 

.005252 

.350 

.099244 

.63 

.815373 

.96 

9.84658 

1.58 

.0000122 

.104 

.000248 

.190 

.005638 

.355 

.107374 

.64 

.858734 

.97 

10.4858 

1.60 

.000282 

.195 

.006047 

.360 

.116029 

.65 

.903921 

.98 

11.1577 

1.62 

.0000134 

.106 

.000320 

.200 

.006478 

.365 

.125233 

.66 

.950990 

.99 

11.8637 

1.64 

.000362 

.205 

.006934 

.370 

.135012 

.67 

1. 

1. 

12.6049 

1.66 

.0000147 

.108 

000408 

.210 

.007416 

.375 

.145393 

.68 

1.10408 

.02 

13.3828 

1.68 

.0000161 

.110 

.000459 

.215 

.007924 

.380 

,156403 

.69 

1.21665 

.04 

14.1986 

1.70 

.0000176 

.112 

.000515 

.220 

.008459 

.385 

.168070 

.70 

1.33823 

.06 

15.0537 

1.72 

.00001  93 

.114 

.000577 

.225 

.009022 

.390 

.180423 

.71 

1.46933 

.08 

15.9495 

1.74 

.0000210 

.116 

.000644 

.230 

.009616 

.395 

.193492 

.72 

1.61051 

.10 

16.8874 

1.76 

.0000229 

.118 

.000717 

.235 

.010240 

.400 

.207307 

.73 

1.76234 

.12 

17.8690 

1.78 

.0000249 

.120 

.000796 

.240 

.011586 

.41 

.221901 

.74 

1.92541 

.14 

18.8957 

1.80 

.0000270 

.122 

.000883 

.245 

.013069 

.42 

.237305 

.75 

2.10034 

.16 

19.9690 

1.82 

.0000293 

.124 

.000977 

.250 

.014701 

.43 

.253553 

.76 

2.28775 

.18 

21.0906 

1.84 

.0000318 

.126 

.001078 

.255 

.016492 

.44 

.270678 

.77 

2.48832 

1.20 

22.2620 

1.86 

.0000344 

.128 

.001188 

.260 

.018453 

.45 

.288717 

.78 

2.70271 

1.22 

23.4849 

1.88 

.0000371 

.130 

.001307 

.265 

.020596 

.46 

.307706 

.79 

2.93163 

1.24 

24.7610 

1.90 

.0000401 

.132 

.001435 

.270 

.022935 

.47 

.327680 

.80 

3.17580 

1.26 

26.0919 

1.92 

.0000432 

.134 

.001573 

.275 

•025480 

.48 

.348678 

.81 

3.43597 

1.28 

27.4795 

1.94 

.0000465 

.136 

.001721 

.280 

.028248 

.49 

.370740 

.82 

3.71293 

1.30 

28.9255 

1.96 

.0000500 

.138 

.001880 

.285 

.031250 

.50 

.393904 

.83 

4.00746 

1.32 

30.4317 

1.98 

.0000538 

.140 

.002051 

.290 

.034503 

.51 

.418212 

.84 

4.32040 

.34 

32.0000 

2.00 

.0000577 

.142 

.002234 

.295 

.038020 

.52 

.443705 

.85 

4.65259 

.36 

36.2051 

2.05 

.0000619 

.144 

.002430 

.300 

.041820 

.53 

.470427 

.86 

5.00490 

.38 

40.8410 

2.10 

.0000663 

.146 

.002639 

.305 

.045917 

.54 

.49*421 

.87 

5.37824 

.40 

45.9401 

2.15 

.0000710 

.148 

.002863 

.310 

.050328 

.55 

.527732 

.88 

5.77353 

.42 

51.5363 

2.20 

.0000754 

.150 

.003101 

.315 

.055073 

.56 

.558406 

.89 

6.19174 

.44 

57.6650 

2.25 

.0000895 

.155 

.003355 

.320 

.060169 

.57 

.590490 

.90 

6.63383 

.46 

64.3634 

2.30 

.000105 

.160 

.003626 

.325 

.065636 

.58 

.624032 

.91 

7.10082 

.48 

71.6703 

2.35 

.000122 

.165 

.003914 

.330 

.071492 

.59 

.659082 

.92 

7.59375 

1.50 

79.6262 

2.40 

JYDRAULICS. 


547 


TABLE  3.    Of  fifth  roots  and  fifth  powers  —  (Continued.) 


Power. 

Iff. 

Root. 

Power. 

No.  or 
Root. 

Power. 

No.  or 
Root. 

Power. 

No.  01 
Root. 

Power. 

No.  or 
Root. 

Power. 

No.  or 

Root. 

88.2735 

2.45 

2824.75 

4.90 

85873 

9.70 

2609193 

19.2 

20511149 

29.0 

459165024 

54. 

97.6562 

2.50 

2971.84 

4.95 

90392 

9.80 

2747949 

19.4 

21228253 

29.2 

503284376 

55. 

107.820 

2.55 

3125.00 

5-00 

95099 

9.90 

2892547 

19.6 

21965275 

29.4 

550731  77b 

56. 

118  814 

2.60 

3450.25 

5-10 

100000 

10.0 

3043168 

19.8 

22722628 

29.6 

601692057 

57. 

130.686 

2.65 

3802.04 

5.20 

110408 

10.2 

3200000 

20.0 

23500728 

29.8 

65635676b 

58. 

143.483 

2.70 

4181.95 

5-30 

121665 

10.4 

3363232 

20.2 

24300000 

30.0 

71492429<J 

59. 

157.276 

2.75 

4591.65 

5-40 

133823 

10.6 

3533059 

20.4 

26393634 

30.5 

777600000 

60. 

172.104 

2.KO 

5032.84 

5-50 

146933 

10.8 

3709677 

20.6 

28629151 

31.0 

844596301 

61. 

188.02W 

2.85 

5507.32 

5-60 

161051 

11.0 

3893289 

20.8 

31013642 

31.5 

916132832 

62. 

205.111 

2.90 

6016.92 

5-70 

176234 

11.2 

4084101 

21.0 

33554432 

32.0 

992436543 

63. 

223.414 

2.95 

6563.57 

5-80 

192541 

11.4 

4282322 

21.2 

36259082 

32.5 

1073741824 

64. 

243.000 

3.00 

7149.24 

5-90 

210034 

11.6 

4488166 

21.4 

39135393 

33.0 

1160290626 

65. 

263.936 

3.05 

7776.00 

6-00 

228776 

11.8 

4701850 

21.6 

42191410 

33.5 

125233257h 

66. 

286.292 

3.10 

8445.96 

6.10 

248832 

12.0 

4923597 

21.8 

45435424 

34.0 

1350125107 

67. 

310.136 

3  15 

9161.33 

6.20 

270271 

12.2 

5153632 

22.0 

48875980 

34.5 

145393356b 

68. 

335.544 

3.20 

91)24.37 

6.30 

293163 

12.4 

5392186 

22.2 

52521875 

35.0 

156403134J; 

69. 

302.591 

3.25 

10737 

6.40 

317580 

12.6 

5639493 

22.4 

56382167 

35.5 

168070000T 

70. 

391.354 

3.30 

11603 

6-50 

343597 

12.8 

5895793 

22.6 

60466176 

360 

1804229351 

71. 

421.419 

3.35 

12523 

6-60 

371293 

13.0 

6161327 

22.8 

64783487 

365 

193491763-2 

72. 

454.354 

3.40 

13501 

6.70 

400746 

13.2 

6436343 

23.0 

H9343957 

37.0 

2073071593 

73. 

488.  7GO 

3.45- 

14539 

6.80 

432040 

13.4 

6721093 

215.2 

74157715 

37.5 

2219006624 

74. 

525.219 

3.50 

15640 

6.90 

465259 

136 

7015834 

23.4 

79235168 

38.0 

2373046875 

75. 

563.822 

3.55 

10807 

.00 

500490 

13.8 

7320825 

23.6 

84587005 

38.5 

2535525376 

76. 

604.662 

3.60 

18042 

.10 

537824 

14.0 

7636332 

23.8 

90224199 

39.0 

2706784157 

77. 

647.835 

3.65 

19349 

.20 

577353 

14.2 

7962624 

24.0 

96158012 

39.5 

2887174368 

78. 

693.440 
741.577 

3.70 
3.75 

-•0731 
22190 

.30 
.40 

619174 
663383 

14.4 
14.6 

8299976 
8648666 

24.2 
24.4 

102400000 
108962013 

40.0 
40.5 

3077056399 
3276800000 

79. 

80. 

792.352 

3.80 

23730 

7.50 

710082 

14.8 

9008978 

24.6 

115856201 

1.0 

3486784401 

81. 

845.870 

3.85 

25355 

7.60 

759375 

15.0 

9381200 

24.8 

123095020 

1.5 

370739843-2 

82. 

902.242 

3.90 

27068 

7.70 

811368 

15.2 

9765625 

25.0 

130691232 

2.0 

3939040643 

83. 

961.580 

3.95 

28872 

7.80 

866171 

15.4 

10162550 

25.2 

l.-iMwT'JlO 

2.5 

4182119424 

84. 

1021.00 

4.00 

10771 

7.90 

923896 

15.6 

10572278 

25.4 

147008443 

3.0 

4437053125 

85. 

1089.62 

4.05 

52768 

8.00 

984658 

15.8 

10995116 

25.6 

155756538 

3.5 

470427017« 

86. 

1158.56 

4.10 

54868 

8.10 

1048576 

16.0 

11431377 

25.8 

164916224 

4  0 

4984209207 

87. 

1230.95 

4.15 

37074 

8.20 

115771 

16.2 

11881376 

26.0 

174501858 

4.5 

5277319168 

88. 

1306.91 

4.20 

39390 

8.30 

186367 

16.4 

12345437 

26.2 

184528125 

45.0 

5584059449 

89. 

1386.58 

4.25 

41821 

8.40 

260493 

16.6 

12823886 

26.4 

195010045 

45.5 

5904900000 

90. 

1470.08 

4.30 

44371 

8.50 

338278 

16.8 

13317055 

26.6 

205962976 

46.0 

6240321451 

91. 

1557.57 

4.35 

47043 

8.60 

419857 

17.0 

13825281 

26.8 

217402615 

46.5 

6590815232 

92. 

1649.16 

4.40 

49842 

8.70 

505366 

17.2 

14348907 

27.0 

229345007 

470 

6956883693 

93. 

1745.02 

4.45 

52773 

8.80 

594947 

17.4 

14888280 

27.2 

241806543 

47.5 

7339040224 

94. 

1845.28 

4.50 

55841 

8.90 

688742 

17.6 

15443752 

27.4 

254803968 

48.0 

7737809375 

95. 

1950.10 

4.55 

59049 

9.00 

786899 

17.8 

16015681 

27.6 

268354383 

48.5 

8153726976 

96. 

2059.63 

4.60 

62403 

9.10 

889568 

18.0 

16604430 

27.8 

282475249 

49.0 

8587340257 

97. 

2174.08 

4.65 

65908 

9.20 

996903 

18.2 

17210368 

2H.O 

297184391 

49.5 

9039207968 

98. 

2293.45 

4.70 

69569 

9.30 

109061 

18.4 

17833868 

28.2 

\\  2500000 

50.0 

9509900499 

99. 

2418.07 

4.75 

73390 

9.40 

2226203 

18.6 

18475309 

28.4 

U5025251 

51. 

2548.04 

4.80 

77378 

9.50 

2348493 

18.8 

19135075 

28.6 

180204032 

52. 

2683.54 

4.85 

81437 

9.60 

2476099 

19.0 

19813557 

28.8 

418195493 

53. 

4|  foil  U01  f(£ll<>  till  1<j 


548 


HYDRAULICS. 


TABLE  6.  Of  the  square  roots  of  the  fifth  powers  of  num- 
bers. In  this  table  the  numbers  and  the  roots  are  supposed  to  be  in  the  same  di- 
mensions ;  that  is,  both  In  inches,  or  both  iu  feet.  &c.  See  the  uext  table.  The  foregoing  formulas, 
a*  given  in  this  book,  do  not  require  the  use  of  this  table,  nor  of  the  next. 


No. 

Sq.  Rt. 
of  5th 

No. 

Sq.  Rt. 
of  5th 

No. 

Sq.  Rt. 
of  5th 

No. 

Sq.  Kt. 
of  5th 

No. 

Sq.  Rt. 
of  5th 

No. 

Sq.  Rt. 
of  5th 

Power. 

Power. 

Power. 

Power. 

Power. 

Power. 

.25 

.031 

7. 

129.64 

17.5 

1281.1 

31. 

5351 

49 

16807 

76  I  50354 

.5 

.177 

7.25!  141.53 

18. 

1374.6 

31.5 

5569 

50 

17678 

77    52027 

.75 

.485 

7.5 

154.05 

18.5 

1472.1 

32. 

5793 

51 

18575 

78 

53732 

1. 

1. 

7.75 

167.21 

19. 

1573.6 

32.5 

6022 

52 

19499 

79    55471 

1.25 

1.747 

8. 

181.02 

19.5 

1679.1 

33. 

6256 

53 

20450 

80 

57243 

1.5 

2.756 

8.25 

195.50 

20. 

1788.9 

33.5 

6496 

54 

21428 

81 

59049 

1.75 

4.051 

8.5 

210.64 

20.5 

1902.8 

34. 

6741 

55 

22434 

82 

60888 

2. 

5.657 

8.75 

226.48 

21. 

2020.9 

34.5 

6991 

58 

23468 

83 

62762 

2.25 

7.594 

9. 

243. 

•21.5 

2143.4 

35. 

7247 

57 

24529 

84 

64669 

2.5 

9.882 

9.25 

260.23 

'22. 

2270.2 

35.5 

7509 

58 

25620 

85 

66611 

2.75 

12.541 

9.5 

278.17 

22.5 

2401.4 

36. 

7776 

59 

26738 

86 

68588 

3. 

15.588 

9.75 

296.83 

23. 

2537. 

36.5 

8049 

60 

27886 

87 

70599 

3.25 

19.042 

10. 

316.23 

23.5 

2677.1 

37. 

8327 

61 

29062 

88 

72646 

35 

22  918 

10.5 

357.2 

24. 

2821.8 

37.5 

8611 

62 

30268 

89 

74727 

3.75 

27232 

11. 

401.3 

24.5 

2971.1 

38.     8901 

63 

31503 

90 

76843 

4. 

32. 

11.5 

448.5 

25. 

3125. 

88.5 

9197 

64 

32768 

91 

78996 

4.25 

37.24 

12. 

498.8 

25.5 

3283  6 

39. 

9498 

65 

34063 

92 

81184 

4.5 

42.96 

12.5 

552.4 

26. 

3446.9 

89.5 

9806 

66 

35388 

93 

83408 

4.75 

49  17 

13. 

609.3 

26.5 

3615.1 

40. 

10119 

67 

36744 

94 

85668 

5. 

55.90 

13.5 

669.6 

27. 

3788. 

41. 

10764 

68 

38131 

95 

87965 

5.25 

63.15 

14. 

733.4 

27.5 

3965.8 

42. 

11432 

69 

39548 

96 

90298 

5.5 

70.94 

145 

800.6 

28. 

4148.5 

43. 

12125 

70 

40996 

97 

92668 

5.75 

79.28 

15. 

871.4 

28.5 

4336.2 

44. 

12842 

71 

42476 

98 

95075 

6. 

88.18 

15.5 

945.9 

29. 

4528.9 

45. 

13584 

72 

43988 

99  !  97519 

6.25 

97.66 

16. 

1024. 

29.5 

4726.7 

46. 

14351 

73 

45531 

100  !  100000 

6.5 

107.72 

16.5 

1105.9 

30. 

4929.5 

47. 

15144 

74 

47106 

6.75 

118.38 

17. 

1191.6 

30.5    5138. 

48.     15963 

75 

48714 

TABLE  6%.  The  following  will  be  found  more  convenient  than  the  foregoing 
in  computing  discharges  through  pipes  :  inasmuch  as  the  diams  are  here  given  in  inches,  and  the  *q 
rts  of  the  5th  powers  in  feet;  in  which  form  the  latter  are  sometimes  required  by  formulas. 

Intermediate  sq  roots  uiay  be  had  near  enough  for  practice,  by  simple  proportion. 


Diam. 

Sq.  Rt.  of 
5th  Pow. 

Diam. 

Sq.  Rt.  of 
5th  Pow. 

Diam. 

Sq.  Rt.  of 
5th  Pow. 

Diam. 

Sq.  Rt.  of 
5th  Pow. 

Diam. 

Sq.  Rt.  of 
5th  Pow. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

M 

.00006 

stf 

.0547 

12. 

1.000 

rtJi 

4.813 

42 

22.92 

9i 

.00017 

4. 

.0641 

^ 

1.108 

23 

5.086 

43 

24.31 

\f 

.00035 

.0731 

13. 

1.221 

X 

5.365 

44 

25.74 

X 

.00062 

i^ 

.0827 

'     « 

1.342 

24 

5.657 

45 

27.23 

x 

.00098 

K 

.0971 

14. 

1.470 

25 

6.264 

46 

28.77 

% 

.00144 

5. 

.1120 

X 

1.605 

26 

6.909 

47 

30.36 

1. 

.0020 

U 

.1271 

15. 

1.747 

27 

7.593 

48 

32.00 

H 

.0027 

X 

.1428 

X 

1.896 

28 

8.316 

49 

33.69 

.0035 

H 

.1590 

16. 

2.053 

29 

9.079 

50 

35.44 

% 

.0044 

6. 

.1768 

1A 

2.217 

30 

9.882 

51 

37.25 

L£ 

.0055 

X 

.2160 

17 

2.389 

31 

10.73 

52 

39.13 

M 

.0067 

7. 

.2599 

2.567 

32 

11.61 

53 

41.02 

1 

.0081 

« 

.3088 

18? 

2.756 

S3 

12.54 

54 

42.96 

% 

.0096 

8. 

.3628 

K 

2.950 

34 

13.51 

55 

44.97 

2. 

.0113 

w 

.4228 

19 

3.155 

35 

14.53 

56 

47.05 

\i 

.0152 

9. 

.4871 

U 

3.365 

36 

15.59 

57 

49.17 

X 

.0198 

X 

.5577 

20 

3.586 

37            16.69 

58 

51.35 

% 

.0252 

10. 

.6339 

H 

3.813 

38       1    17.84 

59 

53.60 

3. 

.0312 

^ 

.7162 

21. 

4.051 

39       ]     19.04 

60 

55.90 

y* 

.0383 

11.               .8043 

^ 

4.297 

40 

20.29 

61 

58.27 

% 

.0459 

X           .8990 

22. 

4.551 

41 

21.58 

Art.  5.  On  the  resistance  which  curved  bends  oppose  to  the 
flow  of  water  through  round  pipes.    Well-rounded  bends  of  large  rad, 


HYDRAULICS. 


Whether  vert  or  nor.  produce  but  little  resistance ;  except 
far  as  the  fir«t  may  cause  accumulations  of  sediment, 
air.     According  to'Weisbach.  the  rules  of 
and  other  authorities,  are  erroneous;  and  he 
in?  one  for  ascertaining  the  additional  head  reqd  to  overcome 
the  resistance  produced  in  a  circular  pipe,  by  a  betid  formed 
by  an  arc  of  a  circle  :  Knowing  the  rad  re  of  the  pipe,  (or.  in 
other  words,  half  its  diam,)  in  feet:  the  rad  r«,  of  the  axis 
rn  o  of  the  bend,  in  feet ;  the  centra!  angle  rsoin  degrees ; 
(which  is  equal  to  the  angle  dbx,  or  cba,)  and  the  reqd  vel 
of  the  water  in  the  pipe,  in  ft  per  sec. 

R u  Ie.  Div  the  central  angle  r  s  o  in  deg,  by  180. 

Call  the  quot  o.  Next,  square  the  reqd  vel.  Div  this  sq  by 
the  constant  number  64.4.  Call  the  quot  b.  Div  the  inner  rad 
re  of  the  pipe,  in  ft,  by  the  rad  rs  of  the  axis  rno  of  the 
bend,  in  ft.  Call  the  quot  c.  Take  from  the  following  Table 
7.  the  number  in  column  d,  which  corresponds  to  c;  (unless 
c  be  less  than  .1,  in  which  case  always  take  .13  as  d.)  Finally,  mult  together  this  number  d.  the 
quot  a,  and  the  quot  b.  The  prod  will  be  the  reqd  head  in  feet;  which  must  either  be  added  to  tlie 
head  previously  calculated  for  the  straight  pipe,  if  the  original  vel  is  required  to  be  maintained  :  or 
must  be  subtracted  from  it,  in  case  the  head  does  not  admit  of  increase,  and  a  new  calculation  made 
to  ascertain  the  diminished  vel  under  the  head  thus  reduced.  If  there  is  more  than  one  bend  of  the 
same  dimensions,  an  equal  alteration  of  head  must  be  made  for  each  ;  or,  if  they  are  of  diff  radii, 
and  with  diff  central  angles  r  s  o.  a  separate  calculation  must  be  made  for  each.  Rennie's  experi- 
ments, at  the  end  of  this  Art,  seem  to  prove  ttiat  this  is  by  no  means  the  case  So  far  as  the  writer 
is  aware,  we  have  no  reliable  data  for  calculating  the  effects  of  a  succession  of  bends. 

In  shape  of  a  formula,  Weisbach's  rule  stands  thus  :  E  being 

the  rad  of  the  axis  of  the  bend,  and  r  the  rad  of  the  pipe : 

_  ,_  .     ftx  7        square  of  vel       central  angle 

""  T  X   *"  ft  pwsw    x     <"  degree* 

64.4  '         180. 

The  expression  ^  means  the  sq  root  of  the  7th  power.     When  the  rad  of  the  bend  exceeds  5  diams 

of  the  pipe,  then  1.847  (7;)^  becomes  inappreciable  in  practice,  and  mny  be  omitted  from  the  for- 
mula.   When  the  pipe  is  square,  instead  of  circular,  the  formula  becomes 


Head  =  .124  -}-  3.104  (  - 

TABL.E  7. 


ve!2        central  angle 
X  64.4  *     '      180. 


c. 

d. 

c. 

d. 

c. 

d. 

c. 

d. 

c. 

d. 

.1 

.15 
.2 
.225 
.25 
.275 
.3 

.131 
.135 
.138 
.145 
.15 
.155 
.16 

.325 
.35 
.375 
.4 
.425 
.45 
.475 

.17 
.18 
.195 
.206 
.225 
.24 
.264 

.5 
.525 
.55 
.575 
.6 
.625 
.65 

.29 
.32 
.85 
.39 
.44 
.49 
.54 

.675 

J25 

.75 
.775 
.8 
.825 

.60 
.66 
.73 
.80 
.88 
.98 
1.08 

.85 
.875 
.9 
.925 
.95 
.975 
1. 

1.18 
1.29 
1.41 
1.54 
1.68 
1.83 
2. 

Ex.  A  straight  pipe  1  mile  long,  and  18  ins  diam.  with  a  total  head  of  20  ft,  will  disch  water  with 
t  vel  of  3.6  ft  per  sec;  but  it  has  been  found  necessary  to  introduce  a  circular  bend  of  90°,  with  a  rnd 
»r.  Fig  2,  of  5  feet.  What  addition  must  be  made  to  the  20  ft  head,  to  compensate  for  the  additional 
resistance  caused  bv  the  bend :  so  that  the  reqd  vel  of  3.6  ft  per  sec  may  still  be  maintained? 

90° 

Here,  —  =  .5  =  a.  Next,  the  square  of  the  reqd  vel  in  ft  per  sec,  is  3.6  X  3.6  =:  12.96.  And 
12  Qfi  180  .75 

^—  =  .2012  =  6.    The  rad  r  e  of  the  pipe  (.75  ft),  div  by  the  rad  rs  of  the  bend  (5  ft),  =:  —  =  .  15 

=  c;  and  opposite  this  .15  in  the  column  c  of  the  foregoing  table,  we  find  d  =  .135.  Finally. 
aX*>Xd  =  .oX  .2012  X  .135  =  .0136  ft,  or  about  %  inch  only,  the  additional  head  reqd.  See  next 
table.  No.  8. 

Du  Btiat's  rule  for  the  additional  head  required  to  over- 
come the  resistance  of  circular  bends  in  water  pipes.  Having 

given  the  diam  of  the  pipe,  in  ft  ;  the  rad  of  the  bend,  in  ft;  and  the  vel  in  ft  per  sec.t  Div  tr.  Fig 
2,  or  half  the  diam  of  the  pipe,  by  the  rad  sw  of  the  outer  side  of  the  bend.  The  quot  will  be  th« 
nbt  versed  sine  of  Du  Buat's  angle  of  reflexion.  Take  this  versed  sine  from  unity,  or 
1.  The  rem  will  be  the  nat  cosine  of  the  same  angle.  From  the  Table  of  Nat  Sin  and  Tang,  take 
both  the  angle  and  the  nat  sine  corresponding  to  this  nat  cosine.  Call  the  angli-  R.  Also,  square  the 
nat  sine,  and  call  this  square  S.  Take  the  angle  w  s  o  from  180°.  Div  the  rem  by  twice  the  angle  R 
*>f  reflexion  just  found.  Call  the  quot  T.  Finally,  mult  together  the  constant  dec  .00375.  the  square 
of  the  vel  in  ft  per  sec,  the  quot  T,  and  the  square  S.  The  prod  will  be  the  reqd  extra  head  in  feet. 


*  This  central  angle,  when  the  bend  is  less  than  a  semicircle,  will  always 
be  equal  to  either  one  of  the  two  exterior  angles,  a  A  c,  or  x  b  d.  Fig  2,  formed  by  the  tangents  a  d 
and  x  c.  Either  of  these  angles  is  called  the  angle  Off  deflection  of  the  bend. 

t  When  we  have  given  the  diam  of  the  pipe  in  ft.  and  the  quantity  of  water  required  in  cub  ft  per 
sec,  the  vel  in  ft  psr  sec  will  be  found  by  dividing  the  quantity  by  the  area  of  the  pipe.  This  area 
may  be  taken  from  Table  3,  Art  2,  p  541. 


550 


HYDRAULICS. 


Ex.  The  same  as  the  foregoing  one  for  Weiabach's  rule ;  that  is,  a  pipe  of  18  ins,  or  1.5  ft  diaia; 
rad  sw  of  outer  side  of  beud,  5  Jo  ft;  vel,  3.6  ft  per  sec.  What  extra  head  will  the  beud  require,  in 
order  that  this  vel  may  not  be  diminished? 

Here,    ~  =  .13043  =  nat  versed  sine  of  angle  of  reflexion.    And  1  —  .13043  =  .86957  =  nat  cos  of 

same  angle.     In  the  Table  of  Nat  Sin  and  Tang,  we  find,  opposite  the  nat  cos  .86957,  the  angle  R  — 
29°  35  ;  and  its  nat  sine  .4937.     The  square  of  .4937  —  .2437  =  S.     Again,  180°  —  90°  =:  90°.    And 
90°      _  5400_mm  __  Qf         Finallv  tne  8qUare  of  the  vel  is  12.96:  hence,  we  have  .00875  X 

59°  10'  ~~  3550  min 

12  96  X  1.52  X  .2437  =  .0172  ft,  the  reqd  extra  head :  or  about  3-  of  an  inch.  Weisbach's  rule  gave 
.0136  ft,  or  about  %  of  an  inch.  Hence  we  see  that  the  resistance  produced  by  well-rounded  beuda  is 
not  great. 

When  the  rad  r  s.  Fig  2,  of  the  bend,  is  less 
than  about  two  diams  of  the  pipe,  which  will  rarely  hap- 
pen, the  resistance  to  the  flow  of  the  water  iucreases  very  rapidly;  while,  on 
the  other  hand,  by  Weisbach's  rule,  as  we  understand  it,  no  advautage  ap- 
pears to  be  gained  by  using  a  rad  greater  than  5  diauirf  of  the  pipe.*  Employ- 
ing  Weisbach's  formula,  the  writer  has  drawn  up  the  following  table  of  head* 
reqd  to  overcome  the  resistance  of  one  bend  of  90°.  for  diff  vels  iu  ft  per  sec ; 
and  for  any  diam  whatever.  This  table  extends  from  a  rad  of  5  diams  down 
to  one  of  %  diam ;  which  is  the  smallest  possible,  inasmuch  as  it  leads  to  a 
bend  like  Fig  3. 

A  vel  of  12  ft  per  sec  is  equal  to  8.18  miles  per  hour;  one  which  will  rarely 
occur,  inasmuch  as  it  requires  a  head  of  about  330  feet  per  mile. 

TABLE  8.    Heads  required  to  overcome  the  resistance  in 
circular  bends  of  9O°.    Original. 


Fin  3 


Velocity  in  feet  per  Second. 

1  ft.   1   2  ft.   1  3  ft.   I  4  ft.   1   5  ft.   j   6  ft. 

7  ft.   I  8  ft.   I  9  ft. 

10  ft. 

12ft. 

HEADS  IN  FEET. 

Rad  —  5  diams  of 

. 

the  pipe. 

.001 

.004 

.009 

.016 

.025   |    .036 

.050 

.065 

.082 

.101 

.145 

Rad  —  3     diams.  . 

.001 

.004 

.010 

.017 

,027 

.038 

.052 

.069 

.086 

.106 

153 

.019 

.029 

.074 

.094 

.167 

Rad  =  2     diams.  .  . 

.001 

.005 

.011 

.012 

.057 

.116 

Rad=  1M  diams... 

.001 

.005 

.012 

.021 

.033 

.048 

.066 

.086 

.108 

.134 

.192 

Rad  -114  diam... 

.002 

.007 

.015 

.026 

.041 

.059 

.080 

.104 

.132 

.163 

.235 

Rad  =  l      diam... 

.002 

.009 

.020 

.036 

.056 

.081 

.110 

.144 

.182 

.225 

.324 

Rad=    *fdiam... 

.005 

.018 

.041 

.072 

.113 

.162 

.   .221 

.288 

.365 

.450 

.649 

Rad  =    ft  diam.  .  . 

.016 

.062 

.140 

.248 

.388 

.559 

.761 

.994 

1.26 

1.55 

2.24 

If  the  central  angle  r  s  o,  Fig  2,  should  be  either  greater,  or 
less  than  9O°,  then  the  heads  given  in  the  table,  must  be  increased,  or  dimin- 
ished directly  in  the  same  proportion. 


Experiments  by  Ronnie,  with  a  pipe  15  ft  long  ;  and  1 
4  ft  head,  gave  the  following  disch  iu  cub  ft  per  sec  : 


inch  bore  ;  with 


t  vertical  bends  near  discharge  end  85.00  sec. 
4  vertical  bends  near  supply  end...  84.00  " 


Straight 00699  cub  ft.  I  One  bend  at  right  angles  near  end 00556 

15  semicircular  bends 00617      "       |  24  bends  at  right  angles 00253 

The  mean  of  many  careful  experiments  tried  at  Liverpool, 
England,  with  a  leaden  pipe,  75  ft  long,  %  inch  bore,  under  8  ft  head,  gave  the 
following  number  of  sees  to  discharge  one  gallon  of  water : 
75  ft  pipe,  straight  and  horizontal..  81.56  sec. 
2  hor  bends  near  dischai-ge  end  ...  83.33    " 

2  hor  bends  near  supply  end 81.80    " 

The  rad  of  the  bends  is  not  stated.     See  Minutes  Trans  Inst  Civ  Eng,  vol  12,  page  501. 

<K.  On  knees,  or  angular  bends  in  water 

.,  pipes,  Fig  4.     The  bends  in  lines  of  water  pipes 

J  should  always  form  circular  arcs ;  because  knees  create  a  much 

greater  resistance.     According  to  Weisbach,  the  head  in  ft  reqd  to 
overcome  the  additional  resistance  caused  by  a  knee  in  a  round 
-F       pipe  is  as  follows  : 
J  Sq  of  vel  in  ft  per  sec 

Head  in  feet  - 64  4 X   constant  in  following 

table  opposite  the  angle  of  deflection,  a  e  o,  or  d  e/.t  Pig  4. 

*  Notwithstanding  this,  we  advise  to  use  as  many  more  than  5  as  can  conveniently  be  done. 
t  The  constant  for  any  angle  of  def,  is  equal  to  .946  times  the  square  of  the  nat  sine  of  half  the 
angle  of  def;  X  2.05  times  the  4th  power  of  the  same  sine. 


HYDKAULICS. 


551 


TABIJB  9. 


Ang.  of 

Aug.  of 

Ang.  of 

Def. 

Constant 

Def. 

Constant. 

Def. 

Constant. 

in  Degs. 

in  Deg. 

in  Degs. 

140° 

2.431 

70° 

.533 

25° 

.049 

130 

2.158 

60 

.364 

20 

.030 

120 

1.861 

50 

.234 

15 

.016 

110 

1.556 

40 

.139 

10 

.007 

100 

1.260 

35 

.102 

5 

.002 

•  90 

.984 

30 

.073 

80 

.740 

Constants  intermediate  of  those  in  the  table  mav  be  obtained  near  enough  by  simple  proportion. 

Ex.  We  see  by  the  first  table  p  539,  that  under  a  total  head  of  20  ft,  a  straight  pipe  1  ft  diam. 
and  a  mile  long,  will  disoh  with  a  vel  Of  2.93  ft  per  sec.  What  additional  head  must  it  have  in  order 
to  maiutaiu  the  same  vel,  if  a  knee  bend  of  90°  be  introduced? 

Here,  -— - *X  .984  —  .1333  X  -984  =  .131  ft,  the  additional  head  reqd.  With  a  circular  arc  of3  ft 
rad,  it  would  be  but  about  .01  ft,  or  but  about  ^3-  as  much  as  with  the  knee. 

The  disch  is  diminished  by  swellings,  or  enlargements  in  pipes, 

as  well  as  by  contractions,  bends,  and  knees;  on  account  of  the  eddies  which  they  produce,  Ac. 

Art.  6.  Inasmuch  as  the  pres  of  quiet  water  against,  and  perp  to,  any  given 
surf,  is  (other  things  being  equal)  in  proportion  to  the  vert  height  of  the  water  above  the  cen  of  grav 
of  the  pressed  surf,  (see  Art  1,  Hydrostatics,)  it  follows  that  in  two  pipes  of  the  same  diams,  aa  a  6, 
and  c  ft,  Fig  5,  the  pres  against,  and  at  right 
augles  to,  the  equal  bases,  mn  of  the  vert  pipe, 
and  op  of  the  inclined  one,  are  equal;  because 
the  vert  heights,  a  b  and  h  g,  of  the  water  above 
the  ceu  of  grav  a  and  c.  of  the  equal  bases,  are 
equal  in  the  two  pipes.  If  the  base  of  the  inclined 
pipe  be  cut  so  that  ty  becomes  the  base,  then  the 
base  is  no  longer  a  circle,  but  an  ellipse;  the 
area  of  which  will  always  be  greater  than  that 
of  the  circular  one ;  and  since  the  vert  height  h  g 
remains  unchanged,  the  pres  against -the  base  ty, 
and  perp  to  it,  will  be  greater  than  that  against 
op,  in  the  same  proportion  as  the  two  areas. 

The  upright  pipe  may  be  but  1  ft  long  ;  and  the 
inclined  one  1  mile,  or  10  miles  long,  still  the  prea 
at  the  base  mn  will  be  the  same  as  that  at  the 
base  op,  so  long  as  the  vert  height  a  b  is  equal  to 
the  vert  height  gh.  The  greater  weight  of  the  water  in  c  h.  does  not  increase  tbepres  at  its  lower  end 

0  p  ;  said  weight  being  sustained  by  the  under  part  p  w  of  the  inclined  pipe.     If,  therefore,  two  steam 
engines  were  employed,  one  to  force  the  water  up  the  1  ft  long  vert  pipe,  and  another  to  force  it 
at  the  same  vel  up  the  10  miles  of  inclined  pipe,  both  engines  would  have  precisely  the  same  amount 
of  resistance  to  overcome,  so  far  as  regards  only  either  the  pres,  or  the  weight  of  the  water  in  the 
respective  pipes.     The  friction,  however,  of  the  water  against  the  much  longer  sides  of  ch,  would 
be  much  greater  than  that  along  a  b;  and  this  excess  of  friction  in  the  inclined  pipe  would  require 
a  proportional  excess  of  power  in  the  engine  to  overcome  It. 

The  amount  of  this  friction  is  easily  ascertained.    It  is  equal 

to  the  weight  of  a  cylindrical  column  of  water,  the  diam  of  which  is  equal  to  the  diam  of  the  pipe; 
and  the  vert  height  of  which  is  equal  to  the  actual  head  of  water  (calculated  by  Art  3 ;  or  Table 
1,  p539)  reqd  to  impart  the  given  vel  to  the  water  flowing  through  the  pipe  at  its  given  inclination. 
For  example  :  it  is  required  to  force  water  at  the  rate  of  4.4  ft  per  sec,  up  a  pipe  1  foot  diam.  3  miles 
long,  and  rising  '21  ft  in  the  3  miles.  What  resistance  will  the  piston  of  the  engine  experience  from 
the  direct  pres  upon  it  of  the  water  in  the  pipe;  and  also  from  the  friction  of  the  water  along  the 
sides  of  the  pipe  ? 

Here  we  have  first,  the  direct  perp  pres  against  the  piston,  equal  to  the  weight  of  a  column  6*f  water 

1  ft  diam,  and  21  ft  high  ;  or  of  1  X  1  X  .7854  X  21  r:  16.5  cub  ft  of  watef,  rr  16.5  X  62.5  =  1031  Its. 
This  assumes  the  diam  of  the  piston  to  be  equal  to  that  of  the  pipe;  if  not.  the  diam  of  the  piston 
must  be  taken  for  that  also  of  the  pressing  column  of  water  against  it.    As  regards  the  resistance 
from  friction,  we  see  by  Table  1,  p539,  that  to  overcome  friction,  so  as  to  impart  a  vel  of  4.4  ft  per 
sec  to  the  water  in  a  pipe  1  ft  in  diam,  we  must  provide  a  head  of  45  ft  per  mile.*    Consequently,  in 
the  case  before  us.  it  would  require  45  ft  X  3  miles  —  135  ft  head  ;  so  that  the  resistance  from  pipe- 
friction  will  be  equal  to  the  weight  of  a  column  of  water  1  foot  diam,  and  135  ft  high ;  or  to  1  X  1  X 
.7854  X  135  X  62  5  =  6627  Ibs.     And  6627  -f  1031  ~  7658  Ibs,  total  resistance  arising  from  the  pres  of 
the  water ;  and  from  its  friction  against  the  sides  of  the  pipe,  when  moving  at  the  rate  of  4.4  ft  per  sec. 

In  the  case  of  pumping  upward  through  the  vert  pipe,  there  is  also  a  slight  friction  to  overcome,  in 
addition  to  the  pres  of  the  water  against  the  piston  ;  and  it  must  be  calculated  by  precisely  the  same 
process  as  in  the  other  case. 

Art.  7.  The  flow  of  water  through  openings,  or  apertures, 
in  the  sides  or  bottom  of  the  containing  vessel,  or  reservoir. 

Theoretically,  the  vel  with  which  water  should  flow  through  such  an  opening,  is  equal  to  that  which 
would  be  acquired  by  a  heavy  body  falling  freely  through  a  height  equal  to  the  head,  or  depth  of 
water,  measured  vert  from  the  level  surf  of  the  water  in  the  reservoir,  to  the  center  of  the  opening; 
or.  more  correctly,  to  its  cen  of  grav.  This  theoretical  vel  is  found  in  ft  per  sec,  by  mult  the  sq  rt 
of  said  head,  or  vert  depth  in  ft,  by  the  constant  number  8.03  ;  or,  mult  the  head  itself  in  ft,  by  64.4 ;  t 

*  By  Weisbach,  or  by  Table  4^,  this  head  would  be  considerably  less. 

f  This  64.4  in  twice  32.2;  which  last  is  the  "  acceleration  of  gravity  "  referred  to  in  Note  to  Art  25 
•f  Force  in  Rigid  Bodies,  p  455. 


552 


HYDRAULICS. 


and  take  th«  sq  rt  of  the  prod.  In  practice,  we  may  use  8  and  61,  as  near  enough.  The  theoretical, 
as  well  as  the  actual  dixcti,  or  the  quantity  iu  cub  ft,  which  flows  out  per  sec,  is  evidently  equal  in 
all  cases  to  the  prod  of  the  theoretical,  or  of  the  actual  vel,  (as  the  case  may  be,)  in  ft  per  sec,  mult 
by  the  area  of  the  opening  in  sq  ft. 

These  theoretical  laws  apply  equally  to  all  fluids,  whatever  may  be  their  sp  grav  ;  thus,  theoretically, 
mercury,  water,  air,  &c,  will  all  flow  with  equal  vels  from  openings  of  equal  sizes,  under  equal  heads. 

Practically,  however,  only  the  mean  vel,  and  the  disch  through  the  veiia  COntraCta,  OF 
Contracted  Veill,  (see  Fig  ll.)  which  forms  itself  just  outside  of  certain  kinds  of  openings, 
(and  which  is  smaller  than  the  openings  themselves,)  are  actually  very  nearly  equal  to  the  theoret- 
ical ones  ;  but  through  the  very  opening  itself  they  are  usually  less.  The  discrepancy  is  greater  in 
some  cases  than  in  others  ;  depending  chiefly  on  the  shape  of  the  opening. 

On  this  account,  the  theoretical  vel  and  disch  found  by  the  foregoing  rule,  must  usually  be  dimin- 
ished by  mult  them  by  certain  decimal  numbers  corresponding  to  the  various  kinds  of  openings  :  and 
called  coefficients  of  discharge.  These  coetfs  have  in  many  cases  been  determined  by  experiment 
very  approximately  ;  and  will  be  found  in  the  following  articles.  It  will  be  seen  iu  Remark  5,  p  554, 
that  by  the  use  of  a  peculiarly  formed  adjutage,  or  attachment,  to  small  openings,  the  actual  disch 
may  even  be  increased  beyond  the  theoretical  one. 

The  following  table  will  save  the  trouble  of  calculating  the  theoretical  vel,  previously  to  mult  it  by 
the  corresponding  coetf  of  disch,  for  obtaining  the  actual  vel.  The  coeffs  for  diff  kinds  of  openings 
will  be  found  further  on. 

TABLE  1O.     Of  the  theoretical  velocities  in  feet  per  sec, 

with  which  water  should  flow  out  into  the  air.  under  diff  heads,  through  openings  in  the  bottom  or 
sides  of  the  containing  reservoir;  the  surf  level  of  which  remains  constantly  at  the  same  height, 

Weisbach  says  (see  third  footnote  to  Art  9)  that  when  water  flows  out  of  an  opening  under  water, 
as  at  n.  Fig  1,  the  vel  and  disch  are  about  fa  part  less  than  when  it  flows  into  the  open  air,  under 
equal  heads.  When  the  diseh  is  made  under  water,  the  vert  dist  a  u.  Fig  1.  between  the  surf  levels 
of  the  two  reservoirs,  must  be  taken  as  the  head.  These  theoretical  vels  are  very  nearly  the  actual 
mean  ones  at  the  contracted  vein;  see  Art  9,  p  554.  Calling  the  head,  H,  then 

=  ^/64Tir  -  8.03  time,  the  sq  rt  of  the  head  in  ft. 


Theoretical  head  _  «eia  _ 

in  feet  ~  T^  ~ 


_  (  square  of  theoret  vei\  x  0155 

^         in  ft  per  sec         )  * 


Head 

Vel, 

Head 

Vel, 

Head  Vel. 

Head  Vel, 

Head  Vel, 

Head 

Vel 

Head  Vel, 

Feet. 

Ft  per 
sec. 

Feet. 

Ft  per 
sec. 

"-fir 

Feet. 

Ft  per 

sec. 

Feet. 

Ft  per 
sec. 

Feet. 

Ftper 
sec. 

Feet. 

Ftper 
sec. 

.005 

.57 

.29 

4.32 

.77    |  7.0t 

1.50 

9.813 

7. 

21.2 

28 

42.5 

76 

69.9 

.010 

.80 

.30 

4.39 

.78 

7.09 

1.52 

9.90 

.2 

21.5 

29 

43.2 

77 

70.4 

.015 

.98 

.31 

4.47 

.79 

7.13 

1.54 

9.96 

.4 

21.8 

30 

43.9 

78 

70.9 

.020 

1.13 

.32 

.54 

.80      7.18 

1.56  1  10.0 

.6 

22.1 

31 

44.7 

79 

71.3 

.025 

1.27 

.33 

.61 

.81       7.22 

J.58     10.  1 

.8 

22.4 

32 

45.4 

80 

71.8 

.030 

1.39 

.34 

.68 

.82 

7  26 

1.60      10.2 

8. 

22.7 

33 

46.1 

81 

72.2 

.035 

1.50 

.35 

.75 

.83 

7.31 

1.65 

10.3 

.2 

23.0 

34 

46.7 

82 

72.6 

.OiO 

1.60 

.36 

.81 

.84 

7.35 

1.70 

105 

.4 

23.3 

85 

47.4 

83 

73.1 

.045 

1.70 

.37 

.87 

.85 

7.40 

1.75 

10.6 

.6 

23.5 

36 

48.1 

84 

73.5 

.05J 

1.79 

.38 

494 

.86 

7.44 

1.80 

10.8 

.8 

23.8 

37 

48.8 

85 

74.0 

.055 

1.88 

.39 

5.01 

.87 

7.48 

1.85 

10.9 

9. 

24.1 

38 

49.5 

86 

74.4 

.060 

1.97 

.40 

5.07 

.88 

7.53 

1.90 

11.1 

.2 

24.3 

39 

50.1 

87 

74.8 

.065 

2.04 

.41 

5.14 

.89 

7.57 

1.95 

11.2 

.4 

24.6 

40 

50.7 

88 

75.3 

.070 

2.12 

.42 

5.20 

.90 

7.61 

2. 

11.4 

.6 

248 

41 

51.3 

89 

75.7 

.075 

2.20 

.43 

5.26 

.91 

7.65 

2.1 

11.7 

.8 

25.1 

42 

52.0 

90 

76.1 

.080 

2.27 

.44 

5.3f 

.92 

7.70 

11.9 

10. 

25.4 

43 

52.6 

91 

76.5 

.085 

2.34 

.45 

538 

.93 

7.74 

2^3 

12.2 

.5 

26;0 

44 

53.2 

92 

76.9 

.030 

2.41 

.46 

5.44 

.94 

7.78 

2.4 

12.4 

11. 

26.6 

45 

53.8 

93 

77.4 

.095 

2.47 

.47 

550 

.95 

7.82 

2.5 

12.6 

.5 

27.2 

46 

54.4 

94 

77.8 

.100 

2.54 

.48 

5.56 

.96 

7.86 

2.6 

12.9 

12. 

27.8 

47 

55.0 

95 

78.2 

.105 

2.60 

.49 

5.62 

.97 

7.90 

2.7 

13.2 

.5 

28.4 

48 

55.6 

96 

78.6 

.110 

2.66 

.50 

5.67 

.98 

7.94 

2.8 

13.4 

13. 

28.9 

49 

56.2 

97 

79.0 

.115 

2.72 

.51 

5.73 

.99 

7.98 

2.9 

13.7 

.5 

29.5 

50 

56.7 

98 

79.4 

.120 

2.78 

.52 

5.79 

IFt. 

8.03 

3. 

13.9 

14. 

30.0 

51 

57.3 

99 

79  8 

.125 

2.84 

.53 

5.85 

1.02 

8.10 

3.1 

14.1 

.5 

30.5 

52 

57.8 

100 

80.3 

.130 

2.89 

.54 

5.90 

1.04 

8.18 

3.2 

14.3 

15. 

31.1 

-53 

58.4 

125 

89.7 

.135 

2.95 

.55 

5.95 

106 

8.26 

3.3 

14.5 

.5 

31.6 

54 

59.0 

150 

98.3 

.140 

3.00 

.56 

6.00 

1.08 

8.34 

3.4 

14.8 

16. 

32.1 

55 

59.5 

175 

106 

.145 

3.05 

.57 

6.06 

1.10 

8.41 

3.5 

15. 

.5 

32.6 

56 

60.0 

200 

114 

.150 

3.11 

.58 

6.11 

1.12 

8.49 

3.6 

15.2 

17. 

33.1 

57 

60.6 

225 

120 

.155 

3.16 

.59 

6.17 

1  14 

8.57 

3.7 

15.4 

.5 

33.6 

58 

61.1 

250 

126 

.160 

3.21 

.60 

6.22 

1.16 

8.64 

3.8 

15.6 

18. 

34.0 

59 

61.6 

275 

133 

.165 

3.26 

.61 

6.28 

1.18 

8.72 

3.9 

15.8 

.5 

34.5 

60 

62.1 

300 

139 

.170 

3.31 

62 

6.32 

1  20 

8.79 

4. 

16.0 

19. 

35.0 

61 

62.7 

350 

150 

.175 

3.36 

.63 

637 

1.22 

8.87 

.2 

16.4 

.5 

35.4 

62 

63.2 

400 

160 

.180 

3.40 

.64 

6.42 

1.24 

8.94 

.4 

16.8 

20. 

35.9 

63 

63.7 

450 

170 

.185 

3.45 

.65 

6.47 

1.26 

9.01 

.6 

17.2 

.5 

36.3 

64 

64.2 

500 

179 

.190 

3.50 

.66 

6.52 

1.28 

9.08 

.8 

17.6 

21. 

36.8 

65 

64.7 

550 

188 

.195 

3.55 

.67 

657 

1.30 

9.15 

5. 

17.9 

.5 

37.2 

66 

65.2 

600 

197 

.2*0 

3.59 

.68 

6.61 

1  .32 

9.21 

.2 

18.3 

22. 

37.6 

67 

65.7 

700 

212 

.21 

3.68 

.69 

6.66 

1.34 

9.29 

.4 

18.7 

.5 

38.1 

68 

66.2 

800 

227 

.22 

3.76 

.70 

6.71 

1.36 

9.36 

.6 

19. 

23. 

38.5 

69 

66.7 

900 

241 

.23 

3.85 

.71 

6.76 

1.38 

9.43 

.8 

19.3 

.5 

38.9 

70 

67.1 

1000 

254 

.24 

3.93 

.72 

6.81 

1.40 

9.49 

6. 

19.7 

24. 

39.3 

71 

67.6 

.25 

4.01 

.73 

6.86 

1.42 

9.57 

.2 

20.0 

.5 

39.7 

72 

68.1 

.26 

4.09 

.74 

6.91 

1.44 

9.63 

.4 

20.3 

25 

40.1 

73 

68.5 

.27 

4.17 

.75 

6.95 

1.46 

9.70 

.6 

20.6 

26 

40.9 

74 

69.0 

M 

i  •>•» 

7fi 

K  .flfl 

1.48 

»  77 

.8 

20.9 

27 

41.7 

75 

69.5 

HYDRAULICS. 


553 


Art.  ,*.  On  the  flow  of  water 
through  vertical  openings  fur- 
nished with  short  tubes.  When  water 

flows  from  a  reservoir,  Fig  6,  through  a  vert  partition 
mm  a  a,  the  thickness  a  TO  of  which  is  about  2^  or  3  times 
the  least  transverse  dimension  of  the  opening,  (whether 
that  dimension  be  its  breadth,  or  its  height;}  or  when,  if 
the  partition  be  very  thin,  as  n  n,  the  water  Hows  through 
a  tube,  as  at  t,  the  length  of  which  is  about  2  or  3  times  its 
least  transverse  dimension,  then  tbe  effluent  stream  wll 
entirely  fill  the  opening,  or  the  tube,  as  shown  in  Fig  6 ;  or, 
in  technical  language,  will  run  with  a  full  flow ;  or  a  full 

bore;  aud  will  disch  more  water  in  a  given  time,  than  if  — ^ ^~ 

the  tube  were  either  materially  longer  or  shorter.     For  if       77^.      />     JL 

longer  than  3  times  the  least  transverse  dimension,  the     Jj  \\\  |) 

How  will  be  impeded  by  the  increased  friction  against  the        "      v 

sides  of  the  tube  ;  and  if  shorter  than  about  twice  the  least 

transverse  dimension,  the  water  will  not  flow  in  a  full  stream,  but  in  a  contracted  one,  as 

Fig  11,  p  554.     This  will  be  the  case  whether  the  tube  be  circular,  or  rectilinear,  in  its  cross 


shown  by 
•section. 


To  find  approximately  the  actual  vel.  and  disch  into  the 
air,  through  a  tube,  or  opening,  either  circular  or  recti- 
linear in  its  outline,  or  cross-section ;  and  whose  length  c  i, 
or  c  e+  in  the  direction  of  the  flow,  is  about  2]4  or  •*  times  its 
least  transverse  dimension  ;  when  the  surface-level.  >-,  Fig  6, 
remains  constantly  at  the  same  height;  and  which  height 
must  not  be  below  the  upper  edge  of  the  tube,  or  opening. 

RULE  1.  Take  out  the  theoretical  vel  from  Table  10,  p  552.  corresponding  to  the  bead  measured  vert 
from  the  center  (or  more  properlv,  the  cen  of  grav)  c,  of  the  opening,  to  the  level  water  surf  «.  Mult 
it  by  the  coeff  of  disch  .81.  The  prod  will  be  the  reqd  vel,  in  ft  per  sec.  Mult  this  actual  vel  by  the 
transverse  area  of  the  opening,  in  sq  ft.  If  circular,  knowing  its  diam.  this  area  will  be  found  in 
Table  3,  p  541.  The  prod  will  be  the  quantity  of  water  dischd,  in  cub  ft  per  sec ;  within,  probably,  3 
or  4  per  cent. 

RULE  2.  Find  the  sq  rt  of  the  head  in  ft.  Mult  this  sq  rt  by  6.5.  The  prod  will  be  the  actual 
vel  in  ft  per  sec. 

Ex.  An  opening  c  o  ;  or  box-shaped  tube  c  f,  Fig  6,  is  3  feet  wide,  by  .25  of  a  ft  high  ;  and  its  length 
in  the  direction  cior  e  e  in  which  the  water  flows  is  about  .62  of  a  ft,  or  about  2^  time.*  its  least 
transverse  dimension,  or  its  height.  The  head  from  the  cen  of  grav  c,  of  the  opening,  to  the  constant 
surf-level  *,  is  4  feet.  What  will  be  the  vel  of  the  water;  and  how  much  will  be  dischd  per  sec? 

By  Ride  1.  The  theoretical  vel  (Table  10.  p  552,)  corresponding  to  a  head  of  4  ft  is  16  ft  per  sec. 
And  16  X  .81  =  12.96  ft  per  sec,  the  actual  vel  reqd.  Again,  the  transverse  area  of  the  opening,  or  of 
the  tube,  is  3  ft  X  .25  ft  =  .75  sq  ft.  And  .75  X  12.96  =  9.72  cub  ft ;  the  quantity  dischd  per  sec. 

By  Rule  2.  The  sq  rt  of  4  is  2.  And  2  X  6.5  =  13  ft  per  sec,  the  reqd  vel,  as  before ;  the  very  slight 
diflf  being  owing  to  the  omission  of  small  decimals  in  the  coeffs. 

•  of  the  vert 

t  case,  use  .71 
or  .7  instead  of  the  .81  of  Rule  1  ;  or  5.7  instead  of  the  6.5  of  Rule  2. 

REM.  2.  When  the  thickness  a  TO  of  the  vert  partition  m  m  a  a  ;  or  the  length  c  e  of  the  tube  t.  Fig 
6,  is  increased  to  about  4  times  the  least  transverse  dimension  of  the  opening  ;  or  of  the  diam.  when 
circular;  then  the  additional  friction  against  its  sides  begins  appreciablv  to  lessen  the  vel  and  disch. 
In  that  case,  or  for  still  greater  lengths,  up  to  100  diams,  they  may  be  found  approximately,  by  using 
instead  of  the  coeff  of  disch  .81  in  Rule  1,  the  following  coeffs,  by  which  to  mult  the  theoretical  veil 
of  Table  10,  p  552.  Or  use  Rule,  p  538. 

TABLE  11.  See  Caution,  p  566. 


REM.  1.    If  the  short  tube  f  projects  partly  inside  of 
partition  n  n,  the  disch  will  be  diminished  about  ys  part.    In  that  ( 


Length  of 
Pipe 
in  Diams. 

Coeff. 

Length  of 
Pipe 
in  Diams. 

Coeff. 

4 

.80 

40 

.62 

6... 

...  .76 

60... 

...  .60 

10 

.74 

60 

.57 

15... 

...  .71 

70... 

...  .55 

20 

.69 

80 

.52 

25... 

...  .67 

90... 

...  .50 

30 

.65 

100 

.48 

•  tube,  in  the  direction  in  which  the  water  flows,  becomes 
icnsion,  the  disch  is  diminished  ;  so  that  for  lengths  from 
plate,  we  may  use  .61,  instead  of  the  .81  of  Rule  1.  For 


RKM.  3.  When  the  length  of  the  opening 
less  than  about  twice  its  least  transverse  d 
IJiJ  times,  down  to  openings  in  a  very  thi 
such  openings,  however,  see  Arts  9  and  10. 

REM.  4.  But  on  the  other  hand,  the  disch  through  such  short  openings  and  tubes  as  are  shown  In 
Fig  6.  may  be  increased  to  nearly  the  theoretical  ones  of  Table  10,  by  merely  rounding  off  neatly  the 
edges  of  the  entrance  end  or  mouth,  as  iu  Fig  7;  which  is  the  shape,  and  half  actual  size  of  one  with 
which  Weisbach  obtained  .975  of  the  theoretical  vel  and  discharge,  when  the  head  was  10  ft;  and  .95* 


554 


HYDRAULICS. 


with  a  head  of  oue  foot;  so  that  in  similar  cases,  .975,  and  .958  may  be  used  instead  of  the  coeff  .81 
in  Rule  1. 


As  much  as  .92  to  .94  may  be  obtained  by  widening  the  opening,  m  n,  toward  its  outer  mouth,  o  », 
Fig,  8,  making  the  divergence,  or  angle  a.  about  5°:  or  by  widening  it  toward  its  inner  mouth,  as  at 
t  c.  Fig  9;  but  increasing  the  angle  of  divergence,  at  b,  to  from  11°  to  16°.  In  all  cases,  we  consider 
the  small  end  as  being  the  opening  whose  area  must  be  multiplied  by  the  vel  to  get  the  discharge. 

In  some  experiments  made  with  large  pyramidal  wooden 
troughs  9.5  ft  long,  with  an  inner  mouth  of  3.2  X  2.-*  ft,  and  a  discharging  one 
of  fi'2  X  .4*  ft;  and  under  a  head  of  9J£  feet,  the  discharge  was  .98  of  the  theoretical  oue.  due  to  the 
smaller  end.  Therefore,  .98  may  be  used  in  such  cases,  instead  of  the  .81  of  Rule  1. 

KKM.  5.    The  discharge  through   a  short  opening  of  small 
transverse  section  may  even   be  made  5O  per  cent  greater 
than  the  theoretical  one,  by  adopting  the  shape,  Fig  10;  where  m  n  is  sup- 
posed to  be  the  diameter  of  the  opening. 

_  The  best  proportions  appear  to  be  about  as 

.  follows:  oy  — 9  inches;  mn~linch;  be 

K   -m ____ J)_  =1.8  inch;  o*=X  inch  ;/<  d~2  ins;  the 

1\. PltsT"       curves,  a  m,  and  d  n,  beiug  quadrants; 

DC  -«s!;:;f;::::----4H--J-S [V^^       the  angle,  z,  of  divergence,  about  5°  6' ; 

' ""IX*- F==^          and  the  tube  of  polished  metal.     In  this 

t \  V  IL  ~ £§§5:5.        case  use  1.55,  or  more  safelv,  1.5.  instead 

rl6  10  v7*^        of  the  .81  of  Rule  1.     The  only  experi- 

-•vlgj  lvr  merits  with  this  form  have  been  on  a  very 

small  scale.     To  what  extent  it  may  be 
applicable  is  unknown. 

So  far  as  regards  the  ordinary  operations  of  the  engineer,  this  subject  is  perhaps  more  curious  than 
useful ;  for  he  will  rarely  have  any  difficulty  in  making  his  openings  large  enough,  without  resorting  to 
such  aids  ;  except,  perhaps,  that  of  a  rounding  off  the  inner  edges,  as  in  Fig  1 ;  whicii  is  usually  done. 

Art.  9.  On  the  disch  of  water  through  openings  in  thin 
vert  partitions,  with  plane  or  flat  faces,  e  e,  or  n  n,  Fig  11.*  If  the 

face  ee,  or  n  n,  instead  of  being  plane,  and  vert,  should  be  curved, 
or  inclining  in  diff  directions  toward  the  opening,  then  the  disch 
will  be  altered.  When  water  flows  from  a  reservoir.  Fig  11,  through 
a  vert  plane  plate  or  partition  nn,  which  is  not  thicker  than  about 
the  least  transversedimension  of  the  opening,  whether  thatdimension 
be  its  breadth,  or  its  height  o  o;  t  or  when,  if  the  partition  e  e  itself 
is  much  thicker,  we  give  the  opening  the  shape  shown  at  b.  (which 
evidently  amounts  to  the  same  thing.)  then  the  effluent  stream  will 
not  pass  out  with  &  full  flow,  as  in  Fig  6,  but  will  assume  the  shape 
shown  in  Fig  11;  forming,  just  outside  of  the  opening,  what  is 
called  the  vena,  contracta,  or  contracted  vein.  In  order  that  this 
contraction  may  take  place  to  its  fullest  extent,  or  become  complete, 
G  H  the  inner  sharp  edges  of  the  opening  must  not  approach  either  the 

.__          .  .  surf  of  the  water,  or  the  bottom  or  sides  of  the  reservoir,  nearer 

\\lfi   11  than  about  1^  times  the  least  transverse  dimension  of  the  opening. 

The  contracted  vein  occurs  at  a  dist  of  about  half  the  smallest  di- 
mension of  the  orifice,  from  the  orifice  itself.     In  a  circular  orifice. 

at  about  half  the  diam  dist;  and  ordinarily  its  area  is  about  .62  or  nenrly  %  that  of  the  orilice  itself. 
At  this  point  the  actual  mean  vel  of  the  stream  is  verv  nearly  (abont  .97)  the  theoretical  vel  given  by 
Table  10,  p  552  ;  and  hence  the  actual  dischs  are  but  ".62,  or  nearly  %  of  the  theoretical  ones. 

Case  1.  To  find  the  actual  disch  into  air.J  through  either  a 
circular  or  rectilinear^  opening  in  a  thin  vert  plane  parti- 


*We  believe  that  these  rules  for  thin  plate  are  also  sufficiently  approximate 
for  most  practical  purposes,  if  the  opening  be  in  the  bottom  of  the  reservoir; 
or  in  an  inclined,  instead  of  a  vert  side. 

t  When  the  side  of  a  reservoir,  or  the  edge  of  a  plank.  &c.  over  which  water 
flows,  has  no  greater  thickness  than  this,  the  water  is  said  to  flow  through, 

or  over,  thin  i>lato.  or  thin  partition. 

J  Should  the  disch  take  place  under  water,  as  in  Fig  12.  both  surf -levels  re- 
levels.  After  making  the  calculation  with  this  head,  we  should,  according  to 
Weisbach,  deduct  the  7-3-  part;  inasmuch  as  he  states  that  the  disch  is  that 
much  less  when  under  water,  than  when  it  takes  place  freely  into  the  air. 
Other  experimenters,  however,  assert  that  it  is  precisely  the  same  in  both  cases. 

§  If  the  shape  of  the  opening  is  oval,  triangular,  or  irregular,  the  head 
must  be  measured  vert  from  its  cen  of  gray. 


HYDRAULICS. 


555 


fioii,  when  the  contraction  is  complete;  and  when  the  surf- 
level,  »,  remains  constantly  at  the  same  height;  water  being- 
supplied  to  the  reservoir  as  fast  as  it  runs  out  at  the  open- 
in^.* 

RULE  1.  When  the  head,  measured  vert  from  the  center  (or  rather  from  the  cen  of  grav)  c,  of  the 
opeuiug,  to  the  surf  level  a  of  the  reservoir,  is  not  less  than  1  ft.  nor  more  than  10  ft ;  and  when  the 
least  transverse  dimension  of  the  opening  is  not  less  than  an  inch,  mult  the  theoretical  vel  in  ft  per 
sec  due  to  the  head,  (Table  10,  p  552,)  by  the  coeff  of  disch.  62.  (See  Kern  2.)  The  prod  will  be  the 
actual  mean  vel  of  the  water  through  the  opening.  Mult  this  vel  by  the  area  of  the  opening  in  sq 
ft;  the  prod  will  be  the  discb  in  cub  ft  per  sec,  approximately. 

When  the  head  is  greater  than  10  ft,  use  .6,  instead  of  .62. 

RULE  2.  Fiud  the  sq  rt  of  the  bead  in  ft.  Mult  this  sq  rt  by  5 ;  the  prod  will  be  the  vel  in  ft  per 
sec ;  which  mult  by  the  area  as  before  for  the  disch. 

Ex.  What  will  be  the  disch  through  an  opening  in  complete  contraction,  whose  dimensions  are  6 
ins,  or  .5  ft  vert ;  and  4  ft  hor ;  the  vert  head  above  the  ceu  of  grav  of  the  opening- being  constantly 

By  Rule  1.  The  theoretical  vel  (Table  10,  p  552)  corresponding  to  6  ft  head,  is  19.7  ft  per  sec.  And 
19.7  X  .62  —  12.214  ft,  the  reqd  vel.  Again,  the  area  of  the  opening  —  .5  X  4  —  2  sq  ft;  and  12  214  X 
2  -  24.428  cub  ft  per  sec ;  the  disch. 

By  Rule  2.  The  sq  rt  of  6  =  2.45 ;  and  2.45  X  5  -  12.25  ft  per  sec,  the  reqd  vel ;  and  12.25  X  2  = 
24.5  cub  ft  per  sec,  the  disch. 

Both  very  approx  even  if  the  orifice  reaches  to  the  surface  of  the  issuing  water. 

Rem.  1.    The  coef  .62  is  a  mean  of  results  of  many  old  experimenters. 

In  1874  Genl.  T.  G.  Ellis  of  Massachusetts  conducted  an  elaborate  series  (Trans  Am  Soc  C  E,  Feb 
1876)  on  a  large  scale,  the  general  results  of  which,  within  less  than  1  per  ct,  are  given  in  the  follow- 
ing table.  See  also  Rem  3.  The  sharp  edged  orifices  were  in  iron  elates  .25  to  .5  inch  thick. 


Orifice. 

2  ft  sq. 
2  "lonK,  1  ft  high 
2  "  long,  .5  high 
2  "  diam. 

Head  above  Center. 

2.    to    3.5  ft. 
1.8  to  11.3" 
1.4  to  17.0  " 
1.8  to    9.6" 

Coef. 

.60  to  .61 
.60  to  .61 
.61  to  .60 
.59  to  .61 

Rem.  2.    Extreme  care  is  reqd  to  obtain  correct  results;  but  for  many 

purposes  of  the  engineer  an  error  of  5  to  10  per  ct  is  unimportant. 

REM,  2.     It  will  rarely  happen  that  greater  accuracy  Is  reqd  than  that  obtained  by  the  foregoing 
rules;  but  when  such  does  occur,  aid  may  be  derived  from  the  following    table    deduced 

from  the  experiments  of  Lesbros  and  Poncelet,  on  openings  8  ins 

wide,  of  diff  heights,  and  with  diff  heads.  Use  that  coeff  in  the  table  which  applies  to  the  case,  in- 
stead of  the  .62  of  Rule  1.  In  some  of  the  cases  in  this  table,  the  upper  edge  of  the  opening  is 
nearer  the  surf-level  of  the  reservoir  than  1>£  times  its  least  transverse  dimension. 

TABLE  12.     Coefficients  for  rectangular  openings  in  thin 
vertical  partitions  in  full  contraction.* 


Head 

Head 

The  breadth  in  all  the  openings  rr  8  inches. 

above  cen. 
of  grav.  of 
opening 

above  cen. 
of  grav.  of 
opening 

Ins. 

H 

Ins. 

EIGHT 

OF  OPENIN 

Ins.     1     Ins. 

GK 
Ins.     I    .Ins. 

in  Feet. 

in  Inches. 

8 

6 

4 

3               2 

1         1       .4 

033 

4 

70 

0666 

8 

65 

69 

0833 

64 

68 

125 

\y>. 

61 

64 

68 

1666 

2 

60 

62 

64 

68 

2083 

2i£ 

59 

.61 

62 

64 

67 

250 

8 

60 

61 

62 

64 

67 

2917 

31^ 

57 

60 

61 

62 

64 

66 

3333 

•PI 

58 

60 

61 

63 

64 

66 

.3750 

4% 

.56 

.59 

.60 

.61 

.63 

.64 

.66 

.4167 

5 

.57 

.59 

.61 

.62 

.63 

.64 

.66 

.6666 

8 

.59 

.60 

.61 

.«2 

.63 

.64 

.65 

1 

12 

.60 

.60 

.61 

.62 

.63 

.63 

.64     1**<J«' 

3 

86 

.60 

.60 

.61 

.62 

.62 

.63 

.63 

5 

60 

.60 

.60 

.61 

.61 

.62 

.62 

.62 

10 

120 

.60 

.60 

.60 

.60 

.60 

61 

.61 

REM.  3.    Careful  experiments  on  openings  4}4  ft  wide,  and  18 

ins  high,  under  heads  of  from  6  to  15  ft,  show  that  the  coeff  .6*2  will  give  results 
correct  within  -i-  part,  for  openings  of  that  size  also,  tinder  large  heads:  although  the  thickness  of 
*he  partition  varied  on  its  diff  sides,  from  12  to  20  ins.  It  must  be  recollected,  however,  that  nothing 
more  than  close  approximations  are  to  be  attained  in  such  matters. 

RRM.  4.  It  has  been  asserted  by  some  writers,  that  when  two  or  more 
contiguous  openings  are  rlisehargino;  at  the  same  time  from  the  same  nser- 
voir,  they  disch  less  in  proportion  than  when  onlv  one  of  them  is  open.  Other  experiments,  how- 
ever, seem  to  show  that  this  is  not  the  case;  it  is  therefore  probable,  at  least,  that  the  diff,  if  any, 
ia_birt  trifling. 

#  S«c  first  footnote  on  preceding  pag«. 


556 


HYDRAULICS. 


m 


Cast  2.  The  discharge  through  thin  vert  partitions  in  com- 
plete contraction,  when  the  surface-level,™,  Fig  13,  descends 
as  the  water  flows  out  into  the  air*  In  this  case,  if  the  reservoir  ia 

voir,  to  supply  the  plac  •  of  that  which  flows  out,  then,  to  find  the  time  reqd  to  disch  the  reservoir. 

RULK.  Inasmuch  as  the  time  in  which  such  a  reservoir  entirely  discharges  itself,  is  twice  that  ia 
which  the  same  quantity  would  flow  out  uuder  a  constant  head,  as  in  Case  1,  p  554;  therefore,  cal- 

tained  in  the  reservoir,  above  the  level  g  of  the  bottom  of  the  opening,  Fig  13,  by 
this  disch  ;  the  quot  will  be  the  number  of  sec  in  which  a  volume  equal  to  that  in 
the  reservoir,  to  the  depth  g.  would  run  out  iu  Case  1,  of  a  constant  head.  And 
twice  this  number  will  be  the  seconds  reqd  to  empty  the  reservoir  ia  Case  2,  of  a 
varying  head. 
REM.  If  it  should  be  reqd  to  find  the  time  in  which  such  a  prismatic  reservoir 

the  above  rule,  the  sees  necessary  to  empty  it  if  it  had  only  been  filled  to  n;  and 
afterward  calculate  as  if  it  had  been  filled  to  m.  The  diff  between  the  two  times 
will  evidently  be  the  time  reqd  to  empty  it  from  m  to  n.  If  the  opening  is  uot  iu 
complete  contraction,  see  Arts  11,  <fec. 

If  the  disch   is  into  a  lower  reservoir,  whose 
17  £  \  0        surf-level  remains  constant,  proceed  in  the  same  manner; 
-l-lv  Itj         only  use  the  diff  of  level  of  the  two  surfs  as  tue  head,  and  afterward  (according 
to  Weisbach)  increase  the  time  ^V  part. 

Art.  1O.  Disch  from  a  reservoir  II,  Fig  14,  the  surf-level,  #, 
of  which  remains  constantly  at  the  same  height;  tiirough 
an  opening,  o,  in  thin  vert  partition;  and  in  complete  con- 
traction;  but  entirely  under  water;  and  into  a  prismatic 
reser%'oir,  m,  in  which  the  surf-level  rises,  as  the  water  flows 
into  it  through  o. 

'  .„  To  find  the  time  reqd  to  fill  the  reservoir  m,  from  any  level  c,  above 

S  j^^j^Mg^  fa  the  top  of  the  opening,  to  any  upper  level,  d. 

-.  3  RULE.     First  find  the  area  iu  sq  ft.  of  a  hor  section  of  the  reservoir 

(1  %  m,  which  is  supposed  to  be  of  uniform  section  throughout  its  depth. 

-n        a^===  Mult  together  this  area,  the  coustaut  number  2,  and  the  sq  rt  of  the 

l£         ^      "  vert  height  oc  in  ft.     Call  the  prod  p.     Mult  together  the  area  of  the 

13  -  opening  o,  in  sq  ft;  the  coeff  of  contraction  (usually  about  .62.  whether 

the  disch  be  into  the  air,  or  uuder  water:)  and  the  constant  8.02.     Call 
the  prod  y.     Div  p  by  y.     The  quot  will  be  the  reqd  time  in  sees. 

Ex.  Let  the  hor  dimensions  of  the  reservoir  m  be  10  ft  by  20  ft;  and 
its  area  consequently  200  sq  ft.  I>et  the  dimensions  of  the  opening  o,  be 
2  ft  by  3  ft;  making  its  area  6  sq  ft.  Now,  c  being  above  the  top  of  the 
opening;  and  cd  being  16  ft;  the  sq  rt  of  which  is  4;  how  long  will  the 
reservoir  m  be  in  filling  from  c  to  of? 
Here  we  have p=  200  X  2  X  4  =  1600 ;  and  y  =  6  X  -62  X  8.02  =  29.83. 

And :  =  53.6  sec,  the  time  reqd. 

REM.  I.  If  it  should  be  reqd  to  find  the  time  of  filling  m,  from 
its  bottom  c,  up  to  rf.  we  may  do  so  very  approximately  by  calculating  by 
the  first  rule  in  Art  9.  the  time  reqd  from  e  to  the  center  of  the  opening  o,  as  if  all  that  portion  of 
the  disch  took  place  into  air;  and  afterward,  from  the  center  of  the  opening  to  d.  by  the  rule  just 
given.  This  case  is  similar  to  that  of  filliug  a  lock  from  the  canal  reach  above,  in  which  the  surf- 
level  mav  be  considered  constant. 

REM.  1  If  the  bottom  of  the  opening  o.  should  coincide  with 
the  bottom  of  the  reservoir,  then  the  coeff  will  become  greater  than  .62. 
See  Art  11,  for  obtaining  coeffs  for  imperfect  contraction. 

R,;M.  3.  If  the  opening,  instead  of  being  in  complete  con- 
traction, ia  of  any  of  the  shapes  Figs  6  to  9,  then  a  reference  to  Art  8  will  show 
what  coeff  must  be  substituted  for  .62. 

Gisr.  3.  Uisch  from  one  prismatic  reservoir,  Fig  15,  W,  into 
another,  X,  of  any  comparative  sizes  whatever,  through  an 
opening  o,  in  a  plane  thin  vert  partition,  and  in  complete 
contraction;  when  the  water  rises  in  X.  while  it  falls  in  W. 

To  find  the  time  in  which  the  water,  flowing  from  W  into  X,  through 
o,  will  fall  through  the  dist  a  s,  so  as  to  stand  at  the  same  level  a  c.  in 
both  reservoirs. 

In  this  case,  the  water  reqd  to  fill  X  from  e  to  d,  (d  being  the  bottom 
of  the  opening  o.)  flows  out  into  the  air;  and  the  time  necessary  for  it 
to  do  so,  must  be  calculated  separately  from  that  reqd  above  d,  which 
flows  into  water. 

RULE.  First  from  e  to  d.  Find  the  hor  area  of  each  reservoir,  in 
sq  ft.  Mult  the  hor  area  of  X,  by  the  vert  depth  de  in  ft,  for  the  cub 
ft  contained  in  that  portion.  Div  these  cub  ft  by  the  hor  area  of  W. 
The  quot  will  be  the  dist  am,  in  feet,  through  which  the  water  in  W 
must  descend,  in  order  to  fill  X  to  d.  Now  calculate  by  Rule  1,  Case  1 , 
the  number  of  sees  in  which  W  would  empty  itself  into  the  air,  under 
a  constant  head  equal  to  an;  also  the  time"  in  which  it  would  empty 
itself  into  air,  under  a  constant  head  equal  to  mn.  Take  the  diff  be- 
tween these  two  times.  This  diff  will  be  the  time  in  which  the  water  in 
•n.  ,  Mf  W  would  fall  from  a  to  m ;  and  that  in  X  rise  from  «  to  d ;  under  a  con- 

_TlU  1 J  *t(ini  hea(1  el"*1  to  an.  Mult  thi»  time  by  2.  for  the  time  rcqd  in  the 

O  actual  eas«  b«fere  us,  u»der  a  haad  varying  from  a  w  to  TO  n.  T«  thii 


W 


X 


HYDRAULICS. 


557 


time  we  mast  add  that  still  reqd  for  the  water  to  fall  from  m  to  a;  filling  from  d  to  c.  To  find  tbti 
time,  take  the  square  root  of  the  remaining  head,  inn,  in  feet.  Mult  together,  this  sq  rt ;  the  hor 
area  of  W;  the  hor  area  of  X;  and  the  constant  number  2.  Call  the  prod  p.  Next,  add  together  the 
hor  areas  of  W  and  X.  Mult  together,  the  sum  of  these  areas  ;  the  constant  number  8.02  ;  the  area 
of  the  opening  o,  in  sq  ft;  and  the  coett'  of  disch  of  the  opening  o  (.which  coelf  tor  opening  in  com- 
plete contraction  will  be  about  .62).  Call  the  prod  y.  Divpbyy.  The  quot  will  be  the  additional 
time  reqd  in  sees,  very  approximately. 

Ex.  Let  the  hor  area  of  W  be  100  sq  ft :  and  that  of  X,  60  sq  ft.  Let  an  be  20  ft ;  and  TO  n  16  ft ; 
and  the  area  of  the  opening  o,  '6  sq  ft.  In  what  time  will  the  water  descend  from  a  to  «,  and  rise 
from  e  to  c  ? 

Inasmuch  as  the  method  of  finding  the  time  for  filling  from  e  to  d,  by  the  water  falling  from  a  to 
m,  requires  no  further  exemplification,  we  will  confine  ourselves  to  the  additional  time  necessary  for 
filling  from  if  to  c,  by  the  water  falling  from  m  to  8.  To  find  this,  we  have,  the  sq  rt  of  the  head 

' 


Fig  16 


20.1  sec;  the  additional  time  reqd,  very  approximately. 

NOTE  1.    If  the  opening,  as  </,  Fig  16,  reaches 
to  the  very  bottom  of  the  reservoirs,  we  may 

consider  all  the  water  flowing  from  R  into  T,  as  flowing  into  water. 
Therefore,  using  the  head  am,  we  at  once  calculate  the  time  necessary 
for  the  water  in  the  two  reservoirs  to  arrive  at  the  same  level  s  c,  by 
the  last  process  of  the  preceding  rule ;  or,  in  other  words,  by  the  pro- 
cess given  in  the  preceding  example.  But  in  this  case  it  must  be  borne 
in  mind  that  the  opening  o  is  no  longer  in  complete  contraction,  inas- 
much as  the  contraction  along  its  lower  edge  is  suppressed. 

The  disch  will  consequently  be  somewhat  increased;  and  a  coeff 
greater  than  .62  becomes  necessary.     The  method  of  finding  this,  is 

Siveu  in  the  following  Case  4.  A  reference  to  Art  8  will  give  the  coeff 
i  case  the  opening  is  shaped  as  Figs  6  to  9. 

Art.  11.  Case  4.  The  discharge  through  openings  in  plane 
thin  vert  partitions ;  but  in  incomplete  contraction. 

The  opening  may  be  such  that  contraction  will  take  place 
along  one  portion  of  its  perimeter,  or  at  the  top  of  the  open- 
ing a,  Fig  17  ;  while  it  is  suppressed  on  another  portion ;  as 
at  the  bottom  and  two  ends  of  the  opening  a;  where  suppres- 
sion is  caused  by  the  addition  of  short  side  and  bottom  pieces 
c,  c,  c.  Or  it  may  be  caused  by  the  bottom,  or  ends,  or  both, 
coinciding  with  the  bottom  and  sides  of  the  reservoir.  In 
such  cases  the  disch  will  be  greater  than  in  those  of  complete 
contraction  j  but  less  than  in  those  of  full  flow  ;  inasmuch  as 
the  opening  now  partakes  somewhat  of  the  character  of  the 
short  tubes  of  Art  8 ;  and  the  coeff  will  rise  from  .62,  or  that 
which  usually  pertains  to  openings  in  full  contraction  ;  and 
will  approach  .8,  or  that  of  full  flow,  in  proportion  to  the  ex- 
tent of  perimeter  along  which  contraction  is  suppressed;  or 
even  to  .9  or  .98  by  the  use  of  such  openings  as  are  shown  by 
Figs  7,  8F  9. 

To  find  approximately  a  new  coeff  of  disch;  and  the  disch 
itself,  in  cases  of  incomplete  contraction. 

RULE.  First  find  by  the  foregoing  rules,  what  would  be  the  disch  in  the  particular  case  that  may 
be  under  consideration,  supposing  the  contraction  to  be  complete.  Then  div  that  portion  of  the 
perimeter  of  the  opening  on  which  contraction  is  suppressed,  by  the  entire  perimeter.  Mult  the  quot 
by  the  dec  .152  if  the  opening  is  rectangular,  or  by  .128  if  circular.  To  the  prod  add  unity,  or  1.  Call 
the  sum,  g.  Then  say,  as  unity,  or  1,  is  to  g,  so  is  the  coeff  for  complete  contraction  in  ordinary  cases 
(usually  .62)  to  the  reqd  new  coeff.  Finally,  repeat  the  original  calculation,  only  substituting  this  new 
copff  in  the  place  of  .62. 

According  to  this  rule,  we  have  the  following  coeff  of  discharge  for  rectangular  openings  within  pro- 
bably 3  or  4  per  cent,  when  contraction  is  not  suppressed  on  more  than  %  of  the  perimeter.  The  theo- 
retical discharge  multiplied  by  the  corresponding  coeff  will  give  the  actual  discharge.  When  the  con- 
traction is  carried  farther,  the  coeff  becomes  extremely  irregular,  and  is  probably  indeterminable. 

For  complete  contraction  (ordinarily) 62 

When  contraction  is  suppressed  on  y±  the  perimeter 64 

X  "  "        67 

"  H  "  "        69 

entirely  around  the,  orifice 80 

Intermediate  ones  can  be  estimated  nearly  enough,  mentally. 

KKM.  1.  When,  instead  of  a  short  spont,  as  in  Fig  17,  the 
opening  is  provided  with  an  indefinitely  long  hor  trough, 

similarly  attached,  and  open  at  top,  there  will  be  no  practically  appreciable  diminution  of  disch  below 
that  through  the  simple  opening  as  at  a,  Fig  11 ;  provided  the  head  measured  above  the  cen  of  grav 
of  the  opening  be  at  least  as  great  as  2  or  2V£  times  the  height  of  the  opening  itself.  Therefore,  under 
such  circumstances  the  disch  may  be  calculated  by  the  rules  in  Art  9.  But  with  smaller  heads  the 
disch  diminishes  considerably  ;  so  when  the  head  above  the  center  becomes  but  as  great  as  the  height 
of  the  opening,  it  will  be  but  about  ^  of  the  calculated  one.  With  still  smaller  heads,  the  flow 
becomes  less  much  more  rapidly ;  but  has  not  been  reduced  to  any  rule. 

REM.  2.    If,   instead  of  being  hor,  the  trough  is  ItfCI.IX EI» 

36 


Fig  17 


558 


HYDRAULICS. 


as  much  as  1  in  1O,  the  disch  will  be  increased  very  slightly,  (some  3  or  4  per 

cent)  over  that  calculated  by  the  rules  in  Art  9,  for  the  plain  opening.  These  results  were  obtained 
by  experiments  on  a  very  small  scale  ;  aud  should  be  considered  as  mere  approximations. 

Art.  12.  In  a  case  like  Fig  18,  where  contraction  is  supposed 
to  be  suppressed  at  the  bottom,  and  at  both  vert  sides  of  the 
opening1  o,  in  consequence  of  their  coinciding 
with  the  bottom  aud  sides  of  the  reservoir:  but  where  the 
front  of  the  reservoir,  instead  of  being  vert,  is  sloped  as  at/; 
and  when  the  water,  after  leaving  the  opening,  flows  away 
over  a  slightly  sloping  apron,  y,  then  the  disch  iu  cub  ft  per 
sec  may  be  approximately  found  by  Rule  1,  Case  I,  Art  9, 
only  substituting  .8  in  place  of  .62,  when  /  slopes  back  45°, 
or  i  to  1  ;  or  .74  when  /  slopes  back  63°,  or  with  a  base  of  1 
to  a  rise  of  2.  In  such  cases  of  inclined  fronts,  the  height  of 
the  opening  must  be  measured  vert,  or  rather  at  right  anglet 
to  the  floor  of  the  reservoir  ;  and  not  iu  a  line  with  the 
sloping  front. 

REM.  When  the  front,  /,  of  the  reservoir  is  vert,  and  a  sloping 
apron  or  trough,  gr,  is  used,  having  its  upper  edge  level  with  the  bottom 
of  the  opening,  the  disch  is  not  appreciably  diminished  below  that  which  takes  place  freely  into  t?he 
air,  provided  the  head  above  the  cen  of  grav  of  the  opening  is  not  less  than  from 

18  to  24  ins,  for  an  opening  6  to  9  ins  high. 
12  to  16  "       "     "        «•        4  ins  high. 

9          "     "  2  ins  or  less,  high. 

Art.  13.  To  find,  approximately,  the  time  reqd  for  the  emp- 
tying of  a  pond,  or  any  other  reservoir,  as  Fig  19,  which  is 
not  of  a  prismatic  shape;  through  an  opening,  w,  near  the 
bottom. 

RULK.  First  ascertain  the  exact  shape  and  dimensions 
of  the  reservoir.  If  large,  aud  irregular,  it  must  be  care- 
fully surveyed;  and  soundings  taken,  and  figured  upon  a 
correct  plan  and  cross-sections.  Next,  consider  the  entire 
body  of  water  to  be  divided  into  a  series  of  thin  nor  strata, 
A,  B,  C,  D  ;  the  top  line  of  the  lower  one  being  at  least  a 
few  ins  above  the  top  of  the  opening  n.  It  is  not  necessary 
that  these  strata  should  be  of  equal  thickness;  although 
l^e  tn*aner  tneJ  are«  tne  niore  correct  will  the  result  be. 
The  depth  of  the  lower  one,  D,  will  vary  to  some  extent 
with  the  height  of  the  opening  ;  those  next  above  it  should 
not  exceed  about  a  foot  in  thickness,  until  a  depth  of  6  or  8  feet  is  reached;  then  they  may  conve- 
niently. and  with  sufficient  accuracy,  be  increased  to  about  2  ft,  for  6  or  8  ft  more  ;  and  so  on  ;  be- 
coming thicker  as  they  approach  the  surf.  By  aid  of  the  drawings,  calculate  the  content  of  each 
stratum  in  cub  ft.  Now,  since  the  strata  are  thin,  we  may,  without  serious  error,  assume  each  of 
them  to  be  prismatic,  as  shown  by  the  dotted  lines  ;  and  may  assume  that  the  head  under  which  each 
stratum  (except  the  lowest)  empties  itself  through  n,  is  equal  to  the  vert  height  from  the  center  of 
the  opening  to  the  center  of  the  stratum.  Thus,  m  n  will  be  the  head  of  A  ;  w  n.  the  head  of  B  ;  xn, 
the  head  of  C.  Then,  for  the  stratum  A:  by  Rule  1,  Art  9,  (only  using  mn  as  the  head  instead  of  on,) 
and  instead  of  the  coeff  .62  of  that  rule  (which  can  only  be  used  if  n  is  in  complete  contraction)  using 
.64,  or  whatever  other  coeff  near  the  end  of  Art  11  applies  to  the  case,  calculate  the  disch  in  cub  ft 
per  sec.  Div  the  content  of  the  stratum  A  by  this  disch,  and  the  quot  will  be  the  number  of  sec  reqd 
for  discharging  A.  Using  the  head  wn,  proceed  in  precisely  the  same  way  with  the  stratum  B  .;  and 
using  the  head  xn,  do  the  same  with  C.  Finally,  for  the  lower  stratum  D.  find  by  Rule  1,  Art  9,  (with 
the  same  caution  as  before  respecting  the  proper  coeff,)  in  what  time  it  would  empty  itself  under  a 
constant  head  equal  to  y  n,  measured  from  its  surf  to  the  center  of  the  opening.  Double  this  time  will 
be  that  reqd  to  empty  itself  in  the  case  before  us.  under  its  varying  head.  Finally,  add  together  all 
these  separate  times  ;  and  their  sum  will  be  the  entire  time  reqd  to  empty  the  pond,  or  reservoir,  ap- 
proximately enough  for  practical  purposes. 

Art.  14.    On  the  discharge  of  water  over  weirs,  or  overfalls. 

Experiments  and  observations  on  a  grand  scale,  were  made  on  this  subject  at  Lowell,  Mass,  by 
Mr  James  B.  FrailClS,  C  E,  one  of  the  most  accomplished  hydraulicians  of  the  age.  (See 
his  "  Lowell  Hydraulic  Experiments.")  To  apply  the  rule  arrived  at  by  Mr  Francis,  the  following 
conditions  must  exist. 

The  crest  a,  Fig  20,  or  top  of  the  weir  over  which  the  water  discharges,  must  be  a  hor.  sharp  cor- 
ner, in  thin  plate,  or  thin  partition.  (See  p554;  first  footnote.)  Its  inner  side  must  form  a  vertical 
straight  line,  ah.  with  the  inner  face  of  the  dam,  to  a  depth  aft,  not  less  than  twice  the  depth,  or 
head  am,  measured  vert  from  a  to  the  level  om  of  the  hor  portion  of  the  water  surf:  and  not  to  c,  the 
curved  surf  of  the  falling  sheet  of  water.  The  head  am  may  vary  from  6  to  24  ins  in  height.  The 
ends  ah,  ah,  of  the  weir,  Figs  21  and  22,  must  be  vert;  and  its  length,  a,  a,  not  less  than  3  times  the 

These  conditions  being  observed,  we  may  distinguish  2  cases  ;  namely.  Case  1st,  that  in  which,  as 
in  Fig  21,  the  weir  extends  entirely  across'  the  reservoir:  so  that  its  ends  ah.  al>,  coincide  with,  or 
form  portions  of,  the  sides  s.  s.  of  the  reservoir  :  in  which  case  contraction  takes  place  only  along  the 
upper  edge  a,  a,  of  the  weir,  Fig  22,  as  shown  at  a  in  Fig  20;  but  is  suppressed  entirely  at  the  ends. 
so  that  the  water  flows  out  as  shown  in  plan  by  Fig  23.  And  Case  2d,  in  which,  as  in  Fig  22,  the 
vert  ends  ah,  ah.  as  well  as  the  crest  a,  a,  are  formed  with  a  sharp  corner  in  thin  plate;  and  are, 
moreover,  removed  from  the  sides  v.  t>,  of  the  reservoir,  a  dist  equal  at  least  to  the  head  am;  so  that 
contraction  takes  place  at  the  ends  of  the  weir,  as  well  as  along  its  crest;  and  less  water  flows  out, 
»s  shown  in  plan  at  a,  a,  Fig  24. 


__ 

JLlfl    19 


HYDKAULICS. 


559 


To  find  the Htlisch  over  a  weir  in  thin  plate,  when,  as  in  Figs 
21  and  23,  tKere  is  no  contraction  at  its  two  ends. 


a 

a 

RULE.  Find  the  cube  of  the  head  am.  Fig  20.  in  ft.  Take  the  sq  rt  of  this  cube.  Mult  together 
this  sq  rt;  the  length  a,  a.  of  the  wtir,  in  ft;  and  the  constant  uuuiber  3.33.  The  prod  will  be  the 
quantity  disch,  in  cub  ft  per  sec.  Or,  in  shape  of  a  formula, 

The  disch,  in    _   the,  sq  rt  of  the  cube  of    v   the  length  a  a.  of  v   const  number 
cubftpersec,   ~       the  head  am,  in  ft       A    the  over/all,  in  ft   A  3.33. 

Ex.    How  many  cub  ft  per  sec  will  flow  over  such  a  weir  in  thin  plate.  200  ft  long:  having  a  head, 
a  TO.  of  1.5  ft,  measured  to  the  level  suit  o  w*.of  the  reservoir;  and  with  no  contraction  at  either  end? 
Here,  the  cube  of  1.5  =  3  375.     Aud  the  sq  rt  of  3.375  =  1.837.     And  1.837  X  200  X  3.33  =  1223.4 
cub  ft  per  sec,  the  reqd  disch.* 

This  rule  will  also  be  very  approximate  even  when  there 
is  contraction  at  both  ends,  provided  the  length  of  the  wreir 
is  at  'least  1O  times  as  great  as  the  head  a  >n •  and  provided  the 

head  is  not  less  than  2  or  3  ins  in  depth.  Indeed,  it  will  be  within  about  6  per  cent  of  the  truth,  for 
weirs  with  contraction  at  the  ends,  and  whose  lengths  sire  but  4  times  the  head  ;  and  for  the  mnny 
cases  in  which  no  closer  approximation  is  reqd,  the  disch  may  betaken  at  once  from  the  following  table. 

To  find  the  discharge  over  a  weir  in  thin  plate;  when 
contract  ion  takes  place  at  both  ends,  as  shown  by  Figs 
22  and  24. 

RULE.  Proceed  precisely  the  same  as  in  Rule  for  Case  1 ;  except  that  when  there  is  contrac- 
tion at  both  ends,  i  part  of  the  head  am  is  to  be  taken  from  the  length,  before  using  it  as  a  multi- 
plier ;  and  when  there  is  contraction  at  one  end  only,  A,  part  of  a  TO  must  first  be  taken  from  the 
length  a  a. 

Ex.  How  many  cub  ft  per  sec  will  flow  over  such  a  weir  in  thin  plate,  200  ft  long;  having  a  head 
cm,  of  1.5  ft;  with  contraction  at  both  ends? 

Here,  the  cube  of  1.5  =  3.375.  And  the  sq  rt  of  3.375=  1.837.  Again,  ^  of  a  m  =  .3  of  a  ft.  And 
200  —  .3  =  199-7.  Therefore  we  have  1.837  X  199.7  X  3.33  =  1221.6  cub  ft  per  sec,  the  reqd  disch  ;  or 
practically  the  same  as  when  there  is  no  contraction,  in  this  case. 

REM.  If  instead  of  3.33,  we  use  3.41,  the  two  foregoing  rules  will  apply  to  heads  n  m.  as  small  as 
H  HI>  inch  ;  and  coeffs  between  3.33  and  3.41  may  be  used  for  heads  between  about  5  inches,  and 
%  an  inch,  where  more  than  common  accuracy  is  aimed  at.  We  may  also  use  3.3  instead  of  3.33,  for 
heads  greater  than  2  feet. 

*  Eytelwein's  rule  for  weirs,  over  a  thin  edge:   and  extending 

Mult  togothor.  the  sq  rt  of  the  head  :  the  h^nd  iMM' :  the  leneth  of  overfall :  (all  in  feet  ;)  and  the 
constant,  number  3.4.  The  prod  will  be  tho  disoh  in  cnh  ft  ppr  «PC.  It  givps  1249.  instead  of  the  above 
1223  cub  ft,  The  sq  rt  of  the  he»d  is  1.226;  and  1.225  X  1.5  X  200  X  3.4  =  1249.  The  rules  are  in 
fact  identical,  except  in  the  coefficients. 


560 


HYDRAULICS. 


TABLE  13.  Of  actual  discharges  in  ciib  ft  per  see,  for  each 
foot  in  length  of  weir  in  thin  plate:  and  without  contraction 
at  either  end ;  a  />,  Fig:  2O.  being  vert,  and  not  less  than  twice 
the  head  am.  Very  approximate  also,  when  there  is  con- 
traction at  both  ends,  provided  the  length  be  at  least  1O 
times  the  head.  And  but  about  6  per  cent  in  excess  of  the 
truth,  if  the  length  be  but  4  times  the  head.  (Original.) 

The  decimal  .01  of  a  foot,  is  precisely  .12  of  an  inch  :  or  scant  %  inch. 


Head, 
am,  in 

Feet. 

Cub.  ft. 
per 
Second. 

Head, 
am,  in 
Feet. 

Cub.  ft. 
per 
Second. 

Head, 

a  m,  in 
Feet. 

Cub.  ft. 
per 

Second. 

Head, 
am,  in 
Feet. 

Cub.  ft. 
per 

Second. 

Head, 
am,  in 

Feet. 

Cub.  ft. 
pt^r 
Second. 

.03 

.OH 

.22 

.351 

.58 

1.47 

.94 

3.04 

2.4 

12.2 

.04 

.OJ7 

.24 

.401 

.60 

1.54 

.96 

3.14 

2.5 

13.0 

.05 

.038 

.26 

.452 

.62 

1.62 

.98 

3.21 

2.6 

13.8 

.06 

.050 

.2S 

.505 

.64 

1.70 

1. 

3.^3 

2.7 

14.6 

.07 

.063 

.30 

.560 

.66 

1.78 

1.1 

3.85 

2.8 

15.4 

.08 

.077 

.32 

.603 

.68 

1.86 

1.2 

4.38 

2.9 

16.2 

.09 

.092 

.34 

.659 

.70 

1.95 

1.3 

4.94 

3. 

17.1 

.10 

.108 

.36 

.719 

.72 

2.03 

1.4 

5.51 

3.1 

18.0 

.11 

.124 

.38 

.78J 

.74 

2.12 

1.5 

6.11 

3.2 

18.9 

.12 

.142 

.40 

.842 

.76 

2.21 

1.6 

6.73 

3.3 

19.8 

.13 

.160 

.42 

.907 

.78 

2.30 

1.7 

7.37 

3.4 

20.7 

.14 

.178 

.44 

.972 

.80 

238  . 

1.8 

8.04 

3.5 

21.6 

.15 

.198 

.46 

1.04 

.82 

2.47 

1.9 

8.72 

3.6 

22.5 

.16 

.218 

.48 

Lit 

.84 

2.56 

2. 

9.42 

3.7 

23.5 

.17 

.239 

.50 

1.18 

.86 

2.65 

2.1 

10.0 

3.8 

24.4 

.18 

.260 

.52 

1.25 

.88 

2.74 

2.2 

10.8 

3.9 

25.4 

.19 

.282 

.54 

1.32 

.90 

2.84 

2.3 

11.5 

4. 

26.4 

.2 

.305 

.56 

1.40 

.92 

2.94 

Iu  calculating  this  table,  the  coeff  3.41  was  used  for  beads  from  .03  ft  to  .3  ft;  then  3.33  to  2  ft; 
then  3.3  to  the  end. 

From  the  Lowell  experiments,  by  Mr  Francis,  it  appears  that 

when  the  depth,  m  a,  is  1  foot,  and  the  entire  sheet  of  water,  after  passing  over  the  weir,  strikes  a 
bor  solid  floor  placed  only  about  6  ins  below  the  crest  o,  <>f  the  weir,  the  disch  is  thereby  diminished 
bur.  about  the  YTf^yTT  Part '  an<*  tnat  wnen  tne  nead  amia  about  10  ins,  and  falls  into  water  of  con- 
siderable depth,  no  diff  whatever  is  perceptible  in  the  disch,  whether  the  surf  of  that  water  be  about 
4.  or  about  13  in*  below  the  crest  a  ;  and  that  a  fall  below  the  crest  a,  equal  to  one-half  of  the  head 
«m,  is  quite  sufficient. 

If  the  water  in  the  reservoir,  or  in  the  feeding  canal,  in- 
stead of  being   stagnant,  has  a  slight  current  toward  the 

weir,  the  dis<%h  will  be  but  very  little  increased  thereby  when  the  head  a  m  is 
several  ins.  Mr  Francis  observed  that  a  current  of  I  foot  per  sec,  or  nearly  .7  of  a  mile  per  hour  in- 
creased the  disch  but  about  2  per  cent,  when  the  head  was  13  ins;  and  one  of  6  ins  per  sec.  about  1 
per  ct,  when  the  head  was  8  ins.  Whenever  the  effeot  of  the  current,  however,  is  so  great  as  to  require 
notice,  proceed  as  follows  :  Find  in  Table  10,  p552,  the  theoretical  head  A  which  corresponds  to  the 
observed  vel  of  the  approaching  current.  Add  this  head  to  the  head  o  TO.  Fig  20.  Cubethesum.  Take 
the  sq  rt  of  this  cube,  calling  it  a.  Also  cube  the  theoretical  head  A.  Take  the  sq  rt  of  this  cube.  Call 
it  ft.  From  i  take  b.  Find  the  square  of  the  remainder.  Take  the  cube  rt  of  this  square,  for  a  new 
head  H',  to  be  employed  in  Cases  1  and  2,  instead  of  H,  or  o  TO  ;  or  in  shape  of  a  formula, 


—  /i2~)3  r=H';  the  new  head. 


Art.  15.    If  the  inner  face  of  the  weir  and  dam,  instead  of 
being  vert,  as  a  b,  Fig  2O.  is  sloped,  as  a  or  t>.  Fig  25;  tho  contraction 

on  the  crest  will  be  diminished ;  and  consequently  the  di«oh  will  be  in- 
creased. This  will  also  be  the  case  if  the  inner  corner  oredce  of  the  crest 
be  rounded  off,  instead  of  being  left  sharp  ;  or  if  the  .sides  of  the  reservoir 
c  'nverge  more  or  les^as  they  approach  the  weir:  so  as  to  form  wings  for 
guiding  the  water  more  directly  to  it:  or  if  a  fc.  Fig  '20.  be  less  than  twice 
a  m.  Indeed,  so  many  modifying  circumstances  exist  to  embarrass  experi- 
ments on  this,  and  similar  subjects,  that  some  of  those  which  have  hoen 
made  with  great  care,  are  rendered  inapplicable  as  other  than  tolerable 

~I7*£  9  *\  approximations,  in  consequence  of  the  neglect  to  take  into  consideration 

&  some  local  peculiarity,  which  was  not  at  the  time  regarded  as  exerting  an 

appreciable  effect.  Unless,  therefore,  circumstances  admit  of  our  com- 
bining all  the  conditions  mentioned  in  the  first,  part  of  this  article ;  and [thereby  securing  very  ap- 
nroximate  results,  we  must  either  resort  to  an  actual  measurement  of  the  disch  in  a  vessel  of  known 
capacity;  or  else  he  contented  with  rules  which  may  lead  t-o  errors  of  5,  10.  or  more  per  cent,  in  pro- 
portion  as  we  deviate  from  those  conditions.  Frequently,  even  10  per  ct  of  error  may  be  of  little  real 
importance. 

The  following  rule  for  finding  the  discharge  over  an  over- 


HYDRAULICS. 


561 


fall,  or   weir,   approximately,  has    been    prepared   by   the 
author  from  various  data. 


Mult  together  the  length,  a,  a,  of  the  weir.  Figs  21  and  22,  in  feet:  the  head,  am.  Fig  20.  measured 
to  the  lentil  surf  of  the  reservoir,  in  ft;  the  theoretical  vel  in  ft  per  sec,  corresponding  to  the  head  a  m. 
(Table  10.  p  552  :)  and  that  coeff  from  the  following  table,  which  agrees  most  nearly  with  the  case.  The 
prod  will  be  the  reqd  disch  in  cub  ft  per  sec,  near  enough  for  ordinary  purposes  ;  and  probably  quite 
as  close  us  can  be  arrived  at  without  actual  measurement  in  each  case  that  presents  itself. 

Ex.  How  much  water  will  be  dischd  over  a  weir  60  ft  long;  the  crest  of  which  is  level,  smooth,  and 
3  ft  wide,  or  thick  ;  and  over  which  the  head  am,  Fig  20,  is  8  ins,  or  .6666  ft  thick  ?  Here  the  theo- 
retical vel  for  a  head  of  8  ins.  (Table  10,  p  552)  is  6.55  ft  per  sec.  The  coeff  for  a  weir  whose  crest  is 
level,  »u(i  3  ft  wide,  with  a  head  of  8  ins.  is  by  the  following  table  .31.  Consequently,  60X.6666  X  6.56 
X  .31  =  81.21  cub  ft  pe,r  sec  ;  the  reqd  disch  approximately. 

TABLE  14,  Of  coefficients  of  approximate  discharge  over 
weirs  of  different  thicknesses,  varying-  from  a  sharp  edire  to 

Sleet.— (Original.) 


3  Ft  Thick  ; 

smooth  ; 

Head 
a  m 
in  Feet. 

Head 
a  m 
in  Inches. 

Sharp 
Edge. 

2  Inches 
Thick. 

sloping  out- 
ward:  and 
downward. 

3  Ft  Thick  ; 
smooth, 
and  level. 

from  1  in  12 

to  1  in  18. 

.0833 

1 

.41 

£1 

.32 

.27 

.1666 

2 

.40 

.38 

.34 

.30 

.'25 

3 

.40 

.39 

.34 

.31 

.3333 

4 

.40 

.41 

.35 

.31 

.4166 

5 

.40 

.41 

.35 

.32 

.5 

6 

.39 

.41 

.35 

.33 

.5833 

7 

.39 

.41 

.35 

.32 

.6666 

8 

.39 

.41 

.34 

.31 

.8333 

10 

.38 

.40 

.34 

.31 

1. 

12 

.38 

.40 

.33 

.31 

2. 

24 

.37 

.39 

.32 

.30 

3. 

36 

.37 

.39 

.32 

.30 

i  inches.  ^;  and  if  1  inch,  fa  or 


RKM.  1.  When  the  water,  after  pass- 
in«-  over  a  weir.  Fig  26,  instead  of 
falling  freely  into  the  air,  is  carried 
away  by  a  slightly  inclined  apron 
or  trough,  T,  the  floor  of  which  coincides 
with  the  crest,  a.  of  the  weir,  then  the  disch  is  not  appre- 
ciably diminished  thereby  when  the  head  a  m  is  15  ins  or 
more.  But  if  the  head  a  m  is  but  1  ft,  then  the  calculated 
disch  must  be  reduced  about  -^  part;  if  6  inches,  y^ ;  if  ! 
one-half,  as  approximations. 

RF.M.  2.    Professor  Thomson,  of  Dublin,  proposed  the  use  of 
triangular  notches,  or  weirs,  for  measuring  the  disch ;  inasmuch  as  then 

the  periphery  always  bears  the  same  ratio  to  the  area  of 

other  form.  Experimenting  with  a  right-angled  triangu- 
lar notch  in  thin  sheet  iron.  Fig  26^  ;  with  heads  of  from 
2  to  7  ins,  measured  vert,  from  the  bottom  of  the  notch,  to 
the  level  surfoftftf  quiet  water,  he  found  the  disch  in  cub 
ft  per  sec  to  be  as  follows  : 

Find  the  fifth  power  of  the  head  in  inche*,  (Table  5,  pages  546-7.)  Take  the  sq  rt  of  this  5th  pow«r, 
(Table  6,  page  548.)    Mult  this  sq  rt  by  .0051.     Or  by  formula, 

eubfTpe^sec  ~  Head  in  inche*  $  *  -O051- 

RKM.  3.    Fig  36V£  shows  a  singular  effect  observed  at  Clegg's 

dam.  across  Cape  Fear  River,  N  C.  It  is  from  measurements  made  by  Ellwood 
Morris,  C  E  ;  by  whom  they  were  communicated  to  the  writer.  The  dam  is  of  wooden  cribwork  ;  and 
its  level  crest,  8  ft  5  ins  wide,  is  covered  with  plank ;  along  which  the  water  glides  in  a  smooth  sheet, 
6  ins  deep,  (nt  the  time  of  measurement.)  At  the  upper  end  of  this  sheet,  and  in  a  dist  of  about  2  ft, 
a  head  of  9  ins  forms  itself,  as  in  the  fig.  We  have  no  comments  to  offer ;  but  consider  the  fact  to  be 
of  sufficient  interest  to  he  made  more  widely  known. 

Art.  16.    On  the  flow  of  water  throngh  open  channels.    The 

following  rules,  to  the  end  of  Art  '20,  must  be  regarded  merely  as  approximations.  The  subject  Is  in- 
rolved  in  much  uncertainty.  See  Re  m,  p  562,  and  "  Caution,"  p  566. 


562 


HYDRAULICS. 


*  2' * 


FtjtX.fi  i 

To  ascertain  approximately  the  mean  vel  of  all  the  water 
in  any  given  cross-section  of  a  river,  canal,  or  other  water- 
course; having  given  the  greatest  surf  vel  only.* 

When  no  great  accuracy  is  reqd,  this  mean  vel  may  be  deduced  from  the  greatest  surf  vel.  Thus, 
select  a  place  where  the  stream  is  for  some  dist  (the  longer  the  better)  of  tolerably  uniform  cross- 
section;  and  free  from  counter-currents,  slackwater,  eddies,  rapids,  &c.  Observe,  bv  a  seconds- 
watch,  or  pendulum,  how  long  a  time  a  float  (such  as  a  small  block  of  wood)  placed  iu"  the  swiftest 
part  of  the  current,  occupies  in  passing  through  some  previously  measured  dist.  From  50  feet  for 
slow  streams,  to  150  ft  for  rapid  ones,  will  answer  very  well.  This  dist  in  ft,  or  ins,  div  by  the  entire 
number  of  seconds  reqd  by  the  float  to  traverse  it,  will  give  the  greatest  surf  vel  iu  ft  or  ins  per  sec. 
Take  4  of  this  vel;  or,  iu  other  words,  mult  it  by  the  dec  .8.  The  prod  will  be  approximately  tb« 
mean  vel  of  the  entire  body  of  water.  The  result  will  be  somewhat  more  accurate,  if,  instead  of 
mult  by  .8  for  every  surf  ve"l,  we  mult  by  the  following  decimals  in  the  4th  col.* 

Ex.    What  will  be  the  mean  vel  of  all  the  water  moving  through  a  uniform 

channel,  the  greatest  surf  vel  of  which  is  found  to  be  60  ins  per  sec?     Here,  60  X  .83  =  49.8  ins  per 
sec  ;  the  reqd  vel. 

TABLE  15.  See  Caution,  p  566. 


SURFACE  VELOCITY. 

Mult, 
greatest 
Surface 
Vel.,  by 

SURFACE  VELOCITY. 

Mult. 

freatest 
urface 
Vel.,  by 

Inches 
per  Sec 

Feet 
per  Sec. 

Miles 
per  Hr. 

Inches 
per  Sec. 

Feet 
per  Sec. 

Miles 
per  Hr. 

4 
12 

20 
40 
60 

.333 
1. 
1.667 
3.333 
5. 

.227 
.6S2 
1.136 
2273 
3.409 

.76 
.77 
.79 
.81 
.83 

80 
100 
120 
140 
160 

6.667 
8.333 
10. 
11.7 
13.3 

4.515 
5.682 
6.818 
7.955 
9.091 

.85 
.86 
.87 
.88 
.89 

The  surf  vel  should  be  measd  in  perfectly  calm  weather, 

so  that  the  float  may  not  be  disturbed  by  wind  ;  a«d.  for  the  same  reason,  the  float  should  not  projoct 
much  above  the  water.  The  measurement  should  be  repeated  several  times  to  insure  accuracy.  In 
very  small  streams,  the  banks  and  bed  may  be  trimmed  for  a  short  dist,  so  as  to  present  a  uniform 
channel-  way.  The  float  should  be  placed  in  the  water  a  little  dist  above  the  point  for  commencing 
the  observation  ;  so  that  it  may  acquire  the  full  vel  of  the  water,  before  reaching  that  point. 

To  tin  «l  the  discharge  in  the  foregoing  case. 

Measure  the  breadth  of  the  stream  at  the  surface  ;  and  also  a  sufficient  number  of  depths,  taken  in 
a  straight  line  across  ;  in  order  to  calculate  the  area  of  its  cross-section.  This  area  in  sq  ft,  mult  by 
the  mean  vel  in  ft  per  sec,  will  evidently  give  the  disch  in  cub  ft  per  sec. 

Rent.    If  the  channel  is  in  common  earth,  especially  if  sandy,  the 

loss  by  soakage  into  the  soil,  and  by  evaporation,  will  frequently  abstract  so  much  water  that  the  disch 
•will  gradually  become  less  and  less,  the  farther  down  stream  it  is  measured.  Long  canal  feeders 
thus  generally  deliver  into  the  canal  but  a  small  proportion  ot  the  water  that  enters  their  upper  ends 

Art.  17.  To  find  approximately  the  vel  at  the  bottom  of  a 
stream  ;  at  any  part  of  its  cross-section.* 

RULE.     First  measure  the  surf  vel  over  the  same  part.    Mult  it  by  the  corresponding  decimal  \n  the 


The  remainder  will  be  approximately  the  reqd  bottom  vel.     Ex.  Surf  vol  100  ins  per  sec.     And  100  X 
.86  =  86  ins  per  sec,  mean  vel  ;  and  86  X  2  =  172  ;  and  172  —  100  =  72  ins  per  sec,  bottom  vel. 

*  As  we  have  already  stated,  these  rules  give  in  many  cases  very  rude  results  ;  that  in  Art  17  par- 
ticularly. With  the  same  surface  vel,  a  wide  and  deep  stream  will  have  greater  mean  and  bottom 
vels  than  a  small  shallow  one.  There  Is  no  reliable  rule. 


Art.  18.  TABLE 
Incites  per  sec ; 


DRAULICS. 


563 


Of  approximate  velocities  of  streams  in 
the  foregoing  rules.   See  Caution,  p  566. 


**  £  «2 

MeaikVel.  of 
all  the 
Stream,  in 
Ins.  per  Sec. 

Greatest  Bot- 
tom Vel.,  in 
Ins.  per  SeCvj 

Greatest  Sur- 
face Vel..  in 
Ins.  per  Sec. 

o      eg 

£|  Is, 

S       02  0 
S 

Greatest  Bot- 
tom Vel..  in 
Ins.  per  Sec. 

lift! 

Greatest  Bot- 
tom Vel.,  in 
Ins  per  Sec. 

i£& 

Vs! 
§"si| 

lit 

3 

2.28 

1.54 

23 

18.2 

13.3 

48 

394 

30.8 

86 

73.4 

60.9 

4 

3.04 

2.08 

24 

19. 

14. 

50 

41.2 

32.3 

88 

75.3 

62.7 

5 

381 

2.62 

25 

19.8 

14.6 

52 

42.9 

33.8 

90 

77.1 

64.3** 

6 

4.58 

3.16* 

26 

20.6 

15.2 

54 

44.7 

35.3 

92 

78.9 

65.9 

7 

5.36 

3.72 

27 

21.4 

15.9 

56 

46.4 

36.8 

94 

80.8 

67.7 

8 

6.14 

4.2S 

28 

22.3 

16.6 

58 

48.2 

38.4 

96 

82.7 

69.3 

9 

6.92 

4.84 

29 

23.1 

17.2 

60 

49.9 

39.8f 

98 

84.5 

71. 

10 

7.70 

5.40 

30 

23.9 

17.9 

62 

517 

41.4 

100 

86.3 

72.6 

11 

8.49 

5.98| 

31 

24.8 

18.5 

64 

535 

43. 

105 

90.9 

7«.9 

12 

9.29 

6.58 

32 

25.6 

19.3 

66 

55.3 

446 

110 

95.6 

81.2 

13 

10.1 

7.16 

33 

26.5 

19.9 

68 

57.1 

46.1 

115 

100.2 

85.3 

14 

10.9 

7.72; 

34 

27.3 

20.6 

70 

58.9 

47.7 

120 

104.9 

89.8 

15 

11.7 

8.34 

35 

28.1 

21.3 

72 

60.7 

49.4 

125 

109.6 

94.3 

16 

12.5 

8.i»62 

06 

29. 

22. 

74 

62.5 

50.9 

130 

114.3 

98.5 

17 

13.3 

9*.5S 

38 

30.7 

23.5|| 

76 

64.3 

526 

135 

119.1 

103.-H- 

18 

14.1 

10.2 

40 

•32.6 

25. 

78 

66.2 

54.3 

140 

123.9 

108. 

19 

14.9 

10.8 

42 

34.2 

26.5 

80 

67.9 

55.8 

145 

128.8 

112. 

20 

15.7 

11.4 

44 

36. 

2$. 

82 

69.8 

57.6 

150 

133.7 

117. 

21 

16.5 

12.1 

46 

37.7 

29.3 

84 

71.7 

59.3 

160 

142.4 

125. 

22 

17.3 

12.7 

*  .18  of  a  mile  per  hour;  and  begins  to  scour  fine  clay.  This,  and  the  follow- 
ing, are  mere  approximations,  chiefly  deduced  by  Du  Buat  from  experiments  on  a  trifling  scale.  More 
observations  on  this  subject  are  needed.  There  is  reason  to  believe  that  the  scouring  vel  is  consider* 
ably  too  small  for  clay.  The  subject  is  very  intricate. 

T  Ji  of  a  mile  per  hour;  and  just  lifts  fine  sand.  tt  5.85  miles  per  hour;   (surface  vel 

J  .44  of  a  mile  per  hour ;  lifts  sand  as  coarse  as  linseed.  7.67  mile.-*  per  hour)  will  roll  stones 

5  .51  of  a  mile  per  hour ;  moves  One  gravel.  of  a  foot  diam. 

II  1  Jf  miles  per  hour ;  moves  pebbles  about  an  inch  in  diam. 

II  2. '26  miles  per  hour;  moves  pebbles  as  large  as  an  egg. 

**  3J$  miles  per  hour;  begins  to  wear  away  soft  shistus. 

(Scouring:  action  is  supposed  to  be  as  square  of  vel. 

According  to  Smeaton,  a  vel  of  8  miles  an  hour  will  not  derange  quarry  rubble  stones,  not  exceed- 
ing half  a  cub  ft,  deposited  around  piers,  Arc  ;  except  by  washing  the  soil  from  under  them, 
i  inch  per  sec,  -  5  ft  per  min.  -  .056818  of  a  mile,  or  300  ft  per  hour. 
1  foot  per  sec.  =  60  ft  per  min,  =  .681816  of  a  mile,  or  3600  ft  per  hour. 

To  reduce  inches  per  sec,  to  feet  per  minute,  multiply  by  5. 

"      hour,  "  "    800. 

"  "  "  "        "        to  miles  per  hour,  divide  by  17.6. 

One  mile  per  hour  —  88  ft  per  min  =  1.4667  ft,  or  17.6  ins  per  sec. 

Art.  19.  The  simplicity,  and  easy  application  of  the  fore- 
going rules  frequently  lead  to  their  adoption  for  the  graug-- 
injj:  of  streams;  but  it  must  be  stated  that  this  method  is 
involved  in  much  uncertainty.  The  experiments  of  Mr  Francis,  at 

Lowell,  have  shown  its  liability  to  err  at  least  15  per  cent  in  deficiency  ;  while  observations  by  others 
would  seem  to  indicate  that  it  may,  under  diff  circumstances,  be  as  much  (if  not  more)  in  excess  of 
the  truth.  Mr  Francis  found  that  surface  floats  of  wax,  2  ins  diam.  floating  down  the  center  of  a  rec- 
tangular flume  10  ft  wide,  and  8  ft  deep,  actually  moved  about  6  per  cent  slower  than  a  tin  tube  2  ins 
diam,  reaching  from  a  few  ins  above  the  surf,  down  to  within  \%  ins  of  the  bottom  of  the  flume:  and 
loaded  at  bottom  with  lead,  to  insure  its  maintaining  a  nearly  vert  position.  While  the  wax  surf  float 
moved  at  the  rate  of  3.73  ft  per  sec,  the  rate  of  the  tube  (which  was  evidently  very  nearly  the  same 
as  that  of  the  center  vert  thread  of  water)  was  3.98  ft  per  sec.  Also,  that  in  the  same  flume,  with  vels 
of  the  center  tube  varying  from  1.55,  to  4  ft  per  sec,  the  vel  of  the  tube  was  less  than  that  of  the  mean 
vel  of  the  entire  cross  section  of  water  in  the  flume,  about  as  .96  to  1,  for  the  lesser  vel ;  and  .93  to.  1 
for  the  greater  ve!.  While.  In  another  rectangular  flnme  20  ft  wide  and  8  ft  deep,  with  vels  varying 
from  1.16  to  1.84  ft  per  sec.  that  of  the  tubes  was  greater  than  that  of  the  entire  mass  of  water,  about 
as  1 .04  to  1.  In  a  flnme  29  ft  wide,  by  8. 1  ft  deep,  with  vels  of  about  3  ft  per  sec.  it  was  as  1  to  .9 ;  and 
in  a  flume  36^  ft  wide,  by  8.4  ft  deep,  with  vels  of  aboirt  3^  ft  per  sec,  as  1  to  .97. 

C'harles  Filet.  Jr,  C  E.  found  in  the  Mississippi  "at  diff  points  on 

the  river,  in  depths  varying  from  54  to  100  ft:  nnd  in  currents  varying  from  3  to  7  miles  an  hour  that 
the  speed  of  a  float  supporting  a  line  50  ft  long,  is  almost  alwnvs  greater  thnn  that  of  the  surf  float 
alone  "  The  same  results  were  obtained  with  lines  25  and  75'ft  long:  the  excess  of  the  speed  of  the 
line  floats  being  about  2  per  cent  over  that  of  the  simple  floats  :  and  Mr  Ellet  concludes,  therefore,  that 
the  mean  vel  of  the  entire  cross-section  of  the  Mississippi,  instead  of  being  less,  is  absolutely  greater 


564  HYDRAULICS. 

by  about  2  per  cent,  than  the  MEAN  surf  vel.  He,  however,  employed  .8  of  the  greatest  surf  rel  at 
representing  approximately,  in  his  opinion,  the  mean  vel  of  the  entire  cross-section  of  water.  In 
shallow  streams,  he  always  found  the  surf  float  to  travel  more  rapidly  than  a  line  float. 

European  trials  of  the  mean  vel  of  separate  single  verticals,  in  tolerably  deep  rivers,  have  resulted 
in  from  .85  to  .96  of  the  surf  vel  at  each  vertical.  The  mean  of  all  may  be  taken  at  .9. 

Art.  2O.  To  grange  a  stream  by  means  of  its  velocity,  more 
accurately  than  by  the  preceding-  methods. 

The  following  is  perhaps  the  most  accurate 
means  that  can  be  adopted  for  gaujrin^  a  stream 
by  means  of  direct  measurement  of  its  vel.  la 
case  of  a  large  river  it  will  involve  some  trouble, 
time,  and  expense;  but  where  accuracy  is  neces- 
sary, those  considerations  must  yield.  Select  a 
Slace  where  the  cross-section  remains  for  a  short 
ist,  tolerably  uniform,  and  free  from  counter-cur- 
rents, eddies,  still  water,  or  other  irregularities. 
Prepare  a  careful  cross-section,  as  Fig  27.  By 
means  of  poles,  or  buoys,  n,  n,  divide  the  stream 
into  sections,  a.  b,  c,  Ac.  Plant  two  range-poles, 
R,  R,  at  the  upper  end;  and  two  others  at  the 

lower  end  of  the  dist  through  which  the  floats  are  to  pass ;  for  observing  the  time,  by  a  seconds 
watch,  or  a  pendulum,  which  they  occupy  in  the  passage.  Then  measure  the  mean  vel  of  each  section 
a,  o,  c,  &c,  separately,  and  directly,  by  means  of  long  floats,  as  F  L,  reaching  to  near  the  bottom ;  and 
projecting  a  little  above  the  surf.  The  floats  may  be  tin  tubes,  or  wooden  rods;  weighted  in  either 
case,  at  the  lower  end,  until  they  will  float  nearly  vert.  They  must  be  of  diff  lengths,  to  suit  the 
depths  of  the  diff  sections.  For  this  purpose  the  float  may  be  made  in  pieces,  with  screw-joints.  The 
area  of  each  separate  section  of  the  stream  in  sq  ft.  being  mult  by  the  observed  mean  vel  of  its  water 
in  ft  per  sec,  will  give  the  disch  of  that  section  in  cub  ft  per  sec.  And  the  discharges  of  all  the  sepa- 
rate sections  thus  obtained,  when  added  together,  will  give  the  total  disch  of  the  stream.  And  thia 
total  disch,  div  by  the  entire  area  of  cross-section  of  the  stream  in  sq  ft,  gives  the  mean  vel  of  all  the 
water  of  the  stream,  in  ft  per  sec. 

Art.  21.  To  find,  approximately;  the  mean  vel,  and  disch 
in  an  open  canal,  aqueduct,  river,  A-c,  of  uniform  cross-sec- 
tion, and  fall  throughout;  knowing:  its  dimensions,  and  rate 
of  fall,  or  descent  in  feet  per  foot.  See  Caution,  p  566.  Also 
Art  16. 


Fiq28 


Rule  1.  Find  area  of  the  water-way,  abcoa,  Figs  28  29,  30,  in  *q  ft.  Find  the  dist  aft  co,  in 
feet ;  being  that  portion  of  the  cross-section  of  the  aides  and  bottom  of  the  channel  bed  which  is  in 
actual  contact  with  the  water.  This  dist  fa  called  the  WET  PERIMETER,  or  WET  BORDER  of  the  channel. 
Mult  together  the  area  abcoa;  the  fall  in  feet  per  foot ;  and  the  constant  number  8975  ;  (in  practice 
we  may  use  9000.)  Div  the  prod  by  the  wet  perimeter.  Take  the  sq  rt  of  the  quot.  From  this  sq  rt 
subtract  the  constant  dec  .1085).  The  rem  will  be  the  mean  vel  in  ft  per  sec  of  all  the  water  in  the 
entire  cross-section  of  the  stream.  Or  by  formula, 

Mean  vel     __  /Area  of  water-way  in  sq  ft  X  fall,  in  ft  per  ft  X  8975  V 

5      \/     '  wet  perimeter  in  feet.  ~  1  -  .1089. 

Ex.  A  canal  of  rectangular  cross-section,  as  Fig  29,  is  to  be  10  ft  wide ;  and  to  have  3.5  ft  depth  of 
water.  Its  bottom  is  to  have  a  uniform  fall,  slope,  or  descent,  at  the  rate  .246  of  a  ft  in  a  length  of  492 
ft.  "What  will  be  the  mean  vel  of  all  the  water  ;  and  how  many  cub  ft  of  water  will  it  disch  per  sec  ? 
Here,  the  area  of  water-  way  =  10  X  3.5  =  35  sq  ft.  The  fall, '.'246  ft,  div  by  492  ft  (the  dist  in  which 
it  occurs)  gives  .0005  of  a  ft  fall  for  every  foot  of  length.  The  wet  perimeter,  =  a  6  -f-  6  c  -j-  c  o  =  3.5 

-f  10  +  3.5  =  17  ft.    Hence  we  have  — ---005  X  897°  =  9.239.      And  the  sq  rt  of  9.239  =  3.04.      And 

3.04  —  .1089  =  2.9311  ft  per  sec,  the  reqd  vel.   The  disch  will  evidently  be,  this  vel  mult  by  the  area  of 
water-way  in  sqft;  or  2.9311  X  35  =  102.588  cub  ft  per  sec.     See  Remark,  p  562:   also,  Remark  of 

next  Art.  _   

Rule  2.      Vel  in  ft  _  /Area  of  water-way,  in  sq  ft         Twice  the  fall 

per  sec          4     /          ^e7^erim*e^~in~ft  *  in  ft  per  mile. 


Vel  in  ft  __  /Area  of  water-way,  in  sq  ft 

per  sec          4     /          wet  perimeter  in~jt 


For  the  foregoing  example  it  gives  a  vel  of  3.3  ft.  instead  of  2.93  ft;  and  some  experiments  with 
sewers  would  indicate  that  3.3  is  nearer  the  truth;  but  as  before  remarked,  perfect  accuracy  in  these 
matters  must  not  be  looked  for.  The  degree  of  smoothness  of  the  channel,  and  even  the  muddiness 
of  the  water,  will  affect  the  result  appreciably. 

Rule  3.  Mr.  Poole,  C  E.  England,  tested  the  following  on  open 
canal  portions  of  the  Rocquefavonr  aqueduct,  France,  and  found  it  to  accord  closely  with  fact. 


inft     4     /!< 
-  «ec=\  /    Jl 


Vel  in  ft     .       /  innn_  total  fall  in  ft  X  area  of  water-way,  in  sq  ft 

per  tec  =\  /    JUUUU 

total  length  in  ft  X  wetperim  in  ft. 


Our  above  Rale  1  gives  in  the  example  thereto,  a  vel  of  2.93  ft  per  sec  ;  aud  Mr.  Poole's  rule  gives 
8.21  feet.    Rule  2  give*  3.3  ft.    See  Kutter'g  formula,  page  6§Qi 


HYDRAULICS.  565 

Art.  22.  To/ft  iid,  approximately,  what  fall  must  be  given  to 
every  foot  A  n  length  of  a  uniform  canal,  or  other  water- 
course. the  dimensions  of  which  are  known;  to  enable  it 
to  discharge  a  reqd  quantity,  in  a  given  time. 

HULK  1.  Piv  the  number  of  cub  ft  of  water  reqd  per  sec,  by  the  area  in  sqftof  the  water-way  a  b  coa, 
Figs  28,  29,  30,  &c.  The  quot  will  be  the  mean  vel  which  the  water  must  have,  iu  ft  per  sec,  in  order  to 
disch  the  reqd  quantity.  Next  find  the  square  of  this  vel.  Also  find  the  length  of  wet  perimeter,  abco, 
in  ft.  Mult  together  the  square  of  the  vel  ;  the  wet  perim  ;  and  the  constant  dec  .000111*.  Div  the 
prod  by  the  area  of  water-way,  abco  a.  Call  the  quot  M.  Next  mult  together,  the  simple  vel  itself; 
the  wet  perim;  and  the  constant  dec  .00002426.  Div  the  prod  by  the  area  of  water-way,  a  t>  c  b  a.  Add 
the  quot  to  M.  The  sum  will  bo  the  reqd  fall,  iu  decimals  of  a  foot,  which  must  be  given  to  every  foot 
in  length  of  the  canal.  Or,  in  shape  of  a  formula,  See  Caution,  p  566. 

Fall  fn  every  _  Veil  X  wet  perim  X  .0001114         Vel  Xwet  perim  X  .00002426 
foot  of  length  —    area  of  wat  er.way  in  sq  ft    "*"   area  of  water-way  in  #q  ft.' 

Ex.  The  same  as  the  foregoing.  What  fall  per  foot  must  be  given  to  a  rectangular  canal,  or  water- 
oourse,  Fig  29,  10  ft  wide,  and  having  3.5  ft  depth  of  water,  to  enable  it  to  disch  102.55  cub  ft  per  sec  ? 

Here,  the  area  of  water-way,  a  6  c  o  a,  =  35  sq  ft.    And  —  „-—  =  2.93  ft  per  sec,  the  vel.    And  2.93» 
=  8.5849.    Also,  the  wet  perim  a  6  co  =3.  5  +  10  +  3.5  =  17  feet.    Hence, 
8.5849  X  17  X  .0001114^  .016258  _  Q^^  _.M 
35  ~~=       35 

.       2.93  X  17  X  .00002426        .0012084 
Again;  -  ^  -  =  —  —    =  .000035. 

Finally,  .000465  -j-  .000035  =  .0005  ft  ;  the  reqd  fall  ia  every  foot  of  length  of  the  canal  ;  the  same  as 
in  the  preceding  example.  See  Remark,  Art  16,  p  562. 

Rnle  2.  Or  the  following  formula,,  based  on  that  in  Rule 
2,  Art  21,  may  be  used. 

The  aquare  of  the  vel  )  ..  .  .   .  ,    (  ^rea  of  water-way 
*"    edb—— 


For  the  foregoing  example  it  gives  2.085  ft  per  mile  ;  or  very  nearly  .0004  ft  per  ft.  Instead  of  .0005. 

Rule  3.     For  the  total  fall  in  feet  in  the  whole  length  of  the  canal  or 

other  channel,  the  following  is  approximate. 

,         length  of  canal  v  Wet  perim  \       /Sqof  vel  in  \ 

Total  fall  _  /   (W7.         in  feet.  *      in   feet.      )  x  /    ft  per  sec.   ) 

infect    ~\  Tr.  area  in  sq  feet.          7      \        6iT~    / 

This  rule  gives  for  the  above  canal,  assuming  it  to  be  one  mile  or  5280  ft  long,  a  total  fall  of  2.564 
ft,  or  31.68  ins  ;  or  .000486  ft  for  each  ft  of  length  ;  or  nearly  as  Rule  1. 

Rule  4.     Mr.  Poole  says  the  fall  per  mile  in  ins,  reqd  in  a  canal,  to  impart  any 
given  mean  vel,  in  ins  per  sec,  may  be  found  thus  :  Div  the  area  of  water-way  in  sq  ft,  by  the  wet  perim 

in  ft.    The  quot  is  what  is  called  the  hydraulic  mean  depth,  or  mean 

radius,  of  the  canal.*  Mult  this  by  1'2,  to  reduce  it  to  ins  ;  and  call  the  prod  D. 
Then  div  the  given  mean  vel,  by  the  dec  .909.  Square  the  quot.  Div  said  square  by  twice  D.  By 
formula, 


The  read  fall        /given  mean  vel  in  ins  per  see\  2. 
(n  in-  permit^  (  -  co,ut  number  ^  -  )  ^ 


For  the  example  at  the  first  part  of  thin  Art,  Mr.  Poole's  rule  gives  a  fall  of  30.3  ins  per  mile  :  while 
the  rule  at  the  head  of  the  Art,  gives  a  fall  of  .0005  of  a  ft  per  ft  in  length  ;  which  is  equal  to  31.68  ins 
per  mile.  The  two  rules,  therefore,  agree  as  closely  as  can  be  expected  in  cases  of  this  kind. 

REM.  In  designing;  watercourses,  it  is  important  to  bear  in  mind  that  when  a 
canal,  as  a  b,  Fig  31,  either  of  uniform  cross-section,  and 
descent,  or  otherwise,  is  carried  directly  from  a  reservoir, 
without  any  enlargement  of 
the  canal  at  the  point  of  junc- 
tion, a  slight  fall  iorms  itself 

at  a,  somewhat  of  the  shape  shown  on  an  ^T"~-~—  -~  b 

exaggerated  scale  in  Fig  31.  This  fall  will  not  only 
produce  an  excess  of  vel  at  that  point,  beyond  that 
tor  which  the  canal  was  intended;  but  will  also  di- 
minish the  calculated  depth  of  water  in  the  canal, 
throughout  its  entire  length,  to  an  amount  equal  to 

the  fall.     Thus,  if  b  m  were  the  depth  calculated  by  "O-       O  4 

our  rules,  the  actual  depth  would  be  n  m.     In  large  JjlCl  U  J- 

oanals,  as  in  those  intended  for  navigation,  this  fall  J 

at  a  will  be  approximately  equal,  iu  feet,  to 


*  Or  of  anv  stream  ;  or  of  a  pipe.    In  a  cylindrical  pipe  running  full;  or  in  any  open  channel  of 
lemicircular  cross-section,  running  full,  the  hyd  mean  depth  thus  found  ia  equal  to  X  of  the  diam 

of  the  circle. 


566 


HYDRAULICS. 


The  square  of  the  calculated  mean  veil,  in  ft  per  tec  X  .0155, 

.9 
and  in  smaller  canals,  such  as  mill  courses,  to 

The  square  of  the  calculated  mean  vel,  in  ft  per  tec,  X  .0155. 


m 


At  the  great  m^an  vel  of  3  miles  an  hour,  or  4.4  ft  per  sec,  this 
fall  at  a  -would  be  as  much  as  .333  of  a  foot,  or  4  ins,  in  a  large 
canal:  and  .4  of  a  foot,  or  4  8,  in  an  ordinary  mill-course.  At  1^ 
miles  per  hour,  or  2.2  ft  per  sec.  it  would  be  .083  of  a  ft.  or  1  inch, 
in  the  canal:  and  .1  ft,  or  1.2  ins.  in  the  mill  course.  Without  this 
fall  the  quiescent  water  of  the  reservoir  would  not  flow  out  suffi- 
ciently fast  to  maintain  the  calculated  vel  in  the  canal. 

To  destroy  this  fall,  or  rather,  to  reduce  it  to 
an  almost  inappreciable  quantitv.  and  thus  secure  the  calculated 
denth  and  disch  ;  at  the  same  time  providing  a  substitute  for  the 
fall,  it  is  merely  necessary  to  enlarge  the  canal  at  its  junction 
with  the  reservoir:  giving  it  at  that  point  a  shape  corresponding 
to  some  deeree  to  that  of  the  contracted  vein.  Thus,  if  a  b.  Fig  32, 
be  the  width  of  the  canal,  make  c  d  equal  to  .7  of  that  width  :  at 
the  reservoir,  make  e  f—  2  cd\  and  describe  two  arcs  of  a  circle  a  c 
and  bf,  with  a  rad  am  —  \%  a  b.  See  Remark.  Art  16,  p  562. 

Remark.  The  rate  at  which  rain  water  reaches  a 
sewer  or  culvert.  &c,  may.  according  to  Uie  admirable  ••  Report 
ou  European  Sewerage  Systems"  by  Mr  Rudolph  Hering,  Civ. 
and  San.  Eng.  of  Philada,  be  found  approximately  by  the  follow- 
ing formula  by  Mr  Burkli-Ziegler.  See  Trans.  Am.  Soc.  C.  E, 
Nov  1881. 


Canal 


Cub.  ft.  per 
EFlmi=    accordi°g 


A       f 


A      cub  ft  of  ra!nfall 

X  Per  second  per  acre,  X 

to  judgment  during  heaviest  tall. 


*  I  Av.  slope  of  ground 

I    in  feet  per  1000  ft 

^  No.  of  acres  drained 


Ills  coefficient  for  paved  streets  i«  .75:  for  ordinary  cases  ,fi?5  :  and  for  suburbs  with  gardens, 
lawu-,  au-i  macadamized  streets  .31.  His  average  heaviest  fall  is  from  1%  to  2%  ins  per  hour. 

To  this  the  writer  will  add  that  each  Inch  of  rainfall  per  hour,  corresponds  closely  enougu  to  1 
cub  ft  per  SCO  per  acre;  so  that  if  \%e  liberally  allow  for  3  or  4,  &c,  ins  per  hour  of  average  heavi- 
est rainfall,  the  third  term  of  the  above  equation  also  becomes  simply  3  or  4,  &c. 

Example.  If  an  area  of  3100  acres  (nearly  5  sq  miles),  with  an  average  slope  of  5  ft  per  1000  ft, 
receives  a  rainfall  averaging  3  ius  per  hour  when  heaviest,  then,  assuming  a  coefficient  of  .5,  the  rate 
at  which  the  water  would  reach  the  mouth  of  a  sewer  at  the  lower  end  of  the  3100  acres  would  be 


•5  X  3  X 


'5  X  3  X  .203  =  .305  cub  ft  per  sec  per  acre  ; 


or  .305  X  3100  =  945.5  cub  ft  per  sec,  total. 

Now  suppo.se  the  fall  of  the  intended  sewer  to  be  say  4  ft  per  mile  ;  and  that  for  fear  of  the  too  rapid 
wearing  away  of  its  brickwork  by  debris  swept  along  by  the  water,  we  limit  its  vel  to  6.3  ft  per  sec, 
which  may  be  permitted  on  occasions  as  rare  as  rains  of  3  ins  per  hour,  although  for  tolerably  constant 
flow,  where  liable  to  debris,  it  should  not  exceed  about  5  ft  per  sec.  To  find  the  diameter,  look 
in  the  Table  of  Vels  in  Sewers,  p  652,  for  a  diam  corresponding  as  near  as  may  be,  to  a  vel  of  6.3, 
and  to  a  tall  of  4  ft  per  mile.  We  find  this  diam  to  be  14  ft,  the  area  of  which  is  154  sq  ft.  Hence, 
154  X  6.3  =  970  cub  ft  per  sec  =  capacity  of  sewer.  This  is  a  trifle  more  than  our  945.5  cub  ft  per 
sec  of  rainfall  ;  nevertheless,  to  allow  for  deposits  in  the  sewer,  it  would  be  advisable  to  increase  the 
diam  say  to  14.5  or  15  ft.  See  Caution,  below. 

Caution.  (1880.)  It  has  for  some  time  been  known  that  tlie  Rules  given 
in  this  volume  for  the  velocity  and  discharge  through  pipes 
and  channels  are  only  approximate.  In  cases  of  common  occurrence 
with  pipes  of  6  to  24  ins  diam  they  are  liable  to  vary  as  much  as  f  om  5  to  15  per  ct 
from  the  truth,  sometimes  in  exces^,  and  sometimes  in  deficiency:  and  in  unusual 
cases  much  more.  Hence  also  the  Rules  for  diams  and  falls  are  similarly  defective. 
These  remarks  do  not  apply  to  the  Rules  for  orifices  and  weirs  in  thin  plate.  At- 
tempts have  been  made  by  eminent  men.  suoh  as  Weisbach,  Darcy,  Kutter  and  others 
to  devise  rules  that  should  apply  equally  well  to  all  diams,  heads,  and  falls;  but  they 
have  not  succeeded  perfectly,  although  Kutter's  formula  (on  p  650)  is  a  great  advance 
on  all  preceding  ones.  Mr.  J.  T.  Fanning,  Civ.  Eng.  of  Manchester,  New  I  lamp- 
shire,  in  his  "  Practical  Treatise  on  Water  Supply  Engineering,"  (D.  Van  Nostrand,  N. 
Y.,  1882,  Publisher)  appears  to  the  writer  to  have  found  the  source  of  error  of  his  prede- 
cessors, and  has  given  rules  which,  although  much  more  complicated  than  those  in  this 
book,  are  perhaps  as  simple  and  as  correct  as  the  nature  of  the  subject  admits  of. 

Art.  23.  The  writer  has  prepared  Tables  17  and  18  for  the  convenience  of 
forming,  at  the  instant,  an  approximate  idea  of  the  requisite  dimensions  of  channels 
tor  fulfilling  certain  given  conditions,  but  no  such  tables  are  perfectly  reliable.  Stut- 
ter's formula,  p  650,  may  be  used  for  such  cases. 


HYDRAULICS. 


567 


TABLE  17.  -Of  mean  velocities  in  feet  per  second;  and  of 
discharges/411  cubic  feet  per  second,  in  a  rectangular  chan- 
nel 1O  1't  wi<ie;  but  with  different  depths,  and  falls.  (Original.) 


Vel 

Formula,        in  ft 
per  sec 


/Ar 

V 


of  water-way  ^,  fall  per  foot  ^ 
iu  *q  ^  _in  feet_ 

eter  in  feet 


wet  peri 

The  vels  are  above  the  disch.    See  Caution,  p  56G. 


Fall 
per 
Mile 

Feet. 

Fall  in 

out-  Foot 

Length. 
Ft. 

.25' 

Vel. 
and 
Dis. 

_50_! 

Vel. 
and 
Dis. 

.75 

Vel. 
and 
Dis. 

Vel. 
and 
Dis 

DBF 
1.5 

Vel. 
and 
Dis. 

THS  IN  FEET. 

Vel. 
aud 
Dis. 

Vel. 
and 

Di8. 

Vel. 
Dis. 

Vel. 
and 
Dis. 

Vel. 
and 
Dis. 

Vel. 
and 
Dis. 

Vel. 
and 
Dig. 

.1 
.2 

.000019 
.000038 

.091 
.227 

.174 
.435 

.168 
.840 

.223 
1.67 

.361 
2.71 

.266 
2.66 

.333 
5.00 

.516 

7.74 

.383 
7.66 

.422 
10.5 

.643 
16.1 

.455 
13.6 

.688 
20.6 

.505 
20.2 

.542 
27.1 

.812 
40.6 

.570 
34.2 

.852 
51.1 

.612 
49.0 

.911 
72.9 

.283 
1.42 

.422 
4.22 

.587 
11.7 

.760 
30.4 

.3 

.000057 

.238 
.595 

.371 
1.85 

.467 
3.50 

.542 
5.42 

.657 
9.86 

.743 
14.9 

.812 
20.3 

.868 
26.0 

.954 
38.2 

1.01 

50.5 

1.07 
64.2 

1.14 
91.2 

A 

.5 
.6 
.7 
.8 
,9 
1. 

.000076 
.000095 

.292 
.730 

.446 
2.23 

.511 
2.55 

.571 

2.85 

.556 
4.17 

.643 
643 

.731 
7.31 

.812 
8.12 

-776 
11.6 

.880 
13.2 

.975 
14.6 

ill? 

.992 
19.8 

1.10 

22.0 

.950 
23.7 

1.08 
27.0 

1.19 
29.7 

1.02 
30.6 

Ti5~~ 

34.5 

1.27 
38.1 

1.12 
44.8 

1.26 
50.4 

1.19 
59.5 

1.35 
675 

1.49 
74.5 

1.25 
75.0 

1.41 
84.6 

1.56 
936 

1.33 
106 

1.51 
121 

1.66 
133 

.340 
.850 

.635 
4.76 

705 
5.29 

.0001H 

.383 
.957 

1.40 
56.0 

.000133 
.000152 

.422 
1.05 

.459 
1.15 

.494 
1  23 

.626 
3.13 

.676 
3.38 

.724 
3  62 

.770 
3.85 

.771 
5.78 

.832 
6.24 

.889 
6.67 

.943 
7.07 

.886 

.954 
9.54 

1.02 
10.2 

1.08 
10.8 

1.06 
15.9 

1.14 
17.1 

1.22 
183 

1.29 
19.3 

1.19 
23.8 

I.tt 

25.6 

1.37 
27.4 

1.45 
29.0 

1  30 
32.5 

1  39 
34.7 

1.49 
37.2 

1.57 
39.2 

1.38 
41.4 

1.48 
44.4 

1.52 
608 

1.63 
65.2 

1.73 

69.2 

1.83 
73.2 

1.61 
80.5 

1.69 
101 

1.80 
144 

1.94 
155 

2.06 
165 

2.17 
174 

1.73 
86.5 

1.81 
109 

1.93 
116 

2.04 
122 

.000170 

1.57 
47.1 

1.85 
92.5 

.000189 

.526 
1.31 

1.68 
50.4 

1  95 
97.5 

1.5 
2. 

.000284 

.670 
1.67 

.790 
1.97 

.970 

4.85 

1.18 
8.85 

1.35 
13.5 

1.60 
24.0 

1.80       1.95 
36.0      48.7 

2.09       2.27 
41.8       56.7 

2.08 
62.4 

2.41 
72.3 

227 
90.8 

2.64 
106 

2.41 
120 

2  80 
140 

2.52 
151 

2.93 
175 

2.70 
216 

3.14 
251 

3.53 

282 

3.86 

4.18 
334 

4.47 
357 

4.74 
379 

500 
400 

5.93 

474 

7.12 
570 

.000379 

1.13 
5.65 

1.38 
10.3 

1.58 
15.8 

1.87 
28.0 

2.5 
3. 
3.5 
4. 

4.5 
5. 

7. 
10. 

.000473 

.896 
2.24 

1.25 
6.25 

1.56 
11.7 

1.71 

12.8 

1.86 
14.0 

1.77 
17.7 

1.95 
19.5 

2.12 
21.2 

2.10 
31.5 

2.35 
47.0 

2.59 
51.8 

2.84 
56.8 

2.55 
63.7 

2.80 
70.0 

3.04 

7fiO 

2.71 
81.3 

2.96 
118 

3.15 
157 

3.46 
173 

3.29 
197 

.000568 

.992 

2.48 

1.08 
270 

1.41 

7.05 

1.53 
7.65 

2.32 
348 

2.51 
37.6 

2.98 
89.4 

3.23 
96.9 

3.26 
130 

8.53 
141 

3.61 
217 

3.91 
235 

.000663 

3.75 
187 

4.01 
200 

.000758 

1.16 
2.90 

1  65 

8.25 

1.76 
8.80 

1.99 
14.9 

2.27    j   2.69 
22.7       40.3 

3.01 

60.2 

3.26 
81.5 

3.46 
86.5 

365 
91.2 

4.34 
108 

5.21 
130 

3.46 
104 

3.79 

148 

4.01 
160 

4  24 

170 

5.03 
201 

6.04 
242 

4.19 

251 

.000852 

1.24 
3.10 

2  12 
159 

2.24 
16.8 

2.68 
20.1 

3.22 
24.1 

2.41 
24.1 

2.55 
25.5 

3.04 

no.  4 

3.65 
36.5 

2.86 
42.9 

3.02 
45.3 

3.59 
53.8 

4.32 
64.8 

3.19 
63.8 

3.37 

1.7.4 

4.01 
80.2 

4.82 
96.4 

3.68 
110 

3.88 
116 

4.61 
138 

5.54 
166 

4  26 
213 

4.50 
225 

4.45 
267 

4.70 
282 

.000947 

1.31 
3.27 

1  57 
3.92 

1.90 
4.75 

1.85 
9.25 

2.22 
11.  1 

2.67 
13.3 

.001326 
.001894 

534 

267 

6.41 
320 

5.58 

335 

6.69 
401 

For  mean  vel  in  other  rectangular  chain 

Below  will  he  seen  how  K litter's  vels  differ  from 


Fall  .1  ft  per  mile. 


See  his  formula,  p  650. 
'Table,     j    Kutter.     Depth  Ft. 


iels,  see  Table  18. 

the  above,  assuming  his 


.068 
.216 
.678 


.25 
1.00 
8.0Q 


Fall  10  ft  per  mile.  ! 

•«         "  " 


Table. 
1.90 
8.65 


Kutter. 
1.03 
2.73 
C.86 


568 


HTDRAULICS. 


TABLE  18.    Of  multipliers  for  vels  only  in  canals  of  widths 
other  than  1O  feet.    (Original.) 


Depths. 

Feet. 

5ft. 
Wide. 

15ft. 
Wide. 

20ft. 
Wide. 

25ft. 
Wide. 

40ft. 
Wide. 

60ft. 

Wide. 

80ft. 
Wide. 

100ft. 
Wide. 

8 

.77 

1.13 

1.22 

1.23 

1.40 

1.47 

1.51 

1.54 

6 

.79 

l.il 

1.19 

1.25 

1.33 

1.39 

1.42 

1.45 

6 

.80 

1.10 

1.17 

1.22 

1.2tf 

1.34 

1.37 

1.39 

1 

.81 

1.0!* 

1.15 

1.18 

1.25 

1.2d 

1.30 

1  32 

3 

.83 

1.07 

1.12 

1  15 

1.21 

1.23 

1.25 

1.26 

2 

.86 

1.05 

l.Oil 

l.ll 

1.15 

1.16 

1.17 

1.18 

1 

.91 

1.03 

l.uti 

1.U7 

1.0* 

1.0* 

1.10 

1.10 

TO.  a 


R333 


Art.  24.  HEM.  1.  In  calculations  of  this  kind,  if  the  channel 
is  not  of  masonry,  or  timber,  care  must  be  taken  that  the 
bottom  vel  is  not  so  great  as  to  wear  away  the  soil.  A  reference 

to  the  table  on  p  563  Art  18,  will  show  if  there  is  any  such  danger.  If  so,  artificial  means  must  be 
applied  to  protect  the  channel- way  ;  or  it  may  be  advisable  to  reduce  the  rate  of  fall;  aud  iucreasa 
the  cross-section  of  the  channel ;  so  as  to  secure  the  same  disch,  but  with  less  vel. 

A  liberal  increase  should  also  be  made  in  the  dimensions 
of  such  channels,  to  compensate  for  obstructions  to  the  flow, 
arising  from  the  growth  of  aquatic  plants,  or  deposits  of  mud  trom 

rain-washes,  &c  ;  or  even  from  very-  strong  winds  blowing  against  the  current.  See  also  Rem,   p  562. 

RKM.  2.  The  two  preceding  tables  will  also  serve  for  giving, 
approximately,  tiie  vel  and  disch  in  a  channel  of  trapezoidal 
Cross-section,  as  rnnop.  Fig  33.  For,  in  a  trapezoidal  channel,  of  any 
such  form  and  proportions  as  would  be  likely  to  be  employed  in  prac- 
tice, the  vel  will  not  be  more  than  from  2  to  6  per  cent  te'xs  than  in  a 

\4-^  r—  **i~7          equal  to  the  average  width  of  the  trapezoidal  one.     If  we  assume  it 

p^1 —     •          — --Y  at  4  per  cent  less,  we  will  be  within  2  or  3  per  cent  of  the  truth  in  auy 

.  I  \  /!..  ordinary  case.     It  is  rarely  that  a  nearer  approximation  is  required. 

**     p  0     U.          Now,  suppose  we  wish  to  know  approximately  the  mean  vel  of  all  the 

*   —     —  water  in  a  channel,  mnop,  of  any  ordinary  trapezoidal  form;  the 

mean  width  of  which,  z z,  is  50  ft;  its  depth,  oo,  6  ft:  and  which 
falls  at  the  rate  of  1.5  ft  per  mile.  Here  we  see  by  Table  17.  that  in  a 
rectangular  canal  10  ft  wide,  6  ft  high,  and  falling  at  the  rate  of  1.5 
ft  p«r  mile,  the  vel  Is  2.52  ft  per  sec.  And  by  Table  18,  we  see  that  in  a  rectangular  channel,  abed, 
50  ft  wide,  and  6  ft  deep,  the  vel  is  about  1.36  times  that  of  the  first;  or  2.52  X  1.36  =  3.43  ft  per  sec. 
But  in  the  trapezoidal  canal,  mnop,  the  vel  will  be  a  trifle  less  than  in  the  rectangular  one,  abed, 
of  the  same  area  of  water-way;  because  the  wet  border  of  the  former  is  greater  than  that  of  the 
latter,  and  therefore  opposes  more  resistance  to  the  flow  of  the  water.  But  as  we  now  desire  only 
an  approximation,  we  will  assume  it  to  be  4  per  cent  less  than  in  the  rectangular  one;  or,  in  other 
words,  that  it  is  only  .96  of  the  rectangular  one.  This  will  give  us  for  the  vel  in  the  trapezoidal 
channel,  3.43  X  .96  =  3.29  ft  per  sec. 

Calculation  by  the  formula,  gives  3.32  ft,  if  the  side-slopes  are  2  to  1 ;  and  3.40  ft,  if  they  are  as 
•teep  as  1  to  1 . 

KKM.  3.  Experiments  have  not  yet  accurately  determined  the  dist  through  which  either  a  canal, 
or  a  pipe  of  given  uniform  dimensions,  head,  aud  slope,  must  be  extended  before  uniform  m»au  vel 
will  be  obtained,  by  the  resistance  arising  from  the  friction  of  the  bed.  and  the  viscidity  of  the  water 
itself,  becoming  equal  to  the  impelling  force  of  gravity.  When  uniformity  of  mean  vel  has  com- 
menced, then  the  water  in  the  pipe,  or  other  channel,  is  said  to  be  in  train,  or  to  have  uniformity 
of  motion.  It  is  said  to  have  permanency  of  motion,  when  equal  quantities  flow  in  equal  times 
through  all  its  cross-sections,  whether  those  sections  have  equal  areas,  or  not.  In  the  latter  case, 
the  mean  vel  through  diff  sections  must  evidently  vary  inversely  as  the  areas  of  the  respective  section*. 

REM.  4.  Water  running  in  a  channel  with  a  horizontal  bed, 
or  bottom,  cannot  have  a  uniform  vel,  or  depth,  throughout 
its  course;  because  the  action  of  gravity  due  to  the  inclined  plane  of  a  sloping 
bottom,  is  wanting  in  this  case ;  and  the  water  can  flow  only  by  forming  its  surface  into  an  inclined 
plane;  which  evidently  involves  a  diminution  of  depth  at  every  successive  dist  from  the  reservoir. 

REM.  5.  Mr  Wicksteed,  an  experienced  English  hydrauli- 
cian,  gives  the  following  table  of  the  least  vels  and  grades 
or  falls,  to  be  given  to  drain-pipes  and  sewers  in  cities,  in  order 

that  they  may  under  ordinary  circumstances  keep  themselves  clean,  or  free  from  deposits.  He  re- 
commends that  no  drain  pipe,  even  for  a  single  common  dwelling,  shall  be  less  than  6  ins  diam. 


Diam. 
in  Inches. 

Vel.  in  ft. 
per  Min. 

Grade, 
1  in 

Grade. 
Feet  per 
Mile. 

Diam. 
in  Inches. 

Vel.  in  ft. 
per  Min. 

Grade, 
1  in 

Grade. 
Feet  per 
Mile. 

4 
6 
7 
8 
9 
10 
11 
12 
15 

240 
220 
220 
220 
220 
210 
200 
190 
180 

36 
65 
76 
87 
98 
119 
145 
175 
244 

146.7 
81.2 
695 
60.7 
53.9 
44.4 
36.7 
30.2 
21.6 

18 
21 
24 
30 
36 
42 
48 
54 
60 

180 
180 
180 
180 
180 
180 
180 
180 
180 

294 
343 
392 
490 

588 
686 
784 
882 
980 

18  0 
15.4 
13.5 
108 
9.0 
7.7 
6.8 
6.0 
5.4 

HYDRAULICS. 


569 


The  following  table  was  prepared  by  Mr  John  Roe,  Chief  Sur- 
veyor to  the  Metropolitan  Commission  of  Sewers,  London.  It  is  based  upon  20  years  of  close  practical 
observation  of  sewers  themselves;  and  confirmed  by  special  observations  since;  including  experi- 
mental gaugiugs  of  sewers  during  both  short  and  long  storms:  and  aided  bv  rain-gauges,  and  every 
precaution  to  insure  accuracv.  The  table  is  approved  by  the  celebrated  hydraulic  engineer,  Mr  Hawks- 
ley,  by  Mr  Bidder,  aud  other  practical  English  engineers.  See  Cull  1 1O1I.  p  566. 

Table  showing*  the  number  of  acres  of  covered  (as  in  cities) 
surface,  from  which  circular  sewers,  with  junctions  properly  connected  by  easy 
curves,  will  convey  away  the  water  fall  of  a  rain  of  one  inch  per  hour;  with  house 
drainage  besides.  This  is  a  very  heavy  rain.  See  Rain,  p.  618. 


Inclination, 
Fall,  or  Slope 
of  Sewer. 

Inner  Diameter,  or  Bore  of  Sewer  in  Feet, 

2 

2K 

3 

4 

5 

6 

7 

8 

9 

10 

Level  

39 
43 
50 
63 

78 
90 
115 

67 
75 
87 
113 
143 
165 
182 

120 
135 
155 
203 
257 

318 

277 
308 
355 
460 
590 
670 
730 

570 
630 
735 
950 
1200 

1500 

1020 
1117 
1318 
16D2 
2180 
2486 
2675 

1725 
1925 
2225 

2875 
3700 
4225 
4550 

2850 
3025 
3500 
4500 
5825 
6625 
7125 

4125 
4425 
5100 
6575 

7850 

5825 
6250 
7175 
9250 
11050 

K  iuch  in  10ft;  or  lin  480.. 
%  inch  in  10  ft  ;  or  1  in  240.  . 
94  inch  in  10ft;  or  1  in  160.. 
1  inch  in  10  ft;  or  1  in  120.. 
X  inch  in  10ft;  or  lin  80... 
2  ins  in  10  ft;  or  1  in  60.... 

A  12-ft  culvert  will,  within  the  limits  of  the  table,  discharge  nearly  \%  times  as  much  as  the  10-ft  one. 
A  square  mile  contains  640  acres. 
Mr  Roe  gives  also  the  following 

Table  of  number  of  acres  of  town  area  that  may  be  drained 
by  small  circular  drains.*  Calculated  for  a  rainfall  of  two  ins  per  hour, 
as  based  on  his  own  actual  observations ;  also,  the  rates  of  fall  or  inclination  which 
the  drains  must  have.  Such  a  rain  occurs  very  rarely.  See  Caution,  p  566. 


Acres. 

Bore  of  Drain  in  Ins, 

Acres. 

Bore  of  Drain  in  Ins, 

3 

4 

5 

6 

7 

8 

8 

9 

12 

15 

18 

X 

tit 

/ 

l!2 
1.5 
1.8 
2.1 

Fall 
lin 
120 
20 

Fall 
1  in 

Full 
1  in 

Fall 
lin 

Fall 
lin 

Fall 
lin 

2.1 
2  5 

Pall 

60 

Pull 

1  in 

120 
80 

Fall 
1  in 

F;ill 
1  in 

Fall 
1  in 

120 

40 

2  75 

60 

30 
20 

80 
60 
20 

4  5 

120 

5  3 

80 

!'!.. 

60 
40 
20 

5.8 

7  8 

60 

210 

20 

« 

120 

120 
80 
60 

9. 
10. 

17. 

80 

60 

240 
120 

| 

.    . 

*  Weight   and   price  per  foot  rnn   of  terra  cotta  vitrified 
pipes  for  drains,  Are;  Philada,  188O. 


Bore. 

Wt. 

Price. 

Bore. 

Wt. 

Price. 

Bore. 

Wt. 

Price. 

Bore. 

Wt. 

Price. 

Ins. 

Lbs. 

Cts. 

Ins. 

Lba. 

Cts. 

Ins. 

Lbs. 

Cts. 

Ins. 

Lbs. 

Cts. 

2 

4 

10 

6 

18 

28 

15 

45 

100 

30 

150 

400 

3 

7 

14 

8 

22 

40 

18 

65 

140 

36 

195 

4 

5 

10 
12 

18 
24 

10 
12 

30 
33 

60 

21 
24 

89 
100 

180 
240 

42 

48 

203 

230 

order. 

Up  to  bores  of  12  ins,  a  bend,  or  a  branch,  costs  about  as  much  as  5  ft  of  pipe.  These  pipes,  together 
with  branches,  elbows,  <fec;  also,  chimney-flues,  chinmey-tops,  roofing-tiles,  vases,  and  other  articles 
in  great  variety  and  excellent  quality,  are  made  at  the  extensive  Moorhead  Clay  Works,  Spring  Mill: 
William  I,  Wilson,  proprietor;  office  No  11  South  Seventh  St,  Phila.  Importers  also. 

The  ends  of  the  pipes  arc  shaped  much  as  the  iron  one  in  Fig  38;  and  the  joints  are  filled  with 
cement  mortar ;  or  merely  with  clay  when  used  for  drainage  only.  Up  to  12  ins  bore,  the  pipes  are 
about  yt  inch  thick ;  48  inch  bore,  about  2  ins  thick. 

Similar  pipes  are  also  made  of  hydraulic  cement. 


570 


HYDRAULICS. 


Art.  25 


>.   Obstructions  by  piers.   When  the  area  of  section  of  a  channel 
of  running  water,  is  diminished  at  any  point  by 


narrowing  the  stream;  or  by  placing  in  it  piers, 
such  as  those  of  a  bridge.  &c,  then  the  same 
quantity  of  water  which  formerly  flowed  through 
the  wider  channel,  has  to  force  its  way  through  the 
narrower  one.  This  it  can  do  only  bv,  as  it  were, 
heaping  itself  up.  at  a  short  dist  above  the  obstruc- 
tion, so  as  there  to  form  a  head,  as  co.  Fig  34,  suffi- 
ciently great  to  overcome  the  increased  resistance; 
and  thereby  f  >rce  along  the  same  quantity  of  water 
as  before;  but  of  course  at  an  increased  vel.  If 
the  straight  Hue  ab  represent  the  original  water 
surf,  then  in  ordinary  oases,  the  curved  line  a  *  c  b 
will  approximately  represent  the  new  one,  (but  la 
an  exaggerated  manner,)  as  produced  when  the 
ends  of  the  pier  are  properly  rounded. 


ends  or  tne  pier  are  properly  rounded. 

REM.  1.    It  is  important  to  bear  in  mind,  that  before  the  erection 

of  an  obstruction,  the  vel  of  the  original  stream  is  greatest  at  the  surf;  aud  diminishes  gradually 
toward  the  bottom;  but  when  a  pier.  &c,  is  built,  the  increased  vel  produced  by  the  head  thereby 


created,  is  actually 
much  greater  we 


the  surf;  but  is  nearly  uniform  throughout  the 
action  upon  the  bottom  than  before  existed,  or  tha 


Architecture  ;  and  must  be  looked  upon  merely  as  probable  approximations.     They  suppose  the  pier, 
&c,  to  be  properly  rounded  or  pointed  at  their  upstream  ends,  so  as  to  give  as  free  a  passage  as  pos- 


TABLE  19.    Of  heads  produced  by  obstructions  to  streams. 


Original  Vel. 
of  Stream.* 

Kind  of  Bottom 
which  begins  to 
wear  away  under 
Bottom  Vel.  equal 
to  those  in  the 
first  three  cols. 

I 

TV 

Vopoi 

iV 

tion  o 
occnp 

i 

f  Are 

ed  ty 

i 

a  Of  0] 

ftheO 

>igina 
bstrtu 

i 

I  Water-w£ 
jtions, 

i|  t 

T' 
f 

*"•-•  *£. 

Ooze,  and  Mud... 
Clay  

He 

.0003 
.0011 
.0045 
.0182 
.0409 
.0728 
.1137 
.1638 
.4550 

id  of 

.0004 
.0014 
.0056 
.0225 
.0507 
.0902 
.1410 
.2030 
.56*0 

Water 

.0004 
.0017 
.0039 
.0276 
.0621 
.1104 
.1725 
.2484 
.6901 

prod 

.0006 
.OSI23 
.0091 
.0364 
.0819 
.1456 
.2275 
.3276 
.9100 

nc^d 
Fee 

.001 

.0<H 
.015 
.060 
.135 
.240 
.375 
.540 
1.50 

attht 

»t, 
.0014 
.0058 
.0231 
.0924 
.2079 
.3696 
.5775 
.8316 
2.310 

>  Obsti 

.0033 
.0133 
.0532 
.2128 
.4788 
.8412 
1.320 
1.915 
5.280 

notion 

.0067 
.0267 
.1069 
.4276 
.9621 
1.710 
2.672 
3.848 
10.69 

sj  in 

.0162 
.06*6 
.2584 
1.036 
2.326 
4.144 
64T5 
9.304 
25.9 

Ins. 
3 
6 
12 
24 
36 
48 
60 
72 
120 

Ft. 

M 
K 

1 

2 
3 
4 
5 
6 
10 

Miles. 
.170 
.341 
.681 
1.36 
2.04 
2.72 
3.41 
4.09 
6.81 

Sand  

Gravel  
Small  Shingle.... 
Large       "       .... 
Soft  Shistiis  
Stratified  Rocks.. 
Hard  Rocks  

See  Table,  Art  18. 

TABLE  2O.  Increased  velocities  produced  at  and  by 
rounded,  or  pointed  obstructions.  If  square,  these  vels  must,  accord- 
ing to  Nicholson,  be  increased  %  part. 


of  Stream.* 

Proportion  of  Area  of  Watei 

A  1  A  1    i    1    * 

'-way,  occnpied  by  the  Obstructions, 

i  i  i  I  i  I  f  I  f 

Per  Sec. 

Per 
Hour. 

V 

.28 
.56 
1.13 
2.27 
3.39 
4.54 
5.60 
6.78 
11.3 

Blocity 

.29 
.58 
1.16 
2.33 
3.48 
4.66 
5.80 
6.96 
11.6   - 

jroduce 

.30 
.60 
1.20 
2.40 
3.60 
4.80 
6.00 
7.20 
12.0 

d  at  the 

.32 
.64 
1.26 
2.52 
3.78 
5.04 
640 
7.56 
12.6 

Obstrn 

.35 
.70 
1.40 
2.80 
4.20 
5.60 
7.00 
8.40 
14.0 

ction  in 

.394 
.788 
1.58 
3.16 
4.74 
6.32 
7.88 
9.48 
15.8 

Feet  p< 

.52 
1.05 
2.1 
4.2 
«.3 
8.4 
10.5 
12.6 
21.0 

jr  Fecon 

.7 
1.4 
2.8 
5.6 
8.4 
11.2 
14.0 
16.8 
28.0 

d, 

1.05 
2.1 
4.2 
8.4 
12.6 
1*5.8 
21.0 
25.2 
42.0 

Ins. 
3 
6 
12 
24 
36 
48 
60 
72 
120 

Ft. 

u 

i 
• 

6 
10 

Miles. 
.170 
.3(1 
.681 
1  36 
2.04 
2.72 
3  41 
4.09 
6.81 

*  A  very  vagne  expression.    Does  it  refer  to  the  greatest  surface  vel  at  mid-channel ;  or  to  the  me*a 
rel  of  the  entire  cross  section? 


HYDRAULICS.  571 

Art.  Sft^l'he  resistance  of  water  against  a  flat  surface  mov- 
ing- ttfrrousrli  it  at  rig-tit  angles,  is  nearly  as  the  squares  of  the  vel :  and, 
according  to  Hutton,  its  amount  in  tbs  per  sq  ft  approx  =  Square  of  vel  in  ft  per 
sec.  Or  like  the  pres  of  a  running'  stream  against  a  perp  tixed  fiat 

surface,  it  is  =  wt  of  a  col  of  water  whose  base  —  pressed  surf,  and  whose  ht  — head  due  to  the  vel  as 
per  table  p  552. 

The  resist  of  a  sphere  is  to  that  of  its  great  circle  about  as  1  to  2.9. 

When  the  moving  surf,  instead  of  being  at  right  angles  to  the  direction  in  -which  it  moves,  forms 
another  angle  with  it,  the  resistance  becomes  less  in  about  the  following  proportions.  Therefore, 
when  the  surf  is  inclined,  first  calculate  the  resistance  as  if  at  right  angles:  and  then  mult  by  the 
following  decimals  opposite  the  angle  of  inclination  : 


90°.. 

..1.00 

60°... 

.  .88 

40°.  .  . 

.  .58 

20°.  . 

.  .16 

80  .. 

..  .98 

55  ... 

.  .83 

35  ... 

.  .46 

15  ... 

.  .10 

70  .. 

..  .95 

50  ... 

.  .76 

30  ... 

.  .34 

10  .. 

.  .06 

65  .. 

..  .92 

45  ... 

.  .68 

25  ... 

.  .24 

5  .. 

.  .02 

The  scour,  or  abrading-  power  of  moving  water  is  considered  to 

be  as  the  square  of  its  vel. 

Art.  27.    To  calculate  the  horse-power  of  falling*  water,  on 

the  ordinary  assumption  that  a  horse-power  is  equal  to  33000  Bbs  lifted  1  foot  vert  per  min.  That  of 
average  horses  is  really  but  about  %  as  much,  or  '22000  R>s,  1  foot  high  per  min.  Mult  together  the 
number  of  cub  ft  of  water  which  fall  per  min  ;  the  vert  height  or  head  in  feet,  through  which  it  falls ; 
and  tbe  number  62.3,  (the  wt  of  a  cub  ft  of  water  in  Bbs  ;)  and  div  the  prod  by  33000.  Or,  by  formula, 

cub  ft     v          vert          v    ll»s 

9  The  number  of  __  per  min   *   height  in  ft  A   62.3 

horae-poivera  '  33000 

Ex.  Over  a  fall  16  ft  in  vert  height,  800  cub  ft  of  water  are  disckd  per  min.  How  many  horse- 
powers does  the  fall  afford  ? 

•  cub  ft       ft         Ibs 

800  X  16  X  «2. 3         797440 

HerC'    33000 =    33000- 

vFater-wheelS  do  not  realize  all  the  power  inherent  in  the  water,  as  found  by  our 
rule.  Thus,  undershots  realize  but  from  y±  to  % ;  breast-wheels.  ^ ;  overshots,  from  %  to  %  ;  tur- 
bines, %  to  .85  of  it ;  according  to  the  skill  of  design,  and  the  perfection  of  workmanship.  Even  when 
the  wheel  revolves  in  a  close-fitting  casing,  or  breast,  elbow  buckets  give  considerably  more  power 
than  plain  radial  or  center-buckets.  Of  the  power  actually  received  by  a  wheel,  part  is  expended  in 
friction,  <tc ;  while  the  remainder  does  the  useful  or  paying  net  work  of  raising  water,  grinding 
grain,  sawing,  &c. 

Observations  by  Oeiil  Haiipt,  in  1866,  gave  the  following  results  for  a 
small  hydraulic  ram.  Head  of  water  to  ram  =  8.812  It;  diam  of  drive-pipe  = 
1%  ins;  length  15  ft.  Diam  of  delivery-pipe  -  %  inch;  length  200  ft.  Vert  height  to  which  the 

which  worked"  the  ram  =  768  cub  ins,  =  3.31  gails,"r:  2T.78  ft/per  min.     Quantity  raised  63.4  ft  high 

Ibs  water         ft  ft-fos 

per  min,  =r  48  cub  ins,  =  1.736  fts.     Hence  the  power  expended  per  min,  waa  27.73  X  8.812  ir  244.35. 

Iba  water        ft  ft-fta 

And  the  useful  effect,  was  1.736  X  63.4  =  110.06.    Hence  the  ratio  which  the  useful  effect  bears  to  the 

power  in  this  instance,  is  ^jr^  °r  -*5.  The  actual  power  of  the  ram  is,  however,  greater  than  this, 
inasmuch  as  it  has  to  overcome  the  friction  of  the  water  along  the  delivery-pipe.* 

To  find  the  horse-power  of  a  running-  stream.    Water-wheels 

with  simple  float- boards, t  instead  of  buckets,  are  sometimes  driven  by  the  mere  force  of  the  ordinary 
natural  current  of  a  stream,  without  any  appreciable  fall  like  that  in  the  foregoing  case.  In  such 
cases,  we  must  substitute  the  virtual  or  theoretic,  head  ;  which  is  that  which  would  impart  to  it  the 
same  vel  which  it  actually  has.  This  virtual  head  may  be  taken  at  once  from  Table  10.  Thus,  a 
stream  has  a  vel  of  2.386  miles  per  hour;  or  210  ft  per  min  ;  or  3>£  ft  per  sec  ;  and  in  the  column  of 
heads  in  Table  10,  opposite  to  3.5  vel  per  sec,  we  fiurl  the  reqd  head  .190  of  a  ft.  Having  thus  found 
the  head,  we  must  now  h'nd  the  quantity  of  water  which  passes  any  given  area  of  the  stream  in  a 
aiin.  Thus,  suppose  that  the  immersed  part  of  a  float  when  vert  is  5  ft  Ion?,  and  1  ft  wide  or  deep  ; 
then  the  area  of  this  part  which  receives  the  force  of  the  current,  is  5  X  1  :~  5  square  feet.  Hence, 

area  vel 

5  sq  ft  X  210  —  1050  cub  ft  per  min.  Having  now  the  cub  ft  per  min,  and  the  vert  height  or  head, 
the  number  of  horse- powers  of  the  stream  of  the  given  area,  is  found  by  the  foregoing  rule,  or  formula. 


*  A  committee  of  the  Franklin  Institute,  in  ISflO.  gave  .71  as  the 

coefficient  for  a  ram  at  the  Girard  College,  in  which  the  diam  of  drive-pipe  was  2^  ins :  its  length, 
160  ft;  fall.  14  ft.  Delivery-pipe,  1  inch  diam  ;  2260  ft  long:  vert  rise,  or  height  to  which  the  water 
was  raised.  93  ft.  No  details  of  the  experiment  are  given.  Some  large  rams  in  France  give  it  usrful 
effect  of  from  .6  to  .65  of  the  whole  power  expended.  It  is  an  excellent  machine  for  many  purposes; 
and  is  somettrnes  used  for  filling  railway  tanks  at  water  stations. 

t  Such  wheels,  for  floating-  mills,  in  Europe,  rarely  exceed  15  ft 

diam.  Whatever  the  diam,  they  may  have  about  18  to  20  floats.  The  floats  are  from  8  to  16  ft  long; 
and  about  ^  to  ^  as  deep  as  the  diam  of  the  wheel.  They  should  not  dip  their  entire  depth  into 
the  water,  but  nearly  so.  They  should  not  be  in  the  same  straight  line  with  the  radii ;  but  should 
incline  from  them  30°  up  stream,  to  produce  their  full  effect.  All  these  remarks  applv  to  wheels 
moving  freely  in  a  wide  or  indefinite  channel ;  as  in  the  case  of  a  floating  mill,  built  on  a  scow,  and 
anchored  out  in  a  stream  :  but  not  to  wheels  for  which  the  water  is  dammed  up,  and  acts  with  a  prac- 
tical fall.  No  great  exactness  is  to  he  expected  in  rules  on  this  subject.  The  best  vel  for  tht  wheel 
if  about  .4  that  of  the  stream. 


572 


HYDRAULICS. 


cub  ft  per  min 
1050 


ert  ht  in  ft 


Ibt 
62.3 


33000  ~  33000  ~  ' 

But  in  practice  the  wheels  actually  realize  but  about  -j^j  of  this  power  of  the  stream,  when  working 
In  an  open  channel ;  and  still  less  when  the  water  flows  with  the  same  vet  through  a  narrow  artificial 
chanuel,  but  little  wider  than  the  wheel.  Therefore,  the  actual  power  of  our  wheel  will  be  but  .377  X 
.4  r:  .1508;  or  about  y  of  a  horse-power;  or  33000  X  .1508  =  4976  ft-fts  per  min.  Making  a  rough 
allowauce  for  the  friction  of  the  machine  at  its  journals,  &c,  we  should  have  say  about  4400  ft-fts  of 
useful  power ;  that  is,  the  wheel  would  actually  raise  about  440  B)s  10  ft  high  ;  or  44  fts  100  ft  high,  &c, 
per  uiiu.  The  vel  of  the  stream  must  not  be  measured  at  the  surface ;  but  at  about  %  of  the  depth  m 
which  the  floats  are  to  dip,  or  be  immersed.  This,  however,  is  chiefly  necessary  in  shallow  streams, 
in  which  the  depth  of  the  float  bears  a  considerable  ratio  to  that  of  the  water. 

This  power  of  a  running;  stream,  (for  any  given  area  of 
transverse  section,)  increases  as  the  cubes  of  the  vels;  for,  as 

we  have  seen,  the  power  in  ft-Bbs  per  min  is  found  by  mult  together  the  weight  of  water  which  passes 
through  the  section  in  a  miu,  and  the  virtual  head  in  ft;  and  since  this  weight  increases  as  the  vel, 
and  this  head  as  the  square  of  the  vel,  the  prod  of  the  two  (or  the  power)  must  be  as  the  cube  of  the 
vel.  Therefore,  if  the  vel  in  the  foregoing  case  had  been  10.5  ft  per  sec,  or  3  times  3.5  ft,  the  power 
of  the  wheel  would  have  been  27  times  as  great,  or  .1508  X  27  —  4.07  horse-powers. 

Art.  28.  Miscellaneous  on 
city  water-pipes.      The  pipes 

are  laid  to  conform  to  tue  vert  uudulatiou.i  of 
the  street  surfaces.  In  Philadelphia  there  is 
about  one  mile  of  street  pipe  to  each  1300  in- 
habitants. The  tops  of  tiie  pipes  are  laid  not 
less  than  3%  feet  below  the  surface  of  tA 
street;  but  in  3-inch  pipes  the  water  has  at 
times  been  frozen  at  that  depth.  8tOp- 
ValveS,  or  yateS,  opening  vertically 
in  grooves,  are  placed  across  the  street  pipes 
at  intervals  of  from  100  to  300  yds.  Their  use 
is  to  shut  off  the  water  from  any  section  dur- 
ing repairs  ;  the  water  of  such  sections  being 
allowed  to  run  to  waste,  and  to  soak  into  the 
ground.  Figs  35,  36,  36>£,  and  36^,  will  ex- 
C  plain  the  general  principle  of  these  valves. 
J  The  details  are  much  varied  by  diff  makers. 
Figs  36J4  and  36^  are  a  form  adopted  by  the 
Chapman  Valve  Co,  of  Boston,  Mass.*  In  it 
the  valve  itself  is  hollow. 

Fig-s  35  and    36    are    two 
styles  used  in  Philada.    The 

valve  itself,  v.  Fig  36,  is  a  circular  plate  of 
cast  iron ;  which,  when  down,  as  in  the  fig, 
closes  the  pipe.  As  in  other  styles,  it  opens 
vertically  by  means  of  a  screw  D;  the  valve 
rising  into  the  cast-iron  case  or  box  B  B.  The 
screw  is  turned  by  a  handle  fitting  on  th« 
square-head  A. 

Whatever  the  style  of  the  valve,  It  Is,  when  in  place,  protected  by  a  surrounding  box,  ii,  Fig  35, 
generally  of  plank  or  cast  iron,  with  four  vert  sides;  a  movable  iron  top,  k,  level  with  the  street; 
and  no  bottom.  This  top  is  taken  off  when  the  valve  needs  inspection.  Two  of  the  opposite  sides 
of  the  box  of  course  have  openings  for  the  passage  of  the  pipes  to  or  from  the  valve. 

Since  the  tops  of  the  pipes  when  laid  are  at  least  3l<4  ft  below  ground,  the  tops  of  the  valve-cases 
for  small  pipes.  Fig  35.  are  also  considerably  below  it;  and  for  such  pipes,  the  lifting-screw,  D,  is 
usually  attached  to  the  top  of  the  valve  itself;  and  extends  above  the  case,  as  in  the  fig;  being  sus- 
tained by  standards  II.  connected  at  top  by  cross-pieces.  On  these  cross-pieces,  is  a  nut  at  D,  so 
confined' that  it  cannot  move  vertically;  and  by  turning  which  the  screw  is  raised,  without  itself 
turning.  In  Fig  35  the  valve  is  down,  or  closed;  when  open,  the  top  of  the  screw  will  reach  nearly 
to  k:  hut  still  will  allow  the  cover  k  to  be  in  place. 

In  large  valves.  Fig  36.  which  is  for  a  30-inch  pipe,  the  screw  D  is  so  long,  that  if  it  rose  when  the 
gate  v  is  lifted  or  open,  its  top  h  would  project  above  the  street  paving.  On  this  account  it  is  not 
allowed  to  rise,  but  merely  to  turn  ;  and  in  so  doing  it  raises  the  gate  v  by  means  of  the  projection  t, 
in  which  is  fitted  a  brass  interior  screw,  or  nut,  through  which  the  lifting-screw  D  works.  The  1'ft- 
iug-screw  foots  into  a  socket  above  c ;  and  passes  through  the  top  of  the  case  at  o:  resting  upon  it  by 
means  of  the  small  enlargement  or  collar  shown  in  the  fig.  Immediately  above  this  collar,  the  screw- 
stem  is  surrounded  br  a  loose  ring,  kept  in  place  by  the  screws  ««;  and  thus  preventing  the  lifting- 
screw  from  rising  when  it  is  turned  by  means  of  a  key  or  lever,  applied  to  its  square-head  ft.  When 
so  turned,  the  valve  v  rises  up  the  screw  ;  and  enters  the  top  d  D  o  of  the  cover.  In  a  front  view,  the 
top  casting  gEg  of  the  cover,  would  present  the  upper  half  of  a  circle. 

The  three  principal  castings  which  compose  a  valve  box  or  cover,  are  bolted  together  by  means  or 
continuous  fl-inges,  as  at  q  g  g.  The  ioint  faces  of  these  castings  are  carefully  smoothed ;  and  a  thin 
strip  of  lead  is  inserted  between  them,  as  a  precaution  asain^t  leaks.  The  gates,  as  well  as  the 
grooves  in  which  they  slide,  are  usually  made  very  slightly  tapering  toward  the  bottom,  so  as  to  in 

*  No.  79  Water  St.  Boston.  The  Ludlow  Valve  Manufacturing  Co,  193  River  St.  Troy.  New  York: 
and  the  Boston  Machine  Co.  office  20  State  St,  Boston,  are  also  noted  for  their  excellent  valves.  The 
latter  make  the  Coffin's  patent  double  disk  valve,  besides  other  patterns;  and  many  styles  of  hy- 
drants, or  fireplugs.  &c;  also,  locomotive  aud  stationary  engines,  dredging  machines,  iron  bridge*, 
and  heavy  tools  and  machinery  in  general. 


Pig35 


HYDKAULIOS. 


573 


gure  closing  tightly.    The  small  oblong  rec«ss  seen  below  the  gate,  admits  small  particles  of  foreign 
matter  which  might  otherwise  prevent  the  gate  from  closing  perfectly.   The  grooves  a  a  a  a,  ID  which 


Fig  36J 


the  gate  slides,  as  well  as  those  parts  of  the  gate  which  come  into  contact  with  them,  are  carefully 
faced  with  brass,  to  diminish  friction.  At  the  top  of  the  cover,  the  screw-stem  passes  through  a 
stuffing-box,  which  prevents  leaking  at  that  point.  Very  careful  workmanship  is  required  throughout. 
The  gates,  especially  of  large  mains,  must  be  closed  very  slowly  ;  otherwise,  the  too  sudden  arrest- 
ing of  the  momentum  of  the  flowing  water  would  be  apt  to  break  either  them  or  the  covers ;  or  burst 
the  pipes.  As  a  precaution  against  this,  the  covers  for  very  large  valves  are  cast  with  outside 


about  50  ft.  Fire-plug's,  Figs  40,  are  placed  as  much  as  possible  at  summits,  so  as  to  serve 
also  for  washing  the  streets ;  and  for  the  escape  of  accumulated  air.  They  average  about  8  in  num- 
ber to  each  mile  of  pipe ;  or  1  to  each  block  of  buildings.  No  g-al VRIliC  action  has  been 
observed  where  lead  pipes  or  brass  unite  with  cast-iron  ones.  Rust  has  not  very  ma- 
terially affected  the  pipes  in  30  to  35  years  ;  even  without  any  preservative  meas- 
ures. See  Art  34.  A  coat  of  whitewash  on  the  outside,  impedes  rust  quite  appreciably.  Some  22- 
inch  ones  which  have  laid  in  Philada  for  50  years,  and  4-inch  ones  for  65  years,  are  still  in  use,  in 

fair  condition.!  No  pipe  as  small  as  3  ins  diam  is  carried  farther 

thail  4OO  feet,  without  being  connected  with  larger  ones  at  both  ends.  Even  this  dist  is  at 
times  too  great  for  so  small  a  pipe.  None  less  than  4  inches  diam  should  \>e  laid  in  cities  ;  and  even 
they  only  for  lengths  of  a  few  hundred  feet.  Their  insufficiency  is  chiefly  felt  in  case  of  fire.  Six  ins 
would  be  a  better  minimum.  No  more  leakage  occurs  in  winter  than  in  summer;  except  from  the 
bursting  of  private  service-pipes  by  freezing. 

The  service-pipes  for  supplying:  single  dwellings,  are  of  lead; 

*  The  following  table  gives  a  tolerable  average  of  the  weights  of  valves,  for  pipes  of,various  bores, 
by  different  makers ;  with  an  approximation  to  average  retail  prices  in  1873. 


Bore. 
Ins. 

Wt. 

Lbs. 

72 
105 
173 

Price. 

Bore. 
Ins. 

Wt. 
Lbs. 

Price. 

$ 

Bore. 
Ins. 

Wt. 

Lbs. 

Price. 

$ 

Bore. 
Ins. 

Wt. 
Lbs. 

Price. 

400 
450 
950 

3 
4 
6 

21 

28 
42 

8 
10 
12 

310 
575 
800 

56 
70 
96 

14 
16 
20 

1200 
1600 
2100 

175 
230 
350 

22 
24 
30 

2200 
2500 
5200 

t  To  compact  the  earth  thoroughly  against  the  pipes  excludes  air,  and  greatly  impedes  rust.    Pipes 
lay  be  corroded  by  the  leakage  of  gas  through  the  body  as  well  as  the  joints  of  adjacent  gas  pipes. 

37 


574 


HYDRAULICS. 


and  of  %  to  %  inch  bore.  They  are  connected  with  the  street 
mains  »n,  Fig  37,  by  a  brass  ferrule/,  which  is  here 
shown  at  ^  real  size.  The  dotted  lines  show  its  ^  inch  bore. 
The  ferrule  is  merely  hard  driven  into  a  slightly  conical  hole 
reamed  out  of  the  main,  as  at  «.  The  lead  pipe  o  is  attached  to 
the  other  end  of  the  ferrule  ;  overlapping  it  about  1%  ins;  and 
the  joint  soldered,  t.  The  extra  thickness  near  /,  is  for  giving 
v^  proper  shape  and  strength  for  hammering  the  ferrule  into  the 

fC-\  ft  O7  y\  main.     The  pipe  and  solder  are  shown  iu  section.     Besides  the 

stopcocks  attached  to  each  service  pipe,  and  to  its  branches 
through  the  house,  there  is  an  underground  one  by  which  the 
city  authorities  can  stop  off  the  water  in  case  ot  delinquency  iu 

payment  of  dues  :  and  another  by  which  the  plumber  can  stop  it  off  when  so  required  during  indoor 
repairs.  Of  late  years  galvanized  ll'OIl  tubes  are  being  much  used  for  service-pipes, 
especially  for  hot  water;  being  less  subject  to  contraction  and  expansion,  which  produce  leaks.  See 
near  bottom  of  page  517,  for  such  water  pipes.  Brass  service  pipes  are  now  much  used  in  Boston. 
See  bottom  of  page  517 ;  also  table,  page  3(55,  and  foot  of  377. 

In    IMiilada,   the    faucets    e  d  o, 

Fig  38.  of  the  city  pipes,  laid  until 

of  late  years,  are  of  the  following  dimensions.  The 
width  of  the  joint,  or  clear  distance  between  the  spigot 
and  the  faucet,  is  the  same  in  all ;  namely,  %  inch. 


Bo  3  8 


Inner  Diam. 

Inner  Diam. 

Dpth.mnJ        Thickness 

QO,  of  Pipe. 

ii.ofFaucet. 

of  Faucet.]  ate.  i  at  d. 

ate. 

Ins. 

Ins. 

Ins.       jlns. 

Ins. 

Ins. 

30 

32?^ 

6 

1 

l>a 

2« 

20 

22* 

6 

H 

1 

IK 

16 

18 

6 

% 

H 

IK 

11 

14 

534 

9-16 

X 

*K 

10 

11M 

5H 

H 

% 

i55 

8 

9K 

5tf 

« 

H 

i« 

6 

IX 

7-16 

% 

IM 

4 

5^ 

^A 

7-16 

% 

IK 

3 

4^ 

4 

H 

% 

1*4 

The  small  beads  at  «  and  m,  on  the  spigot  end  of  the  pipe,  project  about  }4  inCQ  :  and  are  to  prevent 
tae  calking  material  from  entering  the  pipe.  The  calking  consists  of  about  1  to  2  ins  in  depth  of  well- 
rammed  untarred  gasket,  or  ropeyarn  :  above  which  is  poured  melted  lead,  confined  from  spreading 
by  means  of  clay  plastered  around  the  joint.  The  lead  is  afterward  compacted  by  a  calking  hammer. 

The  weight  of  lead  used  per  joint,  averaged  about  as  follows :  30  ins 
diam,  107  tbs  ;  20  ins,  60  fts  ;  18  ins,  50  fts  ;  16  ins,  35  Its ;  12  ins,  24  fts  ;  10  ins,  16  fts ;  8  ins,  13  fts  ; 
6  ins,  10  fts;  4  ins,  8  fts ;  3  ins,  5  fts.  Hydraulic  engineers  are,  however,  now  pretty  generally  re- 
ducing the  depth  of  faucets  about  %  part;  and  the  depth  of  the  lead  joint  to  2  or  2^£  inches,  and  its 
wt  to  2  Bis  per  inch  of  diam  for  pipes  of  any  diam  under  30  ins.  Faucets  are  now  frequently  made 
as  shown  at  W  ;  leaving  for  the  lead  joint  a  thickness  of  %  inch  at  top,  and  -j-^g  inch  below.  At  the 
average  »f  .4  inch,  and  a  depth  of  2^  ins,  we  have  about  1}^  fts for  every  inch  of  outer  diam  of  pipe;  and 
for  an  entire  joint  as  follows,  the  diams  being  the  inner  ones:  2  ins,  3.67  fts:  3  ins,  5  fts;  4  ins,  6.7  fts; 
6  ins,  9.3  fts  ;  8  ins,  12  fts ;  10  ins,  15  fts  ;  12  ins.  17.7  fts  ;  14  ins,  20.7  fts  ;  16  ins.  23.3  fts ;  18  ins,  26  fts ; 
20  ins,  28.6  fts  ;  24  ins,  34.3  fts  ;  30  ins,  43  fts.  The  form  of  joint  at  W  is  said  to  prevent  to  some  extent 
the  tendency  of  the  lead  to  creep,  or  work  loose  by  contraction  and  expansion.  Pipes  are  now  often 
made  12  ft  long,  instead  of  9  ft  as  formerly ;  thus  requiring  only  about  452  joints  11  ft  8  ins  apart  per 
mile,  instead  of  about  610  joints  8  ft  8  ins  apart ;  thereby  reducing  the  expense  for  lead  very  materially. 

Mr  Thomas  Wicksteed,  engineer  of  the  East  London  water-works,  England,  says  that  more  than 
50  years'  experience  proves  that  slightly  tapering  wedges  of  pine,  about  4  ins  long,  2  ins  wide,  and  S| 
ins  thick  at  the  butt,  carefully  shaped  to  suit  the  curve  of  the  pipe,  and  well  driven,  answer  all  the 
purposes  of  lead  joints,  at  considerably  less  cost. 

Cost  of  water  pipes  9  ft  long*,  per  mile,  laid  ;  assuming  the  pipes 

to  cost  3  cts  per  ft,  or  $67.20  per  ton,  delivered  along  the  streets;  lead  for  the  joiuts,  of  the  weights 
formerly  used  in  Philada,  at  10  cents  per  ft ;  and  the  laying,  (including  digging  and  refilling  the 
trenches,  in  earth;  making  the  joints,  including  the  gasket  yarn  for  that  purpose.)  according  to  rates 
which  experience  has  shown  to  be  fair  when  common  labor  costs  $1  per  day.  A  small  addition  must 
be  made  if  unpaving  and  repaving  are  to  be  done.  Owing  to  the  constant  fluctuations  in  prices,  such 
tables  answer  but  an  imperfect  purpose.  In  1873  the  laying  would  be  nearly  twice  as  great. 


Diam. 

Thick's. 

Tons 

Cost  per 

Lead. 

Lead. 

Lnvini? 

Laving. 

Total. 

Ins. 

Ins. 

per  Mile. 

Mile.  $ 

Lbs. 

$ 

.erf,  Cts. 

per"  Mile. 

$ 

30 

1 

860 

57792 

66500 

66oO 

100 

5280 

69722 

24 

y» 

570 

38804 

45000 

4500 

65 

3432 

46236 

20 

H 

400 

26880 

37300 

3730 

45 

2376 

32986 

18 

y\ 

358 

24058 

31100 

3110 

40 

2112 

29280 

16 

« 

325 

21840 

21800 

2180 

36 

1901 

25921 

14 

H 

267 

17943 

18000 

1800 

32 

1690 

21433 

12 

% 

205 

13776 

15000 

1500 

28 

1478 

16754 

10 

% 

171 

11491 

10000 

1000 

25 

1320 

13811 

8 

H 

111 

7459 

8100 

810 

22 

1162 

9431 

6 

% 

84 

5645 

6300 

630 

20 

1056 

7331 

4 

% 

58 

3898 

5000 

500 

18 

951 

5349 

3 

H 

33 

2218 

3200 

320 

16 

845 

ssas 

2 

H 

24 

HM3 

1500 

150 

15 

792 

2555 

For  thickness  of 

heads,  see  p  532  of  HydrostaJ 
300  fibs  per  sq  iuch. 


ULICS. 


575 


etal  pipes  to  resist  safely  the  pressures  of  various 

Those  iu  the  last  table  have  been  tested  to  about  700  feet  head,  or 


The  following-  are  the  contract  prices  for  laying-  pipes  in 
the  city  oc  Brooklyn.  N.  York,  in  186<S ;  they  include  hauling,  unpaving 
and  repavklg,  digging  and  refilling  the  trenches,  laying  the  pipe,  lead,  gasket  yarn, 
&c;  but'exclusive  of  cost  of  pipes. 


Diam. 

Length. 

Per  Foot. 

Per  Mile. 

Diam. 

Length. 

Per  Foot. 

Per  Mile. 

Ins. 

Feet. 

Cents. 

$ 

Ins. 

Feet. 

Cents. 

$ 

6 

9 

37 

1954 

16 

12 

90 

4752 

6 

12 

33 

1743 

20 

12 

120 

6336 

8 

9 

46 

2429 

30 

12 

320 

16896 

8 

12 

39 

2059 

48 

12 

520 

27456 

12 

12 

60 

3168 

In  Figs  39,  A,  is  a  doable  branch  ;  which  is  a  pipe  having,  in  addition  to  the 

faucet  c  at  one  end,  two  others,  «  and  i,  to  which  pipes  leading  in  opposite  directions  (as  at  cross- 
streets)  may  be  attached.  If  either  s  or  i  be  omitted,  the  pipe  becomes  a  single  branch. 
The  pipe  is  stronger  when  these  extra  faucets  are  near  its  end,  than  if  they  were  at  its  middle.  In  a 
long  line  of  pipes,  for  the  sake  of  expedition,  different  gangs  of  men  are  frequently  laying  detached 
portions  some  distance  apart ;  and  when  two  ends  of  different  portions  are  brought  near  enough  to- 
gether to  be  united,  as  h  and  r.  Fig  C,  their  junction  cannot  be  effected  by  the  usual  spigot-and-faucet 
joint.  In  this  case  a  cast-iron  Sleeve  1 1  is  used,  which  is  first  slid  upon  one  of  the  pieces 
of  pipe  ;  and  (after  the  other  piece  also  is  laid)  is  slid  back  into  the  position  in  the  fig,  so  as  to  cover 
the  joint.  Sleeves  are  usually  about 
a  foot  long ;  as  thick  as  the  pipe ;  and 
their  diain  is  sufficient  to  allow  the 
usual  joint  of  gasket  and  lead.  There 
is  of  course  such  a  joint  at  each  end 
of  the  sleeve. 

When  a  crack  takes 
place  across  a  pipe,  a  a,  Fig 
B,  already  in  use,  it  is  repaired  by 
means  of  a  cast-iron  sleeve  g  g,  made 
in  two  parts,  bolted  together  by 
means  of  flanges  as  at  n  n.  In  other 
respects  it  is  like  the  preceding 
sleeve.  The  intermediate  white  ring 
is  the  lead  joint.  If  the  crack  is  too 
long,  or  otherwise  too  bad  to  be  rem- 
edied by  a  sleeve,  the  pipe  is  broken 
to  pieces ;  and  the  lead  joints  at  its 
ends  melted  out,  so  as  to  allow  of  its 
removal.  Then,  since  an  entire  new 
pipe  cannot  now  be  inserted,  owing  to  the  overlapping  of  the  spigot-and-faucet  ends,  two  short  pieces 
must  be  substituted  for  it.  One  end  of  each  of  these  is  lead-jointed  to  the  pipes  already  laid  ;  while 
the  other  two  ends,  which  will  probably  be  a  few  inches  apart,  are  covered  by  a  sleeve  1 1,  Fig  C. 

Cracks  may  at  times  be  temporarily  repaired  in  an  emergency,  bv  a  wrapping  of  folds  of  canvas 
thoroughly  saturated  with  white-lead  paint;  and  tightly  confined  to  the  pipe  by  a  spiral  banding  of 
thin  hoop-iron  or  wire. 

John  F.  Ward's  flexible  joint  for  pipes  laid  across  the  irregular  bottoms 
of  streams,  is  shown  at  Fig  39^.*  A  portion,  a  o,  of  the  inside  of  the  socket  or  faucet  F,  is  bored  out 
truly  to  form  the  middle  zone  of  a  sphere  ;  and  the  spigot  end 
e  o  of  the  other  pipe  is  cast  with  two  raised  collars  o  and  e. 
The  inner  collar,  o,  is  of  such  a  height  as  barely  to  allow  it  to 
pass  into  the  faucet.  The  outer  one,  e,  is  a  little  lower,  so  as 
to  allow  melted  lead  (shown  black)  to  be  poured  in  at  a.  The 
outer  edge  or  diam  of  the  spigot  end  at  o  is  carefully  turned 
so  as  to  tit  the  turned  spherical  zone  ;  so  that  the  joint  will 
admit  of  considerable  play  without  danger  of  leaking.  In 
laying  the  pipes  under  water,  the  joints  are  filled  with  melted 
lead,  as  usual,  on  board  of  suitable  vessels  or  floats.  As  fast 
as  they  are  thus  filled,  the  floats  are  moved  forward,  and  the 
pipes,  if  small,  and  the  water  shallow,  are  passed  into  the 
water  without  further  care.  But  for  large  pipes  in  deep  wa- 
ter, suitable  apparatus  is  used  for  lowering  them  without  un- 
due strain  on  the  joints.  Mr  Ward  has  been  perfectly  suc- 
cessful in  laying  this  pipe  under  water,  in  one  case  40  feet 
deep.  One  of  the  mains  of  the  Philada  water-works  is  thus 
laid  across  the  Schuylkill  River. 

In  some  cases  preliminary  dredging  may  be  expedient,  to  diminish  abrupt  irregularities  of  the 
bottom. 

To  attach  a  pipe  <\  Fig*  39,  to  one  /',  already  in  use,  but  in  which 


*  The  patent  is  held  by  Messrs  A.  W.  Craven  and  John  F.  Ward.  Civ  Engs,  38  Broadway,  N.  York. 
The  former  was  for  many  years  chief  engineer  of  the  New  York  water- works. 


576 


HYDRAULICS. 


In  Figs  4O,  t  t  represent  a  common  street  fireplug,  or  hy- 
drant.   It  is  bolted  by  flanges  p,  to  the  g*oose»neck  w;  and  is  connected  with 


4-  FEET 


I  FOOT 


the  street  main  by  a  pipe  x.  The  valve  v  is  (in  Philada)  of  layers  of  well-hammered  sole-leather;  and 
when  closed,  shuts  against  a  brass  ring-seat  o;  which  is  confined  to  its  place  by  a  lead  joint  as  shown 
in  the  fig.  The  layers  of  leather  are  compressed  by  the  nut  u,  and  the  washer  p.  This  valve  is  opened 
by  working  down  "the  screw  a;  which,  by  means  of  the  swivel  joint  at  a  a.  can  revolve  without  turn- 
ing the  valve-rod  y.  This  rod  works  through  a  brass  stuffing-box  at  6  ft  :  the  small  circle  in  which  de- 
notes the  space  occupied  by  the  stuffing.  The  box  is  lead-jointed  to  the  inside  of  the  fireplug  at  1 1. 
Two  standards  ii  sustain  the  cross-piece  c,  through  which  the  screw  s  works.  When  the  valve  v  is 
closed  after  the  plug  has  been  in  use,  the  interior  of  the  plug  is  full  of  water;  and  to  prevent  this 
from  freezing,  and  bursting  the  plug,  a  small  outlet,  or  frost-pipe,  I,  is  provided,  through  which  it  may 
escape  and  soak  into  the  adjacent  soil.  The  closing  of  the  valve  v  by  working  up  the  screw  «  to  the 
position  in  the  figs,  opens  the  frost-pipe  by  means  of  the  rod  f\  the  lower  end  of  which  is  pointed,  so 
as  to  close  the  frost-pipe  when  the  valve  is  open  ;  and  the  upper  end  of  which  is  bent,  and  formed  into 
two  prongs,  which  rest  upon  a  a;  bv  which  means  the  lower  end  is  drawn  out  from  the  frost-pipe 
when  a  a  rises  by  the  working-up  of  the  screw  s  in  closing  the  valve  v.  The  detached  Fig.?',  shows 
how  the  cylindrical  head  of  the  rod  y  is  inserted  into  a  a,  "by  means  of  an  opening  in  one  side.  An 
iron  ring  is  then  shrunk  around  a  a.  At  e  is  the  stopper  of  "the  plug.  This  is  of  iron  ;  but  it?  inner 
portion  is  surrounded  by  a  screw-ring  of  brass,  leaded  to  it ;  and  which  works  into  another,  leaded  to 
the  fireplug.  This  brass  screw  obviates  friction  from  rust ;  it  reduces  the  diam  of  the  opening,  to  n  n. 

The  whole  plug  is  protected  from  injury  and  frost  by  a  cast-iron  box  resting  on  the  flanges  p.  It  is 
shown  by  the  dotted  lines,  its  lid  m  is  hinged  to  the  body  at  d.  The  box  is  in  three  pieces:  two  of 
them  (each  extending  only  half-way  around  the  plug)  reach  from  the  flanges  p,  up  to  e ;  the  upoer 
piece,  from  e  to  TO.  reaches  entirely  around  the  plug,  and  clasps  the  tops  of  the  two  lower  pieces, 
keeping  them  in  place.  The  fig  shows  the  brick  paving  *  of  the  sidewalk,  the  pebble  paving  n  of  the 
carriage  wav,  and  the  curbstone  k.  In  Philada  there  is,  on  an  average,  one  fireplug  to  about  eacli 
150  yards  of  street  pipe. 

A  great  variety  of  strept  hvdrants,  with  from  1  to  6  nozzles,  with  various  styles  of  ornamentation. 
are  made  by  the  Boston  Machine  Co. 


HYDRAULICS.  577 

Wrongs-iron  pipes  nave  been  fonnd  to  corrode  ranch  more 

rapidly  than  cast.  Cast-iron  pipes  laid  in  Philadelphia  in  1804,  are  still  (now  1871)  in  perfectly  good 
condition,  and  in  constant  use. 

A  gutta-percha  pipe,  ^g  inch  thick,  and  %  inch  bore,  has  sustained  safely 
an  internal  pres  of  more  than  250  fts  per  sq  inch  ;  equal  to  nearly  600  feet  head.  It  merely  swelled 
slightly  at  337  fbs.  The  Croton  water  is  carried  to  Blackwell's  Island  by  a  tube  of  that  material, 
sunk  in  the  East  River,  N  York.  It  is  kept  down  by  weights.  It  is  2^  ins  bore,  and  about  &  inch 
thick.  Was  laid  in  1851,  and  is  still  in  perfect  condition  in  1871.  Experience  has  there  shown  that 
on  rocky  bottom  the  pipe  must  be  prevented  from  shifting  with  the  tide,  &c ;  otherwise  it  will  soon 
wear  through.  It  is  frequently  displaced  by  the  anchors  of  dragging  vessels.  A  wrapping  of  can- 
vas, confined  by  spun  yarn,  has  been  used  as  a  protection  against  abrasion. 

Ball's  patent  iron  and  cement  pipe,  made  by  a  company  at  Jersey 
City,  New  Jersey,  is  formed  of  riveted  sheet  iron,  and  lined  with  about  %  inch  thickness  of  hydraulic 
cement.  It  is  laid  in  a  bed  of  cement  mortar,  and  completely  covered  with  the  same.  Suitable 
means  are  provided  for  making  all  the  attachments,  &c,  required  iu  city  pipes  for  water  and  gas. 
More  than  400  miles  of  it  are  iu  use  in  various  towns,  some  of  it  for  25  years  ;  and  it  appears  to  give 
general  satisfaction.  Tubercles  do  not  form  in  these  pipes,  as  they  are  apt  to  do  in  cast-iron  ones. 
There  is  every  reason  to  suppose  that  they  are  durable.  The  trenches  being  dug.  the  Jersey  City  Co 
furnish  pipes  and  lay  them  (including  the  cement)  at  %,  or  less,  of  the  cost  of  cast-iron  ones  laid. 
The  invention  certainly  possesses  much  merit. 

The  WycKoff  patent  pipe  for  water  or  gas,  is  of  wood.   Several  concentric 

Sipes  of  different  diams  are  bored  out  from  the  same  log.  They  are  coated  outside  with  either  l>y- 
raulic  or  aspbaltic  cement.  When  exposed  to  pressure  from  great  heads,  they  are  spirally  wrapped 
or  banded  with  sheet  or  plate  iron.  A  10-inch  pipe  thus  banded  with  iron  of  1  X  %  inch  ;  spirals  4 
ins  apart;  sustained  a  test  pressure  of  406  Ibs  per  sq  inch  ;  equal  to  a  head  of  water  of  935  ft.  Their 
cost  is  but  about  half  that  of  cast-iron  pipes.  They  are  made  by  J.  A.  Woodward,  Williamsport, 
Penna;  and  have  been  quite  extensively  and  successfully  used  for  both  water  and  gas. 

Water  pipes  of  bored  yellow  pine  logs,  laid  in  Philada  50  to  60 

years  ago.  are  frequently  quite  sound,  and  still  fit  for  use,  except  where  outer  sap  wood  is  decayed. 
When  this  is  removed,  many  of  these  old  pipes  have  been  relaid  in  factories,  <fcc.  Clay  well  packed 
around  wooden  pipes,  excludes  the  contact  of  air,  and  thus  contributes  greatly  to  their  durability. 
Loose  porous  soils,  such  as  gravel,  &c,  on  the  contrary,  are  unfavorable. 

Pipes  made  of  bitumiiiized  paper,  prepared  under  great  pressure, 

have  been  used  tor  both  water  and  gas.  They  are  much  less  liable  to  break  than  cast  iron,  and  do 
not  weigh  or  cost  more  than  about  half  as  much.  Pipes  of  5  ins  bore,  and  %  inch  thick,  have  resisted 
test  strains  of  220  tts  per  sq  inch  ;  equal  to  a  water  bead  of  507  ft. 

Ill  Philada,  in  1881,  there  are  about  75O  miles  of  street 
pipes;  or  about  1  mile  to  every  1133  inhabitants.  The  population  is  about 
850,000 ;  residing  in  about  150000  dwellings. 

Art.  29.  Reservoirs.  In  important  reservoirs  of  earth,  for  storing  water 
to  moderate  depths  for  cities,  experience  appears  not  to  sanction  dimensions  bolder  than  10  ft  thick 
at  top  ;  inner  slope  2  to  1 ;  outer  slope  1>£  to  l.#  A  top  width  of  15  ft  to  20  ft,  and  inside  slopes  of 


3  to  1,  are  adopted  in  some  important  cases;  with  outer  slopes  of  2  to  1.  Both  slopes,  however,  are 
at  times  made  only  \%  to  1.  The  level  water  surf  should  be  kept  at  least  3  or  4  feet  below  the  top  of 
the  embkt ;  or  more,  if  liable  to  waves.  In  a  large  reservoir,  a  quite  moderate  breeze  will  raise  waves 


that  will  run  3  ft  (measured  vert)  up  the  inner  slope.  A  low  wall,  or  close  fence,  «?.  Fig  41,  is  s 
times  used  as  a  defence  against  them.  The  top  and  the  outer  slopes  should  be  protected  at  least  by 
sod  or  by  grass.  To  assist  in  keeping  the  top  dry,  it  should  be  either  a  little  rounding,  or  else  sloped 
toward  the  outside.  The  soft  soil  and  vegetable  matter  should  be  carefully  removed  from  under  the 

that  leakage  may  not  take  place  under  them.  To  aid  in  this,  a  double  row  of  sheet  piles,  or  a  sunk 
wall  of  cement  masonry,  carried  to  a  suitable  depth  below  the  bottom,  may  be  placed  along  the  inner 
toe  in  bad  cases.  If  there  are  springs  beneath  the  base,  they  must  either  be  stopped,  or  led  away  by 
pipes.  The  embkt  should  be  carried  up  in  layers,  slightly  hollowing  toward  the  center,  and  not  ex- 
ceeding a  foot  in  thickness  ;  and  all  stones,  stumps,  and  other  foreign  material,  such  as  clean  gravel, 
sand,  and  decomposed  mica  shists,  Ac,  that  may  produce  leakage,  carefully  excluded.  These  layers 
shriuld  be  well  consolidated  by  the  carts;  and  the  easier  the  slopes  are,  the  more  effectively  can  this 
be  done.  The  layers,  however,  should  not  be  distinct,  and  separated  by  actual  plane  surfaces;  but 
each  succeeding  one  should  be  well  incorporated  with  the  one  below.  This  has  sometimes  been  done 
by  driving  a  drove  of  oxen,  or  even  sheep,  repeatedly  over  each  layer:  in  addition  to  the  carting. 
Rollers  are  not  to  be  recommended,  as  they  tend  to  produce  seams  between  the  layers.  This  might 
possibly  be  obviated  by  projections  on  the  circumf  of  the  roller. 

Gravelly  earth  is  an  excellent  material,  perhaps  the  best.  The  choicest  material  should  be  placed 
In  the  slope  next  to  the  reservoir  ;  and  should  be  deposited  and  compacted  with  special  care  in  that 
portion,  so  as  to  prevent  the  water  from  leaking  into  the  main  body  of  the  dam,  and  thus  weakening 
It.  It  is  not  amiss  to  introduce  a  bench  fc,  Fig  41,  in  the  outer  slope,  to  diminish  danger  from  rain- 
wash  by  breaking  the  rapidity  of  its  descent. 

If  the  bottom  of  the  reservoir  itself  is  on  a  leaky  soil,  er  on  fissured  rock,  through  the  seams  of 
which  water  may  escape,  it  must  be  carefully  covered  with  from  1 J^  to  3  feet  of  good  puddle:  which, 
in  turn  should  be  protected  from  abrasion  and  disturbance,  by  a  layer  of  gravel;  or  of  concrete, 
either  paved  or  not.  according  to  circumstances. 

Reservoirs  constructed  with  the  foregoing  dimensions,  and  with  care,  may  remain  safe  for  an  in- 
definite period  ;  but  where  serious  damage  would  result  from  failure,  the  following  additional  pre- 
cautions should  he  taken.  The  inner  slopes  should  be  carefully  faced 
up  to  the  very  top.  with  at  least  a  close  dry  rubble-stone 

pitching*,  not  less  than  about  15  to  18  ins  thick;  as  a  protection  against  wash,  and  against 
muskrats.  These  animals,  we  believe,  always  commence  to  burrow  under  water.  If  the  slopes  are 

*  The  writer  suggests  that  a  top  width  equal  to  2  ft  +  twice  the  sq  rt  of  the  height  in  feet,  will  b« 
3»fe  for  any  height  whatever  of  reservoir  properly  constructed  in  other  respects. 


578 


HYDRAULICS. 


Fig  41 


much  steeper  than  2  to  1,  this  dry  pitching  will  be  apt  to  be  overthrown  by  the  sliding  down  of  the 
softened  earth  behind  it,  if  tue  water  in  the  reservoir  should  for  any  cau.-,e  be  drawn  tiuwu  rather 
suddenly.  It  will  be  much  more  effective,  but  of  course  more  costly,  if  laid  in  hydraulic  ctrneut ;  aud 
still  more  so  if  laid  upon  a  layer  a  few  ins  thick  of  cemeut-and-gravel  concrete;  especially  if  this 
last  be  underlaid  by  a  layer  about  1}^  to  3  ft  thick  of  good  puddle,  spread  over  the  face  of  the  slope ;  tha 
great  object  being  to  protect  the  inner  elope  from  actual  contact  with  the  water.  If  this  can  be  ef- 
fectually accomplished,  slopes  as  steep  as  1^  to  1  will  be  perfectly  secure;  for  the  danger  does  not 
arise  from  any  want  of  weight  of  the  earth  for  resisting  overthrow.  Special  care  should  be  bestowed 
upon  the  inner  toe  of  the  slope,  to  prevent  water  from  finding  its  way  beneath  it,  and  softening  the 
earth  so  as  to  undermine  the  stone  pitching.  Near  the  top,  reference  should  be  had  to  danger  of  de- 
rangement by  ice,  frost,  rain,  and  waves.  Flat  inner  slopes  tend  not  only  to  prevent  the  displace- 
ment of  the  pitching;  but  increase  the  stability  of  the  embankment,  by  causing  the  pressure  of  the 
water  (which  is  always  at  right  angles  to  the  slope)  to  become  more  nearly  vertical ;  and  thus  to  hold 
the  embankment  more  firmly  to  its  base  than  if  there  were  no  water  behind  it.  Sometimes  the  toes  of 
both  the  inner  and  outer  slopes  abut  against  low  retaining- walls  in  cement.  This  gives  a  neat  finish, 
and  tends  to  preservation  from  injury. 

Many  engineers,  in  order  to  prevent 
leaking,  either  through  or  beneath  the 
embankment,  construct  a  putldle- 
Wall,  jt>,  Fig  41,  of  well- rammed 
impervious  soil,  (gravelly  clay  is  the 
best,)  reaching  from  the  top  to  several 
feet  below  the  base.  This  wall  should 
not  be  less  than  6  or  8  ft  thick  on  top, 
for  a  deep  reservoir;  and  should  in- 
crease downward  by  offsets  (and  not  by 
slopes,  or  batters)  at  the  rate  of  about 
1  in  total  thickness,  to  3  or  4  in  depth.  Other  engineers  object  to  these  puddle-walls ;  and  contend 
that  leakage  should  be  prevented  by  making  both  the  inner  slopes,  and  the  bottom  of  the  reservoir, 
water-tight,  by  means  of  puddle,  concrete,  and  stone  facing  in  cement,  as  just  alluded  to.  They 
argue  that  if  the  embankment  is  well  constructed,  it  is  itself  a  puddle-wall  throughout. 

Near  Sail  Francisco,  Cal,  are  two  earthen  reservoir  clams 

built  about  1864,  one  95  ft  high,  26  on  top,  inner  slope  2.75  to  1,  outer  2.5  to  1.  The  other  93  high,  25 
on  top,  inner  slope  3.5  to  1,  outer  3  to  1.  In  each  the  puddle- wall  is  carried  47  ft  deeper  than  the  base. 
No  stone  facing. 

It  is  difficult  to  prevent  water  under  higli  pressure  from 
finding*  its  way  through  considerable  distances  along-  seams 

iron  pipes  laid  under  reservoir  embankments  ;  or  along  the  tie-rods  sometimes  used  through  the  pud- 
dle of  coffer-dams ;  and  the  same  is  apt  to  occur  under  the  bases  of  embankments  which  rest  on 
smooth  rock.  Special  care  should  be  taken  that  the  earth  used  in  such  positions  is  not  of  a  porous 
nature;  ani  that  it  is  thoroughly  compacted  all  along  the  seam;  and  the  straight  continuity  of  the 
sen;a  should  be  interrupted  or  broken  as  frequently  as  possible  by  projections.  Faucets  or  flanges  do 
this  to  a  limited  extent  in  the  case  of  iron  pipes;  and  something  similar,  but  on  a  larger  scale,  should 
at  short  intervals  be  constructed  in  the  shape  of  collars  or  yokes  of  cement  stonework,  in  the  case  of 
rock  or  masonry.  For  more  on  dams,  see  Dams,  p  583,  also  528  &c. 

It  is  usually  advisable  to  divide  reservoirs  into  two  parts,  so  that  while  the  water  in  one  part  is 
being  drawn  off  for  use,  that  in  the  other  may  purify  itself  by  settling  its  sediment.  Also,  one  part 
may  remain  in  use,  while  the  other  is  being  cleaned  or  repaired.  Many  days,  or  even  two  or  three 
weeks,  sometimes,  are  required  for  the  complete  settlement  of  the  very  fine  clayey  particles  in  muddy 
water:  depending  on  the  depth  of  the  reservoir. 

One  or  more  flights  of  steps  to  the  bottom  of  the  reservoir,  should  be  provided. 

Mud  in  Reservoirs.    The  reservoirs  of  the  New  River  Water  Co,  London, 

England,  were  uncleaned  for  100  years,  during  which  mud  8  ft  deep  was  deposited,  or  about  an  inch 
annually.  At  Philadelphia  it  is  about  .25  inch  per  annum  from  the  Schuylkill,  and  1  inch  from  the 
Delaware  Riv.  At  St  Louis,  Missouri,  about  3  to  4  ft  per  year!  In  shallow  reservoirs  vegetation 
takes  place,  and  injures  the  water.  A  depth  of  about  20  ft  appears  to  prevent  this. 

Water  flowing  through  marsh  lands  is  sometimes  unfit  for  arinking  purposes.  That,  for  instance, 
in  some  sections  of  the  Concord  River,  Massachusetts,  was  reported  by  the  eminent  hydraulic  engi- 
neer, Loammi  Baldwin,  of  Boston,  to  be  absolutely  poisonous  from  this  cause. 

The  construction  of  a  large  deep  reservoir  is  not  only  a  very  costly,  but  a  very  hazardous  under- 
taking. With  every  watchfulness  and  care,  it  is  almost  impossible  entirely  to  prevent  leaking ; 
although  this  may  not  manifest  itself  for  months,  or  even  years.  Should  a  break  occur,  especially 
near  a  city,  it  would  probably  be  attended  by  great  loss  of  life  and  property.  If  the  water  once  finds 
its  way  in  a  stream,  either  across  the  unpaved  top,  or  through  the  body  of  the  embankment,  the  rapid 
destruction  of  the  whole  becomes  almost  certain. 

Art.  3O.  STORING  RESERVOIRS.  The  entire  annual  yield  of  a  stream 
mav  be  much  more  than  sufficient  for  supplying  a  certain  population  with  water;  and  yet  in  its  natu- 
ral condition  the  stream  may  not  be  available  for  this  purpose,  because  it  becomes  nearly  dry  in  sum- 
mer, when  water  is  most  needed;  while,  at  other  seasons,  the  rains  and  melted  snows  produce  floods 
which  supply  vastly  more  than  is  reqd ;  and  which  must  be  allowed  to  run  to  waste.  A  storing  reser- 
voir is  intended  to  collect  and  store  up  this  excess  of  water,  so  that  it  may  be  drawn  off  as  reqd  during 
the  droughts  of  summer,  and  thus  equalize  the  supply  throughout  the  entire  year.  This,  when  the  lo- 
cality permits,  is  effected  by  building  a  dam  across  the  stream,  to  form  one  side  of  the  reservoir ;  while 
the  hill-slopes  of  the  vallev  of  the  stream  form  the  other  sides.  The  stream  itself  flows  into  this  roservoir 
at  its  up-stream  end.  "When  the  stream  is  liable  to  become  nearly  dry  during  long  summer  droughts 
experience  shows  that  the  capacity  of  the  reservoir  should  be  equa".  to  "from  4  to  6  months'  supplv.  ac- 
cording  to  circumstances.  During  the  construction  of  the  dam.  a  free  channel  must  he  provided,  to 
pass  the  stream  without  allowing  it  to  do  injury  to  the  work.  If  the  dam  were  built  precisely  Mke  Fig 
41.  entirely  of  earth,  it  would  plainly  be  liable  to  destruction  by  being  washed  away  in  case  the  reser- 
vo'r  sbou'.t  Become  SG  f&'.l  that  the  water  would  begin  to  flow  over  its  top.  To  provide  against  this  we 
may,  by  means  of  masonry,  or  of  cribs  filled  with  broken  stone,  or  otherwise,  construct  either  th« 


HYDRAUllCS. 


579 


(rhole,  or  a  part  of  the  dam,  to  serve  as  ap/bverfall,  or  a  Waste-weir.  Or  a  side  chan- 
nel (an  open  cut,  pipes,  or  a  cul\vrt,J&)ma,y  be  provided  at  one  or  both  ends  of  the  dam,  arid  in  the 
natural  soil,  at  such  a  level  as  to  cajaryaway  the  surplus  flood  water  before  it  cau  rise  h.gh  enough  to 
overtop  the  eartheu  dam.  BesidjjfThese,  and  the  pipes  for  carrying  the  water  to  the  town,  there  should 


be  a 


order  that. 


if  necessary  "for  repairs,  on/fur  cleaning  by  scouring,  all  the  water  may  be  drawn  off.  The  entrances 
to  the  city  pipes  shouldj»eprotected  by  gratings,  to  exclude  fish,  &c. 

To  facilitajre  repairs  or  renewals  of  all  valves,  «fcc,  which 
are  under  Jwater,  the  reservoir  ends  of  the  pipes  or  culverts  to  which  they 
are  attached,  maybe  surrounded  by  a  water-tight  box  or  chamber,  which  will  usually  be  left  open  to  tho 
reservoir ;  but  may  be  closed  when  repairs  are  required.  Access  may  then  be  had  to  them  by  entering 
at  the  outer  end,  after  the  water  has  flowed  away  from  inside.  In  case  the  outlet  is  through  a  long 
line  of  pipes  which  cannot  thus  be  entered,  a  special  entry  for  this  purpose  may  be  cast  in  the  pipe 
itself,  near  the  outer  toe  of  the  embkt;  to  be  kept  closed  except  in  cas«  of  repairs.  Sometimes  a  bet. 
ter,  but  more  expensive  means  of  access  to  such  valves,  is  secured  by  enclosing  them  in  a  Valve- 
tO  WCr  of  masonry.  This  is  a  hollow  vert  water-tight  chamber,  like  a  well ;  but  near  the  toe  of  the 
inner  slope;  having  its  foundation  at  the  bottom  of  the  reservoir;  whence  the  tower  rises  through  the 
water  to  above  its  surf.  This  chamber  is  provided  with  valves  or  gates  usually  left  open  to  the  reser- 
voir;  but  which  may  be  closed  when  repairs  are  needed ;  and  the  water  in  the  tower  allowed  to  escape 
from  it  through  the  open  valves  of  the  outlets.  This  done,  workmen  can  descend  through  the  tower  by 
ladders  from  the  aperture  at  its  top. 

At  times  the  outlets  for  the  discharge  of  surplus  flood  water  are,  like  those  for  scouring,  placed  at, 
or  just  above,  the  level  of  the  bottom  of  the  reservoir.  In  order  that  these  may  work  in  case  of  a  sud- 
den flood  at  night,  &c,  they  must  be  furnished  with  self-acting  valves,  which  will  open  of  their  own 
accord  when  the  flood  is  about  to  rise  too  high.  This  may  be  effected  by  attaching  them  to  floats,  the 
rising  of  which,  when  the  water  is  high,  wiM  pull  them  open.  All  such  outlets  should  be  large  enough 
to  let  men  enter  them  for  repairs.  They  should  by  no  means  be  laid  through  the  artificial  earthen  body 
of  the  dam  itself,  without  being  supported  upon  masonry  reaching  down  to  a  firm  natural  foundation  ; 
otherwise  they  are  very  apt  to  be  broken  by  the  subsidence  of  the  embankment.  It  is  usually  safer  to 
carry  them  through  the  firm  natural  soil  near  one  end  of  the  dam.  Their  valves,  if  only  single,  should 
be  at  their  inner  or  reservoir  end,  so  as  to  leave  the  outlets  themselves  usually  empty,  for  inspection  ; 
but  it  is  better  to  have  two  valves,  so  that  one  may  be  used  when  the  other  needs  repair ;  and  in  this 
case  one  may  be  placed  at  each  end.  Reservoirs  which  are  supplied  by  pumps,  need  no  precautions 
against  overflow;  because  the  pumping  is  stopped  when  they  are  tilled  to  the  proper  height.  Large 
storing  reservoirs  necessarily  submerge  more  or  less  land,  which  has  therefore  to  be  purchased.  They 
frequently  prevent  spring  floods  from  injuring  low  lands  farther  down  stream  ;  inasmuch  as  they  in- 
tercept the  descending  water.  In  case  there  are  mills  down  stream  from  the  reservoir,  they  would 
evidently  be  deprived  of  water  for  driving  them,  unless  a  portion  of  that  stored  in  the  reservoir  be 
devoted  to  that  purpose.  Water  thus  applied  to  compensate  for  the  loss  of  the  natural  stream,  is  called 
Compensation  water  ;  and  the  reservoir,  a  compensating  one. 

Art-  31.   Distributing:  reservoirs.   Frequently  a  valley  fit  for  a  storing 

reservoir  can  be  found  only  at  a  long  dist  (sometimes  many  miles)  from  the  town;  and  it  then  be- 
comes expedient  to  construct  also  an  additional  one  of  smaller  size  than  the  storing  one,  near  the 
town ;  and  at  as  great  an  elevation  above  it  as  circumstances  will  permit ;  but  lower  than  the  storing 
one.  This  is  called,  by  way  of  distinction,  a  distributing  reservoir,  because  from  it  the  water,  after 
having  flowed  into  it  from  the  storing  reservoir,  through  the  long  supply  pipe  which  connects  them,  is 
distributed  in  various  directions  through  the  town,  by  means  of  the  street  mains,  or  pipes.  This 
small  reservoir  should  hold  a  supply  sufficient  at  least  for  a  few  days  ;  a  few  weeks  would  be  better; 
and  the  end  of  the  supply  pipe  which  terminates  in  it,  should  be  p'rovided  with  a  valve  for  shutting 
off  the  supply  from  the  storing  reservoir.  These  precautions  permit  repairs  to  be  made  along  the  line 
of  supply  pipe  without  depriving  the  town  of  water  in  the  mean  time.  With  a  view  to  such  repairs ;  as 
well  as  to  scouring  out  sediment  from  the  supply  pipe,  this  last  should  be  provided  with  Outlet 
Valves  at  various  low  points  along  the  entire  interval  between  the  two  reservoirs  ;  especially  at 
those  at  which  the  valves  may  disch  into  natural  watercourses.  On  opening  these  valves,  the  out- 
rush  of  the  water  carries  away  sediment;  and  leaves  the  pipe  empty  for  inspection. 

Art.  32.  Air  valves.  Air  is  apt  to  collect  gradually  at  the  high  points  of 
vert  curves  along  the  supply  pipes;  and,  unless  removed,  produces  more  or  less  obstruction  to  the 
flow.  This  may  be  prevented  by  air  valves, 
see  Fig  42,  which  is  %  of  the  full  size  of  those 
once  used  in  Philada.  This  simple  device 
consists  of  a  cast-iron  box,  ccdd.  confined 
to  the  main  pipe  m  m,  by  screw-bolts  passing 
through  its  flange  dd.  It  has  a  cover  gng, 
confined  to  it  by  screws  tt;  and  at  the  top 
-»f  which  is  an  opening  n,  for  the  escape  of 
air  from  within.  In  this  box  is  a  float  /, 
which  may  be  a  close  tin  or  copper  vessel, 
or  of  layers  of  cork,  as  supposed  in  the  fig; 
or  &c.  This  float  has  a  spindle  or  stem  8  s. 
fast  to  it;  which  passes  through  openings  in 
the  bridge-bars  a  a.  and  o  ;  thereby  allowing 
the  float  to  rise  and  fall  freely,  but  prevent- 
cg  it  from  moving  sideways.  When  the 
pipe  mm  is  empty,  the  float  i*  down;  its 
base  y  resting  on  the  cross-bar  a  a.  The 
stem  ss  has  fixed  to  it  avalve  «,  which  rises 
and  falls  with  it  and  the  float.  Suppose  the 
pipe  mm  to  be  empty,  and  consequently  the 
float,  and  the  valve  v ,  down.  Then,  if  water 
be  admitted  into  the  pipe,  it  will  rise  and 
fill  also  the  box  as  far  up  as  e;  and  in 
doing  so  will  lift  the  float/,  and  the  valve  v, 
lo  the  position  in  the  fig ;  thus  preventing 


580 


HYDRAULICS. 


egress  to  the  outer  air,  by  closing  the  opening  at  v.  Now,  air  carried  along  by  the  water,  will,  en 
account  of  its  lightness,  ascend  to  the  highest  points  it  meets  with. 

Hence,  when  such  air  arrives  under  the  opening  at  a  a,  it  will  rise  through  it,  and  ascend  to  e;  the 
closed  valve  preventing  it  from  going  farther.  Thus  successive  portions  of  air  uscend,  and  in  time 
accumulate  to  such  an  extent  as  gradually  to  force  much  of  the  water  downward  out  of  the  box. 
When  this  takes  place,  the  float,  which  is  held  up  only  by  the  water,  of  course  descends  also;  and  iu 
doing  so,  pulls  down  with  it  the  valve  v.  The  accumulated  air  then  instantly  escapes  through  the 
openings  at  v  and  n,  into  the  atmosphere ;  and  the  water  in  the  pipe  mm,  immediately  ascends  again 
into  the  box,  carrying  with  it  the  float ;  and  thus  again  closing  the  valve  v.  The  valve,  and  the  valve- 
seat  e,  are  faced  with  brass,  to  avoid  rust,  and  consequent  bad  tit.  The  whole  is  protected  bv  au  iron 
or  wooden  cover,  reaching  to  the  level  of  the  street,  somewhat  as  in  Fig  35,  p  572. 

Air  valves  are  no  longer  used  in  city  pipes;  their  place  being 

supplied  by  the  fireplugs  at  average  distances  of  about  150  yds  apart.  These,  being  placed  as  much 
as  possible  at  the  summits  of  undulations  in  the  lines  of  pipes,  for  convenience  of  washing  the 
streets,  and  being  frequently  opened  for  that  purpose,  permit  also  the  escape  of  accumulated  air. 

The  escape  of  compressed  air  through  an  air  valve,  or  other 
opening,  has  been  known  to  produce  bursting  of  the  main 
pipes;  for  the  escape  is  instantaneous,  and  permits  the  columns  of  water  in  the 
pipes  on  both  sides  of  the  valve,  to  rush  together  with  great  forces,  which  arrest  each  other,  and 
react  against  the  pipes. 

Water  for  city  use  should  not  be  drawn  from  the  very  bottom  of  the  reservoir,  because  it  will  the* 
be  apt  to  carry  along  the  sediment;  which  not  only  injures  the  water,  but  creates  deposits  within 
the  pipes  ;  thus  obstructing  the  flow.  In  fixing  upon  the  necessary  capacity  of  a  reservoir,  this  must 
be  taken  into  consideration  ;  inasmuch  as  all  the  water  below  the  leve!  for  drawing  oft",  must  be  re- 
garded as  lost.  When  circumstances  justify  the  expense,  it  is  well  to  curve  up  the  reservoir  end  of  the 
service  main,  so  as  to  provide  it  with  valves  at  ditf  heights;  for  drawing  off  only  the  purest  stratum 
that  may  be  in  the  reservoir.  With  this  view,  the  valve-tower  before  spoken  of  generally  has  such 
valves  communicating  with  the  water  in  the  reservoir ;  and  by  this  means  only  the  purest  is  admitted 
into  the  tower;  and  from  it,  into  the  city  pipes.  This  refinement,  however,  is  rarely  practicable. 
Such  valves  must  of  course  be  worked  by  watchmen. 

The  quantity  of  water  reqd  in  cities,  has  been  found  by  experience 

to  increase  faster  than  the  population.  About  60  gallons,  or  8  cub  ft,  per  day,  to  each  inhabitant,  is 
usually  considered  a  fair  ample  allowance.  Many  European  cities  have  not  half  as  much  ;  while 
New  York,  and  some  others,  use  and  waste  half  as  much  more.  With  efficient  means  for  preventing 
waste,  60  gallons  would  probably  suffice  for  any  commercial  city  ;  but  inasmuch  as  cleanliness  and 
health  are  promoted  by  its  free  use,  as  few  restrictions  as  possible  should  be  introduced. 

Ill  fixing  upon  the  diams  of  pipes  for  supplying  cities,  it  is  necessary 

to  bear  in  mind,  that  by  far  the  greater  portion  of  the  24  hour*'  yield  is  actually  drawn  from  them 
during  only  8  to  12  hours  of  daylight;  and  therefore  the  capacity  of  the  pipes  must  be  sufficient  to 
furnish  the  daily  supply  in  much  less  than  24  hours.  Again,  during  the  hot  summer  months,  much 
more  water  is  used  than  during  the  winter  ones ;  and  this  consideration  necessitates  a  still  larger  diam. 

Art.  33.    Systems  of  street  pipes  for  supplying  cities.    The 

writer  knows  of  no  practical  rules  for  proportioning  the  diams  for  such  systems.  The  various  com- 
plications involved,  render  a  purely  scientific  investigation  of  little  or  no  service.  With  much  hesi- 
tation, he  ventures  the  following  purely  empirical  rules  of  his  own  ;  based  on  such  limited  observa- 
tions as  have  casually  fallen  under  his  notice. 

RULE  I.  When,  at  no  point  in  a  system  of  city  pipes,  is  the  head,  or  vert  dist  below  the  surface 
of  the  reservoir,  compared  with  the  hor  dist  from  the  reservoir,  less  than  at  the  rate  of  50  ft  per  mile, 
then  the  population  in  the  last  column  of  the  following  Table  A,  may  be  abundantly  supplied,  for  all 
city  purposes,  by  either  one  pipe  of  the  inner  diam  or  bore  in  the  1st  col;  or  by  2,  3,  &c,  pipes  of  the 
diams  in  the  other  cols.  These  diams  are  given  to  the  nearest  safe  %  inch.  The  supply  is  assumed 
to  be  about  60  gallons  per  day  to  each  inhabitant. 

TABLE  A.    (Original.) 


1 

2 

BTTIMBEB, 

3         |         4 

OF  PIP! 
6 

is. 

8 

12 

24 

Population 

Diam. 
Ins. 

Diam. 
Ins. 

Diam. 
Ins. 

Diam. 
Ins. 

Diam. 
Ins. 

Diara. 

Ins. 

Diam. 
Ins. 

Diam. 
Ins. 

6 

4%  • 

3% 

3J^ 

3 

2%~ 

2% 

1% 

1647 

8 

6% 

4% 

4 

8* 

3% 

2% 

3465 

10 

7% 

6* 

5% 

5 

4% 

3% 

3 

5908 

11 

7% 

7 

9324 

14 

10% 

9% 

8% 

7 

6% 

5% 

4% 

1370* 

16 

18 

u* 

13% 

10% 
11% 

8% 

6 
6% 

4% 
5% 

19141 
25677 

20 

r*y* 

13 

11% 

9% 

8% 

5% 

33426 

22 

14J4 

12% 

10% 

9^6 

8$ 

6M 

42433 

24 

1HV 

15  H 

18  H 

11% 

10*       j         9 

6% 

52671 

26 

19% 

1H% 

15 

12% 

11* 

9% 

64447 

28 
30 

»i* 

22% 

19% 

16* 

17% 

13% 
14% 

13% 

MM 

11* 

8 
8% 

77565 
91580 

32 

24%              20% 

18* 

15% 

14 

n% 

9 

108160 

34 
36 

25%              22 
27%              23  « 

19% 
20% 

16% 
17% 

15 
15% 

12% 
18* 

10* 

125840 
144480 

40 

30% 

25% 

23% 

19% 

17% 

15 

uS 

188320 

44 

33J4 

28% 

25% 

12% 

239600 

48 

36% 

31 

27% 

23% 

21% 

If? 

13% 

297600 

54 
60 
66 
72 
80 

41           1       34% 
45%       j       38% 
50%       |       42% 
54%       |       46^ 
60%       •       51% 

81* 

34% 
38% 
41% 

29% 
32% 
35% 
39* 

23% 
26% 
29% 
31% 
85* 

20%       , 

22% 
24% 
26% 
29% 

15% 
18% 
22% 

391200 
511200 
650400 
800000 
106400G 

HYDKAULICS. 


581 


It  is  well  to  ahow  in  addition  from  ^  inch  to  1  inch,  or  more,  (depending  on  the  character  of  the 
water,)  to  each  diam  ;  for  deposits  and  concretions. 

The  water,  after  reaching  the  city  through  one  or  more  large  main  pipes  from  the  reservoir,  must 
be  distributed  through  the  streets  by  means  of  smaller  mains  branching  from  the  large  ones.  The 
diams  of  these  smaller  ones  also  may  be  found  by  Table  A.  Thus,  if  a  street,  with  its  alleys,  &c, 
contains  about  6000  persons,  (the  rate  of  head  being,  as  before,  not  less  than  50  ft  to  a  mile  at  any 
point  of  the  system,)  then  we  see  ly  the  table  that  a  10- inch  pipe  will  answer.  It  would  be  well  to 
lay  BO  city  street  pipes  of  less  than  6  ins  diam. 

Mains  which  cross  each  other  should  be  connected  at  some 
Of  their  intersections,  to  allow  the  water  a  more  free  circulation  through- 
out the  entire  system;  so  that  if  the  supply  at  any  point  is  temporarily  cut  off  from  one  direction  by 
closing  the  valves  for  repairs,  or  is  diminished  by  excessive  demand,  it  may  be  maintained  by  the 
flow  from  other  directions. 

Avoid  dead  ends  when  possible,  as  the  water  in  them  becomes  foul  and  unwholesome. 

RULE  2.  With  the  same  diams,  different  rates  of  head  will  supply  the  proportionate  populations  in 
cof.  3  of  Table  B.  Or,  to  find  the  diams  which  at  different  rates  of  head  will  supply  the  same  popula- 
tions given  in  the  last  col  of  Table  A,  mult  the  diam  given  in  Table  A,  by  the  corresponding  number 
in  col  4  of  Table  B ;  or  (approximately)  do  as  directed  in  col  5. 

TABLE  B.    (Original.) 


COL.  1. 

COL.  2. 

COL.  3. 

COL.  4. 

COL.  5. 

Rate  of  Head, 
in  Feet  per  Mile. 

Rate  of  Head, 
compared  with 
that  in  Table  A. 

Proportionate 
Populations. 

Proportionate 
Diam.  to  supply 
the  Populations 
in  Table  A. 

Remarks. 

5 

.1 

.32 

1.58 

10 

.2 

.45 

1.37 

12}$ 

.25 

.50 

1.32 

Add  one-third. 

15 

.3 

.55 

1.27 

Add  full  one-fourth. 

20 

.4 

.64 

1.20 

Add  one-fifth. 

25 

.5 

.71 

1.14 

Add  one-seventh. 

30 

.6 

.78 

1.11 

Add  one-ninth. 

35 

.7 

.84 

1.07 

Add  one-  fourteenth. 

40** 

.75 

.87 

1.06 

Add  one-sixteenth. 

45 

J 

.'95 

1.05 
1.02 

Add  one-fiftieth. 

50 

1.0 

1.00 

1.00 

75 

1.5 

1.23 

.92 

Deduct  one-thirteenth. 

100 

2.0 

1.41 

.88 

Deduct  one  eighth. 

125 

2.5 

1.59 

.83 

Deduct  full  one-sixth. 

150 

3.0 

1.73 

.80 

Deduct  one-fifth. 

200 
250 

4.0 
5.0 

2.00 
2.25 

.76 
.73 

Deduct  nearly  one-fourth. 
Deduct  nearly  two-sevenths. 

300 

60 

2.46 

.69 

Deduct  three-tenths. 

400 

8.0 

2.83 

.66 

Deduct  full  one-third. 

500 

10.0 

3.18 

.63 

Examp.    By  Table  A  we  see  that  with  the  rate  of  head  of  50  feet  per  mile,  a 

30-inch  pipe  will  supply  a  population  of  91580 ;  but  with  three  times  that  rate  of  head,  or  150  ft  per 
mile,  we  see  by  col  3,  Table  B,  that  the  same  pipe  will  supply  1.73  times  as  many  persons,  or  91580  X 
1.73  =  158433  persons.  But  if,  at  this  greater  rate  of  head,  we  still  wish  to  supply  only  91580  persons, 
then  we  find  in  col  4,  Table  B,  that  we  may  diminish  the  diam  of  the  pipe  from  30,  down  to  30  X  .80 
=  24  ins;  or,  by  col  5,  we  have  30— 6  =  24  ins. 

Again,  after  the  water  has  reached  the  city  by  the  30-inch  pipe  of  Table  A,  if  we  wish  to  distribute  it 
through  the  city  by  say  eight  branches  or  smaller  mains,  we  see  by  col  6,  Table  A,  that  each  of  them 
must  have  at  least  13}^  ins  diam.  From  these  eight,  other  smaller  ones  may  branch  off  into  the 
cross  streets,  alleys,  Ac;  and  in  estimating  the  supply  required  for  any  particular  street  main,  we 
must  evidently  add  what  is  reqd  also  for  such  cross  streets,  &c,  &c,  as  are  to  be  fed  from  said  main. 

If  certain  limited  parts  of  a  city  pipe  system  have  considerably  less  rates  of  head  than  most  of  the 
remainder,  it  may  become  expedient  to  supply  the  former  by  a  special  separate  main  of  Inrger  diam  j 
which  may  start  either  directly  from  the  reservoir  ;  or  as  a  branch  from  the  grand  leading  main  which 
feeds  the  lower  parts,  according  to  circumstances. 

It  must  be  remembered,  that  although  by  increasing  the  diams,  an  abundant  supply  may  be  ob- 
tained under  a  small  rate  of  head,  as  well  as  under  a  great  one.  yet  the  water  will  not  rise  to  as  great 
a  height  in  the  service  pipes  for  supplying  the  different  stones  of  dwellings,  &c.  Even  with  the 
diams  in  Table  A,  the  water,  under  ordinary  use,  will  not  rise  in  these  pipes  to  the  full  height  of  the 
surface  of  the  reservoir;  and  if  an  unusual'dra wing-off  is  going  on  at  the  same  time  at  many  parts 
of  the  system,  as  in  case  of  an  extensive  fire,  or  frequently  during  the  hot  summer  months,  it  may 
»ot  rise  to  even  one-half  of  that  height. 

Art.  84.  The  following:  has  been  found  very  effective  for 
preventing  concretions  in  water  pipes.  Formerly  in  Boston,  cast* 
iron  city  pipes,  4  ins  diam,  became  closed  up  in  7  years :  and  those  of  larger  diam  became  seriously 
reduced  in  the  same  time.  But  during  the  last  8  years,  in  which  this  varnish  has  been  used,  no  con* 
eretions  have  formed. 

Coal-pitch  varnish  to  be  applied  to  pipes  and  castings, 


582 


HYDRAULICS. 


made  for  the  Water  Department  of  Philadelphia,  under 
the  following:  conditions: 


;arth  or  sand  which 
to  remove  the  loose 


First.  Every  pipe  must  be  thoroughly  dressed  and  made  clean,  free  frc 
eliugs  to  the  irou  iu  the  moulds  ;  hard  brushes  to  be  used  in  finishing  the 
dust. 

Second.  Every  pipe  must  be  entirely  free  from  rust  when  the  varnish  is  applied.  If  the  pipe  can- 
not  be  dipped  immediately  alter  being  cleansed,  the  surface  must  be  oiled  witu  huseed  oil  to  preserve 
it  until  it  is  ready  to  be  dipped  :  no  pipe  to  be  dipped  after  rust  has  set  in. 

Third.  The  coal  tar  pitch  is  made  from  coal  tar,  distilled  until  the  naphtha  is  entirely  removed, 
and  the  material  deodorized.  It  should  be  distilled  until  it  is  about  the  consistency  of  wax.  The 
mixture  of  five  or  six  per  cent  of  linseed  oil  is  recommended.  Pitch  which  becomes  hard  and  brittle 
when  cold,  will  not  answer  for  this  use. 

Fourth.  Pitch  of  the  proper  quality  having  been  obtained,  it  must  be  carefully  heated  in  a  suit- 
able vessel  to  a  temperature  of  300  degrees  Fahrenheit,  and  must  be  maintained  at  not  less  than  this 
temperature  during  the  time  of  dipping.  The  material  will  thicken  and  deteriorate  after  a  number 
of  pipes  have  been  dipped;  fresh  pitch  must  therefore  be  frequently  added;  and  occasionally  the 
vessel  must  be  entirely  emptied  of  its  old  contents,  and  refilled  with  fresh  pitch;  the  refuse  will  be 
hard  and  brittle  like  common  pitch. 

Fifth.  Every  pipe  must  attain  a  temperature  of  300  degrees  Fahrenheit,  before  it  is  removed  from 
the  vessel  of  hot  pitch.  It  may  then  be  slowly  removed  and  laid  upon  skids  to  drip. 

AH  pipes  of  20  inches  diameter  and  upward,  will  require  to  remain  at  least  thirty  minutes  in  the 
hot  fluid,  to  attain  this  temperature;  probably  more  in  cold  weather. 

Sixth.  The  application  must  be  made  to  the  satisfaction  of  the  Chief  Engineer  of  the  Water  De- 
partment; and  the  material  be  subject  at  all  times  to  his  examination,  inspection,  and  rejection. 

Seventh.  Payment  for  coating  the  pipes  will  only  be  made  on  such  pipes  as  are  sound  and  suffi- 
cient according  to  the  specifications,  and  are  acceptable  independent  of  the  coating. 

Eighth.  No  pipe  to  be  dipped  until  the  authorized  inspector  has  examined  it  as  to  cleaning  and 
rust;  and  subjected  it  thoroughly  to  the  hammer  proof.  It  may  then  be  dipped,  after  which,  it  will 
be  passed  to  the  hydraulic  press  to  meet  the  required  water  proof. 

Ninth.  The  proper  coaling  will  be  tough  and  tenacious  when  cold  on  the  pipes,  and  not  brittle  or 
with  any  tendency  to  scale  off.  When  the  coating  of  any  pipe  has  not  been  properly  applied,  and  does 
not  give  satisfaction,  whether  from  defect  in  material,  tools,  or  manipulations,  it  shall  not  be  paid 
for;  if  it  scales  off  or  shows  a.  tendency  that  way,  the  pipe  shall  be  cleansed  inside  before  it  can  be 
recoated  or  be  receivable  as  an  ordinary  pipe.* 

FREDERIC  GRAFF,  Chief  Eng  Water  Department. 

Art.  35.  The  syphon,  or  siphon  f  If  one  leg  a  b  of  a  bent  tube  or  pipe  a  be, 
Fig  43,  of  any  diam,  filled  with  water,  and  with  both  its  ends  stopped, 
be  placed  in  a  reservoir  of  water,  as  in  the  fig;  and  if  the  stoppers  be 
then  removed,  the  water  in  the  reservoir  will  begin  to  flow  out  at  c,  and 
will  continue  to  do  so  until  its  level  is  reduced  to  t,  which  is  the  same  aa 
that  of  the  highest  end  c  of  the  pipe  or  syphon.     The  flow  will  then  stop. 
The  parts  a  b  and  6  c  are  called  the  legs  of  the  syphon,  6  being  its  high- 
est point ;  and  this  is  correct  so  far  as  relates  to  it  merely  as  a  piece  of 
tube ;  but  considering  it  purely  with  regard  to  its  character  as  a  hydrau- 
lic machine,  the  part  t  a  below  the  level  of  the  highest  end  c,  may  be  en- 
tirely neglected;  for  the  water  in  the  reservoir  will  not  be  draw'n  down 
below  the  level  of  the  highest  end,  whether  that  be  the  inner  or  the  outer 
one.   Therefore,  if  the  disch  end  be  above  the  water  in  the  reservoir,  as, 
for  instance,  at  w,  no  flow  will  take  place.     The  vert  height  6  o,  from  the 
highest  part  of  the  syphon,  to  the  lowest  level  t,  to  which  the  reservoir 
is  to  be  drawn  down,  must  not,  theoretically,  exceed  about  33  or  34  ft; 
or  that  at  which  the  pres  of  the  air  will  sustain  a  column  of  water. 
Practically  it  must  be  less,  to  allow  for  the  friction  of  the  flowing  water, 
and  for  air  which  forces  its  way  in.  And  still  less  at  places  far  above  sea 
level;  for  at  such  the  reduced  weight  of  the  atmospheric  column  will  not 
balance  so  great  a  height  of  water.     In  order  readily  to  understand,  or 
at  any  time  to  recall  the  principle  on  which  the  syphon  acts,  bear  in 
mind  that  we  may  theoretically  consider  the  end  of  the  inner  leg  to  be 
not  actually  immersed  below  the  water  surf,  but  only  to  be  kept  precisely 
at  it,  as  the  surf  descends  while  the  water  is  flowing  out:  but  may  re- 
gard the  vert  dist  6  o  as  the  length  of  the  outer  leg ;  and  a  varying  dist,  which  at  first  is  b  s,  and  finally 
b  o  (as  the  surf  of  the  reservoir  descends)  as  the  length  of  the  inner  leg;  and  that  the  flow  continues 
only  while  this  outer  leg  is  longer  than  this  inner  one.     The  books  are  wrong  in  saying  that  the  outer 
leg  6  c  mnst  be  longer  than  the  inner  one  b  a,  in  order  that  the  water  may  run  at  all.     The  principle 
then  is  simply  this:  that  both  these  legs  6  c,  and  bi,  being  first  filled  with  water,  (the  part  i a  being 
considered  at  first  as  a  portion  of  the  reservoir,  and  not  of  the  syphon,)  it  follows  that  when  the  stop- 
pers are  removed  from  the  ends  c  and  a,  the  air  presses  equally'against  these  ends ;  but  the  great  vert 
head  of  water  b  o  iu  the  outer  leg  b  c,  presses  against  the  air  at  c,  with  more  force  than  the  small  head 
of  water  6s  in  the  inner  leg  bi,  does  against  the  air  at  a  or  i.l    Consequently,  the  water  in  be  will 
tend  to  fall  out  more  rapidly  than  that  iu  ft  t ;  and  as  it  commences  to  fall,  would  produce  a  vacuum  at 
6,  were  it  not  that  the  pres  of  the  air  against  the  other  end  a  or  t,  forces  the  water  up  i  b,  to  supply 
the  place  of  that  which  flows  out  at  c.     In  this  manner  the  flow  continues  until  the  surf  of  the  water 
in  the  reservoir  descends  to  t.  on  the  same  level  as  c.     The  pressures  of  the  vert  heads  bo,  bo,  in  the 
two  legs  be,  bt,  being  then  equal,  it  ceases. 

The  syphon  principle  may  be  employed  for  draining  ponds  into  lower  ground  at  a  considerable  dist, 
eveia  though  an  elevation  of  several  feet  (in  practice  perhaps  not  exceeding  about  28  ft  above  the  level 
to  which  the  pond  is  to  be  reduced)  may  intervene.  In  such  a  case  an  escape  must  be  provided 
at  ttie  summit  (or  summits,  if  there  are  niore  than  one)  of  the  bends,  for  the  disch  of  free  air,  which 
will  inevitably  enter,  and  soon  stop  the  flow,  unless  this  precaution  be  taken.  The  air-valve  Fig  42 

*  Such  coating  is  said  by  D^rcy  to  increase  the  discharge  materially. 
tThis  subject  belongs  more  properly  to  Pneumatics. 

}  Said  pres  of  the  air  at  a  or  i,  is  of  course  not  direct ;  but  is  transmitted  through  the  water  to  a; 
ftfld  thence  upward  through  the  syphon  to  i. 


[>AMS. 


583 


,/ill  not  answer  for  this,  because  as  soon  as  the  valve  v  opens,  the  syphon  becomes 
n  effect  two  separate  tubes  i#pen  at  top;  and  the  water  will  fall  in  both.  An  ori- 
fice at  the  escape  will  be  weeded  for  filling  the  syphon  at  the  start;  and  to  pre- 
sent the  water  thus  intnwluced,  from  running  out.  stopcocks  must  be  provided  at 
;he  ends,  and  kept  closed  until  the  filling  is  completed. 

The  greatest  pains  must  be  taken  to  make  all  the  joints  perfectly  air-tight. 

The  motive  power  or  head  which  causes  the  flow  in  a  syphon,  is  the 


ng  this  head,  the  entire  length  a  b  c  ot  the  syphon,  and  its  diam,  all  in  It,  the 
disch  may  be  found  approximately  by  either  of  the  rules  given  in  Art  2  for  straight 


pipes.    These  rules  f 
:>y  Col  Crozet's  syph 


approximately  uy  eii/iier  ui  me  i  iues  givt:u  in  /vri  &  101  siraigiit 
give  55%  galls  per  min,  instead  of  the  43%  galls  actually  discnd 
Ion,  with  a  head  of  20  ft,  as  stated  oil  p  661,  which  see. 


DAMS. 


WE  can  devote  but  little  space  to  this  subject,  in  addition  to  what  Is  said  on  earthen  dams  for  re- 
servoirs p  577;  and  on  stone  ones,  p  528  &c.  Those  we  shall  BOW  describe  will  also  answer  for 
uch  reservoirs,  when  the  perishable  nature  of  timber  is  not  an  objection. 

Primary  objects  in  the  erection  of  dams,  are,  a  foundation  suffi- 

jiently  firm  to  prevent  them  from  settling,  and  thus  leaking;  the  prevention  of  leaks  through  their 
backs,  or  under  their  bases ;  and  the  prevention  of  wear  of  the  bottom  of  the  stream  in  front  of  the 
dam,  by  the  action  of  the  falling  water.  For  the  first  purpose,  hard  level  rock  bottom  is  of  course 
'  best;  and  should  be  chosen,  if  possible.  In  that  case,  thick  planks,  tt.  Fig  6,  (single  or  double, 
ihe  case  may  be,)  closely  jointed,  and  reaching  from  the  crest  c,  to  the  back  lower  edge  w,  (where 
hey  should  be  scribed  down  to  the  rock;)  with  a  good  backing,  b,  of  gravel,  will  suffice  to  prevent 
:aks.  Gravel,  or  rather  very  gravelly  soil,  is  far  better  than  earth  for  this  purpose  ;  for  if  the  water 
bould  chance  to  form  a  void" in  it,  the  gravel  falls  and  stops  it.  To  prevent  this  backing  from  being 
disturbed  near  the  crest  of  the  dam,  by  floating  bodies  swept  along  by  freshets,  a  rough  pavement  of 
stones,  ahout  15  to  18  ins  deep,  as  shown  in  Fig  7,  should  be  added  for  a  width  of  about  10  to  20  ft; 
or  until  its  top  becomes  3  to  5  ft  below  the  crest  c  of  the  dam,  according  to  circumstances. 


Rgl 


ROCK 

In  Fig  1,  (a  dam  on  the  Schuylkill  navigation,)  the  upper  timbers,  e,  are  all  close  jointed,  and  laid 
teaching,  so  as  not  to  require  planking  in  addition. 

But  if  the  bottom  of  the  stream  is  gravel  or  earth,  there  must  in  addition  to  these  be  used  twc 
thicknesses  of  sheet  piles,  p,  Fig  2,  &c,  close  driven,  breaking  joint,  to  a  depth  of  several  ft.  to  pre- 
vent leaking  through  the  soil  beneath  the  base  of  the  dam.     Frequently  but  one  thickness  is  used. 
If  the  bottom  is  soft  or  open  for  a  depth  of  only  a  few  ft,  it  is  at  times  better  to  remove  it.  and  base 
the  dam  on  the  firmer  stratum  below;  still,  however,  using  the  sheet  piles.    Old  decayed  timber  and 
other  rubbish  should  be  removed  from  the  base.     In  very  bad  soils  of  greater  depth,  it  may  be  neces- 
iry  to  support  the  dam  entirely  upon  a  platform  resting  on  bearing  piles.     Here  great  p'recautions 
re  necessary  against  leaks  ;  but  the  case  occurs  so  rarely,  that  we  shall  not  stop  to  consider  it. 
As  to  the  wearing  away  of  the  bottom  of  the  stream  by  the  water  falling  over  the  front  of  the  dam, 
precautions  should  be  used  in  all  cases  except  that  of  very  hard  rock,  or  of  medium  rock  protected 

C 


GRAVE  L 


by  a  considerable  depth  of  water.     The  dam,  Fig  1,  was  built  upon  a  tolerably  firm  micaceous  gneiss 
in  nearly  vert  strata,  covered  by  about  2  feet  of  water  in  ordinary  stages.    lii  39  years  the  rock  was 


584 


DAMS. 


Rq3 


worn  away  in  front  of  the  dam,  as  shown  in  the  fig,  to  the  average  depth  of  3  feet;  or  very  nearly  1 
inch  per  year.  The  depth  of  water  on  the  crest  c,  was  usually  from  6  to  18  ins  ;  rarely  5  or  6  ft  dur- 
ing freshets;  and  but  a  few  times  during  the  whole  period,  8  or  9  ft. 

At  Jones's  dam,  on  Cape  Fear  River;  height  of  dam,  16  ft;  front  vert; 

fall,  usually  10  ft,  into  6  ft  depth  of  water;  the  soft  shale  rock,  in  vert  strata,  was,  in  the  course  of 
a  few  years,  worn  away  16  ft  ;  and  the  dam  was  undermined  to  such  an  extent  as  to  fall  into  the  cavity. 
In  another  case,  dam  36  ft  high  ;  front  vert  ;  the  water  falling  upon  nearly  vert  strata  of  hard  shale 
rock,  usually  covered  by  but  about  2  ft  of  water  ;  in  about  20  years  wore  it  to  an  irregular  depth  of 
from  10  to  20  ft  ;  and  extending  from  the  very  face  of  the  dam',  to  70  or  80  ft  in  front  of  it. 

In  Fig  2,  upon  a  stream  subject  to  very  violent  freshets,  the  gravel  was  washed  awav  for  a  consid- 
erable width  and  depth  beyond  the  apron,  as  at  A.  To  prevent  a  repetition,  the  cavity  was  tilled 
with  cribwork  full  of  stone,  clear  across  the  river. 

A  deposit  of  blocks  of  loose  stone,  of  even 
a  ton  weight  or  more,  will  not  serve  as  a  pro- 
tection in  front  of  a  darn  exposed  to  high 
freshets;  but  will  soon  be  swept  away.  A 
common  precaution  against  this  wear,  in  low 
dams,  is  an  apron,  a  a,  Fig  2  ;  or  d  d,  Fig  3  ; 
of  either  rough  round  tree  trunks,  or  of  hewn 
timber,  laid  close  together  ;  extending  under 
the  entire  base  of  the  dam,  and  from  15  to  30 
ft  in  front  of  its  face.  These  are  sometimes 
bolted  to  pieces,  ss,  Fig  2  ;  or  yy,  Fig  3  ;  laid 
under  them  across  the  stream.  In  Fig  3, 
with  very  soft  bottom,  these  pieces  yy  are 
supposed  to  be  bolted  to  short  piles  1  1,  driven 
"  for  that  purpose. 

At  times  a  distinct  wide  low  timber  crib,  filled  with  stone,  and  covered  on  top  with  stout  plank, 
has  been  placed  in  front  of  the  dam,  to  receive  the  fall  of  the  water;  and  is  effective  in  protecting 
the  bottom.  Also,  in  some  cases,  a  dam  of  less  height,  and  of  cheap  character,  has  been  built  at  a 
short  distance  down  stream  from  the  main  one,  in  order  to  secure  at  all  times  a  deep  pool  in  front 
of  the  latter  for  breaking  the  force. 

Another  precaution    Is    to 
substitute  a  sloping  front  like 
cl,  Fig  4,  or  such  as  Figa  1 
and  2  would  form  if  reversed, 
for  the  nearly  vert  one  of  the 
other  figs  ;  thus  to  some  extent 
reducing  the  force  of  the  wa- 
ter.    This,  however,  is  but  a 
partial     remedy,     especially 
for  soft  bottoms  in   shallow 
water;  for  the  sliding  sheet 
still  descends  with  great  force. 
The  best  form  of  dam,  per- 
haps, in  such  cases,  is  that 
shown  in  Fig  5,  in  which  the 
front  consists  of  a  series  of  steps  of 
about  1  vert,  to  3  or  4  hor.     These  ef- 
fectually break  the  force  of  the  water; 
and,  with  the  addition  of  an  apron  o  o, 
secure  a  satisfactory  result.    It  is  ob- 
jected   against    this     form,    as    also 
against  Figs  4  and  6,  that  their  fronts 
are  liable  to  he  torn   by  descending 
trees,   ice,   and   other    bodies    swept 
but  little  weight;  for  when  such 
the  front  timbers.     On  the  Sea 


along  during  freshets  ;  but  experience  shows  that  this  objection  ha 
bodies  pass,  the  sheet  of  water  is  thicker  than  usual  ;  and  protects 
Nav,  the  timbers  cl,  Fig  6,  scarcely  wear  thin  at  the  rate  of  an  inch  in  10  to  15  years. 


Fig  6 


ROCK 


The  forms  of  wooden  dams  are  many;  (see  the  figs,  which  show 

those  most  used  :)  varying  with  the  circumstances  of  the  case,  and  with  the  fancy  of  the  designer. 
In  the  United  States  they  are  usually  of  cribwork.  of  either  rough  round  logs  with  the  bark  on,  or  of 
hewn  timber;  in  either  case  about  a  foot  through.  These  timbers  are  merely  laid  on  top  of  each 


DAMS. 


585 


»ther,  forming  in  plan  a  series  of  rectangles  with  sides  of  about  7  to  12  ft.  They  are  not  notched 
together,  but  simply  bolted  by  1  inch  square  bolts  (often  ragged  or  jagged)  about  2  to  2Jf>  feet  long, 
through  every  timber  at  every  intersection.  These  are  not  found  to  rust  or  wear  seriously,  even  when 
exposed  to  a  current.  Square  bolts  hold  best.  Round  logs  are  flattened  where  they  lie  upon  ench 
other.  Experience  shows  that  firmer  but  more  expensive  connections  are  entirely  unnecessary.  The 
sribs  are  usually,  but  not  always,  tilled  with 
•ough  stone.  In  triangular  dams,  disposed 
is  in  Figs  1,  '2,  and  7,  this  stoue  filling  is 
jot  so  essential  as  in  other  forms  ;  because 
the  weight  of  the  water,  and  of  the  gravel 
backing,  tends  to  hold  the  dam  down  on  its 
base.  Still,  even  in  these,  when  the  lower 
ibers  are  not  bolted  to  a  rock  bottom,  or 
otherwise  secured  in  place,  some  stone  may 
necessary  to  prevent  the  timbers  from 
Lting  away  while  the  work  is  unfinished, 

and  the  gravel  not  yet  deposited  behind  it.  ra     «7 

On  rock,  the  lowest  timbers  are  often  bolted  -FIQ  I 

to  it,  to  prevent  them  from  floating  away 
during  construction;  and  when  the  water 

Is  some  feet  deep,  this  requires  coffer-dams.  Or.  the  cribs  may  be  built  at  first  only  a  few  feet  high  ; 
then  floated  into  place,  and  sunk  by  loading  them  with  stoue;  for  the  reception  of  which  a  rough 
platform  or  flooring  will  be  reqd  in  the  cribs,  a  little  above  their  lowest  timbers.  The  bolting  to  the 
rock  may  then  be  dispensed  with.  The  water  may  flow  through  the  open  cribwork  as  the  building 
higher  goes  on  ;  attention  being  paid  to  adding  stone  enough  to  prevent  it  floating  away  if  a  freshet 
should  happen.  Or,  cribs  shown  in  plan  at  cc,  Fig  8,  loaded  with 
stone,  may  be  sunk,  leaving  one  or  more  intervals,  like  that  at  o  o  o  o, 
between  them,  for  the  free  escape  of  the  water.  These  openings  to 
be  finally  closed  by  floating  into  them  closing-cribs  shaped  like  «. 

The  workmanship  of  a  dam  in  deep  water  can  of  course  be  much 
better  executed  in  coffer-dams,  than  by  merely  sinking  cribs.  The 
joints  can  be  made  tighter:  the  stone  filling  better  packed  ;  the  sheet 
piling  more  closely  fitted,  &c. 

When  a  very  uneven  rock  bottom  in  deep  water,  or  the  introduc- 
tion of  sluices  in  the  dam,  or  any  other  considerations,  make  it  ex- 
pedient to  build  dams  within  coffer-dams,  both  should  be  carried  on 
in  sections ;  so  as  to  leave  part  of  the  channel-way  open  for  the  es- 
cape of  the  water.  Commencing  at  one  or  both  shores,  the  first  section  of  the  coffer-dam  may  reach 
say  quarter  way  or  more  across  the  stream.  In  the  section  of  The  dam  itself  built  within  this  enclos- 
ing coffer-dam,  ample  sluices  should  be  left  for  the  water  to  flow  through  when  we  come  to  build  the 
closing  section  of  the  coffer-dam.  When  the  dam  has  been  finished,  these  sluices  may  be  closed 
by  drop-timbers*.  Before  removing  one  section  of  coffer-dam,  the  outer  end  of  the  enclosed 
section  of  dam  itself  must  be  firmly  finished  in  such  a  manner  as  to  constitute  a  part  of  the  inner 
end  of  the  next  section  of  coffer-dam.  It  is  impossible  to  give  details  for  every  contingency  ;  the  en- 
gineer must  rely  upon  his  own  ingenuity  to  meet  the  peculiarities  of  the  case  before  him.  In  some 
cases  of  shallow  water,  mere  mounds  of' earth  may  answer  for  coffer-dams;  or  rough  stone  mounds, 
backed  with  earth  or  gravel. 

After  the  water  has  passed  beyond  the  crest,  c  in  the  figs,  there  is  no  necessity  for  preventing  its 
leaking  down  among  the  crib  timbers:  on  the  contrary,  the  thick  sheeting  planks,  (or  squared  tim- 
bers, as  occasion  may  require.)  ci.  Figs  4  and  6,  which  form  the  slopes  along  which  the  water  then 
flows  in  some  dams,  are  usually  not  laid  close  together,  but  with  open  joints  of  about  ^>  inch  wide  be- 
tween them,  for  the  express  purpose  of  allowing  part  of  the  water  to  fall  through  them,  so  as  to 
keep  the  timbers  beneath  them  partially  wet:  which,  to  some  extent,  renders  them  more  durable.  In 
Figs  1,  4,  6,  and  7,  the  water  of  the  lower  pool  flows  freely  back  among  the  crib  timbers,  and  rough 
quarry  stones  with  which  the  cribs  are  filled  either  partly  or  entirely.  In  Figs  4  and  6,  these  stones 
are  not  shown.  In  the  dam,  Fig  1,  none  were  used.  In  Fig  2,  they  were  as  shown. 

A  substantial,  and  not  very  expensive  dam  of  the  form  of  Fig  7.  may  be  built  of  rough  stone  in 
cement.  Some  hewn  timbers  should  be  firmly  built  horizontally  into  the  masonry  of  the  sloping 
back  c  n  w,  at  a  few  feet  apart,  with  their  tops  level  with  the  surf  of  the  masonry.  To  these  must  be 
well  spiked  close-jointed  sheeting-plank  cn«>,  for  protecting  the  masonry  from  the  action  of  the 
water,  and  of  floating  bodies.  The  gravel  backing  b,  may  be  omitted  ;  but  the  sheet  piles  p,  and  an 
apron  in  front  of  the  dam,  will  be  as  indispensable  in  yielding  soils,  as  if  the  dam  were  of  timber. 

Figs  1.  2,  4,  6.  and  7,  are  sections  drawn  to  a  scale,  of  existing  dams  in  Pennsylvania  that  have- 
stood  successfully  the  force  of  heavy  freshets  for  a  long  series  of  years. t  These  fres'hets  at  times  carry 
along  large  bodies  of  ice,  trees,  houses,  bridges,  &c. ;  and  have  risen  to  11  ft  above  the  crests.  Fig  1. 
on  the  Sch  Nav,  was  built  in  1819,  and  served  perfectly  for  39  years,  until  in  1858  the  decay  of  much 
of  its  timber,  especially  of  the  close-laid  top  ones,  e,  rendered  it  necessary  to  build  a  new  one  just  in 
front  of  it.  It  was  of  extremely  simple  construction  ;  and  was  never  filled  w'ith  stone.  The  bottom  tim- 
bers, o  o,  10  ft  apart,  were  bolted  to  the  rock  ;  and  immediately  over  each  of  them,  was  such  a  series  of 
inclined  timbers  a,s  is  shown  in  the  fig.  The  top  ones,  e.  however,  were  close  jointed,  and  laid  touching, 
so  as  to  form  the  top  sheeting,  instead  of  thinner  planks.  The  short  pieces  at  t  were  lai-)  in  the  same 
way.  No  coffer-dam  was  used  :  but  the  bottom  pieces  were  first  bolted  to  the  rock  ;  10  ft  apart ;  then 
the  stringers  and  the  sloping  pieces  were  added.  The  close  covering  (e)  was  carried  forward  from 
each  end  of  the  dam,  until  at  last  a  space  of  only  about  60  ft  was  left  in  the  center,  for  the  water  to 
pass.  The  close  covering  for  this  space  heing'then  all  got  ready,  a  strong  force  of  men  was  set  to 
work,  and  the  space  was  covered  so  rapidly  that  the  river  had  not  time  to  rise  sufficiently  high  to 
impede  the  operation. 


*  Timbers  ready  prppared  fnr  closing  an  opening  through  which  water  is  flowing;  and  suddenly 
dropped  into  place  by  means  of  grooves  or  guides  of  some  kind  for  retaining  them  in  position.  Sev- 
eral such  timbers  may  at  times  be  firmly  framed  together,  and  then  be  all  dropped  at  once;  closing 
the  opening  or  sluice  at  one  operation  ;  especially  when  it  is  of  small  size.  In  some  cases,  a  crib 
may  be  sunk  on  the  up-stream  side  of  such  an  opening,  for  closing  it. 

t  Those  on  the  Schuylkill  Navigation  were  obligingly  furnished  by  James  F.  Smith,  Esq.  chief 
engineer  and  superintendent  of  that  work.  Other  valuable  information  from  the  same  source  will 
b«  found  in  different  parts  of  this  volume. 


586 


DAMS. 


Fig  2  is  a  canal  feeder  dam  on  the  Juniata.  Here  s  s  are  timbers  stretching  clear  across  the  stream, 
(about  300 ft,)  and  sustaining  the  apron  aa,  of  stout  hewn  timbers  laid  touching.  This  lam  was  filled 
with  stone,  for  the  retention  of  which  the  front  sheeting  planks  were  added. 

Fig  6  is  on  the  Sch  Nav  ;  was  built  in  1855.  It  is  a  form  much  approved  of  on  that  work,  for  such 
situations;  namely,  firm  rock  foundation,  with  a  considerable  depth  of  water  in  front.  The  highest 
dam  (32  ft)  on  the  Sch  Nav,  is  very  similar  to  it;  built  in  1851.  All  the  dams  on  this  work  are  of 
hewn  timber,  chiefly  white  and  yellow  pine.  The  water  occasionally  runs  from  8  to  12  ;eet  deep  over 
their  crests;  and  then  overflows  and  surrounds  many  of  the  abuts.  The  vertical  back  allows  the 
overflowing  water  to  leak  down  among  all  the  lower  timbers  of  the  dam,  and  thus  tend  to  their 
preservation. 

Fig  4  shows  the  dams  on  the  Monongahela  slackwater  navigation  ;  W.  Milnor  Roberts,  eng.  They 
are  of  round  logs,  with  the  bark  on  j  flattened  at  crossings.  The  longest  ones  in  the  fig  are  10  feet 
apart  along  the  length  of  the  dam.  Experience  shows  that  such  dams  possess  all  the  strength  neces- 
sary for  violent  streams.  On  rock,  the  lowest  timbers  are  bolted  to  it. 

Fig  7  has  been  successfully  used  to  heights  of  40  ft.* 

Fig  3  is  intended  merely  as  a  hint  for  a  very  low  dam  on  yielding  bottom.  Its  main  supports  are 
piles  ii,  from  4  to  8  ft  apart,  according  to  the  height  of  the  dam;  and  other  circumstances  ;  and  tt 
are  short  piles  for  sustaining  the  apron  dd.  It  may  be  extended  to  greater  heights  by  adding  braces 
in  front ;  which  may  be  covered  by  stout  planks,  to  form  an  inclined  slide  for  the  overfalliug  water. 
Many  effective  arrangements  of  piles,  and  sloping  timbers  for  dams  on  soft  ground,  will  suggest  them- 
selves to  the  engineer.  Thus,  at  intervals  of  several  feet,  rows  of  3  or  more  piles  may  be  dnveu  trans- 
versely of  the  dam ;  the  top  of  the  outer  pile  of  each  row  being  left  at  the  intended  height  of  the  crest, 
while  those  behind  are  successively  driven  lower  and  lower;  so  that  when  all  are  afterward  con- 
nected by  transverse  and  longitudinal  timbers,  and  covered  by  stout  planking,  and  gravel,  they  will 
form  a  dam  somewhat  of  the  triangular  form  of  Fig  7.  It  would  be  well  to  drive  the  piles  with  an 
inclination  of  their  tops  up  stream. 

There  is  much  scope  for  ingenuity  both  in  designing,  and  in  constructing  dams  under  various  cir- 
cumstances ;  and  in  turning  tiie  course  of  the  water  from  one  channel  to  another,  by  means  of  ditches, 
pipes,  or  troughs,  &c.,  at  diff  heights;  aided  at  times  by  low  temporary  dams  or  mounds  of  earth  ;  or 
»f  sheet  piles,  &c ;  or  by  coffer-dams  ;  so  as  to  keep  it  away  from  the  part  being  built,  Each  locality 
will  have  its  peculiar  features ;  and  the  engineer  must  depend  on  his  judgment  to  make  the  most  of 
them. 

Abutments  of  clams  as  a  general  rule  should  not  contract  the  natural? 

width  of  the  stream  ;  or,  if  they  must  do  so,  as  little  as  possible ;  for  contractions  increase  the  height, 
and  violence  of  the  overflowing  water  in  time  of  freshets  ;  during  which  a  great  length  of  overfall  is 
especially  desirable.  They  should  be  very  firmly  connected  with  the  ends  of  the  dams;  and  should, 
if  the  section  of  the  valley  admits  of  it,  be  so  high:  and  carried  so  far  inland,  that  the  high  water 
of  freshets  will  not  sweep  either  over  them,  or  around  their  extremities;  and  thus  endanger  under- 
mining, and  destruction.  In  wide,  flat  valleys  they  cannot  be  so  extended  without  too  much  ex- 
pense;  and  the  only  alternative  is  to  found  them  so  deeply  and  securely  as  to  withstand  such 
action  ;  making  their  height  such  that  they  will,  at  least,  be  overflowed  but  seldom.  Their  enda 
adjacent  to  the  dam,  should  be  rounded  off,  so  as  to  facilitate  the  flow  of  the  water  over  the  crest. 

They  are  best  built  of  large  stone  in  cement;  for  although  sufficient  strength  may  be  secured  by 
timber,  that  material  decays  rapidly  in  such  exposures.  If  of  earth  only,  they  are  very  apt  to  be 
carried  away  if  a  freshet  should  overtop  them. 

Sluices  should  be  placed  iu  every  important  clam,  in  order  that 

all  the  water  may  be  drawn  off,  if  necessary,  for  the  purpose  of  repairs  ;  or  of  removing  mud  deposits; 
or  finding  lost  articles  of  importance,  &c.  They  may  be  merely  strong  boxings,  with  floor,  sides,  and 
top  of  squared  timbers ;  and  passing  through  the  breadth  of  the  dam,  just  above  the  bottom.  To  pre- 
vent trees,  &c,  from  entering  and  sticking  fast  in  them,  some  kind  of  strong  screen  is  expedient.  In 
common  cases  a  sluice  should  not  exceed  about  3>£  ft  by  5  ft  in  cross-section;  otherwise  it  becomes 
hard  to  work.  Two  or  more  such  openings  may  be  used  when  much  water  is  to  be  voided.  They 
should  be  near  the  abutments.  The  gates  or  valves  for  opening  and  shutting  them,  should  be  at  the 
up-stream  end;  for  if  at  the  lower  one,  accumulations  of  mud,  <fec,  will  fill  the  sluices,  and  prevent 
them  from  working.  They  are  usually  of  timber;  and  slide  vertically  in  rebates;  being  raised  and 
lowered  by  rack  and  pinion  ;  but  in  very  important  dams  they  may  be  of  cast  iron.  Two  sets  of  sluices 
are  desirable;  that  one  may  be  always  ready  for  use  if  the  other  is  stopped  for  repairs. 

The  part  of  the  apron  in  front  of  the  sluice  should  be  particularly  firm,  so  as  not  te  be  deranged  by 
the  water  rushing  out  under  a  high  head. 

I>ams  are  sometimes,  but  rarely,  built  in  the  form  of  an 
arch  ;  convex  up  stream.  This  form  is  strong;  and  when  the  shores  are  of  rock 
it  may  be  expedient  to  use  it ;  but  if  the  banks  are  soft,  they  will  be  exposed  to  wear  by  the  curreni 
thrown  against  them  at  the  abuts  of  the  arch. 

At  times  dams  are  built  obliquely  across  the  stream,  with 

the  object  of  increasing  the  length,  and  consequently  reducing  the  depth  of  water  over  the  crest  in 
times  of  freshets.  The  argument,  however,  appears  to  the  writer  to  be  of  but  little  weight,  inasmuch 
as  the  reduction  of  depth  would  extend  but  a  trifling  distance  up  stream  from  the  dam;  and  would 
therefore  scarcely  have  an  appreciable  effect  in  diminishing  the  injury  to  the  overflowed  district  above. 
Moreover,  the  increased  expense  is  probably  always  more  than  commensurate  with  any  advantage 
gained. 

Some  dams  are  subject  to  "tremblings,"  which  have  not  been 

satisfactorily  accounted  for.  They  exhibit  themselves  chiefly  as  undulations  of  the  air,  produced  by 
the  falling  water;  and  which  occasionally  cause  a  rattling  of  windows  within  a  distance  of  ^  a  mile 
or  more.  We  have  known  this  to  be  stopped  unintentionally  in  one  case,  by  building  a  well-covered 

*  Cost  of  crib  dams.  Formerly,  with  common  labor  at  $1  per  dry  ;  plank 
$20,  and  other  timber  $10  per  1000  ft  board  measure,  delivered:  stone  for  filling.  $1  per  cub  yard  ; 
gravel  40  cts  per  cub  yd;  iron  for  bolts,  <fec  1?  <Us  per  ft  ;  such  dams  in  shallow  water,  usually  cost 
complete,  from  8  to  12  cts  per  cub  ft ;  or  $2.16  to  $3.24  per  cub  yd-  of  orib. 


DAMS.  587 

wide  crib  apron,  a  fewfeet  b>£n,  against  the  front  of  the  dam,  for  preventing  the  abrasion  tf  the  bot- 
tom. In  other  cases  a  secies  of  oblique  timbers  placed  against  the  front  of  the  dam,  and  part  way  up 
it,  at  a  slope  of  abouL*5$to  1,  and  covered  with  plauk,  has  been  perfectly  effective  in  stopping  it. 

The  proper  time  for  building-  clams  is  of  course  at  the  longest  period 
of  low  stage'of  water. 

To  ascertain  in  advance  approximately,  the  height  to 
which  the  water  will  rise  above  the  crest  of  a  dam  :  or  rather, 

a  little  back  from  it  ;  the  crest  being  above  the  level  of  the  original  water.  This  will  vary  with  the  shape 
of  the  crest,  as  may  be  seen  by  reference  to  Fig  26  Vo  of  Hydraulics,  which,  however,  is  a  very  peculiar 
case.  Still,  until  we  have  more  experiments,  appreciable  deviations  from  the  results  of  such  rules 
must,  be  expected  in  practice.  Square  the  disch  of  the  stream  in  cub  ft  per  sec.  Call  this  square,  s. 
Square  the  length  of  the  overfall  in  ft.  Mult  this  square  by  7.  Call  the  prod  p.  Divides  by  p.  Take 
the  cube  rt  of  the  quot.*  Thvs  cube  rt  will  be  the  reqd  approximate  height  of  rise  in  ft. 

M'hen,  in  times  of  freshets,  the  -water  rises  above  the  crest  to  a  height  equal  to  that  of  the  dam  itself, 
there  is  no  perceptible  fall  at  the  dam  ;  aud  boats  may  pass  in  safety  over  the  crest. 

For  measuring  the  disch  over  dams,  see  arts  14  and  15  of  Hydraulics.  See  also  Art  1  of  Hydro- 
statics. In  shape  of  a  formula,  the  foregoing  rule  will  be, 


Rise  _  cube  n    *  /discharge  in  cub  ft  per  sec^\ 
inft      '  J  \i~(iength  of  overfall  in  Jt^.  ) 


When    the   dam   is  originally  a     Q,...  __    ____ 

submerged,  or  drowned  one,  as  1), 

Fig  9  ;  the  following  is  a  rough  approximation,  probably  O:  _  :  ;•  _  _  ^-  _^2^     ...  -  ~.            n 

somewhat  in  excess;  off  being  the  natural  level  of  the  |»______  _  _  ______  ~                 ---  J 

water  previous  to  building  the  dam;  and  oc  the  natural 
depth  in  ft,  of  the  -water  above  the  intended  crest.  Then 
the  required  depth  ac,  of  the  up-stream  water,  above  the 

crest  when  built,  will  be,  approximately,  ^^^t^ 


ac  =  oc  +  cuberootof  /^^  2\  Fit)    Q 

V7X  length  2/  J    J 

Having  a  c,  deduct  o  c  ;  and  the  rem  will  be  a  o,  or  the  required  rise  produced  by  the  dam.* 

The  rnles  given  for  the  varying  rise  of  surface  for  consid- 
erable distances  lip  stream  from  dams  ;  as  well  as  for  some  allied  sub- 
jects in  hydraulics  ;  are  extremely  complicated  ;  and  require  much  greater  knowledge  of  mathematics 
than  is  usually  found  among  civil  engineers  ;  and  so  far  as  regards  their  application  to  the  actualities 
of  common  occurrence,  they  are  probably  no  less  useless  than  complicated. 

The  short  table  on  page  317  will  at  times  be  of  use  in  finding  the  di- 
mensions of  timbers  for  sustaining  the  pressure  of  different  heads  of  water. 


GKAVITY,  FALLING  BODIES, 

Caution.  Owing  to  the  resistance  of  the  air  none  of  the  follow- 
ing rules  give  perfectly  accurate  results  in  practice,  especially  at  great  vels.  The  greater  the  sp  gr 
of  the  body  the  better  will  be  the  result.  The  latitude,  and  the  height  above  sea  level  also  cause  a 
slight  difference.  The  air  resists  both  rising  and  falling  bodies. 

The  dist  through  which  a  body  falls  freely  in  the  first  second  is  very 

nearly  16  1  ft;  and  it  increases  as  the  squares  of  the  times.     Therefore  to  find  the  number  of  feet 
fallen  in  any  case,  find  the  square  of  the  number  of  sec;  and  mult  it  by  16.1. 

The  <lists  fallen  in  equal  consecutive  times  are  as  1,  3,  5.  7,  &c. 
The  vel  which  a  fallen  body  acquires  at  the  end  of  the  first  sec  is 

.-  very  nearly  32.2  ft  per  sec  ;  and  it  increases  as  the  times.  Therefore  to  find  it  in  any  case,  mult  the 
;  number  of"  seconds  by  3'2.2.  The  average  vel  will  be  half  the  final  one,  as  in  all  other  cases  of  uni- 
i  formly  accelerated  vel.  See  art  12,  p  449.  Also  the  above  caution. 

To  find  the  vel  in  ft  per  sec  acquired  in  falling  through  a  given  dist, 

i    mult  the  dist  in  ft  by  64.4 ;  and  take  the  sq  rt  of  the  prod.     Or  refer  to  table  on  p  552. 

To  find  the  time  in  sec,  required  to  fall  through  a  given  dist,  divide  the 

1    dist  in  ft  by  16.1,  aud  take  sq  rt  of  quot.     Sec  reqd  to  impart  a  given  vel,=:  vel  -r  32.2. 

To  find  the  fall  in  ft  required  to  impart  a  given  vel  in  ft  per  sec;  find  the 

I    square  of  the  vel,  and  mult  it  by  the  dec  .0155,  or  divide  it  by  64.4.     See  Table,  p  552. 

If  a  body  be  thrown  vertically  upwards  with  a  given  vel,  it  will 

;•  rise  to  the  same  height  from  which  it  must  have  fallen  in  order  to  acquire  said  vel ;  and  its  vel  will 

•  be  retarded  at  the  rate  of  32.2  ft  per  sec.     Its  average  ascending  velocity  will  be  half  of  that  with 

I  which  it  started;  as  in  all  other  cases  of  uniformly  retarded  vel.     See  Art  12,  p  449.    In  falling  it 

I  will  acquire  the  same  vel  that  it  started  up  with,  and  in  the  same  time.     See  above  Caution. 

If  a  body  has  a  certain  vel  before  it  begins  to  be  uniformly  accelerated 

or  retarded,  its  average  vel  will  be  half  the  sum  of  its  first  and  final  ones.     See  inclined  plane,  p  172 ; 
also  Art  12,  p  449. 


*  These  two  rules  are  from  Rankine,  who  says  they  apply  to  "crests  either  flat  or  slightly  rounded.' 
But  that,  in  itself,  is  very  vague.  • 


588 


SUSPENSION    BRIDGES. 


SUSPENSION  BRIDGES. 


Art.  1.    Table  of  data  required   for  calculating  the  main 
chains  or  cables  of  suspension  bridges.  Original. 


Tension  on  all 

Tension  at  the 

Deflection 
in  parts 
of  the 
Chord. 

Deflection 
in  Deci- 
mals of 
the  Chord. 

Length  of 
Main  Chains 
between  Sus- 
pension Piers, 
in  parts  of  the 
Chord. 

Chains  at 
either  Suspen- 
sion Pier,  in 
parts  of  the 
entire  Sus- 
pended Wt. 
of  the  Bridge, 
and  its  Load. 

Center  of  all 
the  Main 
Chains  ;  in 
parts  of  the 
entire  Sus- 
pended Wt. 
of  the  Bridge, 
and  its  Load. 

Angle  of 
Dicec- 
tion  of 
theChains 
at  the 
Piers. 

Natural      Natural 
Sineofthe  Cosine  of 
Angle  of    the  Angle 
Direction  I  of  Direc- 
of  the      tion  of  the 
Chains,  at  Chains  at 
the  Piers,  i  the  Piers. 

Deg.  Min. 

1-40 

.025 

1.002 

5.03 

5.00 

5    43 

.0995 

.9950 

1-35 

.0286 

1.002 

4.40 

4.87 

6    31 

.1135 

.9935 

1-30 

.0333 

1.003 

3.78 

3.75 

7    36 

.1322 

.9912 

1-25 

.04 

1.004 

3.16 

3.12 

9      6 

.1580 

.9874 

1-20 

.05 

1.006 

2.55 

2.51 

1    19 

.1961 

.9806 

1-19 

.0526 

1.007 

2.43 

2.38 

1    53 

.2060 

.9786 

1-18 

.0555 

1.008 

2.30 

2.25 

2    32 

.2169 

.9762 

1-17 

.0588 

1.009 

2.18 

2.12 

3     14 

.2290 

.9734 

1-16 

.0625 

1.010 

2.06 

2.00 

4      2 

.2425 

.9701 

1-15 

.0667 

1.012 

1.94 

1.87 

4    55 

.2573 

.9663 

1-14 

.0714 

1.013 

1.82 

1.74 

5    57 

.2747 

.9615 

1-13 

.0769 

1.016 

1.70 

1.62 

7      6 

.2941 

.9558 

1-12 

.0833 

1.018 

1.57 

1.49 

8    33 

.3180 

.9480 

1-11 

.0919 

1.022 

1.46 

1.37 

19    59 

.3418 

.9398 

1-10 

.1 

1.026 

1.35 

1.25 

21     48 

.3714 

.9285 

1-9 

.1111 

1.033 

1.23 

1.12 

23    58 

.4062 

.9138 

M 

.125 

1.041 

1.12 

1.00 

26    33 

.4471 

.8945 

1-7 

.1429 

1.053 

1.01 

.881 

29    45 

.4961 

.8726 

3-20 

.15 

1.058 

.972 

.833 

30    58 

.5145 

.8574 

% 

.1667 

1.070 

.901 

.750 

33    41 

.5547 

.8320 

1-5 

.2 

1.098 

.800 

.625 

38    40 

.6247 

.7808 

•225 

1.122 

.747 

.555 

42      0 

.6690 

.7433 

y± 

.25 

1.149 

.707 

.500 

45    00 

.7071 

.7071 

.3 

.3 

1.205 

.651 

.417 

50    12 

.7682 

.6401 

X 

.3333 

1.247 

.625 

.375 

53      8 

.8000 

.6000 

.4 

.4 

1.332 

.589 

.312 

58      2 

.8483 

,5294 

9-20 

.45 

1.403 

.572 

.278 

60    57 

.8742 

.4855 

K 

.5 

1.480 

.559 

.250 

63     26 

.8944 

.4472 

These  calculations  are  based  on  the  assumption  that  the  curve  formed  by  the  main  chains  is  a 
parabola ;  which  is  not  strictly  correct.  In  a  finished  bridge,  the  curve  is  between  a  parabola  and  a 
catenary  ;  and  is  not  susceptible  of  a  rigorous  determination.  It  may  Save  SOlUe  trou- 
ble  in  making*  tbe  drawings  of  a  suspension  bridge,  to  remember  that  when  the 
deflection  does  not  exceed  about  yV  of  the  span,  a  segment  of  a  circle  may  be  used  instead  of  the 
true  curve ;  inasmuch  as  the  two  then  coincide  very  closely  ;  and  the  more  so  as  the  deflection  be- 
comes less  than  YQ-.  The  dimensions  taken  from  the  drawing  of  a  segment  will  answer  all  the  pur- 
poses of  estimating  the  quantities  of  materials. 

For  some  particulars  respecting  wire  for  cables,  see  pages  369  and  380. 

The  rule  commonly  given  for  the  length  of  a  parabola,  is 
entirely  erroneous.  See  parabola,  in  Mensuration.  Thus,  it  makes  the  last 
length  in  the  third  column  1.666,  instead  of  1.480 :  or  about  12><j  per  cent  too  long.  The  error  dimin- 
ishes as  the  curve  becomes  flatter. 

The  deflection  usually  adopted  by  engineers  for  great  spans  is 

about  yL-  to  T^TT  tne  sPan-  As  much  as  y1^  is  generally  confined  to  small  spans.  The  bridge  will 
be  stronger,  or  will  require  less  area  of  cable,  if  the  deflection  is  greater ;  but  it  then  undulates  more 
readily  ;  and  as  undulations  tend  to  destroy  the  bridge  by  loosening  the  joints,  and  by  increasing  the 
momentum,  they  must  be  specially  guarded  against  as  much  as  possible.  The  usual  mode  of  doing 
this  is  by  trussing  the  hand-railing;  which  with  this  view  may  be  made  higher,  and  of  stouter  tim- 
bers than  would  otherwise  be  necessary.  In  large  spans,  indeed,  it  may  be  supplanted  by  regular 
bridge-trusses,  sufficiently  high  to  be  braced  together  overhead,  as  in  the  Niagara  Railroad  bridge, 
where  the  trusses  are  18  ft  high ;  supporting  a  single-track  railroad  on  top  ;  and  a  common  roadway 
of  19  ft  clear  width,  below.* 

*  The  writer  believes  himself  to  have  been  the  first  person  to  suggest  the  addition  of  very  deep 
trusses  braced  together  transversely,  for  large  suspension  bridges.  Early  in  1851,  he  designed  such 
a  bridge,  with  four  spans  of  1000  ft  each  ;  and  two  of  500;  with  wire  cables  ;  and  trusses  20  ft  high. 
It  was  intended  for  crossing  the  Delaware  at  Market  Street,  Philada.  It  was  publicly  exhibited  for 
several  months  at  the  Franklin  Institute,  and  at  the  Merchants'  Exchange;  and  was  finally  stolen 
from  the  hall  of  the  latter.  Mr  Rooming's  Niagara  bridge,  of  800  ft  span,  with  trusses  18  ft  high,  was 
not  cffmmenced  until  the  latter  part  of  1852j  or  about  18  months  after  mine  had  been  publicly  ex- 


589 


Another  very  iinpoptsfltaid  is  found  in  deep  longitudinal  floor  timbers,  iirmly  united  where  their 
ends  meet  each  other.  These  assist  by  distributing  among  several  suspeuder-rods,  and  by  that 
means  along  a  considejable  length  of  main  cable,  the  weight  of  heavy  passing  loads  ;  and  thus  pre- 
vent the  undue  undulation  that  would  take  place  if  the  load  were  concentrated  upon  only  two  opposite 
suspenders.  With  this  view,  the  wooden  stringers  under  the  rails  on  the  Niagara  bridge  are  made 
virtually  4  ft  deep.  The  same  principle  is  evidently  good  for  ordinary  trussed  bridges. 

Another  mode  of  relieving  the  main  cables  is  by  means  of  cable-stays  :  which  are  bars  of  Iron,  or 
wire  ropes,  extending  like  c  y,  Fig  1,  from  the  saddles  at  the  points  of  suspension  c,  d,  obliquely  down 
to  the  floor,  or  to  some  part  of  the  truss.  In  the  Niagara  bridge  are  04  such  stays,  of  wire  ropes  of 
1%  inch  diam;  the  longest  of  which  reach  more  than  quarter  way  across  the  span  from  each  tower. 
They  transfer  much  of  the  strain  of  the  wt  of  the  bridge  and  its  load  directly  to  the  saddles  at  the  top 
of  the  towers;  thereby  relieviug  every  paitof  the  cable;  as  well  as  diminishing  undulation.  They 
end  at  c  and  d,  where  they  are  attached,  not  to  the  cables,  but  to  the  saddles. 

The  greatest  danger  arises  from  the  action  of  strong*  winds 
striking  below  tlie  floor,  and  either  lifting  the  whole  platform,  and  letting 
it  fall  suddenly  ;  or  imparting  to  it  violent  wavelike  undulations.  The  bridge  of  1010  ft  span  across 
the  Ohio  at  Wheeling,  by  Charles  Ellet,  Jr,  was  destroyed  in  this  manner.  It  is  said  to  have  undu- 
lated 20  ft  vertically  before  giving  way.  It  bad  no  effective  guards  against  undulation  ;  for  although 
its  hand-railing  was  trussed.it  was  too  low  anJ  slight  to  be  of  much  service  in  so  great  a  span. 
Many  other  bridges  have  been  either  destroyed  or  injured  in  the  same  way.  When  the  height  of  the 
roadway  above  the  water  admits  of  it,  the  precaution  may  be  adopted  of  tie-rods,  or  anchor  rods; 
under  the  floor  at  different  points  along  the  span,  and  carried  from  thence,  inclining  downward,  to  , 
the  abutments,  to  which  they  should  be  very  strongly  confined.  In  the  Niagara  Eailroad  bridge  56 
such  ties,  made  of  wire  ropes  1^  inch  diam,  extend  diagonally  from  the  bottom  of  the  bridge,  to  the 
rocks  below.  They,  however,  detract  greatly  from  the  dignity  of  a  structure. 

Mr  Brunei,  in  some  cases,  for  checking  undulations  from  violent  winds  striking  beneath  the  plat- 
form, used  also  inverted  or  up-curving  cables  under  the  floor.  Their  ends  were  strongly  confined  to 
the  abuts  several  ft  below  the  platform ;  and  the  cables  were  connected  at  intervals,  with  the  plat- 
form, so  as  to  hold  it  down. 

Art.  2.  The  angle  adg,  or  act,  Fig  1,  which  a  tang  dg  or  ci  to  the  curve  at 
either  point  of  suspension  c  or  d,  forms  with  the  hor  line  c  d  or  chord,  is  called  the  Rllgle  Of 
direction  Of  the  maill  Chains,  or  cables,  at  those  points.  Frequently  the  ends 
ch,  and  dr.  of  the  chains,  cnlled  the  backstays,  are  carried  away  from  the  suspension  piers 
in  straight  lines  ;  in  which  case  the  angles  I  d  r,  c,  c  h,  formed  between  the  hor  line  e  I  and  the  chain 
itself,  become  the  angles  of  direction  of  the  backstays. 


Art.  3.    To  find  the  iiat  sine  of  the  angle  of  direction,  ady; 

having  the  span  or  chord  cd  ;  and  the  defl  a  b,  at  the  center  of  the  main  chains.    See  preceding  table. 
RULE.    Mult  the  deflection  by  2.     Square  the  prod  ;  and  call  the  square  w.    Div  the  chord  by  2. 
Square  the  quot.     Add  this  square  to  w.    Take  the  sq  rt  of  the  sum,  and  oall  it  y.    Mult  the  deflec- 
tion by  2.     Div  the  prod  by  y.    Or,  by  formula, 

2  deflection 

Nat  sine  —  —  ~ 

y  (2  deflection)*  -j-  (^  chord)!. 

Having  the  nat  sine,  the  angle  itself  may  be  taken  from  the  Table  of  Sines,  page  102. 

Ex.    Let  the  half  chord  ad  be  500  feet;  and  the  defl  a  5  50  feet  ;  then, 

100  100  100  100 

~    ___  -----  -  ==  -  —  —  mr  =  —  zzurr  =  ---  =  .19611,  nat  sine  required  :  and  the  angle 

^1002  -j-  50J2  1/10000  +  250000          |/260000        509<9 

iiself,  as  found  in  the  Table  of  Nat  Sines,  is  11°  19  . 

NOTE  1.  The  direction  of  the  tanp  d  </  or  c  i,  can  be  laid  down  on  a  drawing,  thus:  Continue  the 
line  a  6,  making  it  twice  as  long  as  ab;  then  lines  drawn  from  d  and  e  to  its  lower  end,  will  be 
tangs  to  the  parabolic  curve  at  the  points  of  suspension. 

NOTE  2.    If  the  chord  c  d  be  not  hor,  as  sometimes  is  the  case,  the  anglo 

must  be  measured  from  a  hor  line  drawn  for  the  purpose  at  each  point  of  suspension  ;  as  the  two 
angles  will  in  that  case  be  unequal,  the  piers  being  of  unequal  heights. 

Art,  4.  To  find  the  tension  or  pull  on  all  the  main  chains. 
or  cables,  together,  at  either  one  of  the  piers,  c  or  d9  Fig  1. 

See  preceding  table. 

one-half  of  the  entire  suspended  weight 
_  oL^cU^panandload  _ 

nat  sine  of  angle  of  direction,  adg. 
direction,  11°  19';  its  nat  sine,  .1961;  half  weigh* 


Ex.  Span,  1000  ft:  defl,  50  ft;  angle  ad 
«f  clear  span  and  load,  900  tons.     Then, 

Total  tension  on  all  _ 
cables,  at  either  pier 


900 

=  :T96i  =  4589*teM'- 


38 


590  SUSPENSION   BRIDGES. 

RUTH  2.  Square  the  half  spaa.  Square  the  defl.  Mult  this  last  square  by  4.  Add  the  prod  to  the 
gquare  of  the  half  span.  Take  the  sq  rt  of  the  sum.  Div  this  sq  rt  by  twice  the  deli.  Mult  the  quot  by 
half  the  entire  weight  of  the  clear  *pi;u  and  load. 

Ex.     Same  as  tlie  preceding.     Here,  the  sq  of  half  span  -  250000  ;  the  sq  of  the  defl  =  2500.    And 

2500  X  4  =  10000.     And  10000  +  250000  =r  260000.     And  the  sq  rt  of  260000  =  509.91.     And  -JO'Q-  = 
5.0991.     And  5.0991  X  900  -  458«J<4  tons  tension. 

Art.  5.  To  find  merely  the  proportion  or  ratio,  which  the 
tension  on  the  maan  chains  or  cables  at  either  point  of  sus- 
pension, bears  to  the  entire  suspended  weight  of  the  span, 
and.  its  load.  See  preceding  table. 


of  direction,  adg. 

Ex.  The  defl  a  b  Is  -f^  of  the  span  c  d  ;  what  proportion  will  the  pull  on  the  main  chains  at  either 
point  o-f  suspension,  bear  to  the  entire  suspended  weight  of  the  span  and  its  load  ? 

Here,  the  nat  sine  corresponding  to  a  defl  of  -A-  is  (see  preceding  table)  .3180  ;  therefore,  —  —  — 
1.57,  the  proportion  or  ratio  reqd  ;  as  per  table.  In  other  words,  when  the  defl  of  the  main  cliain  is 
Y1^-  of  the  span,  every  ton  wt  of  either  the  chains  themselves,  or  of  equally  distributed  load,  pro- 
duces as  great  a  strain  at  either  point  of  suspension,  as  1.57  tons  would  do  if  suspended  from  the 
same  chains  hanging  vert. 

Art.  6.  To  find  the  tension  or  pull  at  the  middle,  &,  Fig  1, 
of  all  the  main  chains  of  the  spaai  tog-ether.  Sue  preceding  table. 

one-ha'f  of  the  entire  suspended    v  nat  cosine  of  angle 

RULE  1       Tension  at  middle,  b,  of  _    u;eig/tt  of  the  clear  span  and  load  A    of  direction,  adg 
all  the  cables  toycthtr  -------------  ~  /•  .    —  •  A  — 

nat  sine  of  angle  of  direction,  adg. 

EX.  Span,  erf,  1000ft;  deflection,  a  b,  50ft;  angle  of  direction,  adg  =  11°  19'  ;  its  nat  cos  .9305  ; 
Us  nat  sine  .1961  ;  half  weight  of  clear  span  and  load  —  900  tons.     Then, 
Tension  at  middle  _    900  X   .9805  _ 
of  all  the  cables     ~     --  j^gl  - 

R  Tension  at  middle  _    ^,8Pa_rl^<^entire  «*  °/  clear  *Pan  and  l°ad 

of  aU  the  cables  twice  the  deflection,  ab. 


NOTE.  Finding  the  nat  sine  by  Art  2,  the  nat  cosine  can  be  taken  from  our  Table  of  Cos;  or  from 
our  preceding  table. 

The  diff  between  the  tensions  at  the  middle,  and  at  the  points  of  suspension,  is  so  trifling  with  the 
proportion  of  chord  and  deflection  commonly  adopted  in  practice,  viz,  from  about  -Jg-  to  y^-,  that  it 
(s  usually  neglected  ;  inasmuch  as  the  saving  in  the  weight  of  metal,  -would  be  fully  compensated  for 
by  the  increased  labor  of  manufacture  in  gradually  reducing  the  dimensions  of  the  chains  from  the 
points  of  suspension  toward  the  middle;  and  in  preparing  fittings  for  parts  of  many  different  sizes. 
The  reduction  has,  however,  been  made  in  some  large  bridges  with  wrought-iron  main  chains  ;  but 
In  none  with  wire  cables. 

Art.  7.    Tension  on  the  backstays,  ch,  dr.  Fig  1.    The  chains  are 

jupponed  either  upon  rollers  or  rockers;  or  by  a  link,*  or  by  some  other  contrivance  by  which  the 
tension  upon  them  at  the  points  of  suspension,  is  equally  transmitted  along  the  backstays  also;  and 
ihus  fully  conveyed  to  the  anchors.  The  same  has  been  effected,  when  the  chains  were  upheld  by 
<imple  pillars  of  wood  or  iron,  by  allowing  these  to  have  a  slight  rocking  motion  on  their  bases, 
which  were  hinged  for  that  purpose.  This  tension  along  the  backstays  remains  the  same,  -whether 
'he  angle  Idr,  or  e  ch,  Fig  1,  formed  by  the  backstays,  be  equal  to  that  (pdg  or  loci)  formed  bv  the 
main  chains,  or  not;  on  the  same  principle  that  a  rope  passing  over  a  pulley  is  equally  strained 
throughout,  whatever  may  be  the  diff  of  inclinations  of  that  part  of  it  which  sustains  the  load;  and 
that  part  to  which  the  force  is  applied  tor  raising  the  load.  See  Rem  2,  page  462,  Force  in  Rigid 
Bodies.  If,  however,  these  angles  are  diff,  the  effect  upon  the  piers  or  towers  will  be  diff  also;  fo! 
the  strain  upon  them  will  in  that  case  no  longer  pass  down  them  in  a  vertical,  but  in  an  oblique  di- 
rection ;  producing  a  tendency  to  overturn  them,  as  will  be  explained  in  Art  10. 

Art.  8.  To  find,  approximately,  the  length  of  a  main  chain 

cod;  having  the  span  or  chord  c  d,  and  the  middle  defl  a  o.  See  preceding  table,  Art  1  ;  and  the 
remark  following  it,  on  the  length  of  a  parabola. 

RULE.  Square  the  defl  ab;  mult  the  sq  by  1^.  Call  the  prod  P.  Also  square  one-half  the  chord 
c  d  ;  add  this  last  sq  to  P.  Take  the  sq  rt  of  the  sum,  for  the  length  of  one-half  the  chain  approxi- 
mately. Or  by  formula, 

Half  length  of  main  chains  J/^  (defl2)  +  (^  chord)2. 

Ex.  In  Menai  bridge  the  chord  c  d  is  579.874  ft  ;  and  the  defl  is  43  ft.  What  is  the  length  of  a  main 
chain  between  the  points  of  suspension  ? 

Here,  \%  times  the  sq  of  the  defl  =  432  X  1J£  =  1849  X  1J£  =  2465.33 
And  half  the  chord  is  289.937,  which  squared  gives  84068.46 

Their  sum  ~  86528.79 


*  For  a  link,  «ee  Fig  45,  p  295.     See  also  Fig  1 


sus: 


SIGN    BRIDGES. 


591 


And  the  yfrl  of  86528.79  1^294.15;  which  is  one-half  the  length  of  a  main  chain  ;  consequently,  the 
entire  length  is  588.3  feeC  By  actual  measurement  the  chain  is  precisely  590  feet.  The  approximate 
rule  below  gives  589^)4  ft. 

NOTE.  Ttie  lengths  obtained  by  this  rule,  are  only  approximate,  not  only  because  the  calculation  ia 
based  upon  the  supposition  that  the  chains  form  a  parabolic  curve  ;  but  because  this  rule  for  a  par- 
abola ia  entirely  incorrect  ;  although  quite  close  when  the  defl  is  not  greater  than  -^  the  chord.  See 
Parabola,  in  Mensuration.  In  fact,  the  curve  of  a  finished  bridge  is  neither  precisely  a  parabola,  not 
a  catenary,  but  intermediate  of  the  two. 

Tbe  following  simple  rule  by  the  writer  is  quite  as  approximate  as  the 
foregoing  tedious  one,  when,  as  is  generally  the  case,  the  defl  is  not  greater  than  -J.y  of  the  chord,  or 
span. 

RULB.  Mult  the  defl  by  the  decimal  .23  ;  and  add  the  prod  to  the  chord,  for  the  length  of  the  main 
clutiu  ;  or  as  a  formula, 

Length  of  main  chain  when 


defl  does  not  exceed 


span 


=  chord  +  «23  defl- 


Art.  9.  To  find,  approximately,  the  length  of  the  vert  sus- 
peiidiiig  rods  b  t9  oc  y,  «fcc,  Fig  1 ;  assuming  the  curve  to  be  a 
parabola. 

Let  x,  Fig  1,  be  any  point  whatever  in  the  curve ;  and  let  a;  w  be  drawn  perp  to  the  chord  c  d  ;  and 
xf  perp  toa&;  then  in  any  parabola,  as  «  c  2  .;  aw%  ::  ab  :  bf.  And  bf  thus  found,  added  to  b  t, 
(which  is  supposed  to  be  already  known,  being  the  length  decided  on  for  the  middle  suspending  rod,) 
gives  xy,  the  length  of  rod  reqd  at  the  point  x;  and  so  at  any  other  point. 

Ifbf  thus  found  be  taken  from  the  middle  deflection  a  b,  it 
leaves  w  a/;  and  thus  any  deflection  w  x  of  the  main  chain  or  cable,  may  be 
found,  when  we  know  its  hor  dist,  aw,  from  the  center,  a,  of  the  span. 

In  the  foregoing  rule,  the  floor  of  the  bridge  is  supposed  to  be  straight;  but  generally  it  is  raised 
toward  the  center;  and  in  that  case,  the  rods  must  first  be  calculated  as  if  the  floor  were  straight,  and 
the  requisite  deductions  be  made  afterward.  When  it  rises  in  two  straight  lines  meeting  in  the  cen- 
ter, the  method  ot  doing  this  is  obvious.  When  an  arc  of  a  circle  is  used,  its  ordinates  may  be  calcu- 
lated by  the  rule  given  at  foot  of  page  20,  and  deducted  from  the  lengths  obtained  by  this  rule.  Or, 
having  drawn  the  curve  by  the  rule  given  for  drawing  a  parabola,  the  dimensions  can  be  approxima- 
ted to  by  a  scale.  The  adjustments  to  the  precise  lengths  must  be  made  during  the  actual  construc- 
tion of  the  bridge,  by  means  of  nuts  on  their  lower  screw-ends.  The  rods  require,  therefore,  only  to 
be  made  long  enough  at  first. 

Art.  1O.  Strains  borne  by  the  suspension  piers,  or  towers, 
or  pillars.  In  Figs  2,  3,  and  4,  let  a  id  represent  suspension  piers,  or  towers; 
a  c,  maiu  chains  ;  a  b,  backstays  ;  a.  points  of  suspension  :  m  n,  hor  lines  drawn  through  said  points ; 
m  at,  angles  of  direction  of  the  main  chains,  found  by  Art  3 ;  and  nas,  angles  of  direction  of  the 
backstays. 

The  chains  at  the  points  of  suspension  are  supposed  to  rest  immediately  upon  rollers,  which  have 
no  other  motion  than  that  of  revolving  around  their  axes  ;  the  frame  to  which  they  are  attached  being 
bolted  to  the  top  of  the  piers.  On  these  rollers  the  chains  slide.* 

Take,  with  a  pair  of  divi- 
ders, a  dist  which  represents 
by  scale  the  tension  in  tons 
on  the  main  chain  at  a.  Lay 
off  this  dist,  in  any  of  tho 
three  figs,  from  a,  toxands. 
and  from  x  and  s,  lay  it  off 
to  o;  or,  in  other  "words, 
complete  the  rectangle  xaso; 
audjoinoo.  Then  will  a  o 

be  the  direction  of  the 

strain  or  pres  upon  the  pier ; 
and  if  measured  by  the  same 
sc;ile  of  tons  that  was  used 
for  ax,  as.  &c,  it  will  give 
the  amount  of  the  pres 
in  tons.  When,  as  in  Fig  2,  „} 
the  angles  mat,  and  nas,  '"" 
are  equal,  this  pres  at  each 
end  of  the  span,  will  be  equal 
to  the  entire  weight  of  the 
clear  span  of  the  bridge  and 
its  load. 

In  Fig  3.  the  angles  mat 
and  nas  are  unequal ;  con- 
sequently, the  direction  ao 
of  the  pres  on  the  pier  is  not 
vert.  Its  amount  also  is  in 
this  instsnce  less  than  in 
Fig  2 ;  that  is,  it  is  less  than 

*  Prof  Rankine,  in  his  Civil  Engineering,  says,  that  if  the  ends 
.  c,  d,  of  the  cable  and  backstay  at  the  top  of  the  pier  P,  be  "  made 
fast  to  a  sort  of  wrought-iron  truck,  w.  Fig  I1*,  which  is  supported 
y  rollers  on  a  horizontal  cast-iron  platform  on  the  top  of  the     ' 


-  ,     .  , 

by  rollers  on  a  horizontal  cast-iron  platform  on  the  top  of  the  pier, 
then  the  pres  on  the  pier  will  be  vert,  whether  the  inclinations  of 
c  and  d  be  equal  or  unequal;  and  it  is  only  necessary  that  the 
horizontal  component*  of  their  tensions  should  be  equal." 


592 


SUSPENSION    BRIDGES. 


the    entire    weight    of    Uv 

loaded  span. 

In  Fig  4  also,  tlie  auglek 
are  unequal;  ami.  coii.se- 
queutly,  the  pres  ao  is  not 
vert;  but,  in  this  instance, 
its  amount  is  greater  than 
in  Fig  2.  In  all  these  cases, 
if  the  rollers  upon  which 
the  chains  slide,  were  sup- 
ported by  single  thin  posts 
or  pillars,  instead  of  piers, 
then  a  o  would  show  the  pro- 
per positions  for  those  pil- 
lars, so  that  the  pres  should 
pass  precisely  along  their 
axes.  If  we  suppose  symmetrical  piers,  aid,  to  be  used  in  each  case,  the  base  i  d  of  that  in  Fig  2, 

has  no  tendency  to  overturn  the  pier.  In  Fig  2,  the  masonry  of  the  pier  should  be  laid  in  the  usual 
hor  courses,  in  order  that  its  bed-joints  may  be  at  right  angles  to  the  pres  upon  them. 

But,  in  Figs  3  and  4,  if  the  bases  were  made  as  narrow  as  in  Fig  2,  the  lines  of  direction  ao.  of 
the  pres,  would  fall  outside  of  them;  and  the  piers  would  consequently  be  in  danger  of  overturning. 
Also,  the  stones  of  the  masonry,  if  laid  in  hor  courses,  would  have  a  considerable  tendencv  to  slide 
on  each  other.  To  prevent  this,  the  beds  should  be  at  right  angles  to  ao.  Such  points, "however, 
will  be  found  to  be  more  fully  explained  under  Force  in  Rigid  Bodies. 

in  Fig  3,  the  obliquity  of  the  pres  a  o  would  tend  to  slide  the  base  of  the  pier  outward,  as  shown 
by  the  arrow;  but  in  Fig  4,  inward.  This  tendency  is  produced  by  the  hor  component  of  the  force 
ao.  The  amount  of  this  may  be  found  thus,  in  either  tig:  From  a  downward  draw  a  vert  line;  and 
from  o,  a  hor  one,  meeting  it  in  e.  Then  o  e,  measured  by  the  same  scale  of  tons  as  before,  will  give 
this  hor  force  ;  and  a  e  will  give  the  vert  component  of  the  pres  ao.  The  effect  upon  the  pier  of  the 
one  pres  ao,  is  precisely  the  same  as  would  be  produced  upon  it  by  one  vert  force  equal  to  ae,  and  a 
hor  one  equal  to  o  e,  acting  at  the  same  time  :  as  explained  under "Cornp  and  Res  of  Forces. 

If,  in  either  fig,  we  draw  the  vert  lines  xh  and  s  ft,  then  a  k,  measured  by  the  foregoing  scale,  will 
give  the  tons  of  hor  pull;  and  ks,  the  vert  pres  produced  on  the  pier  by  the  backstay  ;  and  h  a  and 
h  x  will  in  like  manner  give  the  corresponding  forces  produced  by  the  main  chain.  If  we  add  together 
ft  a  and  hx,  they  will  be  found  to  be  equal  to  ae;  and  if  we  subtract  ak  from  ah,  their  diff  will  equal 
oe.  It  is  this  diff  only  that  tends  to  slide,  or  to  upset  the  pier;  the  other  portions  of  aft  and  a  A, 
neutralizing  each  other  in  that  respect. 

The  foregoing  strains  may  all  be  calculated  thus: 

The  hor  pull  inward  by  the  main  chain  =  Tension  X  Nat  Cosine  of  mat. 
"     "       "    outward  by  the  backstay  .  —  Tension  X  Nat  Cosine  of  n  a  s. 

Vert  pres  "by  main  chain =  Tension  X  Nat  Sine  of  m  at. 

"      "      "backstay =  Tension  X  Nat  Sine  of  n  a  s. 

The  towers,  piers,  or  pillars,  which  uphold  the  chains  or 
cables,  admit  of  an  endless  variety  in  desig'ii.  According  to  cir- 
cumstances, they  may  consist  each  of  a  single  vertical  piece  of  timber,  or  a  pillar  of  cast  or  wrought 
iron;  or  of  two  or  more  such,  placed  obliquely,  either  with  or  without  connecting  pieces  ;  like  the 
bents  of  a  trestle,  as  in  any  of  the  figi  on  page  HOT.  Or  they  may.be  made  (with  any  degree  of  or- 
namentation) of  cast-iron  plates;  as  in  iron  house-fronts.  Or  they  may  be  of  masonry,  brick,  or 
concrete ;  or  of  any  of  these  combined. 

Each  of  the  suspeiiding'-rods,  through  which  the  floor  of  the  bridge  is 

upheld  by  the  main  chains,  requires  merely  strength  sufficient  to  support  safely  the  greatest  loud 
that  can  come  upon  the  interval  between  it  and  half-way  to  the  nearest  rod  on  each  side  of  it ;  in- 
cluding the  wt  of  the  platform,  &c,  along  the  same  interval. 

In  anchoring-  the  backstays  into  the  ground,  it  is  necessary  to 

secure  for  them  a  sufficiently  safe  resistance  against  a  pull  equal  to  the  strain,  as,  upon  the  backstay. 
This  safety  should  be  as  great  as  that  of  the  main  chains  against  that  same  amount  of  tension. 

As  to  the  anchorage  of  the  cables  below  the  surface  of  the  ground, 

natural  rock  of  firm  character  is  the  most  favorable  material  that  can  present  itself.  When  it  is  not 
present,  serious  expense  in  masonry  must  be  incurred  in  large  spans,  in  order  to  secure  the  necessary 
weight  to  resist  the  pull  of  the  cables.  Our  Figs  4%  give  ideas  of  the  modes  most  frequently  adopted. 
For  a  very  small  bridge,  such  as  a  short  foot  bridge,  for  instance,  the  backstays  may  simply  be  an- 
chored to  large  stones,  t.  Fig  A,  buried  to  a  sufficient  depth.  Or,  if  the  pull  is  too  great  for  so  simple 
a  precaution,  the  block  of  masonry,  mm.  may  be  added,  enclosing  the  backstay.  A  close  covering 
of  the  mortar  or  cement  of  the  masonry  has  a  protecting  effect  upon  the  iron. 

To  avoid  the  necessity  for  extending  the  backstays  to  so  great  a  dist  under  ground,  they  are  usually 
curved  near  where  they  descend  below  the  surface,  as  shown  at  B,  D,  and  E ;  so  as  sooner  to  reach 
the  reqd  depth.  This  curving,  however,  gives  rise  to  a  new  strain,  in  the  direction  shown  by  the 
arrows  in  Figs  B  and  D.  The  nature  of  this  strain,  and  the  mode  of  finding  its  amount,  (knowing 
the  pull  on  the  backstay,)  are  very  simple;  and  fully  explained  under  the  head  of  Funicular  Ma- 
chine, page  463.  The  masonry  must  be  disposed  with  reference  to  resisting  this  strain,  as  well  as 
that  of  the  direct  pull  of  the  backstay.  With  this  view,  the  blocks  of  stone  on  which  the  bend  rests 
should  be  laid  in  the  position  shown  in  Fig  D  ;  or  by  the  single  block  in  Fig  B.  Sometimes  the  bend 
is  made  over  a  cast-iron  chair  or  standard,  as  at  x.  Fig  F,  firmly  bolted  to  the  masonry. 

Fig  E  shows  the  arrangement  at  the  Niagara  railway  bridge  of  821  &  ft  span.  The  wire  backstays 
end  at  cc;  and  from  there  down  to  their  anchors,  they  consist  of  heavy  chains;  each  link  of  which 
is  composed  of  (alternately)  7  or  8  parallel  bars  of  flat  iron,  with  eve  ends,  through  which  pass  bolts 
as  iu  Figs  41,  page  293.*  Each  of  the  7  bars  of  each  link  is  1.4  ins  thick,  by  7  ins  wide,  near  the 

*  When  chains  of  iron  bars  are  used  instead  of  wire  cables,  they  are  usually  made  as  at  p  293. 
Since  bar  iron  has  but  about  half  the  tensile  strength  of  wire,  the  chains  must  have  a  sectional  area 
fl«ice  as  great  as  that  of  a  cable. 


SUSPENSI 


BRIDGES. 


593 


lowest  part  of/The  chain  ;  but  th£y*gradually  increase  from  thence  upward,  until  at  c,  c,  where  they 
unite  with^rne  wire  cable,U*e^ectioual  area  of  each  link  is  93  sq  ins.  These  chain  backstays  pass  in 
•ugh  the  ma^Slve  approach  walls,  ('J8  ft  high,)  and  descend  vertically  down  shafts  s,  s,  25 
ft  deep  ili  the  solid  rerck.  Here  they  pass  through  the  cast-iron  anchor-plates,  to  which  they  are  con- 
fined below  by  a  bolt  3^  ins  diam.  The  anchor-plates  are  6%  feet  square,  and  '2^  ins  thick  ;  except 
tor  a  space  of  about  20  ins  by  26  iw  "I  the  center  where  the  chains  pass  through,  where  they  are  1 


foot  thick.     Through  this  thick  part  is  a  separate  opening  for  each  bar  composing  the  lowest  link. 

The  shafts  s,  s,  have  rough  sides,  as  thev  were  blasted  ;  and  average  3  ft  by  7  ft  across;  except  at  the 
bottom,  where  they  are  8  ft  square.     They  are  completely  tilled  with  cement  masonry,  with  dressed 


,  . 

beds,  well  in  contact  with  the  sides  of  the  shafts;  and  thoroughly  grouted,  thus  tightly  enveloping 
the  chains  at  every  point;  as  does  also  the  masonry  of  the  approach  wall  wtv;  which  extends  28  ft 
above  ground;  and  is  6  ft  thick  at  top,  and  10^  ft  thick  at  its  base  on  the  natural  rock. 

D,  Figs  4%,  shews  a  mode  that  may  be  used  in  most  cases,  for  bridges  of  any  span.  The  depth 
and  the  area  of  transverse  section  of  the  shaft,  and  consequently  the  quantity  of  masonry  in  it,  will 
depend  chiefly  upon  whether  it  is  sunk  through  rock,  or  through  earth.  If  through  firm  rock,  then 
if  its  sides  be  made  irregular,  and  the  masonry  made  to  fit  securely  into  the  irregularities,  much  re- 
liance  may  be  placed  upon  it  to  assist  the  weight  of  the  masonry  in  resisting  the  pull  on  the  back- 
stays. Earth  also  assists  materially  in  this  respect. 

P  is  the  arrangement  in  the  Chelsea  bridge  of  333  feet  span,  across  the  Thames,  at  London  ;  Thps. 
Page,  eng.  The  space  from  one  wall  b  b,  to  the  opposite  one,  is  45  feet;  and  is  built  up  solid  with 
brickwork  and  concrete;  except  a  passage-way  4  ft  wide,  and  5  ft  high,  along  the  backstay  ;  and  a 
small  chamber  behind  the  anchor-plates.  It  rests  chiefly  on  piles. 

The  arrangement  by  Mr  Brunei,  in  the  Charing  Cross  bridge,  London,  is  very  similar.  In  it  also 
the  entire  abutment  rests  on  piles  ;  and  is  40  ft  high,  30  ft  thick,  and  solid,  except  a  narrow  passage- 
way along  the  chains.  The  backstays  extend  into  it  60  ft.  Span  676  feet.  Defl  50  feet. 

G  is  intended  merely  as  a  geueral'hint,  which,  variously  modified,  may  find  its  application  in  the 
case  of  a  small  temporary,  or  even  permanent  bridge  ;  for  the  number  of  pieces,  i,  t,  &c,  may  be  in- 
creased to  any  necessary  extent  ;  and  they  may  be  mace  of  iron  or  stone,  instead  of  wood. 

lai  estimating-  the  action  of  the  backstays  upon  the  laia- 

soisry,  <s*re,  to  which  they  are  anchored,  it  is  safest  to  consider  the 
tension  along  them  to  continue  undiminished  to  their  very  ends  ;  although,  when  they  are  embedded 
in  masonry,  friction  causes  it  to  diminish  ;  especially  when  they  are  curved,  as  in  E,  Figs  4^,  in 
which  cuse  the  friction  is  greatly  increased,  and  the  tension  thereby  materially  reduced  as  the  ends 
nre  approached.  Frequently,  however,  thev  are  not  so  embedded  ;  for,  although  embedding  preserves 
the  iron,  many  engineers  prefer  to  leave  an'open  spnce  around  the  entire  length  of  the  anchor-chains  ; 
.'!<  well  as  around  the  anchor-plates;  in  order  that  they  may  be  examined  from  time  to  time.  To  this 
etid,  the  masses  mm  of  masonry  in  Figs  4-^,  may  be  made  not  solid,  but  to  consist  of  two  parallel 
wrtlls,  between  which  the  backstay  may  pass;  and  the  anchor-stones,  or  anchor-plates,  will  extend 
across  the  space  between  the  walls,  and  have  their  bearing  against  the  ends  of  the  walls.  In  F,  the 
cable  may  be  supposed  either  to  be  tightlv  surrounded  by  the  masonrv,  and  grouted  to  it;  or  else  to 
be  surrounded  by  a  cylindrical  passage-way  like  a  culvert,  so  as  to  be  at  all  times  accessible.  And 
the  same  with  regard  to  the  cable  in  the  vertical  shafts  at  D  or  F. 

Art  35,  p  465;  Art  49,  p  475  ;  <tec.  of  Force  in  Rigid  Bodies,  will  assist  in  calculating  the  resistance 
which  the  masses  mm  of  anchorage  masonry  oppo««  to  the  pull  of  the  backstays.  Soft  friable  stone 
must  be  carefully  excluded  from  such  parts  of  these  masses  as  are  mopt  directly  opposed  to  this  pull. 

If  blocks  of  stone  large  enough  for  securing  good  u^nd  are  not  procurable,  heavy  bars  of  iron,  or 
I  Imams,  may  be  advantagenuslv  introduced  for  that  pv"oose. 

The  masses  must  be  founded  at  such  a  depth  as  not  v-   *lide  by  the  yielding  of  the  earth,  in  front 

Experience  shows  that  with  due  attention  to  periodical  painting,  and  renewal  of  woodwork,  a 
protv  -ly  dr*i?ned  suspension  bridge  will  be  very  durnbl'  The  transverse  floor  joists  should  be  of 
wrought  iron  ;  to  prevent  interruption  to  travel  while  puti->n«t  in  new  wooden  ones. 

Particular  care  should  be  bestowe*!  upon  the  strength  of 
the  foints  of  the  side  parapets;  for  the  uttdulations  and  lateral  motions 
of  the  "bridge  expose  them  to  violent  deranging  forces  in  every  direction.  The  parapets  should  be 
high  and  stout:  and  not  restricted  to  mere  service  as  hand-rnils.  or  guards. 

As  a  ru'e  of  thumb,  one-half  the  sq  rt  of  the  spnn  will  be  about  a  good  height  for  them  in  ordinary 
cases,  provided  it  is  not  less  than  a  hand-rail  requires. 


594 


SUSPENSION    BRIDGES. 


"We  do  not  think  that  diagonal  horizontal  bracing  should,  as  is  usual,  be  omitted  under  the  floir. 
It  may  readily  be  efioeiml  by  iron  rods. 

All  the  cables  need  not  be  at  the  sides  of  the  bridge.  One  or  more  of  them  may  be  over  its  axis; 
especially  in  a  wide  bridge.  One  wide  footpath  iu  the  center  luay  be  used,  instead  of  two  narrow 
cues  at  the  sides. 

The  platform  or  roadway  should  be  slightly  cambered,  or  curved  upward,  to  the  extent  say  of  about 
2-jg.  of  the  span. 

Art.  11.    The  Niagara  suspension  bridge,  built  in  1852-3,  John  A 

Eoebling,  engineer,  consists  of  a  single  span  of  b_'l^  ft  measured  straight  from  center  to  center  of 
towers ;  and  800  ft  of  clear  suspended  length  of  roadway.  It  has  two  floors  or  roadways  :  the  upper 
one,  for  a  single-track  railway,  is  25  >£  ft;  and  the  lower  one,  for  common  tcavel,  24>g  ft  wide,  out  to  out 
of  everything.  The  lower  one  is  11)  ft  wide  in  the  clear  of  everything.  They  are  17  ft  apart  verti- 
cally. The  trusses  are  18  ft  total  height,  throughout.  They  are  on  the  Pratt  arrangement;  see  p  284  ; 
with  verticals  5  ft  apart  from  cen  to  cen;  and  single  oblique  iron  rods,  1  inch  square,  running  in 
each  direction  across  four  of  the  5  ft  panels.  Where  these  rods  pass  each  other,  they  are  tiad  together 
by  10  or  12  turns  of  -fa  inch  wire.  Each  vertical  consists  'of  two  pieces  of  4)4  by  6>£  timber,  placed 
4}^  ins  apart,  to  allow  the  oblique  rods  to  pass  between  them.  Both  upper  and  lower  floor  girders  are 
in  two  pieces,  of  4  by  16  ins  each.  Pairs  5  ft  apart.  The  tops  and  bottoms  of  the  verticals  pass  be- 
tween the  two  pieces  which  form  each  floor  girder.  No  tenons  or  mortises  are  used  in  the  framing. 

There  are  four  cables  of  iron  wire ;  two  on  each  side  of  the  bridge.  Each 

cable  is  10  ins  diarn.  The  wire  is  scant  No.  9  of  the  Birmingham  wire  gauge,  or  scant  ,14b  inch  diam. 
Sixty  wires  have  a  united  transverse  section  equal  to  one  square  inch  of  solid  iron.  Each  of  the  four 
cables  contains  3(UO  wires,  with  a  united  cross-section  of  60.4  sq  ins  of  solid  metal.  Therefore,  the 
area  of  solid  metal  in  a  section  of  all  the  four  cables  together  is  241.6  sq  ins,  or  1.678  sq  ft ;  weighing 
814  Ibs  per  ft  of  span.  The  wires  of  each  cable  are  first  made  up,  in  place,  into  7  small  strands ;  and 
these  are  firmly  bound  together  throughout  by  a  continuous  close  wrapping  of  wire.  The  strength  of 
each  individual"  wire  is  1640  fts.  or  .73214  of  a  ton.  This  is  equal  to  98400  fts,  or  43.93  tons  per  sq  inch 
of  solid  metal ;  or  to  5943360  fts,  or  2653.3  tons  per  cable  ;  or  to  10613.2  tons  ultimate  strength  of  the 
four  cables  together.  One  cable  on  each,  side  of  the  bridge  deflects  54  ft ;  and  the  other  64  ft;  average 
deflection  59  ft,  or  about  -j*r  of  the  span.  With  this  av  defl  the  tension  on  the  cables  at  the  tops  of  the 
towers  averages  1.82  times  the  total  suspended  wt  of  the  span  and  its  load.  See  table,  Art  1.  The  wt 
of  the  suspended  span  itself  is  about  900  tons ;  and  if  the  greatest  extraneous  load  on  the  two  floors 
together  be  taken  at  1J4  tons  per  ft  run,  we  have  the  total  suspended  wt  900-4-  (800  X  1 J4)  =  1900  tons. 
And  1900  X  1.82  =  3458  tons  tension  at  towers;  or  very  nearly  ^  of  the  ultimate  strength  of  the  cables, 
without  any  allowance  for  momentum,  or  wind.  But  such  loads,  although  possible,  are  not  permitted 
to  come  upon  the  bridge;  and  moreover  a  part  of  the  strain  is  borne  by  the  upper  stays. 

The  wires  were  perfectly  oiled  before  being  made  into  strands;  and  when  the  strands  were  being 
bound  together  to  form  a  cable,  the  whole  was  again  thoroughly  saturated  with  oil  and  paint. 

The  cables  do  not  hang  vertically;  but  the  two  upper  ones  are  about  37  ft  apart  from  center  to  cen- 
ter, where  they  rest  upon  the  towers,  (where  all  four  are  on  the  same  level ;)  and  are  drawn  to  within 
13  ft  of  each  other  at  the  center  of  the  span  ;  and  at  the  level  of  the  railway  track  on  top  of  the 
bridge :  while  the  two  lower  ones  are  about  39  ft  apart  at  the  towers,  and  25  ft  at  the  center  of  the  span, 
and  at  the  level  of  about  halfway  between  the  two  floors. 

This  drawing  in  of  the  cables  contributes  much  to  lateral  stability  ;  as  do  also  the  upper  and  lower 
floor  of  stout  plank.  There  is  no  horizontal  diagonal  floor  bracing. 

There  are  624  suspenders  of  wire  rope,  1%  ins  diam,  and  5  ft  apart,  or  corresponding  with  the  floor 
girders,  which  they  uphold;  and  with  the  wooden  verticals  of  the  trusses.  They  do  not  hang  verti- 
cally ;  but  incline  inward. 

The  masonry  towers  are  all  founded  on  rock.  They  are  78J4  ft  high  above  the  bottom  of  the  bridge; 
and  60,4  ft  above  the  upper  floor.  The  two  at  each  end  of  the  span  are  39  ft  apart  from  center  to  cen- 
ter. At  the  level  of  the  lower  floor  they  are  19  X  20  ft ;  and  21  ft  apart  iu  the  clear.  At  the  level  of 
the  upper  floor  they  are  15  ft  square;  and  24  ft  apart  in  the  clear.  From  there  they  taper  regularly 
to  the  top,  where  they  are  8  ft  square.  They  are  built  of  limestone,  in  heavy  dressed  hor  courses ; 
laid  in  cement ;  vertical  joints  grouted.  The  upper  courses  are  dowelled.  On  top  of  each  tower  is  a 
cast-iron  plate,  8  ft  sq,  and  2}^  ins  thick,  bedded  in  cement.  Part  of  the  top  of  this  plate  is  planed, 
as  upon  it  move  the  rollers  which  support  the  cast-iron  saddles  on  which  the  cables  rest.  At  each 
tower,  each  cable  has  its  separate  saddle  and  rollers.  Each  saddle  rests  on  10  cast-iron  rollers  25^ 
ins  long,  and  5  ins  diam/carefully  planed.  They  lie  loosely,  and  close  together;  and  are  kept  in  place 
by  side  flanges  on  the  bed-plate. 

The  cast  saddles  are  each  5  ft  long,  by  25^  ins  wide.  Their  bottoms,  which  rest  on  the  rollers,  are 
flat,  and  planed.  Their  tops  are  curved  to  a  rad  of  6J4  ft ;  to  suit  the  bend  of  the  cables  over  the  piers  ; 
and  each  saddle  has  a  longitudinal  groove,  in  which  the  cable  lies.  The  passage  of  the  heaviest  trains 
produces  less  than  %  an  inch  of  movement  in  a  saddle. 

The  floors  have  a  camber  of  5  feet. 

A  change  of  100°  Fah  of  temperature  causes  an  average  variation  of  about  2^  ft  in  the  deflection 
of  the  cables,  or  in  the  camber  of  the  roadways;  and  one  of  150°,  (about  the  extreme  to  which  the 
bridge  is  exposed,)  about  3%  ft.  The  passage  of  a  train  weighing  291  tons,  and  covering  the  entire 
length  of  the  span,  caused  a  deflection  of  10  ins  ;  and  an  ordinary  train  deflects  it  only  from  3  to  5 
inches. 

This  bridge  has,  since  the  year  1853,  demonstrated  the  applicability  of  the  suspension  principle  to 
large  span  railway  bridges.  Its  entire  cost  was  not  quite  $400,000.  For  more,  see  top  of  p  589. 

Art.  12.  The  wire  suspension  bridg-e  near  Freybnrg1,  Swit- 
zerland, finished  in  1834,  Mr.  Chaley,  engineer,  and  still  in  full  service,  is  of 
very  simple  construction,  and  has  served  as  the  prototype  for  several  in  this  country.  It  is  for 
common  travel  only  ;  and  is  narrow:  its  entire  width  of  platform  being  but  21%  ft;  and  its  clear 
available  width  but  19  ft.  The  dist  from  cen  to  cen  of  its  towers  is  889  feet;  and  its  clear  spnn  be- 
tween abutments  800  ft ;  or  the  same  as  the  Niagara.  There  are  4  cables,  each  5  ins  diam.  Each  of 
them  consists  of  1056  wires  of  No.  10,  or  full  ^  inch  diam,  (or  71  wires  to  the  sq  inch  of  solid  mptnl ;) 
arranged  in  20  strands  of  about  53  wires  each.  The  four  cables,  therefore,  have  a  united  area  of  but 
40  sq.  ins  of  solid  metal ;  weighing  202  Ibs  or  .09  of  a  ton,  per  ft  run  of  spaa.  All  its  suspenders  are 


SUi 


'SIGN    BRIDGES. 


595 


Tertlcal ;  about  5  ft  anaKf  and  each  upholds  one  end  of  a  transverse  floor  girder.  It  hai  no  std« 
trussing  except  th^yfight  one  of  the  wooden  baud-railing,  which  is  about  6  feet  high ;  and  conse- 
quently, with  Ujf'great  span  it  is  quite  flexible.  The  deflection  of  the  cables  is  ^  of  the  span  ;  hence 
the  strain  upon  them  at  the  top  of  the  towers  at  either  end,  is  1.82  times  (see  table,  Art  1,)  the  wt  of 
the  suspended  span  itself,  and  its  extraneous  load;  and  supposing  the  wire  to  be  as  good  as  that  of 
the  Niagara,  the  breaking  strain  of  the  four  cables  would  be  60  X  44  —  2640  tons ;  and  their  safe 
strain  cannot  be  taken  at  more  than  %  as  much,  or  880  tons.  The  suspended  weight  reqd  to  produce 

this  safe  strain  would  of  course  be  — -  :=  481  tons.    The  suspended  weight  of  the  span  itself  cannot 

•well  be  less  than  .3  of  a  ton  per  ft  run  :  or  240  tons  in  all ;  *  thus,  leaving  484  —  240  —  244  tons  To.* 
the  maximum  safe  extraneous  load.  This  amounts  to  .305  of  a  ton  per  ft  run  of  span  ;  or  36  fts  per 
sq  ft  of  its  platform,  19  ft  wide  in  the  clear.  The  French  allowance  is  41  fts  per  sq  ft;  t  and  since  no 
allowance  is  here  made  for  momentum  or  wind,  it  is  plain  that  this  celebrated  bridge,  on  account  of 
its  slight  cables,  and  its  flexibility,  is  by  no  means  a  strong  one.  In  that  respect,  as  well  as  steadi- 
ness, it  is  much  inferior  to  the  one  next  spoken  of.  It  is  said,  however,  to  have  withstood  very  severe 
tempests;  and  also  to  have  been  occasionally  completely  covered  by  crowds  of  people.  If  so,  their 
lives  were  not  very  secure. 

Art.  13.    The  wire  suspension  bridge  across  the  Schnylkill 

at  Philada,  finished  in  1842,  Chas  Ellet,  Jr,  engineer,  is  somewhat  similar  in  character,  and  in  the 
dimensions  of  its  details,  to  the  preceding;  but  being  of  much  less  span,  is  much  stronger.  Its  span 
from  cen  to  cen  of  towers  is  358  ft;  suspended  platform  between  abuts  342  ft.  It  has  ten  cables  of  3 
ins  diam  ;  five  on  each  side.  Their  united  sections  present  55  sq  ins  of  solid  iron  ;  or  nearly  as  much 
as  the  preceding  bridge  of  800  ft  clear  span.  The  five  cables  on  either  side  have  different  deflections, 
ranging  between  the  y\j-  and  the  -Jj  of  the  span  from  tower  to  tower.  The  dist  from  cen  to  cen  of 
towers  at  either  end  of  the  span  is  35^  ft;  and  on  top  of  each  tower  the  cables  (considerably  flattened 
at  that  point)  lie  side  by  side  on  a  single  roller  about  30  ins  long,  and  6  ins  smallest  diam,  which  has  5 
grooves,  for  their  reception.  Each  cable  is  drawn- in  about  3^  ft  at  the  center  of  the  span.  At  in- 
tervals of  20  ins  the  parallel  wires  of  the  cable  have  a  close  wrapping  of  finer  wire  for  a  distance  of 
3  ins. 

The  suspenders  are  of  wire;  and  are  %  inch  diam;  and  4  ft  apart.  On  any  one  cable  they  are  20 
n  apart.  They  all  incline  slightly  inward. 

The  width  of  the  platform  from  out  to  out  is  27  ft;  and  in  clear  of  hand-rails  25  ft.  Inside  of  the 
hand-rail  is  a  footway,  4  ft  4  ins  wide,  on  each  side  of  the  bridge.  The  remaining  16  ft  4  ins  serves 
for  a  double  carriage  way,  or  double-track  street  railway.  Figs  5  show  the  arrangement  of  the  wood- 
work, on  a  scale  of  ^ 
inch  to  a  ft.  The  trussing 
of  the  parapets  is  on  the 
Howe  system. (see  p  283.) 
which  does  not  appear  to 
be  as  well  adapted  as  the 
Pratt,  to  suspension 
bridges.  The  diagonals 
in  the  Fairmount  bridge 
work  themselves  out  of 
place  laterally,  by  the 
vibrations  of  the  bridge ; 

ly  seen  several  of  them 

almost  on   the  point  of 

falling     out      entirely. 

Being  under  municipal 

charge,  it  is   of  course 

neglected.      The    upper 

chords    u,    are     12    ins 

wide  by  6  ins  deep;  the  lower  ones  I,  and  the  stringer  c,  below  them,  are  each  12  wide,  by  7  deep. 

The  diagonals  i  are  all  4  ins  wide,  by  5  deep.     The  angle-blocks  nt  their  ends  are  of  cast  iron,  hollow, 

and  about  %  inch  thick.     The  vert  iron  rods  v,  (in  pairs,)  are  %  inch  diam  near  the  center  of  the 

span;  and  lj.g  at  its  ends.     The  top  chords  are  spliced  on  each  vert  face  by  an  iron  bar.  of  5  ft  by  3 

ins,  by  y2  inch;  with  4  bolts  passing  through  them.     The  splice  of  the  bottom  chord  has  merely  2 

bolts,  side  by  side  ;  (see  Figs  5  ;)  which  (except  S)  are  to  a  scale  of  %  inch  to  a  ft.    The  floor  girders  g, 

4k  ft  apart  from  cen  to  cen,  are  6  by  14  ins  at  their  ends  ;  and  6  by  16  at  center. 

The  floor  is  of  two  thicknesses  of  2-inch  plank;  except  the  footpaths,  which  are  single  thickness. 

The  wires  were  well  oiled  when  the  cables  were  made  ;  and  afterward  painted. 

At  S  is  shown  the  mode  of  uniting  a  suspender  with  a  cable,  o,  bv  means  of  a  small  cast-iron  yoke 
g,  which  straddles  the  cable  ;  and  on  the  back  of  which  is  a  groove  %  of  an  inch  wide,  in  which  the 
suspender  rests.  The  metal  of  the  yoke  is  about  )£  inch  thick.  Since  the  lower  ends  of  the  wires 
which  compose  a  suspender  cannot  themselves  be  formed  into  a  screw-bolt,  for  upholding  the  floor 
girders,  they  are  passed  through  the  eye  of  a  screw-bolt  of  bar  iron  ;  then  doubled  on  themselves,  and 
held  by  a  wrapping  of  wire.  It  is  well  to  introduce  a  yoke  here  also,  to  prevent  the  wear  of  the  wire§ 
by  friction.  The  small  fig  on  the  right  of  S  is  an  edge  view  of  the  yoke ' g. 


TR    SEC. 


SIDE. 


*  This  is  probably  nearly  its  actual  weieht,  as  obtained  by  comparing  it  with  the  Fairmount  bridge, 
•which,  by  a  careful  estimate  by  the  writer,  weighs  .375  of  a  ton  per  ft  run  ;  but  is  considerably  wider 
than  the  Freyburg ;  and  carries  four  lines  of  light  street-rails.  But  if  the  Frey  burg  has  longitudinal 
joists,  it  will  weigh  about  .03  ton  more  per  ft  run. 

t  The  greatest  load  that  can  come  upon  an  ordinary  bridge,  is  a  dense  crowd  of  people;  and  this 
the  French  pncineers  estimate  at  41  fts  per  sq  ft  of  platform.  This  is  certainly  as  great  as  can  well 
occur  under  ordinary  circumstances ;  but  it  may  be  considerably  exceeded,  the  French  estimate, 
moreover,  includes  no  allowance  for  wind,  or  for  the  crowd  being  in  motinn.  Including  these,  the 
writer  thinks  that  no  suspension  bridge  should  have  a  less  safety  than  3,  against  100  fts  per  sq  ft; 
added  to  the  weight  of  the  bridge  itself.  A  less  coeff  of  safety  is  admissible  in  a  wire  bridge  than 
in  an  iron  trussed  one,  on  accoun;  of  the  greater  reliability  of  the  material.  See  foot-note  p  297. 


596 


SUSPENSION    BRIDGES. 


There  is  no  transverse  bracing  under  the  floor ;  nor  are  there  longitudinal  floor  joists  resting  on  the 
girders.  Owing  to  the  waut  of  the  distributing  effect  of  these ;  and  to  the  use  of  so  many  small  cables 
instead  of  but  2  or  4  larger  ones  ;  as  well  as  to  the  inefficient  trussing  of  the  hand-railing  or  para- 
pets, the  bridge  is  much  less  steady  than  it  would  otherwise  be.  Still  it  is  very  rarely  (as  during 
the  trotting  of  a  herd  of  cattle)  that  the  trembling  or  undulating  of  the  bridge  becomes  seriously 
objectionable.  Ordinarily  it  cannot  be  said  to  be  at  all  so ;  and  but  few  persons  would  notice  it.  It 
must  be  greatly  less  than  on  the  Freyburg  bridge. 

With  wire  of  the  same  quality  as  the  Niagara,  (or  44  tons  per  sq  inch  breaking  strength,)  the  Fair- 
mount  bridge  would,  with  a  safety  of  3,  (omitting  momentum  and  wind.)  sustain  an  extraneous  load 
of  31:6  tons;  which  is  equal  to  1.01  ton  per  ft  run  of  span ;  or  90  Ibs  per  sq  ft  of  its  clear  platform. 
This  last  is  2.5  times  as  great  as  the  strength  of  the  Freyburg,  with  the  same  quality  of  wire.  The 
Fairmount  is,  however,  we  believe,  built  with  wire  of  but  36  tons  per  sq  irich  ultimate  strength.  If 
so,  its  greatest  extraneous  load  becomes  reduced  to  260  tons  ;  or  .76  ton  per  ft  run ;  or  68  Bbs  per  sq  ft 
of  platform,  or  nearly  twice  that  of  the  Freyburg. 

The  towers  are  of  cut  granite,  in  heavy  courses.  They  are  8*4  ft  square  at  the  ground  line,  or  level 
of  the  floor;  about  5  ft  sq  at  the  top;  and  about  30  ft  high.  The  backstays  have  the  same  angle  of 
direction  as  the  main  cables. 

Art.  14.  The  Wheeling:  bridge  across  the  Ohio  at  Wheeling,  Vir- 
ginia, also  by  Mr  Ellet.  had  a  span  of  1010  ft  between  the  towers  ;  and  960  feet  clear  span  between  the 
abuts  ;  and  was  26  ft  wide  from  out  to  out.  Its  mode  of  construction  was  much  the  same  even  in  de- 
tail as  that  of  the  Fairmount  bridge;  except  in  having  12  cables  instead  of  10.  The  12  cables  con- 
sisted of  6600  wires  of  No.  10  Birmingham  gauge,  presenting  a  sectional  area  of  93  sq  ins  of  solid  metal, 
weighing  313  fts,  or  .14  of  a  ton.  per  foot  of  span.  The  weight  of  the  woodwork  was  about  the  same 
per  foot  run  of  span  as  in  the  Fairmount.  Although  its  clear  span  was  2.8  times  as  great  as  the 
Fairmount,  yet  its  cables  had  but  1.7  times  as  great  area  of  solid  metal.  The  entire  suspended  wt 
between  towers,  is  stated  at  but  440  tons;  therefore,  with  an  average  deflection  of  yV  of  the  span, 
for  a  safety  of  3  against  100  Tbs  per  sq  ft  of  platform  of  24  ft  clear  width  ;  or  1.07  tons  per  ft  run  of  span, 
the  area  of  solid  metal  in  the  cables  should  have  been  173  sq  ins,  with  44  ton  wire  like  that  of  the  Ni- 
agara :  or  214  sq  ins,  with  36  ton  wire,  which  we  believe  was  the  quality  actually  used. 

Art.  15.    The  suspension  canal  aqueduct  at  Pittsburgh  Perm, 

built  in  1845,  John  A  Roebling,  Esq,  engineer,  has  seven  spans  of  160  fc  each.  Deflection  14)6  ft;  or 
about  yy-  of  the  span.  It  has  but  two  cables,  each  7  ins  diam.  The  two  together  contain  3800  No.  10 
wires,  making  53  sq  ins  of  solid  metal  section.  Ultimate  strength  of  each  wire  1100  B>s  ;  equal  to  35.2 
tons  per  sq  inch  of  solid  metal;  and  making  the  ultimate  strength  of  the  two  cables  together  1866  tons. 
The  prism  of  water  in  the  wooden  aqueduct  is  4  ft  deep  ;  by  14%  ft  average  width  ;  and  weighs  265 
tons  per  span.  The  wt  of  one  span  of  the  structure  itself  is  about  111  tons ;  making  the  total  sus- 
pended wt  at  each  span  376  tons.  The  tension  on  the  two  cables  at  either  end  of  a  span,  with  a  dett 
of  y^,  is  1.46  times  the  total  suspended  weight ;  see  table,  p  588.  Hence  it  is  in  this  case  376  X  1.46 

=  549  tons  ;  and  the  strength  of  the  cables  is    —  =  3.4  times  the  constant  strain  upon  them. 

On  one  side  of  the  water  is  a  towpath  for  horses  ;  and  on  the  other  a  footpath  ;  each  7  ft  clear  width. 
With  these  occupied  by  horses  and  people,  the  foregoing  safety  would  be  reduced  to  about  3.  The 
loaded  boats  do  not  add  materially  to  the  weight,  inasmuch  as  they  displace  a  bulk  of  water  equal  to 
their  own  wt;  and  but  little  of  the  displaced  water  remains  on  a  span  at  the  same  time  with  the  boat. 

The  great  wt  of  the  water  prevents  undulations  ;  and  the  aqueduct  is  therefore  very  steady.  On  this 
account  a  less  coeff  of  safety  is  admissible  than  on  a  common  bridge.. 

The  aqueduct  leaked  badly  along  its  lower  corners. 

Art.  16.  In  1796.  Mr  James  Finley.  of  Fayette  County,  Penn, 
introduced  suspension  bridges  in  the  U.  S. ;  and  built  several  with 

spans  of  200  feet  and  less.  Many  of  them  were  very  primitive  structures  ;  but  answered  sufficiently 
well  for  the  times.  They  had  usually  either  two  or  four  chains,  composed  of  links  from  7  to  10  feet 
long,  formed  by  bending  about  1^-inch  square  bars  of  iron,  and  welding  their  ends  together.  At 
each  link-end,  "was  a  vertical  suspender  rod  of  2  ins  by  fo  inch  iron;  which,  at  its  lower  end,  was 
bent  and  welded  into  a  stirrup  for  upholding  one  end  of  a  transverse  floor  beam.  On  these  beams 
rested  longitudinal  joists  supporting  the  floor  plank.  Finley  used  deflections  as  great  as  7,  or  even 
%  of  the  span  ;  and  his  piers  were  frequently  single  wooden  posts ;  the  two  at  each  end  being  braced 
together  at  top.  Such  were  used  in  a  span  of  151 J^  ft  clear,  across  Will's  Creek,  Alleghany  Co,  Penn. 
It  had  two  chains.  The  defl  was  %  of  the  span.  The  double  links  of  1%  inch  sq  iron,  were  10  feet 
long.  The  center  link  was  horizontal,  and  at  the  level  of  the  floor;  and  at  its  ends  were  stirrnped 
the  two  central  transverse  girders.  From  the  ends  of  this  central  link,  the  chains  were  carried  in 
straight  lines  to  the  tops  of  the  single  posts.  25  ft  high,  which  served  as  piers  or  towers.  The  back- 
stays' were  carried  away  straight,  at  the  same  angle  as  the  cables ;  and  each  end  was  confined  to  four 
buried  stones  of  about '  %  a  cub  yard  each.  The  floor  was  only  wide  enough  for  a  single  line  of  ve- 
hicles. All  the  transverse  girders  were  10  ft  apart;  and  supported  longitudinal  joists,  to  which  the 
floor  was  spiked.  There  were  no  restrictions  as  to  travel  ;  but  lines  of  carts  and  wagons  in  close  MIC- 
cession,  and  heavily  loaded  with  coal,  stone,  iron.  &c,  crossed  it  almost  daily  :  together  with  droves 
of  cattle  in  full  run.  The  slight  hand-railing  of  iron  was  tinged,  so  as  not  to  he  be.nt  J>y  the  undu- 
lation* of  the  bridge.  Six-horse  wagons  were  frequently  driven  across  in  a  rapid  trot.  It  was  built 
in  1820;  ami  an  observant  engineer  friend,  who  in  1838  took  the  sketch  and  measurements  upon 
which  this  description  is  based,  informed  the  writer  that  the  iron  was  as  perfect,  and  as  sharp  on  all 
its  edges,  as  on  the  day  it  was  built.  The  iron  was  the  old-fashioned  charcoal,  of  full  30  tons  per  sq 
inch  ultimate  strength.  The  united  cross-section  of  the  two  double  links  was  7.56  sq  ins ;  which,  at 
30  tons  per  sq  inch,  gives  227  tons  for  their  ultimate  strength  :  or  say  76  tons,  with  a  safety  of  3.  Now, 
with  a  defl  of  %  span,  the  tension  on  the  cables  (see  table,  p588)  is  but  .9  of  the  suspended  total  wt. 
The  two  chains  in  plane  would  therefore  sustain,  with  a  safety  of  3.  a  quiet  suspended  load  of  76  X 
.9  =  68  tons ;  and  as  the  span  itself  did  not  weigh  more,  than  15  tons,  we  have  53  tons  for  the  safe  ex- 
traneous weight,  omitting  all  consideration  of  wind  and  momentum.  This  is  equal  to  .35  ton  per  ft 
of  span  ;  equal  to  .7  ton  for  a  bridge  wide  enough  for  two  vehicles  to  pass.  This  primitive 


FRICTION. 


597 


bridge  wopJUTflierefore  safely  sustain  a  greater  load  per  foot 
run  of  span,  than  the  Freyburg. 

These  otdoridges  frequently  failed  by  the  rotting  of  the  end  posts ;  or  were  carried  away  by  fresh- 
ets ;  but^we  have  never  heard  of  a  failure  from  the  breaking  of  the  chains.  Many  of  them  were  built 
in  a  much  more  perfect  style  than  the  one  just  described ;  and  on  the  most  used  roads  in  the  Union. 

Art.  17.  As  it  is  sometimes  convenient  to  form  a  rough  idea  at  the  moment,  of 
the  size  of  cables  required  for  a  bridge,  we  suggest  the  following  rule  for  finding  approximately  the 
area  in  sq  ins  of  solid  iron  iu  the  wire  required  to  sustain,  with  a  safety  of  3,*  the  weight  of  the  bridge 
itself,  together  with  an  extraneous  load  of  1.205  tons  per  foot  run  of  span  ;  which  corresponds  to  100 
E>s  per  sq  ft  of  platform  of  27  ft  clear  available  width.  This  suffices  for  a  double  carriage-way,  and 
two  footways.  The  deflection  is  assumed  at  yV  °^  tne  8Pan  J  atl(i  tne  wire  to  have  an  ultimate 
strength  of  36  tons  per  solid  square  inch,  as  per  table,  page  369;  and  which  can  be  procured  without 

difficulty.    For  spans  of  1OO  ft  or  more, 

RULE.  Mult  the  span  in  feet,  by  the  square  root  of  the  span.  Divide  the  prod  by  100.  To  tha 
quot  add  the  sq  rt  of  the  span.  Or,  as  a  formula, 

span  X  sq  rt  of  span 

-f-  sqrt  of  span. 


Area  of  SOLID  metal  of  all 

the  cables  ;  in  square  ins  ;    — 


for  spans  over  100  feet  100 

For  spans  less  than  100  feet,  proportion  the  area  to  that  at  100  ft. 
If  a  defl  of  j\j- is  adopted  instead  of  ^,  the.  area  of  the  cables  may  be  reduced  very  nearly  ^-  part- 

The  following  table  is  drawn  up  from  this  rule.    The  3d  col 

gives  the  united  areas  of  all  the  actual  wire  cables,  when  made  up,  including  voids.     (Original.) 


Span 
Feet. 

Solid  Iron 
in  all   the 
Cables. 

Areas  of 
all  the 
Finished 
Cables. 

Span 
in 
Feet. 

Solid  Iron 
in  all  the 
Cables. 

Areas  of 
all  the 
Finished 
Cables. 

Span 
Feet. 

Solid  Iron 
in  all  the 
Cables. 

Areas  of 
all  the 
Finished 
Cables. 

Sq.  Ins. 

Sq.  Ins. 

Sq.  Ins. 

Sq.  Ins. 

Sq.  Ins. 

Sq.  Ins. 

1000 

348 

446 

400 

100 

128 

150 

30.6 

39.2 

900 

300 

385 

350 

84 

108 

125 

25.2 

32.3 

800 

254 

326 

300 

69 

89 

100 

20 

25.6 

700 

212 

272 

250 

55 

71 

75 

15 

19.2 

600 

171 

219 

200 

42 

54 

50 

10 

12.8 

500 

134 

172 

175 

36.4 

46.7 

25 

5 

6.4 

Having  the  areas  of  all  the  actual  cables,  we  can  readily  find  their  cliam.  Thus,  suppose  with  a 
gpan  of  500  ft,  we  intend  to  use  four  cables.  Then  the  area  of  each  of  them  will  be  —  =43  sq  ins  ; 

and  from  the  table  of  circles,  page  18,  we  see  that  the  corresponding  diam  is  full  7%  ins. 

The  above  areas  are  supposed  to  allow  for  the  increased  wt  of  a  depth  of  truss,  and  other  additions 
necessary  to  secure  the  bridge  from  violent  winds,  and  from  undue  vibrations  from  passing  loads. 

When  these  considerations  are  neglected,  and  a  less  maximum  load  assumed,  the  foregoing  descrip- 
tions of  the  Wheeling  and  Freyburg  bridges  show  what  reductions  arc  practicable.  Weight,  suffi- 
ciently provided  for,  is  of  great  service  in  reducing  undulation. 


FRICTION. 


Art.  1.     'See  Arts  15,  60,  61,  62,  &c,  of  Force  in  Rigid  Bodies.)    Friction  is  that 

resistance  (produced  bv  an  interlocking  of  the  roughnesses  of  surfaces  in  contact  with  each  other) 
which  opposes  the  sliding  of  one  body  along  another.  It  occurs  in  all  machinery  :  in  water,  gas.  or 
air,  moving  along  pipes,  or  other  channels;  and  in  all  cases  whatever  where  one  body  mores  along 

another  in  contact  with  it.    It  is  the  principal  cause  of  what  is  called 

'*  loss  Of  power."  in  all  these  cases  ;  which  means  simply  that  the  whole  of  any  moving  force 
(such  as  steam,  gravity,  moving  water,  wind,  muscular  force,  &c)  cannot  be  directly  applied  through 
machinery,  to  give  motion  to  other  bodies,  because  a  portion  of  it  is  lost  in  overcoming  friction  in  the 

friction  were  not  first  overcome,  there  could  be  no  motion  at  all.  In  hydraulics,  this  loss  of  power  is 
usually  termed  "loss  of  head,"  meaning  that  portion  of  the  head  of  water,  which  by  its  pres  counter- 
acts the  fric  of  the  water  in  the  pipes,  &c,  and  thus  permits  the  other  portion  to  produce  motion  or 
flow.  In  nearly  all  the  cases  which  come  under  the  notice  of  the  engineer,  the  surfaces  in  contact 
are  rendered  more  or  less  smooth  by  art;  and  the  smoother  they  are  made,  the  less  is  the  friction, 
because  the  roughnesses  are  diminished ;  and  of  course  take  less  hold  on  each  other.  But  in  even  the 
most  highly  polished  surfaces,  there  is  sufficient  roughness  to  produce  some  friction.  Although  the 

*  The  writer  must  not  be  understood  to  advocate  a  safety  of  3  against  100  fts  per  sq  ft,  in  addition 
fo  the  weight  of  the  bridge,  in  all  cases.  He  believes  that  limit  to  be  about  a  sufficient  one  for  a  pro- 
perly designed  wire  suspension  bridge  for  ordinary  travel ;  but  for  an  important  railroad  bridge,  he 
would  (according  to  position,  exposure,  <frc)  adopt  a  safety  of  at  least  from  4  to  6  against  the  greatest 
possible  load,  added  to  the  wt  of  the  bridge.  A  train  of  cars  opposes  a  great  surface  to  the  action  of 
side  winds:  and  trains  must  run  during  violent  storms,  as  well  as  during  cnlms  ;  but  a  large  open 
bridge  for  common  travel  is  not  likely  to  be  densely  crowded  with  people  during  a  severe  storm. 


598 


FRICTION. 


roughness  may  not  be  visible  to  the  naked  eye,  it  becomes  very  apparent  under  a  powerful  microscope. 

Could  i»  be  entirely  removed,  there  would  be  no  t'rictum ;  aud  then  the  smallest  conceivable  force 
would  slide  hor  tue  greatest  conceivable  weight ;  nor  could  the  most  powerful  vise  prevent  the  lightest 
body  from  falling  out  of  its  jaws. 

So  long  as  the  force  waich  presses  any  given  sliding  surfaces  together,  is  considerably  less  than  would 

to  overcome  it)  bears  a  certain  proportion,  o>  ratio,  to  the  pressing1  force  ;  and  is  entirely 
independent  of  the  Size  Of  tlie  Surfaces  themselves.  A  brick  will  slide  just  as 
reauily  when  lying  on  its  broad  side,  as  wneu  on  a  narrow  one.  This  proportion,  however,  is  airi"  for 
diff  materials;  and  for  diff  degrees  of  smoothness;  and  moreover,  is  less  when  the  surfs  are  clean 
and  well  lubricated,  than  when  dirty  aud  dry.  Whatever  this  proportion  may  be  in  any  particular 
case,  it  is  called  the  coefficient  of  friction  for  that  case.  Thus,  if  we  wish"  to  slide  hor  a  block  of  dry- 
cut  stone  weighing  1  ton,  upon  a  surf  of  the  same  kind,  we  shall  have  to  exert  a  force  of  about  % 
of  a  ton ;  or  if  we  Grst  place  a  weight  of  2  tons  upon  the  moving  stone,  making  the  total  pres  upon 
its  base  3  tons,  we  must  exert  a  sliding  force  of  %  of  3  tons.  Because  the  fric  of  dry  cut  stoue  upou 
dry  cut  stone  averages  about  %  of  the  force  which  presses  the  two  surfs  together;  or  in  other  woius, 
the  coeff  of  fric  of  dry  cut  stone  on  dry  cut  stone,  is  about  %,  or  .66, 

But  if  the  pres  upon  the  sliding  surfs  is  sufficient  to  produce  abrasion  (indeed,  while  it  is  much 
less,)  the  fric  becomes  greater:  but  no  precise  law  has  yet  been  discovered  for  estimating  it.  Tne 
fric  between  surfs  actually  in  motion,  is  less  than  when  they  have  been  for  some  time  quiescent  or 
stationary.  Hence  it  is  usual  to  consider  it  under  the  heads  of  moving,  and  quiescent  fric.  Expe- 
rience, however,  has  shown  that  slight  jarring  suffices  to  remove  this  diff ;  aud  since  all  structures, 
even  the  heaviest,  are  subject  to  occasional  jarring,  (as  a  bridge,  or  a  neighboring  building,  or  even 
a  hill,  during  the  passage  of  a  train :  or  a  large  factory,  by  the  motion  of  its  machinery  ;  or  in  num- 
berless cases,  by  the  action  of  the  wind,)  it  is  considered  expedient,  in  construction,  not  to  rely  on 
fric  for  stability,  any  further  than  the  coeff  for  moving  friction  may  justify. 

Beside  the  foregoing  subdivisions  of  sliding  fric,  we  have  rolling,  aud  journal  fric.  At  present  we 
shall  confine  ourselves  to  sliding. 

The  coeff  of  moving*  fric  is  the  same  at  all  vels;*  and  this,  in 

connection  with  the  foregoing  statements  that  friction  increases  as  the  pres,  so  long  as  this  is  con- 
siderably below  the  abrading  point;  and  that  it  is  independent  of  the  amount  of  surf ;  are  considered 
the  three  grand  laws  of  fric.  Unfortunately,  the  experiments  on  this  subject  by  Morin,  which  are 
those  by  which  practical  men  are  chiefly  guided;  were  made  with  slight  pressures;  not  exceeding 
about  30  Ibs  per  sq  inch  of  contact-area  of  the  bodies,  even  with  the  strongest  bodies,  such  as  iron, 
&c  ;  which,  in  machinery  very  often  have  to  move  under  far  greater  pres.  It  is  well  known  that  fric, 
under  much  heavier  pres,  increases  very  considerably  beyond  the  extent  assigned  by  Morin  ;  as  will 
be  seen  by  our  table  of  results  found  by  Rennie  ;  Art  3.  Locomotives  would  frequently  be  unable  to 
draw  the  loads  they  do,  if  their  fric  (miscalled  adhesion)  on  the  rails  did  not  greatly  exceed  the 
limits  of  iron  on  iron  assigned  by  Morin.  The  fact  is,  that  although  it  is  customary  to  consider  our 
knowledge  of  the  laws  of  fric  to  be  very  complete,  it  is  extremely  defective.  The  common  assertion 
that  "fric  is  the  same  at  all  vels,"  may  lead  to  mistakes,  unless  we  clearly  distinguish  between  the 
coeff,  and  the  amount.  If  one  man  drags  a  sled  one  mile  in  one  hour  ;  and  another  man  drags  it  one 
mile  in  %  of  an  hour;  then,  inasmuch  as  both  the  coeff,  and  the  dist,  are  the  same  in  both  cases: 
each  man  has  overcome  the  same  amount  of  fric.  But  if  one  drags  it  4  miles,  in  the  same  time  that 
the  other  drags  it  1  mile,  then,  although  the  coeff,  and  the  time,  remain  unchanged,  the  amount  of 
fric  is  4  times  as  great  in  the  first  case  as  in  the  last ;  and  has  reqd  the  man  to  expend  4  times  as 
much  force  to  overcome  it  alone;  without  any  regard  to  the  force  expended  in  moving  himself  so 
rapidly.  In  other  words,  the  quantity,  or  amount  of  fric;  and  the  total  amount  of  power  reqd  to 
overcome  it,  is  in  proportion  to  the  space  passed  over;  without  any  reference  to  the  time  reqd.  So 
a  train  moving  at  60  miles  per  hour,  overcomes  just  as  great  an  amount  of  fric  in  1  mile,  as  does  a 
train  moving  at  10  miles  per  hour;  but  the  fast  one  overcomes  6  times  as  much  per  hour,  min,  or 
sec,  as  the  slow  one.  So  with  pivots,  journals,  &c,  in  machinery ;  the  more  rapidly  they  revolve,  thu 
greater  is  the  amount  of  friction  they  must  overcome  in  a  given  time;  although  the  degree,  or  coeff, 
or  intensity  of  the  fric  remains  the  same  at  all  vels.* 

Art.  2.  To  find  the  proportion,  or  ratio,  of  fric.  to  the  pres 
which  produces  it,  (or  the  coeffof  fric.)  for  cliff  materials.  This 

may  be  ascertained  by  making  of  one  of  them  an  inclined  plane;  and  finding  by  trial,  what  slope  it 
must  have,  to  enable  the  other  just  to  begin  to  slide  down  it.  Div  the  vert  height  of  this  slope,  by 
its  hor  base  ;  aud  the  quot  will  be  the  reqd  coeff.  It  will  also  be  the  nat  tang  of  the  angle  which  the 
slope  makes  with  the  hor;  so  that  by  looking  for  the  coeff  in  a  table  of  nat  tane,  the  angle  itself  will 
be  found  opposite  to  it.  If  the  weight  of  the  sliding  body  be  mult  by  the  coeff,  the  quot  will  be  the 
force  reqd  to  drag  it  horizontally  along  such  a  surf  as  that  of  the  inclined  plane;  or  more  correctly, 
the  force  required  to  barely  balance  the  fric  ;  still  more  force  must  be  added  to  produce  motion.  The 

angle  of  slope  is  usually  called  the  angle  of  fric ;  sometimes  the  angle  of  re- 
pose ;  and  the  limiting  angle  of  resistance.  See  Force  in  Rigid  Bodies, 
Art  63,  p  486. 

Experiments  on  fric  with  unguents  cannot  well  be  made  on  a  small  ncale  bv  means  of  an  inclined 
plane;  on  account  of  the  cohesion,  or  the  stickiness  of  the  ungnent;  which  when  the  sliding  body  is 
very  light,  requires  a  proportionally  greater  slope  thnn  when  it  is  heavy. 

In  trying  experiments  on  fric  for  himself,  the  student  must  not  expect  by  any  means  to  arrive  at 
the  same  results  as  the  following;  because  the  slightest  deviation  in  point  of  hardness,  smoothness, 
dryness.  dust,  <fcc,  will  make  a  very  appreciable  diff  in  the  coeff.  Even  dampness  in  the  air  will  effect 
this.  Where  fric  is  to  b«  depended  upon  for  stability,  it  is  best  to  assume  it  at  less  than  in  the  table*  ; 

be  taken  at.  considerably  more  than  the  tabular  statement.  When  not  otherwise  stated,  the  fibres  are 
parallel  to  the  motion. 

*  There  is  reason  to  doubt  this.    Prof.  R.  H.  Thurstori  has  proved  it 

untrue  with  regard  to  axle  friction,  which  see. 

The  whole  subject  needs  revision  badly. 


FRICTION. 


599 


Table  o|>inovin^  friction,  of  perfectly  smooth,  clean,  and 
ry,  plane  surfaces,  chiefly  from  Morin. 


Materials  Experimented  with. 


Coeffof 
Fric  :  or 
Propor- 
tion of 
Fric  to  the 
Pres. 


Oak  ou  oak  ;  all  the  fibers  parallel  to  the  motion .48 

"  "       moving  fibres  at  right  angles  to  the  others;  and  to  the  motion...  .32 

"  «'       all  the  fibres  at  right  angles  to  the  motion.  .34 

"  •«       moving  fibres  on  end  ;  resting  fibres  parallel  to  the  motion .19 

41         cast  iron,  fibres  at  right  angles  to  motion .37 

Elm  on  oak,  fibres  all  parallel  to  motion .43 

Oak  ou  elm,       "  "  "        ., 25 

Elm  ou  oak,  moving  fibres  at  right  angles  to  the  others,  and  to  motion .45 

Ash  on  oak,  fibres  all  parallel  to  motion .40 

Fir  on  oak,         •'     "  "  "        .36 

Beech  ou  oak    l4     "  4t  l(        .36 

Wrought  iron  od  oak,  fibres  parallel  to  motion .62 

Wrought  iron  on  elm,      "  "        "       "      ' .25 

Wrought  iron  on  cast  iron,  fibres  parallel  to  motion .19 

"          "     on  wrought  iron,  fibres  all  parallel  to  motion .14 

Wrought  iron  on  brass .17 

Wrought  iron  on  soft  limestone,  well  dressed .49 

"      "    hard        "  "         "        24 

44          "      '4      "  4<  l4        "         wet 30 

44          4'     or  steel  on  hard  marble,  sawed.    By  the  writer about..  .17 

"          44      4<     "      41   smoothly  planed,  and  rubbed  mahogany,  fibres  par- 
allel to  motion *. .18 

44          4I      4<     "      "   smoothly  planed  wh  pine .16 

Cast  iron  on  oak,  fibres  parallel  to  motion .49 

44      "     "  elm,      "  "       "        "     20 

44       44     4<  cast  iron .15 

44      «'     ll  brass ,15 

Steel  on  cast  iron .20 

Steel  on  steel.     By  the  writer .14 

Steel  on  brass 15 

Steel  on  polished  glass.     By  the  writer about..  .11 

'4    quite  smooth,  but  not  polished;  ou  perfectly  dry  planed  wh  pine,  fibres 

parallel  to  motion about..  .16 

44    quite  smooth,  but  not  polished;  on  perfectly  dry  planed  and  smoothed 

mahogany,  fibres  parallel  to  motion about..  .18 

Yellow  copper  on  cast  iron 19 

onoak 62 

Brass  on  cast  iron.. .22 

44      on  wrought  iron,  fibres  parallel  to  motion .16 

"      on  brass 20 

44      on  perfectly  dry  pluued  wh  pine,  fibres  parallel  to  motion about..  .19 

41       4l          "          "         4<      and  smoothed  mahogany,  fibres  parallel  to  mo- 
tion   about ..  .24 

Polished  marble  on  polished  marble.    By  the  writer average 16 

44  "        ou  common  brick 44       .44 

Common  brick  on  common  brick 44       .64 

Soft  limestone  well  dressed,  on  the  same .64 

Common  brick,  on  well-dressed  soft  limestone .65 

"      "        "        hard        "        60 

Oak  across  the  grain,  on  soft  limestone,  well  dressed .38 

"       "        "  hard 38 

Hard  limestone  on  hard  limestone,  both  4I          '4       .38 

'-  4l  "  soft  "  "      44          4<       67 

Soft          4<  4<  hard  "  4<      4<          "       65 

Wood  on  metal,  generally,  .2  to  .62 mean..  .41 

Wood,  very  smooth,  on  the  same,  generally,  .25  to  .5  44     ..  .38 

Wood,     "         "         on  metal,  "          .2    to  .62 4<     ..  .41 

Metal  on  metal,  very  smooth,  dry        44          .15  to  .22 44     ..  .18 

Masonry  and  brickwork,  dry      "         "          .6    to  .7  "     ..          .65 

44         •»  "  with  wet  mortar about..  .47 

44         <4  4I  "     slightly  damp  mortar "     ..  .74 

44      on  dry  clay .* 4<     ..  .51 

41       44    moist"     '4     ..  .33 

Marble,  sawed  ;  on  the  same  ;   both  dry.     By  the  writer.* average     '     ..  A 

"  ll          "    "         "        both  damp "     ..* 44  4     ..  .55 

41  "        on  perfectly  dry  planed  wh  pine.     4l     ..* 4<  4     ..  .45 

44  4<        on  damp  planed  wh  pine 44     ..* 4     ..  .6 

44      polished,  on  perfectly  dry  planed  wh  pine     4*     '     ..  .26 

White  pine,  perfectly  dry  :  planed;  on  the  same;    all  the  fibres  parallel  to 

motion    about..  .4 

<4         4I      damp,  pinned ;  on  the  same "     ..  .6 

*  But  after  a  few  trials  the  surfaces  become  so  much  smoother  aa  to  reduce  the  angles  as  much  as 
from  iQ  to  5°  ;  tlie  sliding  -blocks  weighing  about  30  fts  each. 


600 


FRICTION. 


Art.  3.    Table  of  coefficients  of  moving-  friction  of  smooth 
plane  surfaces,  when  kept  perfectly  lubricated.    (Morin.) 


Substances. 

Dry 

Soap. 

Olive 
Oil. 

Tal- 
low. 

Lard. 

LardA 
Plum- 
bago. 

Oak  on  oak,  fibres  parallel  to  motion  

164 

075 

067 

""        • 

"     "     "      fibres  perpendicular  to  motion  

083 

072 

'*  on  elm,  fibres  parallel  to  motion  

136 

073 

066 

"  on  cast  iron,  fibres  parallel  to  motion  

080 

"  on  wrought  iron,           "        "        "       .  . 

098 

Beech  on  oak,  fibres             "        •'        " 

055 

Elm  on  oak,         "                 "        "        " 

137 

070 

nun 

"  on  elm,         "                "        "        " 

139 

.060 

"  cast  iron,      "                 "        "        "       

066 

Wrought  iron  on  oak,  fibres  parallel,  greased  and  wet,  .256. 
"          "       "     "      fibres  parallel  to  motion  

214 

085 

"          "     on  elm,      "           ''        "        "      

055 

078 

076 

"          "     on  cast  iron           "        "        "      

066 

103 

076 

070 

082 

081 

"          "     on  brass   fibres  '"         "         " 

078 

103 

Cast  iron  on  oak,  fibres  parallel  to  motion  

.189 

.    10 

'      "     on  elm,     "           "         "        "     

.075 
.061 

.078 
.077 

.075 

091 

"      '*     on  cast  iron,  with  water     314       

197 

064 

100 

070 

"      "     on  brass  

078 

103 

075 

.Uo» 

Copper  on  oak.  fibres  parallel  to  motion  

069 

Yellow  copper  on  cast  iron 

066 

072 

Brass  on  cast  iron. 

077 

086 

,uo8 

"      on  wrought  iron  

072 

081 

089 

"      on  brass  

058 

Steel  ou  cast  iron  

079 

105 

081 

"     on  wrought  iron  

093 

076 

'  '     on  brass  

.053 

.056 

.067 

.133 
191 

.159 
241 

•'            •'       on  oak,  with  water,  .29  

g-  friction  of  the  wooden  frigate  Princeton  was  found  by 

lin  Institute  in  1844.  to  average  about  .067  or  one-fifteenth  of  the  pressure 


The  launching 

a  committee  of  the  Frankli 

during  the  first  .75  of  a  sec,  and  .022  or  one  forty-fifth  for  the  next  4  sees  of  her  motion.  The  slope 
of  the  ways  was  1  in  13,  or  4  dear,  24  mins.  TheV  were  heavily  coated  with  tallow.  Pressure  on  them 
=  15.84  fts  per  sq  inch,  or  2280  5>s  per  sq  ft.  In  the  first  .75  of  a  sec  the  vessel  slid  2.5  ins  j  in  the  4 
next  sees  15  ft,  6.5  ius  ;  total  for  4.75  sec  15.75  ft.  See  footnote,  p  598. 

Coeffs  of  friction  of  dry  surfaces,  under  pressures  grad- 
ually increased  up  to  the  limits  of  abrasion.   (By  G.  Rennie,  C  E  ) 


Pres.  in  Los. 
per 
Square  Inch. 

Wrought  Iron 
Wrought  Iron. 

Wrought  Iron 
on 
Cast  Iron. 

Steel 
on 
Cast  Iron. 

Brass 
on 
Cast  Iron 

32.5 
186 
224 
336 
448 
560 
672 
709 
784 

.140 
.250 
.271 
.312 
.376 
.409 

.174 

!292 
.333 
.365 
.367 
.376 
.434 

.166 
.300 
.333 
.347 
.354 
.358 
.403 

.157 
.225 
.219 
.215 
.208 
.233 
.233 
.234 
232 

The  irregular- 
ities in  this  last 
column  are  re- 
markable. 

821 

.273 

Art.  4.  Pivot  friction.  To  find  the  amount  of  power  consumed  by  the 
fric  of  pivots.  The  base  of  the  pivot  is  supposed  to  be  flat;  that  being  the  best  form.  The  moment 
of  pivot  fric,  or.  in  other  words,  its  tendency  to  prevent  the  pivot  from  revolving  ;  is  found  by  mult 
the  amount  of  the  fric  itself,  by  its  leverage ;  which  last  is  equal  to  %  (two  thirds)  of  the  rad  of  the 
pivot.  Hence,  first  find  the  fric,  by  mult  the  entire  pres  on  the  pivot  in  Ibs.  by  the  corresponding 
coeff  taken  from  one  of  the  foregoing  tables.  Mult  the  fric  so  obtained,  by  %  of  the  circumf  of  the 
pivot,  in  feet.*  The  prod  will  be  the^amount  of  power  consumed  by  fric  at  each  turn  of  the  pivot,  ex- 
pressed in  foot  pounds.  And  these  ft-fts.  mult  by  the  number  of  turns  made  per  min,  will  give  the 
power  expended  per  min  in  overcoming  the  fric. 

Ex.  A  pres  of  22400  Ibs  is  supported  by  a  steel  pivot  6  ins  diam,  revolving  on  a  cast-iron  step  ;  and 
kept  well  lubricated  with  oil.  What  power  must  the  prime  mover  expend  in  overcoming  tho  fric? 

Here,  the  diam  of  the  pivot  being  6  inches,  its  circumf  is  18.84  ins:  and  two-thirds  of  18. 84  is  12.56 
ins,  or  1.05  ft.  Also,  the  corresponding  number  for  steel  on  cast  iron  in  the  table,  is  about  .08.  Con- 
sequently, 

22400  X  .08  X  1.05  -  1881.6  ft-fts  of  fric  per  revolution. 


*  Kqual  to  the  whole  circumf  described  by  ^  of  the  rad. 


601 


Now,  If  the  pivot  makes  say  60  tu/HSper  min,  fric  will  consume  1881.6  X  60  —  112896  ft-fts  of  power 

powers  of  thi 
NOTE.     The>Hlim  ot  the  pivot  should  be  as  small  as  considerations  of  strength  will  admit;  for 

to  motion  Increases  with  the  greater  leverage  of  the  larger  one.     See  Pivot,  in  Glossary. 

\Vtioii  the  pres  on  a  pivot  floes  not  exceed  15O  It**  per  s«| 
inch  lor  east  iron,  or  30U  Ibs  for  steel,  with  good  lubrication,  it  will  revolve 
rapidly  in  machinery  for  a  long  time,  without  sensible  wear. 

When  the  motion  is  very  slow  and  intermittent,  as  in  lock-gates, 

draw-bridges,  turntables,  &c,  from  1  to  even  2.5  tons  per  sq  inch  for  cast  iron  ;  and  from  2  to  4  tons 
for  steel,  are  used.  A  flat  base*  or  foot,  is  best  for  a  pivot ;  and  should  be  kept  well  lubricated,  and 
free  from  dust  or  grit. 

Art.  5.  The  friction  of  the  journals  of  axles,  gudgeons,  or 
trunnions,  in  their  boxes  or  bearings,  is  a  species  of  sliding  fric;  yet  somewhat 

distinct  from  the  foregoing.  It  bears  a  less  proportion  to  the  pres  than  in  the  case  of  flat  surfaces. 
Neither  the  fric  itself,  uor  the  power  reqd  to  overcome  it,  is  affected  by  the  length  of  the  journal  in 
its  bearings;  but  its  resistance  to  motion  increases  with  its  diam  :  and  in  the  same  proportion;  be- 
cause the  leverage  with  which  the  fric  resists  motion,  is  as  the  diam.  Therefore,  it  becomes  impor- 
tant to  employ  strong  materials  for  journals,  as  well  as  tor  pivots,  in  order  to  reduce  the  diam  as  much 
as  possible. 

To  find  the  amount  of  power  consumed  by  the  fric  of  jour- 
nals :  Mult  together,  the  weight  or  the  pres  sustained  by  the  journals,  in  pounds; 
the  corresponding  number  taken  from  the  following  table,  and  the  circumf  of  the  journal  in  feet. 
The  prod  will  be  the  loss  of  power  at  each  rev,  expressed  in  ft-B>s. 

Ex.  A  pres  of  22400  pounds  is  sustained  by  two  journals  of  cast  iron,  6  ins  diam,  and  running 
iu  cast-iron  bearings  well  lubricated  with  lard.  How  much  power  of  the  prime  mover  is  consumed 
by  fric,  at  each  rev  ? 

Here,  we  have  the  pres  on  the  journals,  22400  Ibs  ;  the  corresponding  number  from  the  following 
table,  .054 ;  and  the  circumf  of  the  journals  18.84  ins,  or  1.57  ft;  consequently, 
Power  consumed  =  22400  X  -054  X  1.57  =  1899  ft-fts  per  rev. 

Table  of  journal,  or  axle  friction.* 


For  the  boxes  of  jour- 
nals, cast  iron  perfectly 

smoothed,  is  as  good  a  material  for  light 
pressure  as  any,  if  kept  constantly  well 

£ 

be 

0 

Q 

c 
O 

i 

9 

1 

O 

OIL,  TALLOW, 
OR  LARD. 

Very  soft,  purified 
carriage  grease. 

Continuously. 

the  journal  rapidly.  Brass  or  bronze  is 
much  used.  Being  softer  than  cast  iron 
it  does  not  cut  so  much  when  badly  oiled  ; 
but  is  less  durable.  Babbitt  or  other  soft 
metals  are  useful  under  great  pressures. 

S.cg 
|KJ 

Bell-metal  on  bell-metal  

.097 
.075 
.075 
.075 
.075 
.125 
.100 
.116 

.03  to  .054 
.03  to  .054 
.03  to  .054 
.03  to  .054 

.092 

.070 
.03  to  ,054f 
.03  to  .054 

.03  to   .05 

.065 

.090 

Cast-iron  on  bell-metal 

.194 
.251 

.161 

.189 

.079" 

Wrought-iron  on  bell-metal  
Wrought-iron  on  cast-iron  

Cast-iron  on  cast-iron  

.137 

Wrought-iron  on  lignum  vitae.... 
Cast-iron  on  ligriumvitte  
Lignumvitse  on  cast-iron  

188 
.185 

Lignumvitse  on  lignumvitse  

Cast-iron  on  brass  
Wrought-iron  on  brass  

.190 
.250 

.160 
.190 

.075 
.075 
.100 

Brass  on  cast-iron.... 

lard 
mbag 


.111 


.109 


*  Prof  B.  H.  Thurston  of  the  Stevens 

Inst,,  see  Jour  Franklin  Inst,  Nov  1878.  proves  that  contrary 
to  common  opinion  the  coef  of  journal  fric  diminishes  with 
increase  of  pres  up  to  about  600  Ibs  per  sq  inch,  which  is 
seldom  reached.  It  then  increases.  Also  that  the  coef  ia 
much  affected  by  vel,  and  by  the  temp  of  the  journal. 

Much  depends  on   smoothness  of 

journal  and  bearing. 

t  On  railroads  this  is  certainly  at  times  as 

low  as  .02;  and  probably  even  .015  at  high  vels;  and  much 
less  in  well  made  machinery,  with  polished  journals  and 
bearings.  So  with  the  other  metals  in  the  table. 


602 


FRICTION. 


Art.  6.    Friction  rollers.    If  a  journal  J,  instead  of  revolving  on  ordi- 

narv  bearings,  be  supported  on  friction  rollers  R,  R,  its  fric  will  be  reduced  in  nearly  the  same  pro- 
portion that  the  diam  ot  the  axle  o  or  o  of  the  rollers,  is  less  than  the  diam  of  the  rollers  them- 

Thus,  if  the  fric  of  J  when  in  ordinary  bearings  be  1000  pounds,  it  will,  if  placed  on  rollers  12  ins 
diam,  revolving  on  axles  3  ins  diam,  be  reduced  almost  to  250  pounds,  or  to  one-fourth,  or  as  12  ia 
to  3. 

Art.  7.     Rolling  friction  is  that  which  takes  place  where  the  circumf  of 

a  wheel,  or  of  any  rolling  body,  comes  in  contact  with  the  surf  on  which  it  rolls.  It  appears  to  fol- 
low the  same  law  that  applies'to  sliding  friction,  viz,  that  so  long  as  abrasion  does  not  take  place,  it 
increases  as  the  pres.  It  is  also  inversely  as  the  diams.  In  practice  there  are  usually  two  ways  of 
applying  the  force  reqd  to  overcome  rolling  fric.  The  first  is  at  the  axis  of  the  rolling  body  ;  as  the 
force  of  a  horse  is  applied  at  the  axis  of  a  wheel  in  a  wagon  ;  or  of  a  man,  at  that  of  a  wheelbarrow. 
The  second  is  at  the  circumf  of  the  roller;  as  when  workmen  push  along  a  heavy  timber  laid  on  top 
of  two  or  more  rollers ;  or  as  the  ends  of  an  iron  bridge  play  backward  and  forward  by  contraction 
and  expansion,  on  top  of  rollers,  or  balls  of  metal.  The  fric  is  much  less  in  the  second  case,  than  in 
the  first;  for  although  in  the  second  there  are  two  rolling  fries  to  be  overcome,  (one  at  the  top,  and 
one  at  the  bottom  of  the  roller,)  yet  these  two  are  much  less  than  the  one  rolling,  and  the  one  axle- 
fric  of  the  first  case.  The  few  experiments  that  have  been  made  on  the  coeffs  of  rolling  fric,  discon- 
nected from  axle  fric,  have  been  too  incomplete  to  serve  as  a  basis  for  practical  rules. 

Art.  8.    Rolling,  and  axle  friction  combined ;  as  in  railroad  cars. 

A  body  will  not  slide,  or  roll  down  an  inclined  plane,  unless  the  plane  is  so  steep  that  the  sliding  force 
of  grav  is  sufficient  to  overcome  the  sliding  or  rolling  fric  (as  the  case  may  be)  of  the  body.  See  Art 
62,  of  Force  in  Kigid  Bodies.  The  fric  is  in  proportion  to  the  weight,  or  rather  to  the  pres,  of  the 
body  ;  and  the  sliding  force  of  gravity  is  in  proportion  to  the  height  of  the  plane  as  compared  with 
its  hor  length  or  base.  Therefore,  if  the  height  of  the  plane  has  to  be  i,  y1^,  J^Q.  &c,  of  its 
base,  before  the  body  begins  to  slide,  or  to  roll  down  it,  we  know  that  the  fric  of  the  body  is  4,  y1^, 

1  &c,  of  its  weight;  or,  more  correctly  speaking,  of  its  pres  on  the  plane.  This  pres  is  equal  to 
Ihe  weight  only  when  the  plane  is  level,  or  hor  ;  and  becomes  less  than  the  weight,  as  the  plane  be- 
eomes  steeper;  but  the  diff  is  so  slight  on  moderate  railroad  grades,  that  it  may  be  neglected  in  such 
eases  as  that  now  before  us.  See  Table,  p  486,  Force  in  Rigid  Bodies. 

It  is  found  that  railroad  cars  with  wheels  and  journals  of  the  ordinary  diams,  (about  28  to  32  ins 
for  wheels,  and  about  3  ins  for  journals.)  begin  to  roll  down  a  grade  when  it  is  as  steep  as  from  16  to 
24  ft  per  mile.  This  diff  is  owing  to  a  varying  condition  of  the  rails  as  to  smoothness,  dust,  and  irregu- 
larities; and  to  diff  proportions  between"  the  diams  of  the  wheels  and  journals;  the  kind  of  springs; 
khe  kind  and  quantity  of  oil  used  in  lubricating;  and  to  other  minor  considerations.  There  are  5280 
Ft  in  a  mile ;  therefore,  16  and  24  ft  per  mile  give  for  the  height  of  the  plane  in  proportion  to  its  base, 

0 —  —  1  to  330;  and  ^ —  =  1  to  220.  Therefore,  the  combined  rolling  and  axle  fric  of  the  car  vary 
16  24  2240 

from   -a-A-TT  to   -T-i^  part  of  its  weight;  or  since  there  are  2240  fts  in  a  ton,  it  varies  from  — —  to 

330  220  6oO 

2240 

— •=•  6.8  to  10.2  B)s  per  ton  weight  of  car ;  and  the  coeff  of  its  fric  varies  from  -^\-^  or  .00303,  to 

^Q  or  .00454.     Perhaps  8%  H>s  per  ton,  or  ^y  of  the  weight,  (or  .00379,)  which  corresponds  to 

— —  =  20  ft  grade  per  mile,  (or  to  an  inclination  of  13  mius,)  may  in  practice  be  assumed  as  the 

264 

average  on  American  railroads  :  and  that  about  1  tb  of  this  mav  be  ascribed  to  rolling;  and  7J^  to 

axle  or  journal  friction,  but  much  uncertainty  exists  on  this  subject. 

Under  the  same  circumstances,  this  fric,  and  its  coeff,  remain  unchanged  at  all  vels:  that  is,  if  the 
fric  is  8%  tbs  per  ton  at  a  vel  of  1  mile  per  hour,  it  will  be  8>£  B>s  per  ton  at  60  miles  per  hour ;  and 
precisely  the  same  amount  of  it  will  be  generated  at  each  rev  of  the  wheel  in  either  case  ;  and  the 
same  amount  of  motive  power  will  in  either  case  be  required  to  overcome  it  during  one  rev.  This 
being  the  case,  it  may  at  first  sight  appear  paradoxical  that  the  motive  power  or  traction  reqd  to 
overcome  the  fric,  must  increase  with  the  vel.  But  it  is  evident  that  this  follows  from  the  fact  that 
the  fric  at  each  rev  requires  the  same  amount  of  power  to  overcome  it,  at  whatever  vel ;  consequently, 
since  at  60  miles  an  hour  there  are  60  revs  in  the  same  time  that  there  is  1  at  1  mile  an  hour,  there 
must  be  60  times  more  power  continuously  applied  during  the  high  vel  than  during  the  low  one.  Still 
the  total  amount  of  fric,  and  also  the  total  amount  of  power  called  into  action  in  travelling  any  given 
dist.  will  be  the  same  at  all  vels;  for  whether  the  car  travels  60  miles  in  1  hour,  or  in  60  hours,  its 
wheels  make  the  same  number  of  revs,  and  generate  the  same  total  amount  of  friction.  At  1  mile 
per  hour,  we  apply  our  continuous  power  in  a  small  stream,  but  for  a  long  time;  and  at  60  miles  per 
hour,  we  apply  it  in  a  stream  60  times  larger,  but  for  only  one  60th  as  long  a  time.  The  engine  will 
expend  the  same  total  amount  of  power  against  fric  during  a  60-mile  trip  at  1  mile  an  hour,  as  during 
the  same  60  miles  at  60  miles  an  hour;  but  still,  an  engine  to  perform  the  60  miles  in  1  hour,  must  be 
60  times  as  powerful  as  one  that  can  barely  do  1  mile  per  hour ;  in  other  words,  the  fast  engine  must 
be  able  to  apply  60  times  as  much  force  at  any  one  instant.  The  same  principle  applies  to  the  fric 
of  pivots,  bands.  &c,  at  diff  vels.  The  effect  of  grades  is  not  here  considered. 

In  them  all,  the  coe.ff  of  fric  remains  unchanged  at  all  vels  ;  but  the  <|Iiantity  of  fric  to  be  over- 
come ill  a  giveil  time,  of  course  varies  as  the  vels;  and,  therefore,  as  in  the  case  of  the 
cars,  greater  vel  requires  greater  power  during  the  time  of  motion.* 

When  a  train  moves  rapidly,  other  resistances  are  generated ;  such  as  that  of  the  air,  (which  is 
almost  inappreciable  at  low  speeds;)  jolts  against  irregularities  of  the  rails;  fric  against  the  rails, 
caused  by  the  cars  swaying  from  side  to  side  of  the  track,  &c.  These,  however,  do  not  affect  the  frio 
just  spoken  of. 

A  double  purchase  crane,  with  a  weight  of  7000  Ibs  suspended  from  it,  showed  a  fric  Of  ^  the  weight; 

*The  young  student  should  reflect  well  on  this  subject,  and  familiarize  himself  with  the  broad  dis- 
tinction between  the  unvarying  coeff  of  fric,  and  the  varying  amount  of  frio 


at  diff  vels.     See  footnote,  p  598. 


j?*< 
TKACTIQiW;       \ 

'  VJ 


603 


9r  nearly  800  fts.  One  ton  suspended  a£,  «a*SH  end  of  a  chain  passing  over  2  cast-iron  sheaves  of  2  feet 
diam;  with  wrought-iron  jouruaii*<^'forkiug  iu  brass  bearings,  well  oiled,  gave  y^-  of  the  weight;  or 


er  ton.     The  fric  of  au  unloaded  locomotive,  is  about  12  fts  per  ton  of 


-——•'120  fts; 

its  weight;  with  a  tram  attached,  this  is  increased  about  1  ft  per  ton  of  train. 

Moriu  says  the  fric  of  a  sled  on  dry  ground  is  %  of  the  pres.     Babbage  states  that  a  block  of  stone 

floor,  GO  per  cent  ;  with  both  wooden  surfaces  greased,  only  6  per  cent;  and  with  the  block  on  top  of 
wooden  rollers  3  ins  diam,  only  2.6  per  cent.     Rubble  masonry  on  wet  clay  .2  to  .35. 

For  the  friction  of  Hydraulic  presses  see  p  632. 


TEACTION. 


Traction  oil  common  roads,  and  canals;  or  the  power  reqd  to  draw 
vehicles  and  boats  along  them.    In  connection  with  this  subject  read  the  preceding  and  the  following 


The  following  table 
coach  and  passengers, 
as  ascertained  by  me 
results  are  given  per  s 


shows  tolerable  approximations  to  the  force  in  fts  per  ton,  reqd  to  draw  a  stage 
up  ascents  on  the  Holyhead  turnpike  road  iu  England,  (a  fine  road,)  by  horses  ; 
ns  of  a  dynamometer.  The  entire  weight  was  1%  tons;  but  in  the  table,  the 
ingle  ton.  From  the  nature  of  such  cases,  no  great  accuracy  is  attainable. 


Proportional 
Ascent. 

Ascent  iu  Ft. 
per  Mile. 

At  4  Miles 
per  Hour. 

At  6  Miles 
per  Hour. 

At  8  Miles 
per  Hour. 

At  10  Miles 
per  Hour. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

1  in    15J4 

340.7 

210 

216 

225 

240 

1  "     20 

264. 

196 

202 

212 

229 

1  "     26 

203.1 

155 

160 

166 

175 

1  "     30 

176. 

137 

142 

147 

154 

1  "     40 

132. 

114 

120 

124 

130 

1  "     64 

82.5 

109 

115 

120 

126 

1  "  118 

44.7 

102 

107 

113 

120 

1   "  138 

38.3 

99 

103 

109 

117 

1  "  156 

33.9 

98 

101 

106 

112 

1  "  245 

21.6 

93 

96 

101 

107 

1  "  600 

8.8 

81 

85 

91 

96 

Level. 

0. 

76 

80 

85 

91 

Miles  per  hour. 

y. 
1      . 


The  following  results,  raost  of  them  with  the  same  instrument,  are  also  in  fts  per  ton  ;  with  a  four- 
wheeled  wagon,  at  a  slow  pace,  on  a  level;  and  the  roads  iu  fair  condition. 

On  a  cubical  block  pavement  ...........................     32  fts  per  ton  ...........  to  50. 

"       McAdam  road,  of  small  broken  stone  ..............     6'2  "  "     probably  to  75. 

"       gravel  road  .......................................   140  "      "     " 

"      Telford  road,  of  small  stone  on  a  paving  of  spawls     46  "  "  75. 

"      broken  stone,  on  a  bed  of  cement  concrete  ........     46"      "     '  "75. 

"      common  earth  roads  ..............................  200  to  300.   On  a  plank  road  30,  to  50  fts. 

The  tractive  power  of  a  horse  diminishes  as  his  speed  in- 

creases; and  perhaps,  within  certain  limits,  say  from  %  to  four  miles  per  hour, 
nearly  in  inverse  proportion  to  it.     Thus,  the  average  traction  of  a  horse,  on  a  level,  and  actually 
pulling  for  10  hours  in  the  day,  may  be  assumed  approximately  as  follows: 

Lbs.  Traction.  Miles  per  hour.         Lbs.  Traction. 

333.33  iy±  ..............  111.11 

...  250.  2^  ..............  100. 

200.  2%  ..............    90.91 

13^  .............  166.66  3     ..............    83.33 

1??  ..............  142.86  3^  ..............    71.43 

2     ..............  125.  4     ..............    62.50 

If  he  works  for  a  smaller  number  of  hours,  his  traction  may  increase  as  the  hours  diminish  ;  down 
to  about  5  hours  per  day  and  for  speeds  of  about  from  \y±  to  3  miles  per  hour.  Thus,  for  5  hours  per 
day  his  traction  at  2^  miles  per  hour  will  be  200  fts,  Ac.  When  ascending  a  hill,  his  power  dimin- 
ishes so  rapidly,  from  having  partially  to  raise  his  own  weight,  (which  averages  about  1000  to  1100 
fts,)  that  up  a  slope  of  5  to  1.  he  can  barely  struggle  along  without  any  load.  On  such  an  ascent,  (see 
table,  p  486,  of  Force  in  Rigid  Bodies,)  he  must  exert  a  force  equal  to  439  fts  per  ton.  or  of  15)6  fts  for 
the  1000  fts  of  his  own  weight.  Assuming  that  on  a  level  piece  of  good  turnpike,  he  would  when  h;iul- 
ing  a  cart  and  load,  together  weighing  1  ton,  have  to  exert  a  traction  of  60  fts;  then  on  ascending  a 
hill  of  4°  inclination,  (or  1  in  14.3  ;  or  36»M  ft  per  mile,)  he  would  have  to  exert  156  fts.  against  the 
gravity  of  the  1  ton  :  and  67  fts.  against,  that  of  his  own  weight:  or  223  fts  in  all.  He  may,  for  a  few 
inins,  exert  without  injury,  about  twice  his  regular  traction.  This  calculation  shows  that  up  a  hill 
of  4°,  an  average  horse  is  fully  tasked  in  drawing  a  total  load  of  one  ton  ;  and  should,  therefore,  be 
allowed,  in  such  a  case,  to  choose  his  own  gait  ;  and  to  rest  at  short  intervals.  A  fair  load  for  a  single 

road  in  good  order,  is  about  half  a  ton,  in  addition  to  the  cart,  which  will  be  about  half  a  ton  more. 
With  two  horses  to  this  same  cart,  the  load  alone  may  be  about  1  %  tons. 

REM.     Since  the   action  of  gravity  is   the   same  on    good  roads   and  bad   ones,   it  follows   that 

ascents  become  more  objectionable  the  better  the  road  is. 

Thus,  on  an  ascent  of  2°,  or  184.4  ft  pe'r  mile,  gravity  alone  requires  a  traction  of  78  fts  per  ton  ; 


604 


TRACTION. 


which  is  about  10  times  that  ou  a  level  railroad  at  6  miles  an  hour  ;  but  only  about  equal  to  that  ou  & 
level  common  turnpike  road,  at  the  same  speed.  Therefore,  (to  speak  somewhat  at  random,)  it  would 
require  10  locomotives  instead  of  1 ;  but  only  2  horses  instead  of  1.  A  grade  of  I  in  35  ;  or  150  ft  to  a 
mile;  or  1°  'M' ,  is  about  the  steepest  that  permits  horses  to  be  driven  down  a  hard  smooth  road,  in  a 
fast  trot,  without  danger.  It  should,  therefore,  not  be  exceeded  except  when  absolutely  necessary, 
especially  ou  turnpikes. 

On  eajtuis)  ami  other  waters,  the  liquid  is  the  resisting  medium  that 

takes  the  place  of  friction  on  level  roads.  But  unlike  friction,  its  resistance  varies  as  the  squares  ol' 
the  vels;  (see  Art  '26  of  page  571.)  at  least  from  the  vel  of  2  ft  per  sec,  or  1.361  miles  per  hour;  u 
that  of  11  ^  ft  per  sec,  or  7.81  m  per  h.  As  the  speed  falls  below  1  %  in  per  h,  the  resistance  varies  less 
and  less  rapidly;  and  this  is  tue  case  whether  the  moved  body  floats  partly  above  the  surface;  or  is 
entirely  immersed.  In  towing  along  stagnant  canals,  <fec,  the  vel  is  usually  from  1  to  2>£  m  per  h; 
for  freight  most  frequently  from  1J4  to  2.  Less  force  is  required  to  tow  a  boat  at  say  2  m  per  h,  where 
there  is  no  current,  than  at  say  \%  m  per  h,  against  a  current  of  %  m  per  h.  because  in  the  last  case 
the  boat  has  to  be  lifted  up  the  very  gradual  inclined  plane  or  slope  which  produces  the  current. 
Therefore,  a  steamboat  which  has  its  speed  increased  say  3  or  4  m  per  h  by  a  descending  current, 
will  have  it  retarded  more  than  'A  or  4  m  per  h  when  returning  against  the  current. 

The  force  required  to  tow  a  boat  along  a  canal  depends  greatly  upon  the  comparative  transverse 
sectional  areas  of  the  channel,  and  of  the  immersed  portion  of  the  bout.  When  the  width  of  a  canal 
at  water-line  is  at  least  4  times  that  of  the  boat;  and  the  areaof  its  transverse  section  asgreatas  at  least 
6}^  times  that  of  the  immersed  transverse  section  of  the  boat,  the  towing  at  usual  canal  vels  will  be 
about  as  easy  as  in  wider  and  deeper  water.  With  less  dimensions,  it  becomes  more  difficult.  (D'Au- 
buisson.)  Much  also  depends  on  the  shape  of  the  bow  and  other  parts  of  the  boat ;  and  on  the  propor- 
tion of  its  length  to  its  breadth  and  depth.  Hence  it  is  seen  that  the  mere  weight  of  the  load  is  by  no 
means  so  controlling  an  element  as  it  is  ou  laud.  The  whole  subject,  however,  is  too  intricate  to  be 
treated  of  here.  Morin  states  that  naval  constructors  estimate  the  resistance  to  sailing  and  steam 
vessels  at  sea,  at  but  from  about  .5  to  .7  of  a  R>  for  every  sq  ft  of  immersed  transverse  section,  when 
the  vel  is  3  ft  per  sec,  or  2.046  miles  per  hour.  It  is  far  greater  on  canals. 

On  the  Sell uy  lit  ill  Ufavig-ation  of  Pennsylvania,  of  mixed  canal 

and  slackwater,  for  108  miles,  the  regular  load  for  3  horses  or  mules,  is  a  bout  of  very  full  build ;  and  no 
keel :  100  ft  long,  17^  ft  beam  ;  and  8  ft  depth  of  hold  ;  drawing  ;VA  ft  when  loaded.*  Weight  of  boat 
about  65  tons;  load  175  tons  of  coal,  (2240  Us:)  total  weight  240  tons,  or  80  tons  per  horse  or  mule. 
On  the  down  trip  with  the  loaded  boats,  for  4  days,  the  animals  are  at  work,  actually  towing,  (except 
at  the  locks.)  for  18  hours  out  of  the  24;  thus  exceeding  by  far  the  limits  of  time  usually  allowed  for 

Ou  the  canal  sections,  (which  have  60  ft  water-line  ;  and  6  ft  depth,)  thespeed  is  1%  miles  per  hour ; 
and  on  the  deep  wide  pools,  2  miles. 

On  the  up  trip  with  the  empty  65-ton  boats,  the  average  speed  is  about  2U  miles  per  hour.  The 
empty  boats  draw  16  to  18  ins  water  ;  and  frequently  keep  on  without  stopping  to  rest  day  or  night 
through  the  entire  distance  c<f  108  miles.  The  animals  generally  have  2  or  3  days'  rest  at  each  end  of 
the  trip ;  but  are  materially  deteriorated  at  the  end  of  the  boating  season. 

If  our  preceding  assumption  of  143  fits  traction  of  a  horse  at  1%  miles  per  hour,  is  correct,  the 

traction  of  the  loaded  boats  on  the  canal  sections  is  — ~  1.83  B>s  per  ton. 

The  intelligent  engineer  and  superintendent  of  the  Sch  Nav,  Jumes  F  Smith,  gives  as  the  results 
of  his  own  extensive  observation,  that  one  of  these  large  boats  loaded  (240  tons  in  all)  may,  without 
distressing  the  animals,  be  drawn  along  the  canal  sections,  for  10  hours  per  day,  as  fellow's :  By  one 
average  horse  or  mule,  at  the  rate  of  1  mile;  by  two  animals,  at  13^  miles;  and  "by  three,  atl%  miles 
per  hour.  When  four  animals  are  used  the  gain  of  time  is  very  trifling.  At  a  time  of  rivalry  among 
the  boatmen,  one  of  them  used  8  horses  ;  but  with  these  could  not  exceed  2J6  miles  per  hour  in  the 
canal  portions.  Two  or  more  horses  together  cannot  for  hours  pull  as  much  as  when  working  sepa- 
rately. 

If  our  preceding  short  table  of  the  traction  of  a  horse  at  diff  vels  for  10  hours  is  correct,  then  the 
traction  of  the  above  loaded  conl  boats  (240  tons)  on  the  canal  sections  of  the  navigation,  is  as  follows : 
The  last  column  shows  the  traction  in  fts  per  sq  ft  of  area  of  immersed  transverse  section  where  largest ; 
viz,  about  95  sq  ft. 

Horses.  Miles  per  Hour.  Lbs.  per  Ton.  Lbs.  per  Sq  Ft. 


3                               \% 

42*      ^ 

.  1  78 

4  50 

3  on  pools  2     

3  75 

1.56  

3  95 

8....                   ....2K... 

8  0  0. 

...3.33.... 

...8.42 

3up-trip 2^ %Y 4.61 12.50 

Laehiiie  Canal,  Canada,  120  ft  wide  at  water-line ;  80  ft  at  bottom ;  depth 

on  mitre  sills  9  ft ;  6  horses  tow  loaded  schooners  with  ease. 

Before  the  enlargement  of  the  Erie  <'«MUll,f  its  dimensions  were  40  ft  water-line  ;  28  ft  bottom  ; 
4  ft  depth  of  water.  The  average  weight  of  the  boats  was  about  30  tons.  With  75  tons  of  load,  or  105 
tons  total,  they  were  towed  by  2  horses,  at  the  rate  of  about  2  miles  per  hour  ;  which  by  our  table  gives 
a  traction  of  nearly  2.4  fts  per  ton.  The  boats  were  about  80  ft  long  ;  14  ft  beam  ;  full  83/4  ft  draught 
loaded  ;  hence  the  traction  by  our  table  >vould  be  about  5.7  fts  per  sq  ft  of  immersed  transverse  section. 

*  Cost  of  boats  in  1872  averaged  about  $2500  ;   arid  their  repairs  about  $150 

per  annum.  They  last  from  9  to  12  years.  Before  the  canal  was  enlarged,  boats  weighing  about  22 
tons,  and  carrying  60  tons  of  coal,  were  used.  Length  70  ft;  beam  13;  draft  empty,  about  1  foot; 
•when  loaded,  4  ft ;  cost  $700.  These  are  approximately  the  dimensions  of  the  freight  boats,  on  our 
ordinary  canals.  With  2  horses,  speed  \%  miles. 

t  Length  363  miles ;  cost  $19680  per  mile.  The  enlarged  canal  has  70  ft ;  42  ft ;  and  7  ft  of  water ; 
and  cost  $90800  per  mile  for  the  enlargement  only.  The  cost  of  the  several  canals  in  Pennsylvania 
has  ranged  between  $23000  and  $50000  per  mile. 


5WER.  605 

While,  for  the  82-ton  loaded  boatajp*«?fie  footnote,  on  the  smaller  canal,  (the  boats  nearly  touch- 
ing bottom,)  the  tractiona£-i£fjS«es,  would  be  3%  fts  per  ton  ;  or  about  twice  as  great  as  the  above 
1.78  fts.  It  also  wouJA^SirjJreper  sq  ft  of  immersed  section. 

Traction  on^c'fevel  straight  railroad,  at  speeds  not  exceeding  about 

12  miles  per  hour^Wrrom  5  to  10  fts  per  ton,  depending  on  diam  of  wheels  and  journals ;  lubrication, 
condition  of  t 


ANIMAL  POWEE, 


Art.  1.     So  far  as  regards  horses,  this  subject  has  been  partially  considered 

under  the  preceding  head,  Traction.  All  estimates  on  this  subject  must  to  a  certain  extent  be  vague, 
owing  to  the  diff  strengths  aud  speeds  of  animals  of  the  same  kind ;  as  well  as  to  the  extent  of  their 
training  to  any  particular  kind  of  work.  Authorities  on  the  subject  differ  widely  ;  and  sometime!* 
express  themselves  iu  a  loose  manner  that  throws  doubt  on  their  meaning.  We  believe,  however, 
that  the  following  will  be  found  to  be  as  close  approximation  to  practical  averages  as  the  nature  of 
the  case  admits  of  with  our  present  imperfect  knowledge.  We  suppose  a  good  average  trained  horse, 
•weighing  not  less  than  about  J^  a  ton,  well  fed  aud  treated.  Such  a  one,  when  actually  walking  for 
10  hours  a  day,  at  the  rate  of  2  J^  miles  per  hour,  on  a  good  level  road,  such  as  the  tow-path  of  a  canal, 

or  &  circular  horse  -path,  *  can  exert  a  continuous  pull,  draught,  power, 
or  traction,  of  1OO  Ibs. 

Now,  2Ji  miles  per  hour,  is  220  ft  per  min.  or  3%  ft  per  sec;  and  since  10  hours  contain  600  min, 
his  day's  work  of  actual  hauling  on  a  level,  at  that  speed,  amounts  to 
min          ft  Ibs 

600  X  220  X  100  —  13200000ft-ftsperday. 

Or,  22000  ft-Ibs  per  min.  or  366%  ft-Ibs  per  sec.t  Which  means  that  he  exerts  force  enough  during  the 
day  to  lift  13  200000  fts  1  foot  high;  or  1  320000  Rs  10  feet  high;  or  132  000  fts  100  ft  high,  &c.  He  may 
exert  this  force  either  in  traction  (hauling)  or  in  lifting  loads.  If  he  has  to  raise  a  small  load  to  a 
great  height,  the  machinery  through  which  he  does  it  must  be  so  geared  as  to  gain  speed,  at  the  losa 
(commonly  but  improperly  so  expressed)  of  power.  Whether  he  lifts  the  great  weight  through  a 
small  height,  or  the  small  weight  through  a  great  height,  he  exerts  precisely  the  same  amount  of 
force  or  power.  In  connection  with  this  subject,  the  student  should  read  Arts  5,  9,  11,  &c,  of  Force 
in  Rigid  Bodies.  Also,  see  Hauling  by  Horses  and  Carts. 

Experience  shows  that  within  the  limits  of  5  and  10  hours  per  day,  (the  speed  remaining  the  same,/ 

the  draft  of  a  horse  may  be  increased  in  about  the  same  pro- 
portion as  the  time  is  diminished  ;  so  that  when  working  from  5  to  10  hour! 
per  day,  it  will  be  about  as  shown  in  the  following  table.  Hence,  the  total  amount  of  13  200  000  ft-ftf 
per  day  may  be  accomplished,  whether  the  horse  is  at  work  5,  6,  or  8,  &c,  hours  per  day.+  This,  of 
course,  supposes  him  to  be  actually  lifting  or  hauling  all  the  time;  and  makes  no  allowance  for  stop- 
pages for  any  purpose. 

Table  of  draft  of  a  horse,  at  2^£  miles  per  hour,  on  a  level. 

Hours  per  day.  Lbs.  Hours  per  day.  Lbs. 

10  ....'. 100  7  142J 

9 lllj  6 166% 

8  125  5  200 

Experience  also  shows  that  at  speeds  between  %  and  4 
miles  an  hour,  his  force  or  draught  will  be  inversely  in  pro* 
portion  to  his  speed.  Thus,  at  2  miles  an  hour,  for  10  hours  of  the  day,  hif 

draught  will  be 

miles     miles  ftg  9>s 

2    :    iyz    :  :    100    :     125  draught. 

At  m  miles,  it  would  be  166%  fts ;  at  3  miles,  83  &  fts;  and  at  4  miles,  62^  B>s  ;  as  per  table  i» 
Traction. 
Therefore,  in  this  case  also,  the  entire  amount  of  his  day's  work  remains  the  same  ;  §  and  within 

*  To  enable  a  horse  to  work  with  ease  in  a  circular  horse- walk,  its  diam 

should  not  be  less  than  25  ft;  30  or  35  would  be  still  better. 

f  A  nominal  horse-power  is  33000  ft-fts  per  minute;  this  being  the  rate 
assumed  by  Boulton  and  Watt  in  selling  their  engines ;  so  that  purchasers  wishing  to  substitute 
steam  for  horses,  should  not  be  disappointed.  Their  assumption  can  be  carried  out  by  a  very  Btrong 
horse  day  after  day  for  8  or  10  hours ;  but  as  the  engine  can  work  day  and  night  for  months  without 
stopping,  which  a  horse  cannot,  it  is  plain  that  a  one-horse  engine  can  do  much  more  work  than  any 
one  such  horse.  Hence  many  object  to  the  term  horse-power  as  applied  to  engines  ;  but  since  every- 
body understands  its  plain  meaning,  and  such  a  term  is  convenient,  it  is  not  in  fact  objectionable. 
Boulton  and  Watt  meant  that  a  one-horse  engine  would  at  any  moment  perform  the  work  of  a  very 
strong  horse.  An  average  horse  will  do  but  22000  ft-fts  per  min. 

+  It  is  plain  that  although  the  day's  labor  will  be  the  same,  that  of  an  hour,  or  of  a  min,  will  vary 
with  the  number  of  hours  taken  as  a  day's  work.  It  must  be  remembered  that  a  working  day  of  a 
given  number  of  hours,  by  no  means  implies,  in  every  case,  that  number  of  hours  of  actual  work; 
but  includes  intermissions  and  rests. 

\  This  remark  about  speed  will  not  apply  to  loads  towed 
through  the  water.  Thus,  if  his  draught  at  2  miles  an  hour  be  125  Ibs ;  and 

at  4  miles, "62  H  fts :  he  will  on  land  draw  loads  in  these  proportions ;  but  in  hauling  a  boat  through 
the  water  at  the  greater  speed,  he  has  to  encounter  the  increased  resistance  of  the  water  itself ;  which 
resistance  at  4  miles  is  much  more  than  twice  as  great  as  at  2  miles ;  probably  4  times  as  great. 
Therefore,  at  4  miles  on  a  canal,  his  draught  of  62>$  fts  would  not  suffice  for  a  load  half  as  great  as 
he  could  tow  with  his  draft  of  125  fts  at  2  miles. 

39 


606 


ANIMAL   POWER. 


all  the  foregoing  limits  of  hours  and  speed,  may  be  practically  taken  to  be  about  13200000  ft-Ibs  per 
day ;  or  22000  ft-B>s  per  min  of  a  day  of  10  hours.  But  it  does  uot  follow  that  the  horse  can  always 
in  practice  actually  lift  loads  at  that  rate;  because  generally  a  part  of  his  power  is  expended  in 
overcoming  the  friction  of  the  machinery  which  he  puts  in  motion  ;  aud  moreover,  the  nature  of  the 
work  may  require  him  to  stop  frequently  ;  so  that  in  a  working  day  of  8  or  10  hours,  the  horse  may 
not  actually  be  at  work  more  than  5,  6,  or  7  hours. 

As  a  rough  approximation,  to  allow  for  the  waste  of  force  in  overcoming  the  friction  of  hoisting 
machinery,  and  the  weight  of  the  hoisting  chains,  buckets,  &c,  we  may  say  that  the  Useful 

or  paying  daily  net  work  of  a  horse,  in  hoisting-  by  a  com- 
mon gill,  is  about  10000000  t't-fts.  That  is,  he  will  raise  equivalent  to  10000000  tt>s  net  of 
water,  or  ore,  &c,  1  foot.  The  load  which  he  can  raise  at  once,  including  chains,  bucket,  aud  aa 
allowance  for  friction,  will  be  as  much  greater  than  his  own  direct  force,,  as  the  diam  of  the  horse - 
walk  is  greater  than  that  of  the  winding  drum;  aud  it  wiii  jiove  that  much  slower  than  he  does. 
His  own  direct  force  will  vary  according  to  the  number  of  hours  per  day  that  he  may  be  required  to 
work,  as  io  the  foregoing  table.  With  these  data,  the  size  of  the  buckets  can  be  decided  on  ;  and  of 
these  there  should  be  at  least  two,  so  that  the  empty  one  at  the  bottom  may  be  filled  while  the  full  one 
at  top  is  being  emptied ;  so  as  to  save  time.  The  same  when  the  work  is  done  by  men. 

Art.  2.  A  practised  laborer  hauling  along:  »  level  road,  by 
a  rope  over  his  shoulders;  or  in  a  circular  path,  pushing  before  him  a 
hor  lever,  at  a  speed  of  from  1  ^  to  3  miles  per  hour,  exerts  about  %  fl>  part  as  much  force  as  a  horse; 
or  2  200000  ft-B>s  per  day ;  or  3666%  ft-Ibs  per  min  of  a  day  of  10  hours  of  actual  hauling  or  pushing. 

But  laborers  frequently  have  to  work  under  circumstances  less  advantageous  for  the  exertion  of 
their  force  than  when  haulingor  pushingin  the  manner  just  alluded  to  ;  and  in  such  cases  they  cannot 
do  as  much  per  day.  Thus  in  turning  a  winch  or  crank  like  that  of  a  grindstone,  or  of  a  crane,  the 
continual  benditig  of  the  body,  and  motion  of  the  arms,  is  more  fatiguing.  The  size  of  a 
WillCll  Should  not  exceed  18  illS,  or  the  rad  of  acircle  of  3  ft  diam;  and  against 
it  a  laborer  can  exert  a  force  of  about  16  fts,  at  a  vel  of  '2%  ft  per  sec,  or  150  ft  per  min,  making  very 
nearly  16  turns  per  min ;  for  8  hours  per  day.  To  these  8  hours  an  addition  must  be  made  of  about 
%  part,  for  short  rests.  Or  if  a  working  day  is  taken  at  8,  or  10,  &c,  hours,  -^  part  must  generally  be 
taken  from  it  for  such  rests.  On  the  foregoing  data  an  hour's  work  of  60  min  of  actual  hoisting 
would  be 

Hw         ft         min 

16  X  150  X  60  =  144000  ft-ftg; 

or,  deducting  ^  part  for  rests,  115200  ft-fts  per  hour  of  time,  including  rests.  In  practice,  however, 
a  further  deduction  must  be  made  for  the  fric  of  the  machine,  and  for  the  wtof  the  hoisting  chains  j 
and  in  case  of  raising  water,  stone,  ore,  &c,  from  pits,  for  the  wt  of  the  buckets  also.  As  a  rough 
average  we  may  assume  that  these  will  leave  but  100000  ft-lb.s  of  paying,  or  useful  work  per  hour; 

that  is,  that  a  man  at  a  winch  will  actually  lift  equivalent  to 
100OOO  Ibs  of  water,  ore,  «fcc,  1  foot  high  per  hour's  time,  in- 
cluding rests.  This  is  equal  to  1666%  ft-Tbs  per  min  of  a  day  of  10  hours,  including  rests. 
Therefore,  in  a  day  of  10  working  hours  he  would  raise  1  000000  fts  net.  1  foot  high  ;  Or  jllSt  ^ 

part  Of  What  a  horse  WOllld  do  With  a  gin  in  the  same  time.  We  have 
before  seen  that  in  hauling  along  a  level  road,  he  can  at  a  slow  pace  perform  about  %  of  the  daily 
duty  of  a  horse.  He  may  also  work  the  winch  with  greater  force,  say  up  to  30  or  even  40  fts;  but 
he  will  do  it  at  a  proportionately  slower  rate;  thus,  accomplishing  only  the  same  daily  duty. 
With  a  gin,  like  tKtise  for  horses,  but  lighter,  with  2  or  more  buckets,  a  prac- 
tised laborer  will  in  a  working  day  of  10  hours,  raise  from  1  200000  to  1  400000  ft-fts  net  of  water,  ore, 
&c.  With  a  shallow  well  or  pit,  more  time  is  lost  in  emptying  buckets  than  in  a  deep  one;  but  the 
deep  one  will  require  a.greater  wtof  rope.  To  save  time  in  all  such  operations  on  a  large  scale,  there 
should  be  at  least  two  buckets;  the  empty  one  to  be  fi.led  while  the-full  one  is  being  emptied.  It  is 
also  best  to  employ  2  or  more  men  to  hoist  at  the  same  time,  by  winches,  at  both  ends  of  the  axis ; 
and  the  men  will  work  with  more  ease  if  the  winches  are  at  right  angles  to.each  other.  Each  winch 
handle  may  be  long  enough  for  2  or  3  men.  An  extra  man  should  be  employed  to  emptv  the  buckets. 
He  may  take  turns  with  the  hoisters.  The  same  remarks  apply  in  some  of  the  following  cases. 

On  a  treadwheel  a  practised  laborer  will  do  about  40  per  cent  more  daily 
duty  than  at  a  winch  ;  or  in  a  working  day  *  of  10  hours,  including  rests,  he  will  do  about  1  400000  ft- 
Ibs.  And  he  can  do  this  whether  he  works  at  the  outer  circumf  of  the  wheel,  stepping  upon  foot- 
boards, or  tread-boards,  on  a  level  with  its  axis ;  or  walks  inside  of  it,  near  its  bottom.  In  both  cases 
he  acts  by  his  wt,  usually  about  130  to  140  Ibs ;  and  not  by  the  muscular  strength  of  his  arms.  When 
at  the  level  of  the  axis,  his  wt  acts  more  directly  than  when  he  walks  on  the  bottom  of  the  wheel; 
but  in  the  first  case  he  has  to  perform  a  slow  and  fatiguing  duty  resembling  that  of  walking  up  a 
continuous  flight  of  steps ;  while  in  the  second  he  has  as  it  were  "merely  to  ascend  a  very  slightly  in- 
clined plane;  which  he  can  do  much  more  rapidly  for  hours,  with  comparatively  little  fatigue:  and 
this  rapidity  compensates  for  the  less  direct  action  of  his  wt.  Therefore,  in  either  case,  as  experience 
has  shown,  he  accomplishes  about  the  same  amount  of  daily  duty.  Treadwheels  may  be  from  5  to  25 
ft  in  diam,  according  to  the  nature  of  the  work.  They  are 'generally  worked  by  several  men  at  once  , 
and  may  at  times  be  advantageously  used  in  pile-driving,  as  well  as  in  hoisting  water,  stone,  &c. 

By  a  good  common  pump,  properly  proportioned,  a.  practised  laborer 

will  in  a  day  of  10  working  hours,  raise  about  1000000  ft-'fts  of  water,  net.t 

Bailing  with  a  light  bucket  or  scoop,  he  can  accomplish  about 

200000  ft- Ibs  net  of  water.  By  a  bucket  and  SWape,  (a  long  lever  rocking  vertically  ; 
and  weighted  at  one  end  so  as  to  balance  the  full  bucket  hung  from  the  other;  often  seen  at  country 

*The  working  day  must  be  understood  to  include  necpssary  rests,  and  such  intermissions  as  th« 
nature  of  the  work  demands  :  but  does  not  include  time  lost  at  meals.  A  working  <#ay  of  10  hours 
may,  therefore,  have  but  8,  7,  or  6,  <fcc  hours  of  actual  labor.  This  will  be  understood  when  we  here- 
after speak  of  a  working  day,  or  simply  a  day. 

t  Desagulier's  estimates  of  daily  work  of  men  and  horses  exceed  the  above,  but  are  entirely  too  great. 


ANIMAL   POWER.  607 

ells,)  60Q4XJO  to  800000.    In  the  last  he  has  only  to  pull  down  the  empty  bucket,  and  thereby  raise  the 

Counterweight.    By  2  buckets  at  the  ends  of  a  rope  suspended  over 

a  pulley,  500  000  to  600000.     Here  he  works  the  buckets  by  pulliug  the  rope  by  band. 

By  a  ty  in  pan,  or  tympanum,*  worked  by  a  treadwheel,  about  1 200000 

to  1400000. 

By  a  Persian  wheel.-j  a  chain-pump,  a  chain  of  buckets,!  or 
ail  Archimedes  screw,  all  worked  by  a  treadwheel,  from  800000  to  1000000 
rt-fts.  Or  these  four,  the  tirst  three  lose  useful  effect  by  either  spilling,  leaking,  or  the  necessity  for 
raising  the  water  to  a  level  somewhat  higher  than  that  at  which  it  is  discharged. 

When  any  of  the  five  foregoing  machines  are  worked  by  men  at  winches,  the  result  will  be  about 
%  less  than  by  treadwheels.  They  are  all  frequently  worked  also  by  either  steam, water,  or  horse-power. 

By  walking-  backward  and  forward,  on  a  lever  which  rocks 
Oil  its  center,  a  man  may,  according  to  Robison's  Mech  Philosophy,  perform  a 
much  greater  duty  than  by  any  of  the  preceding  modes.  He  states  that  a  young  man  weighing  135 
fts,  and  loaded  with  30  fts  in  addition,  worked  in  this  manner  for  10  hours  a  day  without  fatigue; 
and  raised  9J4  cubic  feet  of  water,  11}^  ft  high  per  min.  This  is  equal  to  3  984  000  ft-fts  per  day  of  10 
hours ;  or  6640  ft-fts  per  min ;  or  nearly  -j^y  of  the  net  daily  work  of  a  horse  in  a  gin. 

A  laborer  standing:  still,  can  barely  sustain  for  a  few  min,  a  load  of  100 
Ibs,  by  a  rope  over  his  shoulder,  and  thence  passing  off  hor  over  a  pulley.  And  scarcely  as  much, 
when  (facing  the  load  and  pulley)  he  holds  the  end  of  a  hor  rope  with  his  hands  before  him.  He  can- 
not push  hor  with  his  hands  at  "the  height  of  his  shoulders,  with  more  than  about  30  fts  force. 

Weisbach  states  from  his  own  observation,  that  4  practised  men  raised  adolly  (a  wooden  beetle 
or  rammer,  of  wood;  with  4  hor  projecting  round  bars  for  handles)  weighing  120  fts.  4  ft  high,  at  the 
rate  of  34  times  per  min,  for  4%  min  ;  and  then  rested  for  4}£  min  ;  and  so  on  alternately  through 
the  10  hours  of  their  working  day.  Therefore,  5  of  these  hours  were  lost  in  rests ;  and  the  duty  per- 
formed by  each  man  during  the  other  5  hours,  or  300  nuns,  was 


In  the  old  mode  of  driving  piles,  where  the  ram  of  400  to1200ft>s 

suspended  from  a  pulley,  was  raised  by  10  to  40  men  pulling  at  separate  cords,  from  35  to  40  fts  of  the 
ram  were  allotted  to  each  man,  to  be  lifted  from  12  to  18  times  per  min,  to  a  height  of  3%  to  4%  feet 
each  time,  for  about  3  min  at  a  spell,  and  then  3  min  rest.  It  was  very  laborious ;  and  the  gangs  had 
to  be  changed  about  hourly,  after  performing  but  K  an  hour's  actual  labor. 

Hauling  by  horses.   See  Traction.  When  working  all  day,  say  10  working 

hours,  the  average  rate  at  which  a  horse  walks  while  hauling  a  full  load,  and  while  returning  with 
the  empty  vehicle,  is  about  2  to  2^  miles  per  hour;  but  to  allow  for  stoppages  to  rest,  &c,  it  is  safest 
to  take  it  at  but  about  1.8  miles  ;>er  hour,  or  160  ft  per  miu.  The  time  lost  on  each  trip,  in  loading 
and  unloading,  may  usually  be  taken  at  about  15  min.  Therefore,  to  find  the  number  of  loads  that  can 
be  hauled  to  any  given  dist  in  a  day,  first  find  the  time  in  min  reqd  in  hauling  one  load,  and  return- 
ing empty.  Thus :  div  twice  the  dist  in  ft  to  which  the  load  is  to  be  hauled ;  or  in  other  words,  div 
the  length  in  ft,  of  the  round  trip,  by  160  ft.  The  quot  is  the  number  of  min  that  the  horse  is  in  mo- 
tion during  each  round  trip.  To  this  quot  add  15  min  lost  each  trip  while  loading  and  unloading  ;  the 
sum  is  the  total  time  in  min  occupied  by  each  round  trip.  Div  the  number  of  min  in  a  working  day 
(600  min  in  a  day  of  10  working  hours)  by  this  number  of  min  reqd  for  each  trip ;  the  quot  will  be  the 
number  of  trips,  or  of  loads  hauled  per  day. 
Ex.  How  many  loads  will  a  horse  haul  to  a  dist  of  960  ft,  in  a  day  of  10  working  hours,  or  600  min? 

Here,  960  X  2  =  1920  ft  of  round  trip  at  each  load.     And  -'--  =  12  min,  occupied   in  walking.    And 

Jb()         goo          min  in  10  hours 

12  +  15  in  loading,  &c)r:,27  min  reqd  for  each  load.     Finally,  —   = -    -  22.2,  or 

27  mm  per  trip 

Bay  22  trips;  or  loads  hauled  per  day. 

Table  of  number  of  loads  hauled  per  day  of  1O  working; 
hours.  The  first  col  is  the  distance  to  which  the  load  is  actually  hauled  ;  or  half 
the  length  of  the  round  trip.  The  cost  of  hauling  per  load,  is  supposed  to  be  for  one-horse  carts  ;  the 
driver  doing  the  loading  and  unloading;  rating  the  expense  of  horse,  cart,  and  driver  at  $2  per  day. 
See  Cost  of  Earthwork,  page  437. 

*  The  tympan  revolves  on  a  hoi-  shaft:  and  is  a  kind  of  large  wheel,  the  spokes,  arms,  or  radii  of 
which  are  gutters,  troughs,  or  pipe?,  which  at  their  outer  ends  terminate  in  scoops,  which  dip  into 
the  water.  As  the  water  is  gradually  raised,  it  flows  along  the  arms  of  the  wheel  to  its  axis,  where 
it  is  dischd.  The  scoop  wheel  is  a  modification  of  it.  It  is  an  admirable  machine  for  raising  large 
quantities  of  water  to  moderate  heights.  We  cannot  go  into  any  detail  respecting  this  and  other 
hydraulic  machines. 

t  A  kind  of  large  wheel  with  buckets  or  pots  at  the  ends  of  its  radiating  arms  ;  revolves  on  a  hor 
axis  ;  discharges  at  top.  The  buckets  are  attached  loosely,  so  as  to  hang  vert,  and  thus  avoid  spill- 
ing until  they  arrive  at  the  proper  point,  where  they  come  into  contact  with  a  contrivance  for  tilting 
and  emptying  them.  The  noria  is  similar,  except  that  the  buckets  are  firmly  held  in  place,  and  thus 
spill  much  water.  It  is  therefore  inferior  to  the  Persian  wheel. 

1  An  endless  revolving  vert  chain  of  buckets.  D'Aubuissou  and  some  others  erroneously  call  thii 
tic  noria.  It  is  an  effective  machine. 


608 


ANIMAL   POWEU. 


Dist. 

Feet. 

No.  of 
Loads. 

Cost  per 
Load. 

Dist. 
Feet. 

No.  of 
Loads. 

Cost  per 
Load. 

Dist. 

Miles. 

No.  of 
Loads. 

Cost  per 
Load. 

Cts. 

Cts. 

Cts. 

50 

38 

5.26 

1500 

18 

11.11 

1 

7 

28.57 

100 

37 

5.H 

2000 

15 

13.33 

IK 

6 

33.33 

2'K) 

34 

5.88 

2500 

13 

15.39 

1% 

5 

40.00 

300 

32 

6.23 

3000 

11 

18.18 

2 

4 

50.00 

400 

30 

6.67 

3500 

10. 

20.00 

3 

3 

66.67 

600 

27 

7.41 

4000 

9 

22.22 

4 

2 

100.00 

1000 

22 

9.09 

5000 

7 

28.57 

9 

1         I    200.00 

i 

If  the  loading  and  unloading  is  such  as  cannot  be  done  by  the  driver  alone ;  but  requires  the  help 
of  cranes,  or  other  machinery,  an  addition  of  from  10  to  50  cts  per  load  may  become  necessary.  Haul- 
ing can  generally  be  more  cheaply  done  by  using  2  or  3  horses,  and  one  driver,  to  a  vehicle.  The  neat 
load  per  horse,  in  addition  to  the  vehicle,  will  usually  be  from  J^  to  1  ton,  depending  on  the  condition, 
and  grades  of  the  road.  From  13  to  15  cub  ft  of  solid  stone ;  or  from  23  to  27  cub  feet  of  broken  stone, 

make  i  ton.    In  estimating*  for  hauling  rough  quarry  stone  for 

drains,  CUlvertS,  «feC,  bear  in  mind  that  each  cub  yard  of  common  scabbled  rubble 
masonry,  requires  the  hauling  of  about  1.2  cub  yds  of  the  stone  as  usually  piled  up  for  sale  in  the 
quarry  ;  or  about  %  of  a  cub  yd  of  the  original  rock  in  place.  A  Cllb  yd  Of  Solid  Stone, 

when  broken  into  pieces,  usually  occupies  about  1.9  cub  yt9s 

perfectly  lOOSe  t  or  about  1%  when  piled  up.  A  strong  cart  for  stone  hauling,  will  weigh 
about  %  ton  ;  or  1500  fts  ;  and  will  hold  stone  enough  for  a  perch  of  rubble  masonry  ;  or  say  1.2  pers 
of  the  rough  stone  in  piles.  The  average  weight  of  a  good  working  horse  is  about  %  a  ton. 

Mori  11  gives  the  following  results  from  careful  experiments  made  by 
him  for  the  French  Government.  The  draft  of  the  same  wheeled  vehicle  on  a  road,  may  in  practice 
be  considered  to  be, 

1st.    On  hard  turnpikes,  and  pavements;  in  proportion  to  tho 

loads  ;  Inversely  as  the  diams  of  the  wheels ;  and  nearly  independent  of  the  width  of  tire.  It  increases 
to  uncertain  extents  with  the  inequalities  of  the  road  ;  the  stiffness  (want  of  spring)  of  the  vehicle; 
and  the  speed;  (considerably  less  than  as  the  square  roots  of  the  last.) 

2d.  On  soft  roads,  the  tlraft  is  less  with  wide  tires  than 
with  narrower  ones;  and  for  farming  purposes  he  recommends  a  width  of 
4  ins.  With  speeds  from  a  walk  to  a  fast  trot,  the  draft  does  uot  vary  sensibly. 


CHORDS  TO  A  RADIUS  1, 


M. 

0° 

1° 

2° 

3° 

46 

5° 

6° 

7° 

8° 

9° 

1O° 

M. 

0' 

.0000 

.0175 

.03*9 

.0524 

.0698 

.0872 

.1047 

.1221 

.1395 

.1569 

.1743 

0' 

2 

.0006 

.0180 

.0355 

.0529 

.0704 

.0878 

.1053 

.1227 

.1401 

.1575 

.1749 

2 

4 

.0012 

.018!) 

.0361 

.0535 

.0710 

.0884 

.1058 

.1233 

.1407 

.1581 

.1755 

4 

6 

.0017 

.0192 

.0566 

.0541 

.0715 

.0890 

.1064 

.1238 

.1413 

.1587 

.1761 

6 

8 

.0023 

.0198 

.0372 

.0547 

.0721 

.0896 

.1070 

.1244 

.1418 

.1592 

.1766 

8 

10 

.0029 

.0204 

.0378 

.0553 

.0727 

.0901 

.1076 

.1250 

.1424 

.1598 

.1772 

10 

12 

.00:55 

.0209 

.0384 

.0558 

.0733 

.0907 

.1082 

.1256 

.1430 

.1604 

.1778 

12 

14 

.0041 

.0215 

.0390 

.0564 

.0739 

.0913 

.1087 

.1262 

.1436 

.1610 

.1784 

14 

1« 

.0047 

.0221 

.0396 

.0570 

.0745 

.0919 

.1093 

.1267 

.1442 

.1616 

.1789 

16 

18 

.0052 

.0227 

.0401 

.0576 

.0750 

.0925 

.1099 

.1273 

.1447 

.1621 

.1795 

18 

20 

.0058 

.0233 

.0407 

.0582 

.0756 

.0931 

.1105 

.1279 

.1453 

.1627 

.1801 

20 

22 

.0064 

.0239 

.0413 

.0588 

.0762 

.0936 

.1111 

.1285 

.1459 

.1633 

.1807 

22 

24 

.0070 

.0244 

.0419 

.0593 

.0768 

.0942 

.1116 

.1291 

.1465 

.1639 

.1813 

24 

2fi 

.0076 

.0250 

.0425 

.0599 

.0774 

.0948 

.1122 

.1296 

.1471 

.1645 

.1818 

26 

28 

.0081 

.0256 

.0430 

.0605 

.0779 

.0954 

.1128 

.1302 

.1476 

.1650 

.1824 

28 

30 

.0087 

.0262 

.0436 

.0611 

.0785 

.0960 

.1134 

.1308 

.1482 

.1656 

.1830 

30 

32 

.0093 

.0-268 

.0442 

.0617 

.0791 

.0965 

.1140 

.1314 

.1488 

.1662 

.1836 

32 

34 

.0099 

.0273 

.0448 

.0622 

.0797 

.0971 

.1145 

.1320 

.1494 

.1668 

.1842 

34 

36 

.0105 

.0279 

.0454 

.0628 

.0803 

.0977 

.1151 

.1325 

.1500 

.1674 

.1847 

36 

38 

.0111 

.0285 

.0460 

.0634 

.0808 

.0983 

.1157 

.1331 

.1505 

.1679 

.1853 

38 

40 

.0116 

.0291 

.0465 

.0640 

.0814 

.0989 

.1163 

.1337 

.1511 

.1685 

.1859 

40 

42 

.0122 

.0297 

.0471 

.0646 

.0820 

.0994 

.1169 

.1343 

.1517 

.1691 

.1865 

42 

44 

.0128 

.0303 

.0477 

.0651 

.0826 

.1000 

.1175 

.1349 

.1523 

.1697 

.1871 

44 

46 

.0134 

.0308 

.0483 

.0657 

.0832 

.1006 

.1180 

.1355 

.1529 

.1703 

.1876 

46 

48 

.0140 

.0314 

.0489 

.0663 

.0838 

.1012 

.1186 

.1360 

.1534 

.1708 

.1882 

48 

50 

.0145 

.0320 

.0494 

.0669 

.0843 

.1018 

.1192 

.1366 

.1540 

.1714 

.1888 

50 

52 

.0151 

.0326 

.0500 

.0675 

.0849 

.1023 

.1198 

.1372 

.1546 

.1720 

.1894 

52 

54 

.0157 

.0332 

.0506 

.0681 

.0855 

.1029 

.1204 

.1378 

.1552 

.1726 

.1900 

54 

56 

.0163 

.0337 

.0512 

.0686 

.0861 

.1035 

.1209 

.1384 

.1558 

.1732 

.1905 

56 

58 

.0169 

.0343 

.0518 

.0692 

.0867   .1041 

.1215 

.1389 

.1513 

.1737 

.1911 

58 

60 

.0175 

.0349 

.0524 

.0698 

.0872 

.1047 

.1221 

.1395 

.1569 

.1743 

.1917 

60 

TABLE   OP  CHORDS. 


609 


Table  of  0iorcls,  in  parts  of  a  ratl  1;  for  protracting— Continued. 


M. 

11° 

12° 

13° 

14° 

15° 

16° 

17° 

18° 

19° 

20° 

M. 

0' 
2 
4 
6 
8 
10 

.1917 
.1923 
.1928 
.1934 
.1940 
.1946 

.2091 
.2096 
.2102 
.2108 
.'2114 
.2119 

.'22G4 
.2270 
.2276 
.2281 
.2287 
.2293 

.2437 
.2443 
.2449 
.2455 
.2460 
.2466 

.2611 
.2616 
.2622 
.2628 
.2634 
.2639 

.2783 
.2789 
.2795 
.2801 
.2807 
.2812 

.2956 
.2962 
.2968 
.2973 
.2979 
.2985 

.3129 
.3134 
.3140 
.3146 
.3152 
.3157 

.3301 
.3307 
.3312 
.3318 
.3324 
.3330 

.3473 
.3479 
.3484 
.3490 
.3496 
.3502 

0' 
2 

4 
6 
8 
10 

12 
14 

16 

18 
20 

.1952 
.1957 
.1963 
.1969 
.1975 

.2125 
.2131 
.2137 
.2143 
.2148 

.2299 
.2305 

.2310 
.2316 

.2322 

.2472 
.2478 
.2484 
.2489 
.2495 

.2045 
.2651 
.2G57 
.2662 
.2668 

.2818 
.2824 
.2830 
.2835 
.2841 

.2991 
.2996 
.3002 
.3008 
.3014 

.3163 
.3169 
.3175 
.3180 
.3186 

.3335 
.3341 
.3347 
.3353 
.3358 

.3507 
.3513 
.3519 
.3525 
.3530 

12 
14 
16 
18 
20 

'M: 

26 
28 
30 

.1981 
.1986 
.1992 
.1998 
.2004 

.2154 
.2160 
.2166 

.2172 
.2177 

.2328 
.2333 
.2339 
.2345 
.2351 

.2501 
.2507 
.2512 
.2518 

.2524 

.2674 
.2680 
.2685 
.2691 

.'2697 

.2847 
.2853 
.2858 
.2864 
.2870 

.3019 
.3025 
.3031 
.3037 
.3042 

.3192 
.3198 
.3203 
.3209 
.3215 

.3364 
.3370 
.3376 
.3381 
.3387 

.3536 
.3542 
.3547 
.3553 
.3559 

22 
24 
26 
28 
30 

32 
34 
36 
38 
40 

.2010 
.2015 
.2021 
.2027 
.2033 

.2183 
.2189 
.2195 
.2200 
.2206 

.2357 
.2362 
.2368 

.2374 
.2380 

.2530 
.2536 
.2541 
.2547 
.2553 

.2703 
.2709 
.2714  . 
.2720 
.2726 

.2876 
.2881 
.2887 
.2893 
.2899 

.3048 
.3054 
.3060 
.3065 
.3071 

.3221 
.3226 
.3232 
.3238 
.3244 

.3393 
.3398 
.3404 
.3410 
.3416 

.3565 
.3570 
.3576 
.3582 
.3587 

32 
34 
36 
38 
40 

42 
44 
46 
48 
50 

.2038 
.2044 
.2050 
.2056 
.2062 

.2212 
.2218 
.2224 
.2229 
.2235 

.2385 
.2391 
.2397 
.2403 
.2409 

.2559 
.2564 
.2570 
.2576 

.2582 

.2732 
.2737 
.2743 
.2749 

.2755 

.2904 
.2910 
.2916 
.2922 
.2927 

.3077 
.3083 
.3088 
.3094 
.3100 

.3249 
.3255 
.3261 
.3267 
.3272 

.3421 
.3427 
.3433 
.3439 
.3444 

.3593 
.3599 
.3605 
.3610 
.3616 

42 
44 
46 
48 

50 

52 
54 
56 
58 
60 

.2067 
.2073 
.2079 
.2085 
.2091 

.2241 
.2247 
.2253 
.2258 
.2264 

.2414 
.2420 
.2426 
.2432 
.2437 

.2587 
.2593 
.2599 
.2605 
.2611 

.2760 
.2766 
.2772 
.2778 
.2783 

.2933 
.2939 
.2945 
.2950 
.2956 

.3106 
.3111 
.3117 
.3123 
.3129 

.3'278 
.3284 
.3289 
.3295 
.3301 

.3450 
.3456 
.3462 
.3467 
.3473 

.3622 
.3628 
.3633 
.3639 
.3645 

52 
54 
56 
58 
60 

M. 

21° 

22° 

23° 

24° 

25° 

26° 

27° 

28° 

29° 

30° 

M. 

0' 
2 
4 
6 
8 
10 

.3645 
.3650 
.3656 
.3662 
.3668 
.3673 

.3816 
.3822 
.3828 
.3833 
.3839 
.3845 

.3987 
.3993 
.3999 
.4004 
.4010 
.4016 

.4158 
.4164 
.4170 
.4175 
.4181 
.4187 

.4329 
.4334 
.4340 
.4346 
.4352 
.4357 

.4499 
.4505 
.4510 
.4516 

.4522 

.45'27 

.4669 
.4675 
.4680 
.4686 
.4692 
.4697 

.4838 
.4844 
.4850 
.4855 
.4861 
.4867 

.5008 
.5013 
.5019 
.5024 
.5030 
.5036 

.5176 
.5182 
.5188 
.5193 
.5199 
.5204 

0' 
2 

4 
6 
8 
10 

12 
14 
16 

18 
20 

.3679 
.3685 
.3690 
.3696 

.3702 

.3850 
.3856 
.3862 
.3868 
.3873 

.4022 
.4027 
.4033 
.4039 
.4044 

.4192 
.4198 
.4204 
.4209 
.4215 

.4363 
.4369 
.4374 
.4380 

.4386 

.4533 
.4539 
.4544 
.4550 
.4556 

.4703 
.4708 
.4714 

.4720 
.4725 

.4872 
.4878 
.4884 
.4889 
.4895 

.5041 
.5047 
.5053 
.5C58 
.5064 

.5210 
.5216 
.5221 
.5227 
.5233 

12 
14 
16 
18 
20 

22 
24 
26 
28 
30 

.3708 
.3713 
3719 
.3725 
.3730 

.3879 
.3885 
.3890 
.3896 
.3902 

.4050 
.4056 
.4061 
.4067 
.4073 

.4221 
.4226 
.4232 
.4238 
.4244 

.4391 
.4397 
.4403 
.4408 
.4414 

.4561 
.4567 
.4573 
.4578 

.4584 

.4731 
.4737 
.4742 
.4748 
.4754 

.4901 
.4906 
.4912 
.4917 
.4923 

.5070 
.5075 

.5081 
.5086 

.0092 

.5238 
.5244 
.5249 
.5255 
.5261 

22 
24 
26 

28 
30 

32 
34 
36 
88 
40 

.3736 
.3742 
.3748 
.3753 
.3759 

.3908 
.3913 
.3919 
.3925 
.3930 

.4079 
.4084 
.4090 
.4096 
.4101 

.4249 
.4255 
.4261 
.4266 
.4272 

.4420 

.4  H>5 
.44,",! 
.4437 
.4442 

.4590 

.1  •-)!».•> 
.4601 
.4607 
.4612 

.4759 
.4765 
.4771 
.4776 

.4782 

.4929 
.4934 
.4940 
.4946 

.4951 

.5098 
.5103 
.5109 
.5115 
.5120 

.5266 
.5272 
.5277 
.5283 
.5289 

32 
34 
36 
38 
40 

42 
44 
46 

48 
50 

.3765 
.3770 
.3776 

.3782 
.3788 

.3936 
.3942 
.3947 
.3953 
.3959 

.4107 
.4113 
.4118 
.4124 
.4130 

.4278 
.4283 
.4289 
.4295 
.4300 

.4448 

.4i:>» 

.4459 
-.4*65 
.4471 

.4618 
.4(i:?4 
.4629 
.4635 
.4641 

.4788 
.4793 
.47C9 
.4805 
.4810 

.4957 
<49flS 
.4968 
.4974 

.4979 

.51-26 
.5131 
.5137 
.5143 

.5148 

.5294 
.5300 
.5306 
.5311 
.5317 

42 
44 
46 

48 
50 

52 
54 

56 
58 
60 

.3793 
.3799 
.3805 
.3810 
.3816 

.3965 
.3970 
.3976 
.3982 
.3987 

.4135 
.4141 
.4147 
.4153 
.4158 

.4306 
.4312 
.4317 
.4323 
.4329 

.4476 
.4482 

.4488 
.4493 
.4499 

.4646 
.4652 
.4658 
.4663 
.4669 

.4816 
.4822 
.48-27 
.4833 
.4838 

.4985 
.4991 
.4996 
.5002 
.5008 

.5154 
.5160 
.5165 
.5171 
.5176 

.5322 
.53-28 
.5334 
.5339 
.5345 

52 
54 
56 
58 
60 

610 


TABLE  OF   CHORDS. 


Table  of  chords,  in  parts  of  a  rad  1;  for  protracting: — Continued 


M. 

31° 

32° 

33° 

34° 

35° 

36° 

37° 

38° 

39° 

40° 

M. 

0' 
2 
4 
6 
8 
10 

.5345 
.5350 
.5356 
.5362 
.5367 
.5373 

.5513 
.5518 
.5524 
.5530 
.5535 
.5541 

.5680 
.5686 
.5691 
.5697 
.5703 
.5708 

.5«47 
.5853 
.5859 
.5864 
.5870 
.5875 

.6014 
.6020 
.6025 
.60151 
.6036 
.6042 

.6180 
.6186 
.6191 
.6197 
.6202 
.6208 

.6:546 
.6352 
.6357 
.6363 
.6368 
.6374 

.6511 
.6517 
.6522 
.6528 
.6533 
.6539 

.6676 
.66S2 
.6687 
.6693 
.6698 
.6704 

.6840 
.6846 
.6851 
.6857 
.6862 
.6868 

0' 

2 
4 
6 
8 
10 

12 
14 
16 

18 
20 

.5378 
.5384 
.5390 
.5395 
.5401 

.5546 
.5552 
.5557 
.5563 
.5569 

.5714 
.5719 
.5725 
.5730 
.5736 

.5881 
.5886 
.5892 
.5897 
.5903 

.6047 
.6053 
.6058 
.6064 
.6070 

.6214 
.6219 
.6225 
.6230 
.6236 

.6379 
.6385 
.6390 
.6396 
.6401 

.6544 
.6550 
.6555 
.6561 
.6566 

.6709 
.6715 
.6720 
.6725 
.6731 

.6873 
.6879 

.6884 
.6890 
.6895 

12 
14 
16 
18 
20 

22 
24 
26 
28 
30 

.5406 
.5412 
.5418 
.5423 
.5429 

.5574 
.5580 
.5585 
.5591 
.5597 

.5742 
.5747 
.5753 
.5758 
.5764 

.5909 
.5914 
.5920 
.5925 
.5931 

.6075 
.6081 
.6086 
.6092 
.6097 

.6241 
.6247 
.6252 
.6258 
.6263 

.6407 
.6412 
.6418 
.6423 
.6429 

.6572 
.6577 
.6583 

.6588 
.6594 

.6736 
.6742 
.6747 
.6753 
.6758 

.6901 
.6906 
.6911 
.6917 
.6922 

22 
24 
26 
28 
30 

32 
34 
36 
38 
40 

.5434 
.5440 
.5*46 
.5451 
.5457 

.5602 
.5608 
.5613 
.5619 
.5625 

.5769 
.5775 
.5781 

.5786 
.5792 

.5936 
.5942 
.5947 
.5953 
.5959 

.6103 
.6108 
.6114 
.6119 
.6125 

.6269 
.6274 
.6280 
.6285 
.6291 

.6434 
.6440 
.6445 
.6451 
.6456 

.6599 
.6605 
6610 
.6616 
.6621 

.6764 
.6769 
.6775 
.6780 
.6786 

.6928 
.6933 
.6939 
.6944 
.6950 

32 
34 
36 
38 
40 

42 
44 
46 

48 
50 

.5462 
.5468 
.5174 
.5479 
.5485 

.5630 
.5636 
.5641 
.5647 
.5652 

.5797 
.5803 
.5808 
.5814 
.5820 

.5964 
.5970 
.5975 
.5981 
.5986 

.6130 
.6136 
.6142 
.6147 
.6153 

.6296 
.630*2 
.6307 
.6313 
.6318 

.6462 
.6467 
.6473 
.6478 
.6484 

.6627 
.6632 
.6638 
.6643 
.6649 

.6791 
.6797 
.6802 
.6808 
.6813 

.6955 
.6961 
.6966 
.6971 
.6977 

42 
44 
46 
48 
50 

52 
54 

56 
58 
60 

.5490 
.5496 
.5502 
.5507 
.5513 

.5658 
.5664 
.5669 
.5675 
.5680 

.5825 
.5831 

.5836 

.5842 
.5847 

.5992 
.5997 
.6003 
.6009 
.6014 

.6153 
.6164 
.6169 
.6175 
.6180 

.6324 
.6330 
.6335 
.6341 
.6346 

.6489 
.6495 
.6500 
.6506 
.6511 

.6654 
.6660 
.6665 
.6671 
.6676 

.6819 
.6824 
.6829 
.6835 
.6840 

.6982 
.6988 
.6993 
.6999 
.7004 

52 
54 
56 
58 
60 

M. 

41° 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

50° 

M. 

0 
2 
4 
6 
8 
10 

.7004 
.7010 
.7015 
.7020 
.7026 
.7031 

.7167 
.7173 
.7178 

.7184 
.7189 
.7195 

.7330 
.7335 
.7341 
.7346 
.7352 
.7357 

.7492 
.7498 
.7503 
.7508 
.7514 
.7519 

.7654 
.7659 
.7664 
.7670 
.7675 
.7681 

.7815 

.7820 
.7825 
.7831 
.7836 

.7841 

.7975 
.7980 
.7986 
.7991 
.7996 
.8002 

.8135 
.8140 
.8145 
.8151 
.8156 
.8161 

.8294 
.8299 
.8304 
.8310 
.8315 
.8320 

.8452 
.8458 
.8463 
.8468 
.8473 
.8479 

0' 
2 
4 

6 

8 
10 

12 
14 
16 

18 
20 

.7037 
.7042 
.7048 
.7053 
.7059 

.7200 
.7205 
.7211 
.7216 
.7222 

.7362 
.7368 
.7373 
.7379 
.7384 

.7524 
.7530 
.7535 
.7541 
.7546 

.7686 
.7691 
.7697 
.7702 
.7707 

.7847 
.7852 
.7857 
.7863 
.7868 

.8007 
.8012 
.8018 
.8023 
.8028 

.8167 

.8172 
.8177 
.8183 
.8188 

.8326 
.8331 
.8336 
.8341 
.8347 

.8484 
.8489 
.8495 
.8500 
.8505 

12 
14 
16 
18 
20 

22 
24 
26 
28 
30 

.7064 
.7069 
.7075 
.7080 
.7086 

.7227 
.7232 
.7238 
.7243 
.7249 

.7390 
.7395 
.7400 
.7406 
.7411 

.7551 
.7557 
.7562 
.7568 
.7573 

.7713 
.7718 
.7723 
.7729 
.7734 

.7873 
.7879 
.7884 
.7890 
.7895 

.8034 
.8039 
.8044 
.8050 
.8055 

.8193 
.8198 

.8204 
.8209 
.8214 

.8352 
.8357 
.8363 
.8368 
.8373 

.8510 
.8516 
.8521 
.8526 
.8531 

22 
24 
26 
28 
30 

32 
34 
86 

38 
40 

.7091 
.7097 
.7102 
.7108 
.7113 

.7254 
.7260 
.7265 

.7270 
.7276 

.7417 
.7422 
.7427 
.7433 
.7438 

.7578 
.7584 
.7589 
.7595 
.7600 

.7740 
.7745 
.7750 
.7756 
.7761 

.7900 
.7906 
.7911 
.7916 
.7922 

.8060 
.8066 
.8071 
.8076 

.8082 

.8220 
.8225 
.8230 
.8236 
.8241 

.8378 
.8384 
.8389 
.8394 
.8400 

.8537 
.8542 
.8547 
.8552 
.8558 

32 
34 
36 
3& 
40 

42 
44 
46 

48 
50 

.7118 
.7124 
.7129 
.7135 
.7140 

.7281 
.7287 
.7292 
.7298 
.7303 

.7443 

.7449 
.7454 
.7460 
.7465 

.7605 
.7611 
.7616 
.7621 
.7627 

.7766 
.7772 
.7777 

.7782 
.7788 

.7927 
.7932 
.7938 
.7943 
.7948 

.8087 
.8092 
.8098 
.8103 

.8108 

.8246 
.8251 
.8257 
.8262 
.8267 

.8405 
.8410 
.8415 
.8421 

.8426 

.8563 
.8568 
.8573 
.8579 
.8584 

42 
44 

46 

48 
50 

52 
54 
56 

58 
60 

.7146 
.7151 
.7156 
.7162 
.7167 

.7308 
.7314 
.7319 
.7325 
.7330 

.7471 
.7476 
.7481 
.7487 
.7492 

.7632 
.7638 
.7643 
.7648 
.7654 

.7793 
.7799 
.7804 
.7809 
.7815 

.7954 
.7959 
.7964 
.7970 
.7975 

.8113 
.8119 
.8124 
.8129 
.8135 

.8273 
.8278 
.8283 
.8289 
.8294 

.8431 
.8437 
.8442 
.8447 
.8452 

.8589 
.8594 
.8600 
.8605 
.8610 

52 
54 
56 
58 
60 

TABLE   OF   CHORDS. 


611 


Table  j»f  chords,  in  parts  of  a  rad  1 ;  for  protracting1  — 


Continued. 


M. 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

59° 

60° 

M. 

0' 

1 
4 
6 
8 
10 

.8610 

.8615 
.8621 
.8626 
.8631 
.8636 

.8767 
.8773 
.8778 
.8783 
.8788 
.879* 

.8924 
.8929 
.8934 
.8940 
.8945 
.8950 

.9080 
.9085 

.9oyo 

.9095 
.9101 
.9106 

.9235 
.9240 
.9245 
.9250 
.9256 
.9261 

.9389 
.9395 
.9400 
.9405 
.9410 
.9415 

.9543 

.9548 
.9553 
.9559 
.9564 
.9569 

.9696 
.9701 

.9706 
.9711 
.9717 
.9722 

.9848 
.9854 
.9859 
.9864 
.9869 
.9874 

1.0000 
1.1005 
1.0010 
1.0015 
1.0020 
1.00* 

0' 
2 
4 
6 
8 
10 

12 
14 
16 
18 
20 

.8642 
.8647 
.8652 
.8657 
.8663 

8799 
.8804 
.8809 
.8814 
.8820 

.8955 
.8960 
.8966 
.8971 
.8976 

.9111 
.9116 
.9121 
.9126 
.9132 

.9266 
.9271 
.9276 
.9281 
.9287 

.9420 
.9425 
.9430 
.9436 
.9441 

.9574 
.9579 
.9584 
.9589 
.9594 

.9727 
.9732 
.9737 
.9742 
.9747 

.9879 
.9884 
.9889 
.9894 
.9899 

1.0030 
1.0035 
1.0040 
1.0045 
1.0050 

12 
14 
16 
18 
20 

22 
24 
26 

28 
30 

.8668 
.8673 
.8678 
.8684 
.8689 

.8825 
.8830 
.8835 
.8841 
.8846 

.8981 
.8986 
.8992 
.8S97 
.9002 

.9137 
.9142 
.9147 
.9152 
.9157 

.9292 
.9297 
.9302 
.9307 
.9312 

.9446 
.9451 
.9456 
.9461 
.9466 

.9599 
.9604 
.9610 
.9615 
.9620 

.9752 
.9757 
.9762 
.9767 
.9772 

.9904 
.9909 
.9914 
.9919 
.9924 

1.0055 
1.0060 
1.0065 
1.0070 
1.0075 

22 
24 
26 

28 
30 

32 
34 
36 

38 
40 

.8694 
.8699 
.8705 
.8710 
.8715 

.8851 
.8856 
.8861 
.8867 
.8872 

.9007 
.9012 
.9018 
.9023 
.9028 

.9163 
.9168 
.9173 
.9178 
.9183 

.9317 
.9323 

.9328 
.9333 

.9388 

.9472 
.9477 
.9482 
.9487 
.9492 

.9625 
.9630 
.9635 
.9640 
.9645 

.9778 
.9783 
.9788 
.9793 
.9798 

.9929 
.9934 
.9939 
.9945 
.9950 

1.0080 
1.0086 
1.0091 
1.0096 
1.0101 

32 
34 
36 
38 
40 

42 
44 
46 
48 
50 

.8720 
.8726 
.8731 
.8736 
.8741 

.8877 
.8882 
.8887 
.8893 
-.8898 

.9033 
.9038 
.9044 
.9049 
.9054 

.9188 
.9194 
.9199 
.9204 
.9209 

.9343 
.9348 
.9353 
.9359 
.9364 

.9497 

.9502 
.9507 
.9512 
.9518 

.9650 
.9655 
.9661 
.9666 
.9671 

.9803 
.9808 
.9813 
.9818 
.9823 

.9955 
.9960 
.9965 
.9970 
.9975 

1.0106 
1.0111 
1.0116 
1.0121 
1.0126 

42 
44 
46 
48 
50 

52 
54 
56 
58 
60 

.8747 
.8752 
.8757 
.8762 
.8767 

.8903 
.8908 
.8914 
.8919 
.8924 

.9059 
.9064 
.9069 
.9075 
.9080 

.9214 
.9219 
.9225 
.9230 
.9235 

.9369 
.9374 
.9379 
.9384 
.9389 

.9523 
.9528 
.9533 
.9538 
.9543 

.9676 
.9681 
.9686 
.9691 
.9696 

.9828 
.9833 
.9838 
.9843 
.9848 

.9980 
.9985 
.9990 
.9995 
1.0000 

1.0131 
1.0136 
1.0141 
1.0146 
1.0151 

52 
54 
56 
58 
60 

M. 

61° 

62° 

63° 

64° 

65° 

66° 

67° 

68° 

69° 

7O° 

M. 

0 
2 
4 
6 
8 
10 

1.0151 
1.0156 
1.0161 
1.0166 
1.0171 
1.0176 

1  .0301 

i.o;;o6 

1.0311 
1.0316 
1.0321 
1.0326 

1.0450 
1.0455 
1.0460 
1.0465 
1.0470 
1.0475 

1.0598 
1.0603 
1  .0608 
1.0C13 
1.0618 
1.0623 

1.0746 
1.0751 
1.0756 
1.0761 
1.0766 
1.0771 

1.0893 
1.0898 
1.0903 
1.0907 
1.0912 
1.0917 

1.1039 
1.1044 
1.1048 
1.1053 
1.1058 
1.1063 

1.1184 
1.1189 
1.1194 
1.1198 
1.1203 
1.1208 

1.1328 
1.1333 
1.1338 
1.1342 
1.1347 
1.1352 

.1472 
.1476 
.1481 
.1486 
.1491 
.1495 

0' 
2 
4 
6 
8 
10 

12 
14 
16 

18 
20 

1.0181 
1.0186 
1.0191 
1.0196 
1.0201 

1.0331 
1.0336 
1.0341 
1.0346 
1.0351 

1.0480 
1.0485 
1.0490 
1.0495 
1.0500 

1.0628 
1.0633 
1.0638 
1.0643 
10648 

1.0775 
1.0780 
1.0785 
1.0790 
1.0795 

.0922 
.0927 
.0932 
.0937 
.0942 

1.1068 
1.1073 
1.1078 
1.1082 
1.1087 

1.1213 
1.1218 
1.1222 
1.1227 
1.1232 

1.1357 
1.1362 
1.1366 
1.1371 
1.1376 

.1500 
.1505 
.1510 
.1514 
.1519 

12 
14 
16 

18 
20 

22 
24 
26 
28 
30 

1.0206 
1.0211 
1.0216 
1  .0221 
1.0226 

1.0356 
1  .0361 
1.0366 
1.0370 
1.0375 

1.05t4 
1.0509 
1.0514 
1.0519 
1.0524 

1.0653 
1.0658 
1.0662 
1.0667 
1.0672 

1.0800 
1.0805 
1.0810 
1.0815 
1.0820 

.0946 
.0951 
.0956 
.0961 
.0966 

1.  092 
1.  097 
1.  102 
1.  107 
1.  Ill 

1.1237 
1.1242 
1.1246 
1.1251 
1.1256 

1.1381 
1.1386 
1.1390 
1.1395 
1.1400 

.1524 
.1529 
.1533 
.1538 
.1543 

22 
24 
26 
28 
30 

32 
34 
36 

38 
40 

1.0231 
1.0236 
1.0241 
1.0246 
1.0251 

1.0380 
1.0385 
1.0390 
1.0395 
1.0400 

1.0529 
1.0534 
1.0539 
1.0544 
1.0549 

1.0677 
1.0682 
1.0687 
1.0692 
1.0697 

1.0824 
1.0829 
1.083* 
1.0839 
1.0844 

1.0971 
1.0976 
1.0980 
1.0985 
1.0990 

1.1116 
1.1121 
1.1126 
1.1131 
1.1136 

1.1261 

1.1266 
1.1271 
1.1275 
1.1280 

1.1405 
1.1409 
1.1414 
1.1419 
1.1424 

.1548 
.1552 
.1557 
.1562 
.1567 

32 
34 
36 
38 
40 

42 
44 
46 
48 
50 

1.0256 
1.0261 
1.0266 
1.0271 
1.0276 

1.0405 
1.0410 
1.0415 
1  .0420 
1.0425 

1.0554 
1  .0559 
1  0564 
.0569 
.0574 

1.0702 
1  .0707 
1.0712 
1.0717 
1  .0721 

1.0849 
1.0854 
1  .0859 
1.0863 
1  .0868 

1.0995 
1.1000 
1.1005 
.1010 
.1014 

1.1140 
1.  145 
1.  150 
1.  155 
1.  160 

1.1285 
1.1290 
1.1295 
1.1299 
1.1304 

1.1429 
1.1433 
1.1438 
1.1443 
1.1448 

.1571 
.1576 
.1581 
.1586 
.1590 

42 
44 
46 

48 
50 

52 
54 
56 
58 
60 

1  .0281 
1.0286 
1.0291 
1.0296 
1.0301 

1  .0430 
1.0435 
1.0440 
1.0445 
1.0450 

.0579 
.0584 
.0589 
.0593 
.0598 

1.0726 
1.0731 
1.0736 
1.0741 
1.0746 

1.0873 
1.0878 
1.0883 
1.0888 
1.0893 

.1019 
.1024 
.1029 
.1034 
.1039 

1.  165 
1.  169 
1.  174 
1.  179 
1.  184 

1.1309 
1.1314 
1.1319 
1.1323 
1.1328 

1.1452 
1.1457 
1.1462 
1.1467 
1.1472 

.1595 
.1600 
.1605 
.1609 
1.1614 

52 
54 
56 
58 
60 

612 


TABLE   OF   CHORDS. 


Table  of  Chords,  in  parts  of  a  rad  1 ;  for  protracting  — Continued. 


M. 

71° 

72° 

73° 

74° 

75° 

76° 

77° 

78° 

79° 

80° 

M. 

0 

2 
4 
6 

8 
10 

12 
14 
16 
18 
20 

1.161* 
1.1619 
1.1324 
1.1628 
1.1333 
1.1638 

1.16*2 
1.16A7 
1.1652 
1.1657 
1.1661 

1.1756 
1.1760 
1.1765 
1.1770 
1.1775 
1.1779 

1.1784 
1.1789 
1.1793 
1.1798 
1.1808 

1.1896 
1.1901 
1.1906 
1.1910 
1.1915 
1.1920 

1.1924 
1.1929 
1.1934 
1.1938 
1.1943 

1.2036 
1.2041 
1.2046 
1.2050 
1.2055 
1.2060 

1.2064 
1.2069 
1.2073 
1.2078 
1.2083 

1.2175 
1.2180 
1.2184 
1.2189 
1.2194 
1.2198 

1.2313 
.2318 
.2322 
.2327 
.2332 
.2336 

.2341 
.2345 
.2350 
.2354 
.2359 

1.2450 
1.2455 
1.2459 
1.2464 
1.2468 
1.2473 

1.2586 
1.2591 
1.2595 
1.2600 
1.2604 
1.2609 

1.2722 
1.2726 
1.2731 
1.2735 
1.2740 
1.2744 

1.2748 
1.2753 
1.2757 
1.2762 
1.2766 

1.2856 
1.2860 
1.2865 

1.2869 
1.2874 
1.2878 

1.2882 
1.2887 
1.2891 
1.2896 
1.2900 

0' 
2 
4 
6 
8 
10 

12 
14 
16 

18 
20 

1.2203 
1.2208 
1.2212 
1.2217 
1.2221 

1.2478 
1.2482 
1.2487 
1.2491 
1.2496 

1.2614 
1.2618 
1.2623 
1.2627 
1.2632 

22 
24 
26 
28 
30 

1.1668 
1.1671 
1.1676 
1.1680 
1.1685 

1.1690 
1.1694 
1.1699 
1.1704 
1.1709 

1.1807 
1.1812 
1.1817 
1-1821 

1.1826 

1.1831 
1.1836 
1.1840 
1.1845 
1.1850 

1.1948 
1.1952 
1.1957 
1  1962 
1.1966 

1.1971 
1.1976 
1.1980 
1.1985 
1.1990 

1.2087 
1.2092 
1.2097 
1.2101 
1.2106 

1.2226 
1.2231 
1.2235 
1.2240 
1.2244 

.2364 
.2368 
.2373 
.2377 

.2382 

1.2500 
1.2505 
1.2509 
1.2514 
1.2518 

1.2636 
1.2641 
1.2645 
1.2650 
1.2654 

1.2771 
1.2775 
1  2780 
1.2784 
1.2789 

1.2905 
1.2909 

1.2914 
1.2918 
1.2922 

22 
24 
26 

28 
30 

32 
34 
36 
38 
40 

32 
34 
36 
38 
40 

42 
44 
46 
48 
50 

52 
54 
56 

58 
60 

1.2111 
1.2115 
1.2120 
1.2124 
1.2129 

1.2249 
1.2254 
1.2258 
1.2263 
1.2267 

.2386 
.2391 
.2396 
.2400 

.2405 

1.2523 
1.2528 
1.2532 
1.2537 
1.2541 

1.2659 
1.2663 
1.2668 
1.2672 
1.2677 

1.2793 
1.2798 
1.2802 
1.2807 
1.2811 

1.2927 
1.2931 
1.2936 
1.2940 
1.2945 

1.1713 
1.1718 
1.1723 
1.1727 
1.1732 

1.1854 
1.1859 
1.1864 
1.1868 
1.1873 

1.1994 
1.1999 
1.2004 
1.2008 
1.2013 

1.2134 
1.2138 
1.2143 
1.2148 
1.2152 

1.2272 
1.2277 
3.2281 
1.2286 
1.2290 

1.2295 
1.2299 
1.2304 
1.2309 
1.2313 

.2409 
.2414 
.2418 
.2423 
.2428 

.2432 
.2437 
.2441 
.2446 
.2450 

1.2546 
1.2550 
1.2555 
1.2559 
1.2564 

1.2568 
1.2573 

1.2577 
1.2582 
1.2586 

1.2681 
1.2686 
1.2690 
1.2695 
1.2699 

1.2704 
1.2-08 
1.2713 
1.2717 

1.2722 

1.2816 
1.2820 
1.2825 
1.2829 
1.2833 

1.2838 
1.2842 
1.2847 
1.2851 
1.2856 

1.2949 
1.2954 
1.2958 
1.2962 
1.2967 

1.2971 
1.2976 
1.2980 
1.2985 
1.2989 

42 
4* 
46 

48 
50 

52 
54 
56 
58 

60 

1.1737 
1.1742 
1.1746 
1.1751 
1.1758 

1.1878 
1.1882 

i.issr 

1.1892 
1.1896 

1.2018 
1.2022 
1.2027 
1.2032 
1.2036 

1.2157 
1.2161 
1.2166 
1.2171 
1.2175 

M. 

81° 

82°  1    83° 

84° 

85° 

86°  |    87° 

88° 

89°  |  M. 

0' 
2 
4 
6 
8 
10 

12 
14 

16 
18 

20 

22 
24 
26 
28 
30 

32 
34 
36 
38 
40 

42 
44 
46 
48 
SO 

la" 

54 
56 
58 
60 

1.2989 
1.2993 
1.2998 
1.3002 
1.3007 
1.3011 

1.3121 
1.3126 
1.3130 
1.3134 
1.3139 
1.3143 

1.3252 
1.3-257 
1.3261 
1.3265 
1.3270 
1.3274 

1.3383 
1.3387 
1.3391 
1-3396 
1.3400 
1.3404 

1.3409 
1.3413 
1.3417 
1.8421 

1.3426 

1.3430 
1.8434 

1.3439 
1.3443 
1.3447 

1.3512 
1.3516 
]  .3520 
1.3525 
1.3529 
1.3533 

1.3640 
1.3644 
1.3648 
1.3653 
1.3657 
1.3661 

1.3767 
1.3771 
1.3776 
1.3780 
1.3784 
1.3788 

1.3792 
1.3797 
1.3301 
1.3805 
1.3809 

1.3893 
1.3897 
1.3902 
1.3906 
1.3910 
1.3914 

1.4018 
1.4022 
1.4026 
1.4031 
1  4035 
1.4039 

Ov 
2 
4 
6 
8 
10 

12 
14 
16 
18 

20 

22 
24 
•26 
28 
30 

32 
34 
36 
38 
40 

42 
44 
46 
43 
50 

~52~ 
54 
56 
58 
60 

1.3015 
1.3020 
1.3024 
1.3029 
1.3033 

1.3038 
1.3012 
1.3016 
1.3051 
1.3055 

1.3147 
1.3152 
1.3156 
1.3161 
1.3165 

1.3279 
1.3283 
1.3287 
1.3292 
1.3296 

1.3538 
1.35f2 
1.3546 
1.3550 
1.3555 

1.3559 
1.3563 
1.3567 
1.3572 
1.3576 

1.3665 
1.3670 
1.3674 
1.3078 
1.3682 

1.3687 
1.3691 
1.3695 
1  .3(599 
1.3704 

1.3918 
1.3922 
1.3927 
1.3931 
1.3935 

1  .4043 
1.4047 
1.4051 
1.4055 
1.4060 

1.3169 
1.3174 
1.3178 

1.3183 
1.3187 

1.3300 
1.3305 
1.3309 
1.3313 
1.3318 

1.3813 
1.3818 
1.3822 
1.3H26 
1.3830 

1.3834 
1.3839 
1.3843 
1.3847 
1.3851 

1.3939 
1.3943 
1.3947 
1.3952 
1.3956 

1.4064 
1.4068 
1.4072 
1.4076 
1  .4080 

1.3060 
1.3064 
1.3068 
1.3073 

1.3077 

1.3191 
1.3196 
1.3200 
1.3204 
1.3209 

1.3322 
1.3326 
1.3331 
1.3335 
1.3339 

1.3152 
1.3456 
1.3460 
1.3465 
1.3469 

1.3580 
1.35S5 
1.3589 
113593 
1.3597 

t.3708 
1.3712 
1.3716 
1  .3721 
1.3725 

1.3960 
1.3964 
1.3968 
1.3972 
1.3977 

1.4084 
1.4089 
1.4093 
1.4097 
1.4101 

1.3086 
1.3090 
1.3035 
1.3099 

1.3218 
1.3222 
1.3226 
1.3231 

1.3348 
1.3352 
1.3357 
1.3361 

1.3365 
1.3370 
1.3374 
1.3378 
1.3383 

1.3477 
1.3482 
1.3486 
1.3490 

1.3495 
1.3499 
1.3503 
1.3508 
1.3512 

1.3606 
1.3610 
1.3614 
1.3619 

1.3623 
1.3627 
1.3631 
1.3636 
1.3640 

1.3729 
1.3733 
1  .3738 
1.3742 
1.3746 

1.3750 
1.3764 

1.3759 
1.3763 
1.3767 

1.3855 
1.3860 
1.3864 
1  .3868 
1.3872 

1  .3876 
]  .3881 
1.3885 
1.3889 
1.3893 

1.3981 
1.3985 
1.3989 
1.3993 
1.3997 

1.4105 
1.4109 
1.4113 
1.4117 
1.4122 

1.3104 
1.3103 
1.3112 
1.3117 
1.3121 

1  .3235 
1  .3239 
1.3244 
1.3248 
1.3252 

1.4002 
1.4006 
1.4010 
1  .4014 
1.4018 

1.4126 
1.4130 
1.4134 
1.4138 
1.4142 

BLE   OF   LOGARITHMS. 


613 


Mims  of  Numbers,  from  O  to  1OOO.* 


No. 

A  - 

2 

3 

4 

5 

6 

7 

8 

0 

Prop. 

0 

0 

00000 

30103 

47712  60206 

69897 

77815 

84510 

90309 

95424 

10 

00000 

00432 

00860 

01283  01703 

02118 

02530 

02938 

03342 

03742 

415 

11 

04139 

04532 

04921 

05307 

05690 

06069 

06445 

06818 

07188 

07554 

379 

12 

07918 

08278 

08636 

08990 

09342 

09691 

10037 

10380 

10721 

11059 

349 

13 

11394 

11727 

12057 

12385 

12710 

13033 

13353 

13672 

13987 

14301 

323 

14 

14613 

14921 

15228 

15533 

15836 

16136 

16435 

16731 

17026 

17318 

300 

15 

17609 

17897 

18184 

18469 

18752 

19033 

19312 

19590 

19865 

20139 

281 

16 

20412 

20682 

20951 

21218 

21484 

21748 

22010 

22271 

22530 

22788 

264 

17 

23045 

23299 

23552 

23804 

24054 

24303 

24551 

24797 

25042 

25285 

249 

18 

25527 

25767 

26007 

26245 

26481 

26717 

26951 

27184 

27415 

27646 

236 

19 

27875 

28103 

28330 

28555 

28780 

29003 

29225 

29446 

29666 

29885 

223 

20 

30103 

30319 

30535 

30749 

30963 

31175 

31386 

31597 

31806 

32014 

212 

21 

32222 

32428 

32633 

32838 

33041 

33243 

33445 

33646 

33845 

34044 

202 

22 

34242 

34439 

34635 

34830 

35024 

35218 

35410 

35602 

35793 

35983 

194 

23 

36173 

36361 

36548 

36735 

36921 

37106 

37291 

37474 

37657 

37839 

185 

24 

38021 

38201 

38381 

38560 

38739 

38916 

39093 

39269 

39445 

39619 

17T 

25 

39794 

39367 

40140 

40312 

40483 

40654 

40824 

40993 

41162 

41330 

171 

26 

41497 

41664 

41830 

41995 

42160 

42324 

4248S 

42651 

42813 

42975 

164 

27 

43136 

43206 

43156 

43616 

43775 

43933 

44090 

44248 

44404 

44560 

158 

28 

44716 

44870 

45024 

45178 

45331 

45484 

45636 

45788 

45939 

46089 

153 

29 

46240 

46389 

46538 

46686 

46834 

46982 

47129 

47275 

47421 

47567 

148 

30 

47712 

47856 

48000 

48144 

48287 

48430 

48572 

48713 

48855 

48995 

143 

31 

49136 

49276 

49415 

49554 

49693 

49831 

49968 

50105 

50242 

50379 

138 

32 

50515 

50650 

50785 

50920 

51054 

51188 

51321 

51454 

51587 

51719 

134 

33 

51851 

51982 

52113 

52244 

52374 

52504 

52633 

52703 

52891 

53020 

130 

34 

53148 

53275 

53402 

53529 

53655 

53781 

53907 

54033 

54157 

54282 

126 

35 

54407 

54530 

54654 

54777 

54900 

55022 

55145 

55266 

55388 

55509 

122 

36 

55630 

55750  55870 

55990 

56110 

56229 

56348 

56466 

56584 

56702 

119 

37 

56S20 

569371  57054 

57170 

57287 

57403 

57518 

57634 

57749 

57863 

116 

38 

57978 

58092 

58206 

58319 

58433 

58546 

58658 

58771 

58883 

58995 

113 

39 

59106 

59217 

59328 

59439 

59549 

59659 

59769 

59879 

59988 

60097 

110 

40 

60206 

60314 

60422 

60530 

60638 

60745 

60852 

60959 

61066 

61172 

107 

41 

61278' 

61384 

61489 

61595 

61700 

61804 

61909 

62013 

62117 

62221 

104 

42 

62325 

62428 

62531 

62634 

62736 

62838 

H2941 

63042 

63144 

63245 

102 

43 

63347 

63447 

63548 

63648 

63749 

63S48 

63948 

64048 

64147 

64246 

99 

44 

64345 

64443  64542 

64640 

64738 

64836 

64933 

65030 

65127 

65224 

98 

45 

65321 

65417 

65513 

65609 

65705 

65801 

65896 

65991 

660*6 

66181 

96 

46 

66276 

66370 

66464 

66558 

666ol 

06745 

66838 

66931 

67024 

67117 

94 

47 

67210 

67302 

67394 

67486 

67577 

67669 

67760 

67851 

67942 

68033 

92 

48 

68124 

68214 

68304 

68394 

68484 

68574 

6S663 

68752 

68842 

68930 

90 

49 

69020 

69108 

69196 

69284 

69372 

69460 

69548 

69635 

69722 

69810 

88 

50 

69897 

69983 

70070 

70156 

70243 

70329 

70415 

70500 

70586 

70671 

86 

51 

70757 

70842 

70927 

71011 

71096 

71180 

71265 

71349 

71433 

71516 

84 

52 

71600 

71683 

71767 

71850 

71933 

72015 

72098 

72181 

72263 

72345 

82 

53 

72428 

72509 

72591 

72672 

72754 

72835 

72916 

72997 

73078 

73158 

81 

54 

73239 

73319 

73399 

73480 

73559 

73639 

73719 

73798 

73878 

73957 

80 

55 

74036 

74115 

74193 

74272 

74351 

74429 

74507 

74585 

74663 

74741 

78 

56 

74818 

74896 

74973 

75050 

75127 

75204 

752S1 

75358 

75434 

75511 

77 

57 

75587 

75663 

75739 

75815  J75891 

75966 

76042 

76117 

76192 

76267 

75 

58 

76342 

76417 

76492 

76566 

76641 

76715 

76789 

76863 

76937 

77011 

74 

59 

77085 

77158 

77232 

77305 

77378 

77451 

77524 

77597 

77670 

77742 

73 

60 

77815 

77887 

77959 

78031 

78103 

78175 

78247 

78318 

78390 

78461 

72 

61 

78533 

78604 

78675 

78746 

78816 

78887 

78958 

79028 

79098 

79169 

71 

62 

79239 

79309 

79379 

79448 

79518 

79588 

79657 

79726 

79796 

79865 

70 

63 

79934 

80002 

80071 

80140 

80208 

80277 

80345 

80413 

80482 

80550 

69 

64 

80618 

80685 

80753 

80821 

80888 

80956 

81023 

81090 

81157 

81224 

68 

65 

81291 

81358 

81424 

81491 

81557 

81624 

81690 

81756 

81822 

81888 

67 

*Each  log  is  supposed  to  have  the  decimal  sign  .  before  it. 


614 


TABLE   OF   LOGARITHMS. 


Logarithms  of  Numbers,  from  O  to  10OO*— (Continued.) 


So. 

f 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. 

66 

81954 

82020 

82085 

82151 

82216 

82282 

82347 

82412 

82477 

82542 

66 

67 

82607 

82672 

82736 

82801 

82866 

82930 

82994 

83058 

83123 

83187 

65 

68 

83250 

83314 

8337S 

83442 

83505 

83569 

83632 

83695 

83758 

83821 

64 

69 

83884 

83947 

84010 

84073 

84136 

84198 

84260 

84323 

84385 

84447 

63 

70 

84509 

84571 

84633 

84695 

84757 

84818 

84880 

84941 

85003 

850ti4 

62 

71 

85125 

85187 

85248 

85309 

85369 

85430 

85491 

85551 

85612 

85672 

61 

72 

85733 

85793 

85853 

85913  1  8597  3 

86033 

86093 

86153 

86213 

86272 

60 

73 

86332 

86391 

86451 

86510 

86569 

86628 

86687 

86746 

86805 

86864 

59 

74 

86923 

86981 

87040 

87098 

87157 

87215 

87273 

87332 

87390 

87448 

58 

75 

87506 

87564 

87621 

87679 

87737 

87794 

87852 

87909 

87966 

88024 

57 

76 

88081 

88138 

88195 

88252 

88309 

88366 

88422 

88479 

88536 

88592 

56 

77 

88649 

88705 

88761 

88818 

88874 

88930 

88986 

89042 

89098 

89153 

56 

78 

89209 

89265 

89320 

89376 

89431 

89487 

89542 

89597 

89652 

89707 

55 

79 

89762 

89817 

89872 

89927 

89982 

90036 

90091 

90145 

90200 

90254 

54 

80 

90309 

90363 

90417 

90471 

90525 

90579 

90633 

90687 

90741 

90794 

54 

81 

90848 

90902 

90955 

91009 

91062 

91115 

91169 

91222 

91275 

91328 

53 

82 

91381 

91434 

91487 

91540 

91592 

91645 

91698 

91750 

91803 

91855 

53 

83 

91907 

91960 

92012 

92064 

92116 

92168 

92220 

92272 

92324 

92376   52 

84 

92427 

92479 

92531 

92582 

92634 

92685 

92737 

92788 

92839 

92890 

51 

85 

92941 

92993 

93044 

93095 

93146 

93196 

93247 

93298 

93348 

93399 

51 

86 

93449 

93500 

93550 

93601 

93651 

93701 

93751 

93802 

93852 

93902 

50 

87 

93951 

94001 

94051 

94101 

94151 

94200 

94250 

94300 

94349 

94398 

49 

88 

94448 

94497 

94546 

94596 

94d45 

94694 

94743 

94792 

94841 

94890 

49 

89 

94939 

94987 

95036 

95085 

95133 

95182 

95230 

95279 

95327 

95376 

48 

90 

95424 

95472 

95520 

95568 

95616 

95664 

95712 

95760 

9580S 

95856 

48 

91 

95904 

95951 

95999 

96047 

96094 

96142 

96189 

96236 

96284 

96331 

48 

92 

96378 

96426 

96473 

96520 

96507 

96614 

96661 

96708 

96754 

96801 

47 

93 

96848 

96895 

96941 

96988 

97034 

97081 

97127 

97174 

97220 

97266 

47 

94 

97312 

97359 

97405 

97451 

97497 

97543 

97589 

97635 

97680 

97726 

46 

95 

97772 

97818 

97863 

97909 

97954 

98000 

98045 

98091 

98136 

98181 

46 

96 

98227 

98272 

98317 

98362 

98407 

98452 

98497 

98542 

98587 

98632 

45 

97 

98677 

98721 

98766 

98811 

98855 

98900 

98945 

98989 

99033 

99078 

45 

98 

99122 

99166 

99211 

99255 

99299 

99343 

99387 

99431 

99475 

99519 

44 

99 

99563 

99607 

99651 

99694 

99738 

99782 

99825 

99869 

99913 

99956 

44 

*  Each  log  is  supposed  to  have  the  decimal  sign  .  before  it. 


The  log  of  2870  is  3.45788 
"  "  "  287  is  2.45788 
"  "  "  28.7  is  1.45788 
"  "  "  2.87  is  0.45788 


The  log  of  .287  is  —  1.45788 

"      "     "  .028  is  —  2.44716 

"      "     "  .002  is  —  3.30103 

«      «     "  .0002  is  —  4.30103 


What  is  the  log  of  2873  ? 
Here,  log  of  2870  =  3.45788 
And  prop  153  X  3  =          459 


3.458339 

To  find  roots  divide  the  log  (with  its  index)  of  the  given  number,  by  that 

number  which  expresses  the  kind  of  root.     The  quotient  will  be  the  log  of  the  required  root. 

Example.     What  is  the  cube  root  of  2870? 
Here,  the  log  of  2870,  with  its  index,  is  3.45788.     And  '-• =  1.15263.    Hence  the  cube  root  is  14.1 

The  Hyperbolic,  or  Napierian  logarithm  is  the  common  log  of 

the  table  multiplied  by  2.3025851. 


GLOSSARY   OF   TERMS.  615 

GLOSSAEY  OF  TEEMS. 


Abacus ;  the  flat  square  member  on  top  of  a  column. 

Absciss  or  abscissa  ;  any  portion  of  the  axis  of  a  curve,  from  the  vertex  to  any  point  from  which 
a  Hue  leaves  the  axis  at  right  angles,  and  extends  to  meet  the  curve  itself;  said  line  being  called  an. 
ordinate.  An  absciss  and  ordinate  together  are  called  co-ordinates. 

Acclivity  ;  an  upward  slope,  or  ascent  of  ground,  &c. 

Adit ;  a  Horizontal  passage  into  a  mine,  &c. 

Adze;  a  well-known  curved  cutting  instrument,  for  dressing  or  chipping  horizontal  surfaces. 

Air-vessel.  Motion  is  imparted  to  the  water  in  a  line  of  pipes,  by  the  forward  stroke  of  the  piston 
of  a  single-acting  pump;  but  during  the  backward  stroke,  this  motion  is  stopped;  and  the  water  in 
the  pipes  comes  to  rest.  Therefore,  at  the  next  forward  stroke,  all  the  water  has  to  be  again  set  in 
motion  ;  and  the  force  that  must  be  exerted  by  the  pump  to  do  this  is  much  greater  than  would  be 
required  if  the  motion  previously  imparted  had  been  maintained  during  the  time  of  the  backstroke. 
The  addition  of  an  air-vessel  secures  this  maintenance  of  motion,  and  thus  effects  a  great  saving  of 
power;  besides  diminishing  the  danger  of  bursting  the  pipes  at  each  forward  stroke.  It  is  merely  a 
tall  and  strong  air-tight  iron  box,  usually  cylindrical,  strongly  bolted  on  top  of  the  pipes  just  beyond 
the  pump,  and  communicating  freelv  with  them  through  an  opening  in  its  base.  It  is  full  of  air. 
The  forward  stroke  of  the  piston  then  forces  water  not  only  along  the  pipes,  but  also  into  the  lower 
part  of  the  air-vessel,  through  the  opening  in  its  base  ;  thus  compress. ng  its  contained  air.  But 
during  the  backstroke,  this  compressed  air,  being  relieved  from  the  pressure  of  the  pump,  expands; 
and  in  so  doing  presses  upon  the  water  in  the  pipes,  and  thus  keeps  it  in  motion  until  the  next  for- 
ward stroke;  and  so  on.  An  air-vessel  also  acts  as  an  air-cusliion ;  permitting  the  piston  to  apply 
its  force  to  the  water  in  the  pipes  gradually  ;  thus  preserving  both  the  pipes  and  the  pump  from  vio- 
lent shocks.  The  air  in  the  vessel,  however,  becomes  by  degrees  absorbed  and  taken  away  by  the 
water;  and  its  action  as  a  regulator  then  ceases.  To  prevent  this,  fresh  air  must  be  forced  into  the 
vessel  from  time  to  time  by  a  condenser,  or  forcing  air-pump.  A  double-acting  pump  does  not  so 
much  need  an  air-vessel.  There  is  no  particular  rule  for  the  size  or  capacity  of  air-vessels.  In  prac- 
tice it  appears  to  vary  from  about  5  to  50  times  that  of  the  pump  ;  with  a  height  equal  to  two  or  more 
times  the  diam.  A  stand-pipe,  which  see,  is  sometimes  used  instead  of  an  air-vessel. 

Alternating  motion;  up  and  down,  or  backward  and  forward,  instead  of  revolving,  &c. 

Angle-bead,  or  plaster  bead;  a  bead  nailed  to  projecting  angles  in  rooms,  to  protect  the  plaster  on 
their  edges  from  injury. 

Angle-block;  a  triangular  block  against  which  the  ends  of  the  braces  and  counters  abut  in  a  Howe 
bridge. 

Angular  velocity.     See  footnote  to  Art  6  of  Force  in  Rigid  Bodies.     Page  447. 

Anneal;  to  toughen  some  of  the  metals,  glass,  Ac,  by  first  heating  them,  and  then  causing  them  to 
cool  very  slowly.  This  process  however  lessens  the  tensile  strength. 

Anticlinal  axis ;  in  geology  ;  a  line  from  which  the  strata  of  rocks  slope  away  downward  in  oppo- 
site directions,  like  the  slates  on  the  roof  of  a  house ;  the  ridge  of  the  roof  representing  the  axis. 

Apron;  a  covering  of  timber,  stone,  or  metal,  to  protect  a  surface  against  the  action  of  water  flow- 
ing over  it.  Has  many  other  meanings. 

Arbor.    See  Journal. 

Architrave ;  that  part  of  an  entablature  which  is  next  above  the  columns.  Applies  also  when  there 
are  no  columns.  Also,  the  mouldings  around  the  sides  and  tops  of  doors  and  windows,  attached  to 
either  the  inner  or  outer  face  of  the  wall. 

Arris ;  a  sharp  edge  formed  by  any  two  surfaces  which  meet  at  an  angle.  The  edges  of  a  brick  are 
arrises. 

Ashler ;  a  facing  of  cut  stone,  applied  to  a  backing  of  rubble  or  rough  masonry,  or  brickwork, 

Astragal;  a  small  moulding,  about  semi-circular  or  semi-elliptic,  and  either  plain  or  ornamented  by 
carving. 

Axis ;  an  imaginary  line  passing  through  a  body,  which  may  be  supposed  to  revolve  around  it;  as 
the  diam  of  a  sphere.  Any  piece  that  passes  through  and  supports  a  body  which  revolves  ;  in  which 
case  it  is  called  an  axle,  or  shaft. 

Axle-box.     See  Journal-box. 

Axletree;  an  axle  which  remains  fixed  while  the  wheel  revolves  around  it,  as  in  wagons,  &c. 

Azimuth.  The  azimuth  of  a  body  is  that  arc  of  the  horizon  that  is  included  between  the  meridian 
circle  at  the  given  place,  and  another  great  circle  passing  through  the  body. 

Backing ;  the  rough  masonry  of  a  wall  faced  with  finer  work.  Earth  deposited  behind  a  retainlng- 
wall,  &c. 

Balance-beams;  the  long  top  beams  of  lock-gates,  by  which  they  are  pushed  open  or  shuts 

Balk ;  a  large  beam  of  timber. 

Baliast;  broken  stone,  sand  or  gravel,  &c,  on  which  railroad  cross-ties  are  laid. 

Ball-cock;  a  cistern  valve  at  one  end  of  a  lever,  at  the  other  end  of  which  is  a  floating  ball.  Th« 
ball  rises  and  falls  with  the  water  in  the  cistern ;  and  thus  opens  or  shuts  the  valve. 

Ball-value.     See  Valve. 

Bargeboards ;  boards  nailed  against  the  outer  face  of  a  wall,  along  the  slopes  of  a  gable  end  of  a 
house,  to  hide  the  rafters,  &c  ;  and  to  make  a  neat  finish. 

Batter,  (sometimes  affectedly  batir.)  or  talus  ;  the  sloping  backward  of  a  face  of  masonry. 

Bay ;  on  bridges,  &c,  sometimes  a  panel ;  sometimes  a  span. 

Bead ;  an  ornamenfeither  composed  of  a  straight  cylindrical  rod  ;  or  carved  or  cast  in  that  shape 
on  any  surface. 

Bearing  ;  the  course  by  a  compass.  The  span  or  length  in  the  clear  between  the  points  of  support 
of  a  beam,  &e.  The  points  of  support  themselves  of  a  beam,  shaft,  axle,  pivot,  &c. 

Bed- moulding » ;  ornamental  mouldings  on  the  lower  face  of  a  projecting  cornice,  &c. 

Bed-plate ;  a  large  plate  of  iron  laid  as  a  foundation  for  something  to  rest  on. 

Beetle ;  a  heavy  wooden  rammer,  such  as  pavers  use. 

Bell-crank.     See  Crank. 

Bench-mark;  a  level  mark  cut  at  the  foot  of  a  tree  for  future  reference,  as  being  more  permanent 
than  a  stake. 

Herm,,  or  berme;  a  horizontal  surface,  as  if  for  a  pathway,  and  forming  a  kind  of  step  along  the  face 


616 


GLOSSARY   OF   TEEMS. 


of  sloping  ground.  In  canals,  the  level  top  of  the  embankment  opposite  and  corresponding  to  the 
towpath  is  called  the  berm. 

Bessemer  steel  is  formed  by  forcing  air  into  a  mass  of  melted  cast  iron  ;  by  which  means  the  excess 
of  carbon  in  the  iron  is  separated  from  it,  until  only  enough  remains  to  constitute  cast  steel.  The 
carbon  is  chemically  united  with  the  steel,  but  mechanically  with  the  iron. 

Beton;  concrete  of  hydraulic  cement,  with  broken  stone  and  bricks,  gravel,  &c. 

Bevel;  the  slope  formed  by  trimming  away  a  sharp  edge,  as  of  a  board,  &c.  Edges  of  common 
drawing  rulers  aud  scales  are  usually  bevelled.  See  13,  Figs  42,  of  Trusses,  p  294. 

Bevel  gear;  cog-wheels  witn  teeth  so  formed  that  the  wheels  can  work  into  each  other  at  an  angle. 

Bilge;  the  nearly  flat  part  of  the  bottom  of  a  ship  on  each  side  of  the  keel.  Also,  the  swelled  part 
of  a  barrel,  &c.  To  bilge  is  to  spring  a  leak  in  the  bilge,  or  to  be  broken  there. 

Bitts ;  the  small  boring  points  used  with  a  brace. 

Blast-pipes;  in  a  locomotive;  those  through  which  the  waste  steam  passes  from  the  cylinder  into 
the  smoke-pipe,  and  thus  creates  an  artificial  draft  in  the  chimney,  or  smoke-pipe. 

Boasting ;  dressing  stone  with  a  broad  chisel  called  a  boaster,  aud  mallet.  The  boaster  gives  a 
gmoother  surface  after  the  use  of  the  point,  or  the  narrow  chisel  called  a  tool. 

Bond;  the  disposing  of  the  blocks  of  stone  or  brickwork  so  as  to  form  the  whole  into  a  firm  struc- 
ture, by  a  judicious  overlapping  of  each  other,  so  as  to  break  joint.  Applies  also  to  timber,  &c,  in 
various  ways. 

Bonnet;  a  cap  over  the  end  of  a  pipe,  &c.  A  cast-iron  plate  bolted  down  as  a  covering  over  an 
aperture. 

Bore ;  inner  diameter  of  a  hollow  cylinder. 

Boss ;  an  increase  of  the  diameter  at  any  part  of  a  shaft  for  any  purpose.  A  projection  in  shape 
of  a  segment  of  a  sphere,  or  somewhat  so,  whether  for  use  or  for  ornament;  often  carved,  or  cast. 

Box-drain;  &  square  or  rectangular  drain  of  masonry  or  timber,  under  a  railroad,  &c. 

Brace ;  a  kind  of  curved  handle  used  for  boring  holes  with  bitts.  The  head  of  the  brace  remains 
stationary,  being  pressed  against  by  the  body  of  the  person  using  it,  while  the  other  part  with  the 
bitt  is  turned  round  by  his  haud.  Also,  an  inclined  beam,  bar,  or  strut,  for  sustaining  compression. 

Bracket;  a  projecting  piece  of  board,  &c,  frequently  triangular,  the  vertical  leg  attached  to  the 
face  of  a  wall,  and  the  horizontal  one  supporting  a  shelf,  &c.  Often  made  in  ornamental  shapes  for 
supporting  busts,  clocks,  &c.  Also,  the  supports  for  shafting ;  as  pendent,  wall,  and  pedestal  brackets. 

Brake;  an  arrangement  for  preventing  or  diminishing  motion  by  means  of  friction.  The  friction 
is  usually  applied  at  the  circumference  of  a  revolving  wheel,  by  means  of  levers.  On  railroads,  the 
car-brakes  should  be  worked  by  steam,  as  those  of  Loughridge,  Westiughouse,  and  Creamer.  Also, 
such  a  handle  as  that  of  a  common  pump. 

Brass  is  composed  of  copper  and  zinc. 

Brasses;  fittings  of  brass  in  many  plummer-blocks,  and  in  other  positions,  for  diminishing  the 
friction  of  revolving  journals  which  rest  upon  them. 

Braze;  to  unite  pieces  of  iron,  copper,  or  brass,  by  means  of  a  hard  solder,  called  spelter  solder, 
and  composed,  like  brass,  of  copper  aud  zinc,  but  in  other  proportions. 

Break  joint;  to  so  overlap  pieces  that  the  joints  shall  not  occur  at  the  same  place,  and  thus  pro- 
duce a  bad  bond. 

Breast-summer;  a  beam  of  wood,  iron,  or  stone,  supporting  a  wall  over  a  door  or  other  opening; 
a  kind  of  lintel. 

Breast-wall;  one  built  to  prevent  the  falling  of  a  vertical  face  cut  into  the  natural  soil;  in  dis- 
tinction to  a  retaining-wall  or  revetment,  which  is  built  to  sustain  earth  deposited  behind  it. 

Breech;  the  hind  part  of  a  cannon,  &c. 

Bridge,,  or  Iridge-piece,  or  bridge  bar ;  a  narrow  strip  placed  across  an  opening,  for  supporting 
something  without  closing  too  much  of  the  opening. 

Bronze  is  composed  of  copper  and  tin. 

Bulkhead ;  on  ships,  &c,  the  timber  partitions  across  them.    Also,  a  long  face  of  wharf  parallel 

Buoy ;  a  floating  body,  fastened  by  a  chain  or  rope  to  some  sunk  body,  as  a  guide  for  finding  the 
latter,  'sometimes  also  used  to  indicate  channels,  shoals,  rocks,  &c. 

Burnish  ;  to  polish  by  rubbing;  chiefly  applies  to  metals. 

Bush  •  to  line  a  circular  hole  by  a  ring  of  metal,  to  prevent  the  hole  from  wearing  larger.  Also, 
when  a  piece  is  cut  out,  and  another  piece  neatly  inserted  into  the  cavity,  the  last  piece  is  sometimes 
said  to  be  bushed  in ;  sometimes  it  is  called  a  plug. 

Buttress ;    a    vertical    projecting    piece  of  brickwork  or  masonry,   built  in  tront  or   a  wall  to 

*  cSLmVa  large  wooden  box  with  sides  that  may  be  detached  and  floated  away. 

Caliber;  the  inner  diameter,  or  bore. 

Calipers  •  compasses  or  dividers  with  curved  legs,  for  measuring  outside  diameters. 

Calk,  or  caulk;  to  fill  seams  or  joints  with  something  to  prevent  leaking. 

Calking  iron ;  a  tool  for  forcing  calking  into  a  joint. 

Camb.  or  cam,  or  wiper;  a  piece  fixed  upon  a  revolving  shaft  in  such  a  manner  as  to  produce  an 
alternating  or  reciprocation  motion  in  something  in  contact  with  the  cam.  An  eccentric. 

Camber  •  a  slight  upward  curve  given  to  a  beam  or  truss,  to  allow  for  settling. 

Camel;  a  kind  of  barges  or  hollow  floating  vessels,  which,  when  filled  whir  water,  are  fastened  to 
the  sides  of  a  ship  ;  and  the  water  being  then  pumped  out,  they  rise  by  their  buoyancy  ;  and  lift  the 
ship  so  that  she  can  float  in  shallower  water. 

Cantilevers  ;  projecting  pieces  for  supporting  an  upper  balcony,  &c 

Cants,  rims,  or  shroudings;  the  pieces  forming  the  ends  of  the  buckets  of  water-wheels,  to  prevent 

t  ^nns^n  «  "ing  'hollow  rope^drum  surrounding  a  strong  vertical  pivot,  upon  the  head  of  which  it 
rests  and  around  which  it  turns.  Its  top  is  a  thick  projecting  circular  piece,  having  holes  around  its 
outer  edge  or  circumference,  for  the  insertion  of  the  ends  of  levers  ;  or  capstan-bars.  It  is  a  kind  oi 

VeeL««-»ar«taftrto  convert  the  outer  surface  of  wrought  iron  into  steel,  by  heating  it  while  in  contact 
w'th  charcoal. 

Casemate;  in  fortification  ;  the  small  apartment  m  which  a  cannon  stands. 

Castors  ;  rollers  usually  combined  with  swivels;  as  those  used  under  heavy  furniture,  &c. 

Causeway  ;  a  raised  footway  or  roadway. 

Cavetto  ;  "a  moulding  consisting  of  a  receding  quadrant  of  a  circle. 

Cementation  •  the  process  of  converting  wrought  iron  into  steel,  by  heating  it  in  contact  with  cnar- 


'GLOSSARY   OF   TERMS.  617 


eoal.  This  process  produces  blisters  on  the  steel  bars ;  hence  blister  steel.  These  are  removed,  and 
the  steel  compacted,  by  reheatiug  it,  aud  then  subjecting  it  to  a  tilt-hamiuer.  It  is  then  tilted  steel, 
or  shear  aM&l.  Or  if  the  blister  steel  is  broken  up;  remelted;  and  then  run  into  ingots  or  blocks;  it 
is  called  cast,  or  ingot  steel;  which  is  harder  aud  closer-grained  than  tilted  steel.  It  may  be  softened, 
and  thus  become  less  brittle,  by  annealing.  The  ingots  may  be  converted  into  bars  by  either  rolling 
or  hammering,  the  same  as  shear  and  blister. 

Center ;  the  supports  of  an  arch  while  being  built. 

Canter  of  gravity.    See  Art  56,  of  Force,  p  481.    Also  see  Cen  of  Grav,  p  442. 

Center  of  gyration.  Suppose  a  body  free  to  revolve  around  an  axis  whicu  passes  through  it  in  any 
direction;  or  to  oscillate  like  a  pendulum  hung  from  a  point  of  suspension.  Then  suppose  in  either 
case,  a  certain  given  amount  of  force  to  be  applied  to  the  body,  at  a  certain  given  dist  from  the  axis, 
or  from  the  point  of  suspension,  so  as  to  impart  to  the  body  an  angular  vel;  or  in  other  words,  to 
came  it  to  describe  a  number  of  degrees  per  sec.  Now,  there  will  be  a  certain  point  in  the  body,  such 
that  if  the  entire  wt  of  the  body  were  there  concentrated,  then  the  same  force  as  before,  applied  at  the 
same  dist  from  the  axis,  or  from  the  point  of  su.speusion  as  before,  would  impart  to  the  body  the  .same 
angular  motion  as  before.  This  point  is  the  center  of  gyration  ;  and  its  dist  from  the  axis,  or  from  the 
point  of  su.spension,  is  the  Radius  of  gyration,  of  the  body.  To  find  the  position  of  this  center,  or  the 
length  of  this  rad,  see  Moment  of  Iuertia,,iu  this  glossary.  One  use  of  the  center  of  gyration  is  to 
enable  us  to  calculate  the  momentum  or  moving  force  in  a  revolving  or  oscillating  body.  This  force  in 
foot  pounds  is  found  by  mult  the  wt  of  the  body  in  tts,  by  the  vert  height  in  ft  through  which  it 
would  have  to  fall,  in  order  to  acquire  the  vel  which  its  center  of  gyration  has.  This  height  may  be 
found  in  Table  10,  p  552,  of  Hydraulics.  Also  p  495. 

Center  of  oscillation,  or  of  vibration.    See  Rem  2,  of  Pendulums,  p  173. 

Center  of  percussion,  in  a  moving  body,  is  that  point  which  would  strike  an  opposing  body  with 
greater  force  than  any  other  point  would.  If  the  opposing  body  is  immovable,  it  will  receive  all  the 
force  of  a  rigid  moving  body  which  strikes  with  its  center  of  percussion.  See  Pendulum,  page  173. 

Cesspool;  &  shallow  well  for  receiving  waste  water,  filth,  &c. 

Chamfer;  means  much  the  same  as  bevel ;  but  applies  more  especially  when  two  edges  are  cut  away 
30  as  to  form  either  a  chamfer-groove,  (see  14,  p  294,  of  Trusses,)  or  a  projecting  sharp  edge. 

Cheeks ;  two  flat  parallel  pieces  confining  something  between  tnem.  See  w,  at  15,  of  Figs  21^,  of 
Trusses,  p  265. 

Chilling,  chill- hardening,  or  chill  -cast  ing ;  giving  great  hardness  to  the  outside  of  cast-iron,  by 
pouring  it  into  a  mould  made  of  iron  instead  of  wood.  The  iron  mould  causes  the  outside  or  skin  of 
the  casting  to  cool  very  rapidly  ;  and  this  for  some  unknown  reason  increases  its  hardness.  This  pro- 
cess is  frequently  confounded  with  case-hardening. 

Chock  ;  any  piece  used  for  filling  up  a  chance  hole,  or  vacancy. 

Chuck;  the  arrangement  attached  to  the  revolving  shaft,  arbor,  or  mandril  of  a  lathe,  for  holding 
the  thing  to  be  turned. 

Churn-drill;  a  long  iron  bar.  with  a  cutting  end  of  steel ;  much  used  in  quarrying,  and  worked  by 
raising  it  aud  letting  it  fall  When  worked  by  blows  of  a  hammer  or  sledge  it  is  called  a  jumper. 

Cima,  or  cymn ;  a  moulding  nearly  in  shape  of  an  S.  When  the  upper  part  is  concave,  it  is  called 
a  cirna  recta  ;  when  convex,  a  cima  reversa.  See  page  67. 

Clack  valve.     See  Valve. 

Clamp;  a  piece  fastened  by  tongue  and  groove,  transversely  along  the  end  of  others,  to  keep  them 
from  warping.  A  kind  of  open  collar,  which,  being  closed  by  a  clam-screw,  holds  tight  what  it  sur 
rounds.  See  Cramp. 

Clap-boards;  short  thin  boards,  shingle-shaped,  and  used  instead  of  shingles. 

Claw ;  a  split  provided  at  the  end  of  an  iron  bar,  or  of  a  hammer,  &c,  to  take  hold  of  the  heads  ot 
nails  or  spikes  for  drawing  them  out ;  as  in  a  common  claw-hammer. 

Cleat ;  a  piece  merely  bolted  to  another  to  serve  as  a  support  for  something  else;  as  at  7,  8.  9,  10, 
&c,  p  294,  of  Trusses.  Often  used  on  shipboard  for  fastening  ropes  to,  as  at  11.  Also  a  piece  of 
board  uailed  across  two  or  more  other  boards,  for  holding  them  together,  as  is  often  done  in  temporary 
doors,  &c. 

Clewi*.    See  Shackle. 

Click.     See  Ratchet. 

Clip ;  a  fastening  like  that  on  the  tops  of  the  Y's  of  a  spirit  level ;  being  a  kind  of  half  collar  opening 
by  a  hinge. 

Clutch  ;  applied  to  various  arrangements  at  the  ends  of  separate  shafts,  and  which  by  clutching  or 
catching  into  each  other  cause  both  shafts  to  revolve  together.  A  kind  of  coupling. 

Cock;  a  kind  of  valve  for  the  discharge  of  liquids,  air,  steam,  &c. 

Coefficient;  or  a  Constant  of  friction,  safety,  or  strength,  Ac,  may  usually  be  taken  to  be  a  num- 
ber which  shows  the  proportion  (or  rather  the  ratio)  which  friction, "safety,  tensile  strength,  &c,  bear 
to  a  certain  something  else  which  is  not  generally  expressed  at  the  time,  but  is  well  understood.  Thus, 
when  we  say  that  the  coeff  of  friction  of  one  body  upon  another  is  y1^-,  Ac,  it  is  understood  that  the 
friction  is  in  the  proportion  of  y^-th  of  the  pressure  which  produces  it.  A  coeff  of  safety  of  3,  ni^ans 
that  the  safety  has  a  proportion  or  ratio  of  3  to  I  to  the  theoretical  breaking  loud.  A  coeff  of  500  fts, 
or  of  '20  tons,  Ac,  of  tensile  strength  of  any  material,  denotes  that  said  strength  is  in  the  proportion 
of  500  KM.  or  of  20  tous.  &c,  to  each  square  inch  of  transverse  section.  <fec.  Same  as  Modulus. 

Coffer  dam ;  an  enclosure  built  in  the  water,  and  then  pumped  dry,  so  as  to  permit  masonry  or 
other  work  to  be  carried  on  inside  of  it. 

Coq ;  the  tooth  of  a  cog- wheel. 

Collar;  a  flat  rina;  surrounding  anvthini?  closely. 

Collar-beam;  a  horizontal  timber  stretching  from  one  to  another  of  two  rafters  which  meet  at  top; 
but  above  the  main  tie-beam.  See  21 .  Figs  42.  of  Trusses.  Page  294. 

Concrete;  artificial  stone  formed  by  mixing  broken  stone,  pravel,  &c,  with  common  lime.  «:hen 
hydraulic  cement  is  used  instead  of  lime,  the  mixture  is  called  beton.  The  terms  "  lime  concrete  '" 
and  '-cement  concrete  "  would  be  convenient. 

Connecting-rod;  apiece  which  connects  a  crank  with  something  which  moves  it,  or  to  which  it 
g'ves  motion. 

Console ;  a  kind  of  ornamental  bracket,  somewhat  in  shape  of  an  S ;  much  used  In  cornices,  &c, 
for  supporting  ornamental  mouldings  above  it. 

Coping  ;  flat  plates  of  stone,  iron.  &c,  placed  on  the  tops  of  walls  exposed  to  the  weather. 

Corbel ;  a  horizontal  projecting  piece  which  assists  in  supporting  one  resting  upon  it  which  project* 


618 


GLOSSARY   OF   TERMS. 


Core;  anything  serving  as  a  mould  for  anything  else  to  be  formed  around.  A  term  much  used  Im 
foundries. 

Cornice ;  the  ornamental  projection  at  the  eaves  of  a  building,  or  at  the  top  of  a  pier,  or  of  any  other 
Structure. 

Cotter-bolt,  or  key-bolt ;  a  bolt  which,  instead  of  a  screw  and  nut  at  one  end,  has  a  slot  cut  through 
it  near  that  end,  for  the  insertion  of  a  wedge-suaped  key  or  cotter,  for  keeping  it  in  its  place.  Some- 
times the  ends  of  these  keys  are  split,  so  as  to  spread  open  after  being  inserted,  so  as  not  to  be  jolted 
out  of  place. 

Counterfort;  vertical  projections  of  masonry  or  brickwork  built  at  intervals  along  the  back  of  a  wall 
to  strengthen  it ;  and  generally  of  very  little  use. 

Counter-shaft ;  a  secondary  shaft  or  axle  which  receives  motion  from  the  principal  one. 

Countersunk.     See  iieamiug. 

Counter- weight;  or  counter- balance ;  any  weight  used  to  balance  another. 

Couplings;  a  term  of  very  general  application  to  arrangements  for  connecting  two  shafts  so  that 
they  shall  revolve  together. 

Crab  ;  a  short  shaft  or  axle,  which  serves  as  a  rope-drum  in  raising  weights  ;  and  is  revolved  either 
by  cog-wheels,  a  winch,  or  by  levers  or  handspikes,  inserted  in  holes  around  its  circumference  like  a 
windlass,  or  capstan,  of  which  it  is  a  variety.  It  may  be  either  vertical  or  horizontal.  It  is  often 
•et  in  a  frame,  to  be  carried  from  place  to  place.  Also  the  whole  machine  is  called  a  crab. 

Cradle  ;  applied  to  various  kinds  of  timber  supports,  which  partly  enclose  the  mass  sustained. 

Cramp :  a  short  bar  of  metal,  having  its  two  ends  bent  downward  at  right  angles  for  insertion  into 
two  adjoining  pieces  of  stone,  wood,  &c,  to  hold  them  together.  Much  used  at  the  ends  of  coping-stones. 
Also  a  similar  bent  piece,  with  a  set-screw  passing  through  one  of  the  bent  ends,  for  holding  things 
tight  between  it  and  the  other  end.  This  last  is  also  called  a  clamp. 

Crane;  a  hoisting  machine  consisting  of  a  revolving  vertical  post  or  stalk;  a  projecting  ji b  ;  and 
a  stay  for  sustaining  the  outer  end  of  the  jib.  The  stay  may  be  either  a  strut  or  a  tie.  There  are 
also  cog-wheels,  a  rope  drum  or  barrel,  with  a  winch,  ropes,  pulleys.  &c.  In  a  crane  the  post,  jib, 
and  stay  do  not  change  their  relative  positions,  as  they  do  in  a  derrick. 

Crank;  a  double  bend  at  right  angles,  somewhat  like  a  Z,at  the  end  of  a  shaft  or  axle,  and  forming 
a  kind  of  handle  by  which  the  axle  may  be  made  to  revolve.  Sometimes,  as  in  common  grindstones, 
this  crank  is  formed  of  a  separate  piece  removable  at  pleasure.  That  part  of  this  piece  which  has  the 
square  opening  in  it  for  fitting  it  to  the  square  end  of  the  axle,  is  called  the  crank-arm ;  and  the  other 
part  the  crank- handle.  A  bell-crank  consists  of  4  bends  at  right  angles  at  the  center  of  an  axle,  form- 
ing in  it  a  kind  of  U.  A  double  crank  consists  of  two  bell  cranks  arranged  thus,  ^j^-  Tne  bend  in 
the  U  forms  the  crank-wrist.  The  term  bell  crank  is  applied  also  to  those  used  in  fixing  common  dwell- 
ing house  bells  :  and  to  larger  ones  on  the  same  principle.  A  crank-pin  is  a  pin  projecting  from  a  re- 
volving wheel,  disk,  or  other  body,  and  serving  as  a  crank- haridle.  A  crank-shaft  is  a  shaft,  which 
has  a  crank  in  it.  or  at  its  end.  A  cranked  shaft  has  it  in  it  only.  A  ship  or  other  vessel  is  said  to 
be  crank  when  its  breadth  is  so  small  in  proportion  to  its  depth  as  to  make  it  liable  to  upset  easily ;  or 
when  the  same  liability  is  caused  by  want  of  sufficient  ballast. 

Crest ;  that  top  part  of  a  dam  over  which  the  water  pours. 

Cross-cut  saiv ;  a  large  horizontal  saw  worked  by  two  men,  one  at  each  end. 

Cross-head  ;  a  piece  attached  across  the  end  (or  near  it)  of  another  piece,  and  at  right  angles  to  it, 
so  as  to  form  a  kind  of  T  or  cross.  Often  seen  on  piston  rods,  which  they  serve  to  keep  in  place  by- 
resting  on  the  slides,  or  guides. 

Crowbar  •  a  bar  of  iron  used  as  a  lever  for  various  purposes ;  often  pointed  at  one  end. 

Crown,  or  contrate  wheel ;  a  cog-wheel  in  which  the  teeth  stand  not  upon  its  outer  circumference  as 
nsual.  but  upon  the  plane  of  its  circle. 

Curb  ;  a  broad  flat  circular  ring  of  wood,  iron,  or  stone,  placed  under  the  bottoms  of  circular  walls, 
as  in  a  well,  or  shaft,  to  prevent  unequal  settlement;  or  built  into  the  walls  at  intervals,  for  the  same 
purpose.  Has  manv  other  meanings. 

Cut-off,-  an  arrangement  for  cutting  off  the  steam  from  a  cylinder  before  the  piston  has  made  its 
full  stroke.  Also  a  channel  cut  through  a  narrow  neck  of  land,  to  straighten  the  course  of  a  river. 

Cutwater,  or  starling ;  the  projecting  ends  of  a  bridge  pier,  &c,  usually  so  shaped  as  to  allow  water, 
ice,  &c,  to  strike  them  with  but  little  injury. 

Damper ;  a  floor  or  valve  to  regulate  the  admission  of  air  to  a  furnace,  stove,  &c. 

Dead  load;  the  cars,  engine,  &c,  in  a  train  ;  non -paying  load. 

Dead  points ;  those  two  points  in  the  revolution  of  a  crank,  when  the  crank  arm  is  parallel  w 
the  rod  which  connects  it  with  the  moving  power ;  and  at  which  said  rod  neither  pulls  nor  pushes 

Declination,  of  the  sun,  or  of  a  star,  is  its  latitude,  or  angle  north  or  south  of  the  earth's  equator 
at  the  time  of  observation. 

Declivity;  a  downward  slope  or  descent  of  ground.  «c. 

Dentils;  blocks  constituting  ornaments  in  a  cornice;  placed  at  short  intervals  apart,  they  resemble 
teeth.  When,  instead  of  mere  blocks,  they  are  handsomely  carved  in  various  shapes,  they  are  called 

Derrick';  a  kind  of  crane,  differing  from  common  ones,  chiefly  in  the  fact  that  the  rope  or  chain 
which  forms  the  stay  mav  be  let  out  or  hauled  in  at  pleasure,  thus  raising  or  lowering  the  inclination 
of  a  jib;  therehv  enabling  the  raised  load  to  be  placed  vertically  at  the  required  spot.  This  cannot 
be  done  with  a  crane,  which,  therefore,  is  not  as  well  adapted  for  laying  heavy  masonry,  especially 
at  great  heights. 

Diaphram ;  a  thin  plate  or  partition  placed  across  a  tube  or  other  hollow  body. 

Die ;  that  part  of  a  stamp  that  gives  the  impression.  Dies  are  also  two  flat  plates  of  hardened  steel, 
on  an  edsre  of  each  of  which  is  hollowed  out  a  semicircular  half  of  a  short  female  screw.  When  these 
plates  are  put  in  contact  they  form  a  complete  female  screw,  like  that  in  a  nut:  and  being  strongly 
held  together  by  an  iron  boxing  called  the  die-stocks,  which  have  long  handles  tor  revolving  them,  they 
constitute  a  mould  or  cutter  for  forming  threads  on  a  male  screw.  Also  the  mam  body  of  a  pedestal. 

Dip;  in  geology,  either  the  angle  which  the  slope  of  a  stratum  forms  with  a  horizontal ;  or  the 
direction  by  compass,  toward  which  it  slopes.  In  surveying,  the  inclination  at  which  an  unbalanced 
compass-needle  rests  on  its  pivot  after  being  magnetized. 

Disk ;  a  flat  circular  piece.  .       . 

Dock;  an  artificial  enclosure,  either  partial  or  total,  in  which  ships  and  other  Tessels  are  placed 
for  being  loaded  or  unloaded,  or  repaired.  The  first  is  a  tvet  dock ;  the  last  a  dry  one. 

Dcgiron;  a  short  bar  of  iron,  forming  a  kind  of  cramp,  with  its  ends  bent  down  at  right  angles 
and  pointed,  so  as  to  hold  together  two  pieces  into  which  they  are  driven.  Often  used  for  temporary 
purposes.  It  is  also  called  a  dog-iron  when  only  one  end  is  bent  down  and  pointed  for  drivmg,  tne 


GLflSSA'KY   OF   TERMS.  619 

•ttier  end  being  formed  into  an  eye  or  a  handle  by  which  the  piece  into  which  the  other  end  is  driven 
may  be  hauled  or  towed  away.        / 

Donkey-engine;  a  small  steam Vngine  attached  to  a  large  one,  and  fed  from  the  same  boiler.  It  is 
B,sed  for  pumping  water  into  the'boiler. 

Double  crank.     See  Crank./ 

Double  keys.     See  K,  FigsjR,  of  Trusses  ;  page  294. 

Dovetail;  a  joint  like  20,  Jrigs  42  of  Trusses ;  it  is  a  poor  one  for  timber  when  there  is  much  strain, 
being  then  apt  to  dravy  ouymore  01  less. 

Dowel;  a  straight  pin  o/wood  or  metal,  inserted  part  way  into  each  of  two  faces  which  it  unites. 

Draft;  the  depth  to  which  a  floating  vessel  sinks  in  the  water;  in  other  words  the  water  it  draws. 

Draught;  a  drawing.  A  narrow  level  stripe  which  a  stonecutter  first  cuts  around  the  edges  of  & 
rough  stone,  to  guide  him  in  dressing  off  the  face  thus  enclosed  by  the  draught. 

Draw-plate;  a  plate  of  very  hard  steel,  pierced  with  small  circular  holes  of  different  diameters, 
through  which  in  succession  rods  of  iron  are  drawn,  and  thus  lengthened  out  into  wire.  Sometimes 
the  holes  are  drilled  through  diamond  or  ruby,  &c,  instead  of  steel. 

Drift;  a  horizontal  or  inclined  passage-way,  or  small  tunnel,  in  mines,  &c.  To  float  away  with  a 
current.  Trees,  &<r,  carried  along  by  freshets. 

Drip ;  a  small  channel  cut  under  the  lower  projecting  edge  of  coping,  &c,  so  that  rain  when  it 
reaches  that  point  will  drip  or  fall  off,  instead  of  finding  its  way  horizontally  beneath  to  the  wall, 
which  it  would  make  damp. 

Drop;  short  pieces  of  nearly  complete  cylinders,  placed  at  small  distances  apart,  in  a  row  like 
teeth,  as  an  ornament  to  cornices,  &c. 

Drum;  a  revolving  cylinder  around  which  ropes  or  belts  either  travel  or  are  wound.  "When  nar- 
row and  used  with  belts  they  are  called  pulleys. 

Dry-rot;  decay  in  such  portions  of  the  timber  of  houses,  bridges,  &c,  as  are  exposed  to  dampness, 
especially  in  confined  wanft  situations.  The  timber  in  cellars  and  basement  stories  is  mere  liable  to 
it  than  in  other  parts,  owing  to  the  greater  dampness  absorbed  by  the  brickwork  from  the  ground. 
Contact  with  lime  or  mortar  hastens  dry  rot.  The  ends  of  girders,  joists,  &c,  resting  on  damp  walls, 
may  be  partially  protected  by  placing  pieces  of  slate  or  sheet  iron  under  them.  The  painting  or  tar- 
ring of  unseasoned  timber  exnedites  internal  dry  rot.  A  thorough  soaking  of  timber  in  a  solution  of 
28  grains  of  quicklime  to  1  gallon  of  water  is  said  to  be  a  preventive  of  dry-rot ;  but  the  best  process 
for  that  purpose  is  saturation  with  creosote  or  carbolic  acid  by  Seely's  mode,  pp  358,  359. 

Dyke ;  mounds  of  earth,  &c,  built  to  prevent  overflow  from'rivers  or  the  sea.  A  kind  of  geological 
irregularity  or  disturbance,  consisting  of  a  stratum  of  rock  injected  as  it  were  by  volcanic  action,  be- 
tween or  across  strata  of  rocks  of  another  kind.  A  levee. 

Eccentric;  &  circular  plate  or.  pulley,  surrounded  by  a  loose  ring,  and  attached  to  a  revolving 
shaft,  and  moving  around  with  it,  but  not  having  the  same  center ;  for  producing  an  alternate  motion. 
Often  used  instead  of  a  crank,  as  they  do  not  weaken  the  axle  by  requiring  it  to  be  bent.  There  are 
many  modifications. 

Escarpment ;  a  nearly  vertical  natural  face  of  rock  or  soil. 

Escutcheon;  the  little  outside  movable  plate  that  protects  the  keyhole  of  a  lock  from  dust. 

JSt/e ;  a  circular  hole  in  a  flat  bar,  &c,  for  receiving  a  pin,  or  for  other  purposes. 

Eye  and  strap;  a  hinge  common  for  outside  shutters,  &c,  one  part  consisting  of  an  iron  strap  one 
end  of  which  is  forged  into  a  pin  at  right  angles  to  it;  and  the  other  part,  of  a  spike  with  an  eye, 
through  which  the  pin  passes.  When  the  eye  is  on  the  strap,  and  the  pin  on  the  spike,  it  is  called  a 
hook  and  strap.  Such  hinges  are  sometimes  called  "  backfiaps." 

Eye-bolt;  a  bolt  which  has  an  eye  at  one  end. 

Face-wall;  one  built  to  sustain  a  face  cut  into  natural  earth,  in  distinction  to  a  retaining- wall, 
which  supports  earth  deposited  behind  it. 

Fall;  the  rope  used  with  pulleys  in  hoisting. 

False-works;  the  scaffold,  center,  or  other  temporary  supports  for  a  structure  while  it  is  being 
built.  In  very  swift  streams  it  is  sometimes  necessary  to  sink  cribs  filled  with  stone,  as  a  base  for 
false-works  to  foot  upon. 

Fascines ;  bundles  of  twigs  and  small  branches,  for  forming  foundations  on  soft  ground. 

Fatigue;  of  materials  ;  the  increase  of  weakness  produced  by  frequent  bending;  or  by  sustaining 
heavy  loads  for  a  long  time. 

Faucet ;  a  short  tube  for  emptying  liquids  from  a  cask,  &c;  the  flow  is  stopped  by  a  spigot.  The 
wider  end  of  a  common  cast-iron  water  or  gas  pipe. 

Feather;  a  slightly  projecting  narrow  rib  lengthwise  of  a  shaft,  and  which,  catching  into  a  corre-" 
upondini?  groove  in  any  thin?  that  surrounds  and  slides  along  the  shaft,  will  bold  it  fast  at  any  required 
part  of  the  length  of  the  feather.  Has  other  applications. 

Feather-edge;  when  one  edge  of  a  board,  &c,  is  thinner  than  the  other. 

Felloe,  or  felly ;  the  circular  rim  of  a  wheel,  into  which  the  outer  ends  of  the  spokes  fit ;  and  which 
is  often  surrounded  by  a  tire. 

Felt;  a  kind  of  coarse  fabric  or  cloth  made  of  fibres  of  hair,  wool,  coarse  paper,  &c,  by  pressure, 
and  not  by  weaving. 

Fend».r;  a  piece  for  protecting  one  thing  from  being  broken  or  injured  by  blows  from  another: 
frequently  vertical  timbers  along  the  outer  faces  of  wharves,  to  prevent  injury  from  the  rubbing  of 
vessels. 

Fender-piles ;  piles  driven  to  ward  off  accidental  floating  bodies. 

Ferrule ;  a  broad  metallic  ring  or  thimble  put  around  anything  to  keep  it  from  splitting  or  breaking. 

Fillet;  a  plain  narrow  flat  moulding  in  a  cornice,  &c.     See  Platband. 

Fish;  to  join  two  beams,  &c,  by  fastening  other  long  pieces  to  their  sides. 

Flags ;  broad  flat  stones  for  paving. 

Flange;  a  projecting  ledge  or  rim. 

Flashings;  broad  strips  of  sheet  lead,  copper,  tin,  Ac,  with  one  edge  inserted  into  the  joints  of 
brickwork  or  masonry  an  inch  or  two  above  a  roof,  &c;  and  projecting  out  several  inches,  so  as  to  be 
flattened  down  close  to  the  roof,  to  prevent  rain  from  leaking  through  the  joint  between  the  roof  and 
the  brick  chimney,  &c,  which  projects  above  it. 

Fia«k*;  upper  and  lower;  the  two  parts  of  the  box  which  contains  the  mould  into  which  melted 
Iron  is  poured  ^r  castings. 

Flatting ;  causing  painting  to  have  a  dead  or  dull,  instead  of  a  glossy  finish,  by  using  turpentine 
instead  of  oil  in  the  last  coat. 

Flier*;  a  straight  flight  of  steps  in  a  stairway. 

Floodgate;  a  gate  to  let  off  excess  of  water  in"  floods,  or  at  other  times. 


620 


GLOSSARY   OF  TERMS. 


Flume;  a  ditch,  trough,  or  other  channel  of  moderate  size  for  conducting  water.  The  ditches  or 
culverts  through  which  surplus  water  passes  from  an  upper  to  a  lower  reach  of  a  canal. 

Flush ;  forming  an  even  continuous  line  or  surface.  To  clean  out  a  line  of  pipes,  sewers,  gutters, 
&c,  by  letting  on  a  sudden  rush  of  water.  The  splitting  of  the  edges  of  stones  under  pressure. 

Fluxes;  various  substances  used  to  prevent  the  instantaneous  formation  of  rust  when  welding  two 
pieces  of  hot  metal  together.  Such  rnst  would  cause  a  weak  weld.  Borax  is  used  for  wrought  iron  ; 
a  mixture  of  borax  and  sal  ammoniac  for  steel ;  chloride  of  ziuc  for  zinc ;  sal  ammoniac  lor  copper 
or  brass  ;  tallow  or  resin  for  lead. 

Fly-ivheel;  a  heavy  revolving  wheel  for  equalizing  the  motion  of  machinery. 

Foaming ;  an  undue  amouut  of  boiling,  caused  by  grease  or  dirt  in  a  boiler. 

Follower;  any  cog-wheel  that  is  driven  by  another;  that  other  is  the  leader. 

Forceps;  any  tools  for  holding  things,  as  by  pincers,  or  pliers. 

Forebay,  or  penstock;  the  reservoir  from  which  the  water  passes  immediately  to  a  water-wheel. 

Forge;  to  work  wrought  iron  into  shape  by  first  softening  it  by  heat,  dud  then  hammering  it  into 
the  required  form. 

For ge- hammer ;  a  heavy  hammer  for  forging  large  pieces  ;  and  worked  by  machinery. 

Foxtail;  a  thin  wedge  inserted  into  a  slit  at  the  lower  eud  of  a  pin,  so  that  as  the  pin  is  driven 
down,  the  wedge  enters  it  and  causes  it  to  swell,  and  hold  more  firmly. 

Frame;  to  put  together  pieces  of  timber  or  metal  so  as  to  form  a  truss,  door,  or  other  structure. 
The  thing  so  framed. 

Friction.- rollers ;  hard  cylinders  placed  under  a  body,  that  it  may  be  moved  more  readily  than  by 
sliding.  See  Fig  43  ;  page  295. 

Friction-ichecls ;  wheels  so  placed  that  the  journals  of  a  shaft  may  rest  upon  their  rims,  and  thus 
be  enabled  to  revolve  with  diminished  friction.  See  page  601. 

Frieze;  iu  architecture,  the  portion  between  the  architrave  and  cornice.  The  term  is  often  applied 
when  there  is  no  architrave. 

Fulcrtim;  the  point  about  which  a  lever  turns. 

Furrings;  pieces  placed  upon  others  which  are  too  low,  merely  to  bring  their  upper  surfaces  up  to 
a  required  level ;  as  is  often  done  with  joists,  when  one  or  more  are  too  low  ;  a  kind  of  chock. 

Fuze,  or  fuse;  to  melt.  A  slow  match,  which,  by  burning  lor  some  time  before  the  fire  reaches  the 
powder,  gives  the  uieu  engaged  in  blasting  time  to  get  out  of  the  way  of  flying  fragments  of  store. 

Gasket;  rope-yarn  or  hemp,  used  lor  stuffing  at  the  joints  of  water-pipes,  &c. 

Gearing ;  a  train  of  cog-wheels.    Now  much  supplanted  by  belts. 

Gib;  the  piece  of  metal  somewhat  of  this  shape,  I !,  often  used  in  the  same  hole  with  a  wedge- 

shaped  key  for  confining  pieces  together.  In  common  use  for  fastening  the  strap  to  the  stub-end  of 
the  connecting-rod  of  an  engine. 

Gin;  a  revolving  vertical  axis,  usually  furnished  with  a  rope-drum,  and  having  one  or  more  long 
arms  or  levers,  by  means  of  which  it  is  worked  by  horses  walking  in  a  circle  around  it.  Used  for 
hoisting.  Cotton-gin,  a  machine  for  separating  cotton  from  its  seeds. 

Girder ;  a  beam  larger  than  a  common  joist,  and  used  for  a  similar  purpose. 

Glacis;  in  fortification,  an  easy  slope  of  earth. 

Gland.    See  Stuffing-box.     Also,  a  kind  of  coupling  for  shafts. 

Glue;  a  cement  for  wood,  prepared  chi  fly  from  the  gelatine  furnished  by  boiling  the  parings  of 
hides.  Good  glue  will  hold  two  pieces  of  wood  together  with  a  force  of  from  400  to  750  fts  per  sq  in. 

Governor ;  two  balls  so  attached  to  an  upright  revolving  axis  as  to  fly  outward  by  their  centrifugal 
force,  and  thus  regulate  a  valve. 

Grapnel;  a  kind  of  compound  hook  with  several  curved  points,  for  finding  things  in  deep  water. 

Grillage;  a  kind  of  network  of  timbers  laid  crossing  each  other  at  right  augles;  frequently  placed 
on  the  heads  of  piles,  for  supporting  piers  of  bridges,  and  other  masonry.  See  p  634. 

Groin;  an  arch  formed  by  two  segmeutal  arches  or  vaults  intersecting  each  other  at  right  angles. 
Also,  a  kind  of  pier  built  from  the  shore  outward,  to  intercept  shingle  or  gravel. 

Groove;  a  small  channel,  as  at  17  and  18  of  p  294  of  Trusses.  A  triangular  one  is  called  a 
chamfered  groove. 

Ground-swett;  waves  which  continue  after  a  storm  has  ceased ;  or  caused  by  storms  at  a  distance. 

Grout;  thin  mortar,  to  be  poured  into  the  interstices  between  stones  or  bricks. 

Gudgeons ;  the  metal  journals  of  a  horizontal  shaft,  such  as  that  of  a  water-wheel.  The  diam  of 
a  gudgeon  should  not  be  less  than  its  length;  and  this  being  assumed,  the  diam  in  ins  of  a  cast-iron 
gudgeon  may  be  found  thus :  Divide  the  wt  of  the  wheel  and  water,  or  whatever  wt  is  sustained,  by  2. 
'Find  sq  rt  of  rem.  Divide  this  sq  rt  by  20.  For  wrought  iron,  add  lAjth  part. 

Gun-metal,  or  bronze  ;  a  compound  of  copper  and  tin,  sometimes  used  for  cannon.  Also,  a  quality 
of  cast  iron  fit  for  the  same  purpose. 

Gussets;  plain  triangular  pieces  of  plate  iron,  riveted  by  their  vertical  and  horizontal  legs  to  the 
sides,  tops,  and  bottoms  of  box-girders,  tubular  bridges,  &c,  inside,  for  strengthening  their  angles. 

Guys ;  ropes  or  chains  used  to  prevent  anything  from  swinging  or  moving  about. 

Gyrate;  to  revolve  around  a  central  axis,  or  point.    See  center  of  gyration. 

Halving;  to  notch  together  two  timbers  which  cross  each  other,  so  deeply  that  the  joint  thickness 
shall  equal  onlv  that  of  one  whole  timber. 

Hammer  dress ;  to  dress  the  face  of  a  stone  by  slight  blows  of  a  hammer  with  a  cutting  edge.  The 
patent  hammer  for  such  purposes  has  several  such  edges  placed  parallel  to  each  other,  each  of  which 
may  be  removed  and  replaced  at  pleasure. 

Hand-lever;  in  an  engine,  a  lever  to  be  worked  by  hand  instead  of  by  steam. 

Handspike;  a  wooden  lever  for  working  a  capstan  or  windlass  ;  or  other  purposes. 

Hand-wheel;  a  wheel  used  instead  of  a  spanner,  wrench,  winch,  or  lever  of  »ny  kind,  for  screwing 
nuts,  or  for  raising  weights,  or  for  steering  with  a  rudder,  &c. 

Hangers,  or  pendent  brackets;  fixtures  projecting  below  a  ceiling,  to  support  the  journals  of  long 
lines  of  shafting  ;  and  for  other  purpose.  Should  be  "  self-adjusting." 

Hasp ;  a  piece  of  metal  with  an  opening  for  folding  it  over  a  staple. 

Hatchway;  a  horizontal  opening  or  doorway  in  a  floor,  or  in  the  deck  of  a  vessel. 

Haunches  ;  the  parts  of  an  arch  from  the  keystone  to  the  skewback. 

Head-block ;  a  block  on  which  a  pillow-block  rests. 

Header;  a  stone  or  brick  laid  lengthwise  at  right  angles  to  the  face  of  the  masonry. 

Heading  ;  in  tunnelling,  a  small  driftway  or  passage  excavated  in  advance  of  the  main  body  of  tnt 
tunnel,  but  forming  part  of  it;  for  facilitating  the  work. 

Headway;  the  clear  height  overhead.     Progress. 

Heel-post;  that  on  which  a  lock  gate  turns  on  its  pivot, 

Helve ;  the  handle  of  an  axe. 


ARY    OF   TERMS.  621 


ffinge;  those  epjarmonl/used  on  the  doors  of  dwellings  are  called  butts,  or  butt  hinges.  See  Ey« 
and  Strap.  Rjxftig  hinges  are  such  as  allow  the  door  to  rise  a  little  as  it  is  opened,  and  these  cause 
the  door  tpx«nut  itself. 

Hip  rdy,  or  hipped  roof;  one  that  slopes  four  ways ;  thus  forming  angles  called  hips. 

Hoarding;  a  temporary  close  fence  of  boards, "placed  around  a  work  in  progress,  to  exclud* 
Stragglers. 

Holding-plates,  or  anchors  ;  strong  broad  plates  of  iron  sunk  into  the  ground,  and  generally  sur- 
rouuded  by  masonry  ;  for  resisting  the  pull  of  the  calles  of  suspension  bridges  ;  and  for  other  simi- 
lar purposes. 

Hook  and  strap.     See  Eye  and  strap. 

Horses ;  the  sloping  timbers  which  carry  the  steps  in  a  staircase. 

Housings;  in  rolling  mills,  &c,  the  vertical  supports  for  the  boxes  in  which  the  journals  revolve. 

Hub,  or  nave;  the  central  part  of  a  wheel,  through  which,  the  axletree  passes,  and  from  whick 
the  spokes  radiate. 

Inipoat;  the  upper  part  of  a  pier  from  which  an  arch  springs. 

Ingot ;  a  lump  of  cast  metal,  generally  somewhat  wedge  shaped.    A  pig  of  cast  iron  is  an  ingot. 

Invert;  an  inverted  arch  frequently  built  under  openings,  in  order  to  distribute  the  pressure  more 
evenly  over  the  foundation. 

Jack;  a  raising  instrument,  consisting  of  an  iron  rack,  in  connection  with  a  short  stout  timber 
which  supports  it,  aud  worked  by  cog-wheels  and  a  winch.  A  screw-jack  is  a  large  screw  working 
in  a  strong  frame,  the  base  of  which  serves  for  it  to  stand  on ;  and  which  is  caused  to  revolve  and 
rise,  carrying  the  load  on  top  of  it,  by  turning  a  nut,  or  otherwise. 

Jack-raj tem,  or  common  rafters ;  small  ratters  laid  on  the  purlins  of  a  roof,  for  supporting  the 
shingling  laths,  &c. 

Jag-spike;  a  spike  whose  sides  are  jagged  or  notched,  with  the  mistaken  idea  that  its  holding  power 
is  thereby  much  increased.  If  a  spike  or  bolt  is  first  put  into  its  place  looselv,  and  then  has  melted 
Vead  run  around  it,  the  jaggiug  does  assist;  but  not  when  it  is  driven  into  wood. 

Jambs;  the  sides  of  an  opening  through  a  wall,  &c;  as  door,  window,  and  fireplace  jambs. 

Jamb-linings;  the  facing  of  woodwork  with  which  jambs  are  covered  and  hidden. 

Jaw;  an  opening,  often  V-shaped,  the  inner  edges  of  which  are  for  holding  something  in  place. 

Jettie,  or  jetty ;  a  pier,  mound,  or  mole  projecting  into  the  water;  as  a  wharf-pier,  &c. 

fib ;  the  upper  projecting  member  or  arm  of  a  crane,  supported  by  the  stay. 

Jig-saw;  a  very  narrow  thin  saw  worked  vertically  by  machinery,  and  used  for  sawing  curved 
/laments  in  boards. 

Joggle;  a  joint  like  that  at  3  or  4,  &c,  p  294,  of  Trasses,  for  receiving  the  pressure  of  a  strut  at 
rfght  angles  or  nearly  so.  Also  applied  to  squared  blocks  of  stone  sometimes  inserted  between 
Bourses  of  masonry  to  prevent  sliding,  &c. 

Joist ;  binding  joists  are  girders  for  sustaining  common  joists.  The  common  ones  are  then  called 
bridging  joists.  Ceiling  joists  are  small  ones  under  roof  trusses,  or  under  girders,  and  for  sustain- 
ing merely  the  plastered  ceiling. 

Journal-box;  a  fixture  upon  which  a  journal  rests  and  revolves,  instead  of  a  plummer-block. 

Journals  ;  the  cylindrical  supporting  ends  of  a  horizontal  revolving  shaft.  Their  length  is  usual)/ 
about  1  to  1>6  times  their  diam.  In  lines  of  shafting  4  diams.  To  find  the  diam,  see  Gudgeon. 

Jumper ;  a  drill  used  for  boring  holes  in  stone  by  aid  of  blows  of  a  sledge-hammer. 

Kedge  ;  a  small  anchor. 

Keepers  ;  the  pieces  of  metal  or  wood  which  keep  a  sliding  bolt  In  its  place,  and  guide  it  in  sliding. 

Kerf;  the  opening  or  narrow  slit  made  in  sawing. 

Key-bolt.     See  Cotter-bolt. 

Keystone  ;  the  center  stone  of  an  arch. 

Kibble;  the  bucket  used  for  raising  earth,  stone,  &c,  from  shafts  or  mines. 

King-post,  king-rod  ;  the  center  post,  vertical  piece,  or  rod,  in  a  truss  ;  all  those  on  each  side  of  it 
are  queen-posts,  or  queen-rods.  Frequently  called  simply  kings  and  queens. 

Knee  ;  a  piece  of  metal  or  wood  bent  at  an  angle ;  to  serve  as  a  bracket,  or  as  a  means  of  uniting 
two  surfaces  which  form  with  each  other  a  similar  angle. 

Lagging,  or  sheeting  ;  a  covering  of  loose  plank ;  as  that  placed  upon  centers,  and  supporting  the 
archstones.  Also,  an  outer  wooden  casing  to  locomotive  boilers  and  others. 

Landing  ;  the  resting-place  at  the  end  of  a  flight  of  stairs. 

Lantern  wheel.     See  Trundle. 

Lap;  to  place  one  piece  upon  another,  with  the  edge  of  one  reaching  beyond  that  of  the  other. 

Lap-welding;  welding  together  pieces  that  have  first  been  lapped;  in  distinction  to  butt- welding. 

Lead,  (pronounced  leed ;)  in  locomotives,  a  certain  amount  of  opening  of  the  port-valve  before 
each  stroke  of  the  piston  begins.  The  distance  to  which  earth  is  hauled  or  wheeled. 

Leading-beam;  leading-pili;  one  placed  as  a  guide  for  placing  others. 

Leading-wheels;  iu  a  locomotive,  those  frequently  placed  in  front  of  the  driving-wheels. 

Leaves ;  the  cogs  of  pinions. 

Ledge;  a  part  projecting  over  like  a  shelf;  a  rock  so  projecting.  A  narrow  strip  of  board  nailed 
across  other  boards,  to  hold  them  together,  as  in  temporary  ledge-doors. 

Lewis;  an  arrangement  composed  of  2  or  3  pieces  of  metal  let  into  a  wedge-shaped  hole  in  a  block 
of  stone,  by  which  to  raise  the  block. 

Lighter ;  a  scow,  raft,  or  other  vessel,  used  for  unloading  vessels  out  from  the  shore, 

Linch-pin;  a  pin  near  the  end  of  an  axle,  to  hold  the  wheel  on. 

Link  ;  one  of  the  divisions  of  a  chain :  or  a  piece  shaped  like  one. 

Link-motion;  a  device  for  regulating  the  movement  of  the  main  or  port  valve  in  a  locomotive. 

Lintel;  a  horizontal  beam  across  an  opening  in  a  wall,  as  seen  in  windows,  doors,  &c.  When  of 
wide  span,  and  supporting  heavy  brickwork  or  masonry,  it  is  called  a  breast-summer,  or  bressummer. 

Lock;  those  common  door  locks  which  are  entirely  concealed  within  the  thickness  of  the  door,  are 
called  mortice  locks  ;  those  which  are  screwed  against  the  face  of  a  door,  rim  locks.  It  must  be  remem- 
bered  that  locks  are  "  right  and  left." 

Louvre  ;  a  kind  of  vertical  window,  frequently  at  the  tops  of  roofs  of  depots,  &c,  provided  with  hor- 
izontal slats,  which  permit  ventilation,  and  exclude  rain. 

Lozenqe;  the  shape  of  a  rhomb  ;  often  called  diamond-shaped. 

Luq  •  in  casting,  small  projections  from  the  general  surface,  and  for  various  purposes,  such  as  for 
lifting  the  body ;  or  for  a  flange  for  joining  it  to  another ;  or  for  a  support  for  something  else. 

Mallet ;  the  wooden,  hammer  used  by  stonecutters. 

40 


622 


GLOSSARY   OP   TERMS. 


Mandrel ;  an  iron  rod  used  as  a  core  around  which  a  flat  piece  may  be  bent  into  a  cylindrical  shape 
Also  the  shaft  that  carries  the  chuck  of  a  lathe. 

Manhole;  an  opening  by  which  a  man  can  enter  a  boiler,  culvert,  £c.  to  clean  or  repair  it. 

Mattock;  a  kind  of  pick  with  broad  edges  for  digging 

Maul;  a  heavy  wooden  hammer. 

Mean,  arithmetical;  half  the  sum  of  two  numbers. 

'      ,  geometrical;  the  sq  rt  of  the  product  of  two  numbers. 

Mean-proportional ;  the  same  as  the  geometrical  mean. 

Meridian  ;  a  north  and  south  line.     Noon. 

Mitre-joint;  a  joint  formed  along  the  diagonal  line  where  the  ends  of  two  pieces  are  united  at  an 
angle  with  each  other. 

Mitre-sill;  the  sill  against  which  the  lock  gates  of  a  canal  shut. 

Modulus  ;  a  datum  serving  as  a  means  of  comparison.     Same  as  constant  or  coefficient. 

MoUulus  of  elasticity  ;  see  pp  177  :iud  632.     Modulus  of  Rupture   pp  185,  195. 

Moment ;  tendency  of  force  acting  with  leverage.     See  p  475. 

Moment  of  inertia,  of  a  body  either  revolving  (or  imagined  to  revolve)  around  an  axis,  as  a  grind, 
•tone;  or  oscillating,  like  a  pendulum.  Suppose  that  the  shortest  dist  from  the  axis  (or  from  tne 
point  of  suspension,  as  the  case  may  be,)  to  each  single  individual  particle  of  the  body,  has  been  meas- 
ured; also  that  each  of  these  dists  has  been  squared:  and  that  each  square  has  been  mult  by  the 
weight  of  that  particle  to  which  the  dist  was  measured;  also  that  all  these  last  products  are  added 
into  one  sum.  That  sum  is  the  moment  of  inertia  of  the  body;  and  is  the  I  of  scientific  writers.  In 
practice  we  may  suppose  the  body  to  be  divided  into  portions  of  a  cub  inch,  or  some  other  size ;  and  use 
these  instead  of  the  theoretical  infinitely  small  particles.  When  the  moment  of  inertia  of  a  mere  surface 
is  wanted,  (instead  of  that  of  a  solid,)  the  surf  must  be  supposed  to  be  div  into  an  infinite  number  of 
small  areas,  which  must  be  used  instead  of  the  weights  of  the  particles  of  the  solid  body.  For  an 
example,  see  page  195,  Art  25,  of  Strength  of  Materials.  If  the  moment  of  inertia  is  div  by  the  wt 
of  the  entire  solid  body  ;  or  by  the  area  of  the  entire  surf,  as  the  case  may  be  ;  the  square  root  of  the 
quot  will  be  the  radius  of  gyration,  of  the  body  or  surf.  If  the  length  of  the  rad  of  gyration  be  laid 
off  perp  from  the  axis,  or  from  the  point  of  suspension,  it  will  reach  to  the  center  of  gyration.  See 
Cen  of  Gyration,  in  this  Glossary.  The  moment  of  inertia  of  a  body  is  ~  its  wt  X  sq  of  rad  of  gyr. 

Moment  of  rupture,  or  of  bending :  the  tendency  which  any  load  or  force  exerts  to  break  or  bend  & 
body  by  the  aid  of  leverage.  Its  amount  is  found  in  foot-pounds  by  multiplying  the  force  in  fts,  by 
the  length  of  leverage  in  feet  between  it  and  that  part  of  the  body  upon  which  the  tendency  is  exerted. 

Moment  of  stability.     See  Art  69,  of  Force  in  Rigid  Bodies,  page  489. 

Momentum;  moving  force.     See  page  4W.  Force  in  Rigid  Bodies. 

Monkey ;  the  hammer  or  ram  of  a  pile-driver. 

Monkey-wrench,  or  screw-wrench;  a  spanner,  the  gripping  end  of  which  can  be  adjusted  by  means 
»f  a  screw  to  fit  objects  of  different  sizes. 

Moorings;  fixtures  to  which  ships,  &c,  can  make  fast. 

Mortise;  a  hole  cut  in  one  piece,  for  receiving  the  tenon  which  projects  from  another  piece. 

Muck;  soft  surface  soil  containing  much  vegetable  matter. 

Muntins,  or  mullions ;  the  vertical  pieces  which  separate  the  panes  in  a  window-sash. 

Nailing-blocks ;  blocks  of  wood  inserted  in  walls  of  stone  or  brick,  for  nailing  washboards,  &c,  to. 

Nave;  the  main  body  of  a  building,  having  connecting  wings  or  aisles  on  each  side  of  it.  The  hub 
of  a  wheel. 

Neioel;  the  open  space  surrounded  by  a  stairway. 

Newel-post;  a  vertical  post  sometimes  used  for 'sustaining  the  outer  ends  of  steps.  Also  the  large 
baluster  often  placed  at  the  foot  of  a  stairway. 

Nippers  ;  pincers.     An  arrangement  of  two  curved  arms  for  catching  hold  of  anything. 

Normal;  perpendicular  to.     According  to  rule,  or  to  correct  principles. 

Nosing  ;  the  slight  projection  often  given  to  the  front  edge  of  the  tread  ot  a  step  ;  usually  rounded. 

Nut,  or  burr ;  the  short  piece  with  a  central  female  screw,  used  on  the  end  of  a  screw-bolt,  &c,  for 
keeping  it  in  place. 

Ogee;  a  moulding  in  shape  of  an  S.  the  same  as  a  cima. 

Ordinate  ;  a  line  drawn  at  right  angles  from  the  axis  of  a  curve,  and  extending  to  the  curve. 

Oscillate ;  to  swing  backward  and  forward  like  a  pendulum. 

Out  of  wind,  pronounced  wynd ;  perfectly  straight  or  flat. 

Ovolo ;  a  projecting  convex  moulding  of  quarter  of  a  circle ;  when  it  is  concave  it  is  a  cavetto,  or 
hollow. 

Packing  ;  the  material  placed  in  a  stuffing-box,  &c,  to  prevent  leaks. 

Packing-pieces ;  short  pieces  inserted  between  two  others  which  are  to  be  riveted  or  bolted  together, 
to  prevent  their  coming  in  contact  with  each  other. 

Pall,  or  pawl.     See  Ratchet. 

Parapet;  a  wall  or  any  kind  of  fence  or  railing  to  prevent  persons  from  falling  off. 

Parcel;  to  wrap  canvas  or  rags  round  a  ropp. 

Parge ;  to  make  the  inside  of  a  flue  smooth  by  plastering  it. 

Patent  hammer  ;  a  hammer  with  several  parallel  sharp  edges  for  dressing  stone. 

Pay.     To  cover  a  surface  with  tar,  pitch,  &c.     A  ship  word. 

Pay  out.     To  slacken,  or  let  out  rope. 

Pediment;  the  triangular  space  in  the  face  of  a  wall  that  is  included  between  the  two  sloping  sides 
«f  the  roof  and  a  line  joining  the  eaves. 

Penstock.     See  Forebay. 

Pier;  the  support  of  two  adjacent  arches.  The  wall  space  between  windows,  &c.  A  structure  built 
«ut  into  the  water. 

Pierre  perdue ;  lost  stone  ;  random  stone,  or  rough  stones  thrown  into  the  water,  and  let  find  their 
own  slope. 

Pilaster ;  a  thin  flat  projection  from  the'  face  of  a  wall,  as  a  kind  of  ornamental  substitute  for  a 
column. 

Pile-planks;  planks  driven  like  piles. 

Pillow-block,  or plummer- block;  a  kind  of  metal  chair  or  support,  upon  which  the  journals  of  hor- 
izontal shafts  are  generally  made  to  rest,  and  on  which  they  revolve. 

Pinion;  a  small  cog  wheel  which  gives  motion  to  a  larger  one. 

Pintle  ;  a  vertical  projecting  pin  like  that  often  placed  at  the  tops  of  crane-posts,  and  over  which 
the  holding  rings  at  the  tops  of  the  wooden  guys  fit.  Also,  such  as  is  used  for  the  hinges  of  rudders 
or  of  window-shutters  to  turo  around. 


5ARY    OF    TERMS.  623 

Pitch  ;  the  slope  of  a  roof^/.  The  distance  from  center  to  center  of  the  teeth  of  a  cog-wheel,  or 
the  threads  of  a  screw^-'BoilqA  tar.  Also  the  dist  apart  of  rivets,  &c. 

Pitman;  a  conja«Cung-rod  /or  transmitting  motion  from  a  prime  mover  to  machinery  at  a  distance, 
moved  by  Mu^ 

Pit-s^w ;  a  large  saw  worked  vertically  by  two  men,  one  of  whom  (the  pitman)  stands  in  a  pit. 

Pivot;  the  lower  end  of  a  vertical  revolving  shaft,  whether  a  part  of  the  shaft  hsr.lf,  or  attached  to 
it.  It  should  be  tlat;  and  both  ic  aud  the  step  or  socket  upon  which  it  rests  should  be  of  hard  steel. 
If  a  steel  pivot  has  to  revolve  rapidly  aud  continuously,  it  is  well  to  proportion  its  diaui,  so  as  not  to 
have  to  sustain  more  than  250  fts  per  sq  inch;  otherwise  it  will  wear  quicklv.  Dust  and  grit  should 
for  the  same  reason  be  carefully  guarded  against.  Pivots  which  revolve  but  seldom,  and  slowly,  as 
those  of  a  railroad  turntable,  may  be  trusted  with  half  a  ton.  or  even  a  whole  ton  per  sq  inch.  As  a 
rude  rule,  cast-iron  pivots  should  not  be  loaded  with  more  than  half  as  much  as  steel  ones.  A  steel 
one  may  be  welded  to  the  foot  of  its  cast-iron  shaft;  or  may  be  inserted  part  way  into  it;  and  the 
whole  strengthened  by  iron  bands  shrunk  on. 

Planish;  to  polish  metals  by  rubbing  with  a  hard  smooth  tool. 

Plant;  the  outfit  of  machinery,  &c,  necessary  for  carrying  on  any  kind  of  work. 

Plaster-bead;  a  small  vertical  strip  of  iron  or  wood  nailed  along  projecting  angles  in  rooms,  to 
protect  the  plaster  at  those  parts. 

Platband;  a  plain,  flat,  wide,  slightly  projecting  strip,  generally  for  ornament.  When  narrow,  It 
is  called  a  fillet. 

Pliers  ;  a  kind  of  pincers. 

Plinth  ;  the  square  lowest  member  of  the  base  of  a  column  or  pillar. 

Plug ;  a  piece  inserted  to  stop  a  hole.     Screw-plug,  a  plug  that  is  screwed  into  a  hole. 

Plumb ;  vertical. 

Plummet,  or  plumb-bob ;  a  weight  at  the  lower  end  of  a  string,  for  testing  verticality. 

Plunger  ;  a  kind  of  solid  piston,  or  one  without  a  valve. 

Point;  a  kind  of  pointed  chisel  for  dressing  stone.  To  put  a  finish  to  masonry  by  touching  up  the 
outer  mortar  joints.  To  dress  stone  with  a  point  and  mallet. 

Pole-plate ;  a  longitudinal  timber  resting  on  the  ends  of  tie-beams  of  roofs ;  and  for  supporting 
the  feet  of  the  common  or  jack  rafters,  when  such  are  used. 

Port;  the  opening  or  passage  controlled  by  a  valve. 

Prime ;  to  put  on  the  first  coat  of  paint.  Priming  also  is  when  water  passes  into  a  steam  cylinder 
along  with  the  steam. 

Projection.  If  parallel  straight  lines  be  imagined  to  be  drawn  in  any  one  given  direction,  from 
every  point  in  any  surface  s.  whether  flat,  curved,  or  irregular,  then  if  all  these  lines  be  supposed  to 
be  intersected  or  cut  by  a  plane,  either  at  right  angles  to  their  direction,  or  obliquely,  the  figure 
which  their  cross-section  thus  made  would  form  upon  said  plane,  is  called  the  projection  of  the  sur- 
face s.  If  such  lines  be  supposed  to  be  drawn  from  a  person's  face,  in  a  direction  in  front  of  him, 
and  to  be  cut  by  a  plane  at  right  angles  to  their  direction,  their  projection  on  the  plane  would  be  the 
person's  full-face  portrait.  If  the  lines  be  drawn  sideways  from  his  face,  the  projection  will  be  his 
profile.  The  projection  of  a  globe  upon  a  flat  plane,  will  evidently  be  a  circle  if  the  plane  cuts  the 
lines  at  right  angles  ;  and  an  ellipse  if  it  cuts  them  obliquely.  Shadows  cast  by  the  sun  are  projec- 
tions. 

Proportion.     See  Ratio. 

Puddle;  earth  well  rammed  into  a  trench,  &c,  to  prevent  leaking.  A  process  for  converting  cast 
iron  into  wrought  by  a  puddling  furnace. 

rufi-mill ;  a  mill  for  tempering  clay  for  bricks  or  pottery,  &c. 

Pulley  ;  a  circular  hoop  which  carries  a  belt  in  machinery. 

Puppet;  in  machinery,  a  small  short  pedestal  or  stand.     Puppet-valve.  See  Valves. 

Purlins;  the  horizontal  pieces  placed  on  rafters,  for  supporting  the  roof  covering. 

Piit-loys,  or put-lockft ;  horizontal  pieces  supporting  the  floor  of  a  scaffold;  one  end  being  inserted 
into  put-log  hole*  left  for  that  purpose  in  the  masonry. 

Quay;  a  wharf. 

Quoin;  the  hollow  into  which  a  quoin-post  of  a  canal  lock-gate  fits.  Stones,  usually  dressed,  placed 
along  the  vertical  angles  of  buildings,  chiefly  for  ornament. 

Quoin-pout ;  the  vertical  post  on  which  a  lock-gate  turns.     The  heel-pflst. 

Rabbet,  or  rebate ;  a  half  groove  along  the  edge  of  a  board,  &c.  See  16,  p  294,  of  Trusses,  where 
two  rabbets  are  shown  overlapping  each  other. 

Race;  the  channel  whioh  conducts  water  either  to  or  from  n  water-wheel :  the  first  is  a  head  race ; 
the  last  a  tail  race.  The  waves  produced  by  the  meeting  of  strong  opposing  currents ;  also,  a  rapid 
tidewav,  or  roost. 

Rtirk  and  pinion ;  the  rack  is  a  straight  row  of  cogs  on  a  bar,  and  called  a  rack-bar ;  and  the  pinion 
is  a  small  cog-wheel  working  into  u.  /moment  of  inertia 

Radius  of  gyration.     See  Center  of  gyration.    Kad  of  gyr  is  ~    /  — 

Rag-bolt.     See  Jag-spike.  N     weight  of  body 

Raft-wheel ;  one  with  teeth  or  pins  which  catch  into  the  links  of  a  chain. 

Rails  ;  the  horizontal  pieces  in  a  door. 

Ram  ;  the  hammer  of  a  pile-driver. 

Random  stone ;  rip  rap,  or  rough  stones  thrown  promiscuously  into  the  water,  to  form  a  founda- 
tion, Ac. 

Rasp  ;  a  coarse  file. 

Ratchet  and  pall ;  the  former  is  sometimes  a  straight  bar,  at  others  a  wheel :  in  either  case  it  ii 
furnished  with  teeth  between  which  the  pall  drops  aud  prevents  backward  motion.  Used  for  safety 
in  hoisting  machinery,  &c.  The  pall  is  sometimes  called  a  click. 

Ratio.  Simple  ratio  is  a  number  denoting  how  often  one  quantity  is  contained  in  another.  Thus, 
the  ratio  of  5  to  10  is  J>^,  or  i  ;  and  the  ratio  of  10  to  5  is  \P ,  or  2.  When,  of  four  numbers,  two 
have  to  each  other  the  same  ratio  that  the  other  two  have,  the  numbers  are  said  to  be  in  proportion 
to  each  other.  Thus,  6  has  the  same  ratio  (2)  to  3.  as  100  has  to  50 ;  therefore,  6,  3,  100,  and  50,  are 
said  to  be  in  proportion  ;  or,  as  6  :  3  : :  100  :  50.  In  other  words,  an  equality  of  ratios  ia  called  pro- 
portion. Ratio  and  proportion  are  often  confounded  with  one  -another;  but  the  error  is  one  of  no 
importance.  Duplicate  ratio  is  that  of  the  squares  of  numbers. 

Ream.  A  hole,  wider  at  top  than  at  bottom,  (see  19.  p  '294,  of  Trusses,)  through  which  a  screw, 
bolt,  <ferc,  is  to  be  inserted,  so  that  Its  head  shall  not  project  above  the  general  surface,  is  said  to  be 
reamed,  or  reampd  out ;  or  to  be  countersunk. 

Reciprocal  of  a  number  is  the  quotient  found  by  dividing  1  by  that  number. 

Reciprocating  motion.    See  Alternating  motion. 


624 


GLOSSARY   OF   TERMS. 


Re-entering  angle  ;  an  angle  or  corner  projecting  inward. 

Revetment ;  steep  facing  of  stoue  to  the  sides  of  a  ditcb  or  parapet  in  fortification.  A  retaining- wall. 

Rib  ;  the  curved  pieces  which  form  the  arches  of  irou  or  woodeu  bridges,  &c.  Also,  tuose  to  which 
the  outer  planking  of  a  sailing  vessel,  <fec,  are  fastened. 

Ridge  of  a  roof;  its  peak,  or  the  sharp  edge  along  its  very  top.     Has  various  similar  applications. 

Ridge-pole,  ridge-piece,  or  ridge-plate;  the  highest  horizontal  timber  in  a  roof,  extending  from  top 
to  top  of  the  several  pairs  of  rafters  of  the  trusses ;  for  supporting  the  heads  of  the  jack-rafters. 

Right  and  left ;  a  lock  which  in  its  proper  position  suits  one  flap  of  a  pair  of  folding  doors,  will 
not  suit  if  fastened  to  the  other  flap  ;  nor  even  to  the  same  flap  if  required  to  open  to  the  right  in- 
stead of  to  the  left,  or  vice  versa,  according  to  whether  it  is  a  right  or  a  left-hand  lock.  And  so  with 
many  other  things,  as,  for  instance,  certain  arrangements  for  working  railway  switches,  &c.  Right 
and  left  boots  and  shoes  are  a  familiar  illustration  ;  also,  right  and  left  screws.  Therefore,  in  order- 
ing several  of  anything,  it  is  necessary  to  consider  whether  they  may  all  be  of  the  same  pattern,  or 
whether  some  must  be  right-hand,  and  others  left-hand  ones. 

Right  shore  of  a  river;  that  which  is  on  the  right  hand  when  descending  the  river. 

Right-solid  body  ;  one  which  has  its  axis  at  right  angles  to  its  base;  when  not  so,  it  is  oblique. 

Ring-bolt;  a  bolt  with  an  eye  and  a  ring  at  one  end. 

Rip-rap.    See  Random  stone. 

Roadstead;  anchorage  at  some  distance  from  shore. 

Rock-shaft ;  a  shaft  which  only  rocks  or  makes  part  of  a  revolution  each  way,  instead  of  revolving 
eutirely  around. 

Rockwork;  squared  masonry  in  which  the  face  is  left  rough  to  give  a  rustic  appearance. 

Rubble  ;  masonry  of  rough,  undressed  stones.  Scabbled  rubble  has  only  the  roughest  irregularities 
knocked  off  by  a  hammer.  Ranged  rubble  has  the  stones  in  each  course  rudely  dressed  to  nearly  a 
uniform  height. 

Rundle,  or  round  ;  the  step  of  a  ladder. 

Rustic;  much  the  same  as  rockwork. 

Saddle;  the  rollers  and  fixtures  on  top  of  the  piers  of  a  suspension  bridge,  to  accommodate  ex- 
pansion and  contraction  of  the  cables.  The  top  piece  of  a  stoue  cornice  of  a  pediment.  Has  many 
other  applications. 

Sag  ;  to  bend  downward. 

Salient;  projecting  outward. 

Sandbag  ;  a  bag  filled  with  sand  for  stopping  leaks. 

Scabble;  to  dress  off  the  rougher  projections  of  stones  for  rubble  masonry,  with  a  stone-axe,  or 
scabbling  hammer. 

Scantling ;  the  depth  and  breadth  of  pieces  of  timber ;  thus  we  say,  a  scantling  of  8  by  10  ins,  &c. 

Scarf;  the  uniting  of  two  pieces  by  a  long  joint,  aided  by  bolts,  &c. 

Scarp;  a  steep  slope.     In  fortification,  the  inner  slope  of  a  ditch. 

Scotia ;  a  receding  moulding  consisting  of  a  semi-circle  or  semi-ejlipse,  or  similar  figure. 

Screeds  ;  long  narrow  strips  of  plaster  put  on  horizontally  along  a  wall,  and  carefully  faced  out  of 
wind,  to  serve  as  guides  for  afterward  plastering  the  wide  intervals  between  them. 

Screw-bolt ;  a  bolt  with  a  screw  cut  on  one  end  of  it. 

Screiv-jack.     See  Jack. 

Screw-wrench.     See  Wrench. 

Scribe ;  to  trim  off  the  edge  of  a  board,  &c,  so  as  to  make  it  fit  closely  at  all  points,  to  an  irregular 
surface.  The  lower  edges  of  an  open  caisson  are  scribed  to  fit  the  irregularities  of  a  rocky  river  bottom. 

Scroll;  an  ornamental  form  consisting  of  volutes  or  spirels  arranged  somewhat  in  the  shape  of  S. 

Scupper  nails  ;  nails  with  broad  heads  for  nailing  down  canvas,  &c. 

Scuppers  ;  on  shipboard,  holes  for  allowing  water  to  flow  off  from  the  deck  into  the  sea. 

Scuttle  ;  a  small  hatchway.     To  make  holes  in  a  vessel  to  cause  sinking. 

Sea-wall;  a  wall  built  to  prevent  encroachment  of  the  sea. 

Secret  nailing  ;  so  nailing  down  a  floor  by  nails  along  the  edges  of  the  boards,  that  the  nail-heads 
do  not  show. 

Serve  ;  to  wrap  twine  or  yarn,  Ac,  closely  round  a  rope  to  keep  it  from  rubbing. 

Set-screw,  or  tightening- screw ;  a  screw  for  merely  pressing  one  thing  tightly  against  another  at 
will ;  such  as  that  which  confines  the  movable  leg  of  a  pair  of  dividers  in  its  socket. 

Shackle,  or  clevis;  a  link  in  a  chain  shaped  like  a  U,  and  so  arranged  that  by  drawing  ont  a  bolt 
or  pin,  which  fits  into  two  holes  at  the  ends  of  the  U,  the  chain  can  be  separated  at  that  point. 

Shaft ;  a  vertical  pit  like  a  well.     The  body  of  a  column.     A  large  axle. 

Shank;  the  body  of  a  bolt  exclusive  of  its  bead.  The  long  straight  part  of  many  things,  as  of  an 
anchor,  a  key,  Ac. 

Shears,  or  sheers;  two  tall  timbers  or  poles,  with  their  feet  some  distance  apart,  and  their  tops 
fastened  together;  and  supporting  hoisting  tackle. 

Sheave;  a  wheel  or  round  block  with  a  groove  around  its  circumference  for  guiding  a  rope. 

Sheeting,  or  sheathing ;  covering  a  surface  with  boards,  sheet  iron,  felt,  &c. 

Shingle  ;  the  pebbles  on  a  seashore. 

Shoes;  certain  fittings  at  the  ends  of  pieces  ;  as  the  pointed  iron  shoes  for  piles.  The  wall  shoes 
into  which  the  lower  ends  of  iron  rafters  generally  fit,  &c. 

Shore;  a  prop. 

Shot ;  the  edge  of  a  board  is  said  to  be  shot  when  it  is  planed  perfectly  straight. 

Shrink.  When  an  iron  hoop  or  band  is  first  heated,  and  then  at  once  placed  upon  the  body  which 
it  is  intended  to  surround,  it  shrinks  or  contracts  as  it  cools,  and  therefore  clasps  the  body  more  firmly. 
This  is  called  shrinking  on  the  hoop. 

Shuttle ;  a  small  gate  for  admitting  water  to  a  water-wheel,  or  out  of  a  canal  lock,  *o. 

Siding ;  a  short  piece  of  railroad  track,  parallel  to  the  main  one,  to  serve  as  a  passing-place. 

Silt ;  soft  fine  mud  deposited  by  rivers,  &c. 

Siphon  culvert;  a  culvert  built  in  shape  of  a  U,  for  carrying  a  stream  under  *n  obstaHe.  and  allow- 
ing  it  afterward  to  rise  again  to  its  natural  level.  The  term  is  improper,  inasmuch  as  the  principle 
of  the  siphon  is  not  involved. 

Skewback;  the  inclined  stone  from  which  an  arch  springs. 

Skids;  vertical  fenders,  on  a  ship's  sides.     Two  parallel  timbers  for  rolling  things  upon. 

Skirting;  narrow  boards  nailed  along  a  wall,  as  the  washboards  in  dwellings. 

Sledge';  a  heavy  hammer. 

Sleeper;  any  lower  or  foundation  piece  in  contact  with  the  ground. 

Sleeve;  a  hollow  cylinder  slid  over  two  pieces  to  hold  them  together. 


ARY   OF   TERMS.  625 


Slide-bars,  or  sWdarfbjlrs  for  anything  to  slide  along;  as  those  for  the  cross-heads  of  piston-rods, 
Ac.  Often  called^nides. 

Slings ;  piecerf'of  rope  or  chain  to  be  put  around  stones,  &c,  for  raising  them  by. 

Slip;  th^tidiug  down  of  the  sides  of  earth-cuts  or  banks.  A  long  narrow  water  space  or  dock 
between  two  wharf  piers. 

Slope-wall;  a  wall,  generally  thin  and  of  rubble  stone,  used  to  preserve  slopes  from  the  action  of 
water  in  the  banks  of  cauuls,  rivers,  reservoirs,  &c  ;  or  from  the  action  of  rain. 

Slot;  a  long  narrow  hole  cut  through  anything. 

Sluice:  a  water-channel  of  wood,  masonry,  &c;  or  a  mere  trench.  The  flow  is  usually  regulated 
by  a  sluice-gate. 

Smoke- box;  in  locomotives,  that  space  in  front  of  the  boiler,  through  which  the  smoke  passes  to 
the  chimney. 

Snag  ;  a  lug  with  a  hole  through  it,  for  a  bolt. 

Socket;  a  cavity  made  iu  one  piece  for  receiving  a  projection  from,  or  the  end  of,  another  piece;  as 
that  into  which  the  movable  leg  of  a  pair  of  dividers  fits. 

Soffit;  the  lower  or  underneath  surface  of  an  arch,  cornice,  window,  or  door-opening,  &c. 

Solder;  a  compound  of  different  metals,  which  when  melted  is  used  for  uniting  pieces  of  metal  also 
heated.  Soft  solder  is  a  compound  of  lead  and  tin,  and  is  used  for  uniting  lead  or  tin.  There  are 
various  hard  solders,  such  as  spelter  solder,  composed  of  copper  and  zinc,  for  uniting  iron,  copper,  or 
brass. 

Sole  ;  that  lining  around  a  water-wheel  which  forms  the  bottoms  of  the  buckets. 

Spandrel ;  the  space,  or  the  masonry,  &c,  between  the  back  or  extrados  of  an  arch  and  the  roadway. 

Spanner;  a  kind  of  wrench,  consisting  of  a  handle  or  lever  with  a  square  eye  at  one  end  of  it;  much 
used  for  tightening  up  the  nuts  upon  screw-bolts,  &c.  The  eye  fits  over  or  surrounds  the  nut. 

Spar;  a  beam  ;  but  generally  applied  to  round  pieces  like  masts,  &c. 

Spelter;  zinc. 

Spigot ;  the  pin  or  stopper  of  a  faucet.    The  smaller  end  of  a  common  cast-iron  water  or  gas  pipe. 

Spindle;  a  thin  delicate  shaft  or  axle. 

Splay ;  to  widen  or  Hare,  like  the  jambs  of  a  common  fireplace,  or  those  of  many  windows  ;  or  like 
the  wing- walls  of  most  culverts. 

Splice;  to  unite  two  pieces  firmly  together. 

Spokes ;  the  radii  of  wheels. 

Springer ;  the  lowest  stone  of  an  arch. 

Sp^lr-wheel;  a  common  cog-wheel,  in  which  the  teeth  radiate  from  a  common  cen.like  thoseof  a  spur. 

Square-head;  a  square  termination  like  that  upon  which  a  watch-key  fits  for  winding;  or  that 
upon  which  the  eye  of  the  handle  of  a  common  grindstone  fits  for  turning  it,  &c. 

Staging  ;  the  temporary  flooring  of  a  scaffold,  platform,  &c. 

Stanchion;  a  vertical  prop  or  strut. 

Standing- bolt,  or  stud-bolt ;  a  bolt  with  a  screw  cut  upon  each  end;  one  end  to  be  screwed  perma- 
nently into  something,  and  the  other  end  to  hold  by  means  of  a  nut  something  else  that  may  be  re- 
quired to  be  removed  at  times. 

Stand-pipe;  a  stand-pipe  is  sometimes  used  for  the  same  purpose  as  an  air-vessel ;  which  see.  It 
ts  a  tall  pipe,  open  to  the  air  at  top  ;  and  communicating  freely  at  its  toot  with  the  water-pipe,  in 
the  same  manner  as  in  an  air-vessel.  Its  top  must  be  somewhat  higher  than  that  to  which  the 
pump  has  to  force  the  water  through  the  system  of  pipes  ;  otherwise  the  water  would  be  wasted  by 
Mowing  over  its  top.  The  area  of  its  transverse  section  should  be  at  least  equal  to  that  of  the  pipe  or 
pipes  which  conduct  the  water  from  it;  but  it  is  at  times  better  to  have  it  much  larger,  as  a  stand- 
pipe  may  then  answer,  especially  in  a  small  town,  as  a  reservoir,  if  the  punrping  should  cease  for  a 
few  hours.  A  stand-pipe  should'be  cylindrical,  not  conical ;  for  if  thick  ice  should  form  on  top  of  the 
water  in  a  conical  one,  a  sudden  forcing  of  it  upward  by  the  pump  might  strain  the  stand-pipe  seri- 
ously. There  are  four  or  five  stand-pipes  connected  with  the  Philadelphia  Water- Works.  They  are 
(with  one  exception)  cylindrical,  from  125  to  170  ft  high  ;  5  ft  diam  ;  and  made  of  riveted  boiler-iron 
ibout  %  inch  thick  near  the  base,  and  about  %  inch  near  the  top.  They  have  no  protection  from  the 
weather;  nor  are  thev  braced  in  any  manner:  but  retain  their  positions  by  their  own  inherent 
ureugth,  although  exposed  at  times  to  violent  winds. 

Staple;  a  kind  of  double  pin  in  shape  of  a  U ;  its  two  sharp  points  are  driven  into  timber,  and 
fcurved  part  is  left  projecting,  to  receive  a  hoop,  pin,  or  hasp,  &c. 

Starlings  ;  the  projecting  up  and  down-stream  ends  or  cutwaters  of  a  bridge  pier. 

Stay;  variously  applied  to  props,  struts,  and  ties,  for  staying  anything  or  keeping  it  in  place. 

Stay-bolt*;  long  bolts  placed  across  the  inside  of  a  boiler,  &c,  to  give  it  greater  strength. 

Strum-chest;  the  iron  box  in  locomotive  engines  and  others,  through  which  the  steuiu  is  admitted 
V>  the  cylinders 

Steam-pipe ;  the  one  which  leads  steam  from  a  boiler  to  the  steam-chest. 

Step  ;  a  cavity  in  a  piece  for  receiving  the  pivot  of  an  upright  shaft ;  or  the  end  of  any  upright  piece. 

Stiles ;  the  flat  vertical  pieces  between  and  at  the  sides  of  the  panels  in  doors.  &c. 

Stork;  the  eye  with  handles  for  turning  it,  in  which  the  dies  for  the  cutting  of  screws  are  held. 

Stone.-itp,  or  stoved.  or  upset;  when  a  rod  of  iron  is  heated  at  one  end.  and  then  hammered  end- 
wise so  that  that  part  becomes  of  greater  diameter  or  stouter  than  the  remainder.  The  heads  of  bolts 
are  frequently  made  in  one  piece  with  the  shank  in  this  way  ;  and  the  screw  ends  of  long  screw-rods 
are  often  upset,  so  that  the  cutting  of  the  threads  of  the  screw  mny  not  reduce  the  strength  of  the  bar. 

Strap;  a  long  thin  narrow  piece  of  metal  bolted  to  two  bodies'  to  hold  them   together.     A  strap- 
hinge  is  a  strap  fastened  to  a  shutter.  &c,  and  having  an  eye  or  a  pin  at  one  end  for  fitting  it  to  the 
jther  part  of  the  hinge  which  is  attached  to  the  wall. 
Stratum  ;  a  layer,  or  bed ;  as  the  natural  ones  in  rocks,  &c. 

Stretcher ;  a  brick,  or  a  block  of  masonry  laid  lengthwise  of  a  wall.  A  frame  for  stretching  any 
)hing  upon. 

Stretcher-course ;  a  course  of  masonry  all  of  stretchers,  without  any  headers. 

Strike;  an  imaginary  horizontal  line  drawn  upon  the  inclined  face  of  a  stratum  of  rocks.    Thus, 
If  the  slates  or  shingles  on  a  roof  represent  inclined  strata  of  rocks,  then  either  the  ridge  or  the  eaves 
jf  the  roof,  or  any  horizoutal  line  between  them,  will  represent  their  strike.     The  inclination  ia 
walled  the  dip  of  the  strata;  and  the  strike  is  always  at  right  angles  to  it  by  compass. 
String  ;  variously  applied  to  longitudinal  pieces. 

String  board;  the  boarding  (often  ornamented)  at  the  outer  ends  of  steps  in  staircases.  It  hides 
the  horses,  as  the  inclined  timbers  which  carry  the  steps  are  called. 

String-course;  a  long  horizontal  course  of  brick  or  masonry  projecting  a  little  beyond  the  others  ; 
and  often  introduced  for  ornament. 


626 


GLOSSARY   OF   TERMS. 


Strut ;  a  prop.     A  piece  that  sustains  compression,  whether  vertical  or  inclined. 

Strut-tie,  or  tie-strut ;  a  piece  adapted  to  sustain  both  tension  and  compression. 

Stub-end;  a  blunt  end. 

Stud ;  a  short  stout  projecting  pin.     A  prop.     The  vertical  pieces  in  a  stud  partition. 

Stud-bolt.     See  Standing-bolt. 

Stuffing-box;  a  small  boxing  on  the  end  of  a  steam  cylinder,  and  surrounding  the  piston-rod  like 
a  collar  ;  or  in  other  positions  where  a  rod  is  required  to  move  backward  and  forward,  or  to  revolve, 
In  an  opening  through  any  kind  of  partition,  without  allowing  the  escape  of  steam,  air,  or  water,  &c, 
as  the  case  may  be.  The  box  is  filled  with  greased  hemp  or  other  packing,  which  is  kept  pressed  close 
around  the  moving  rod  by  means  of  a  top-piece  or  kind  of  cover  called  the  gland,  which  may  be 
screwed  down  more  or  less  tightly  upon  it  at  pleasure.  The  rod  passes  through  the  gland  also.  Se« 
Figs  40  of  Hydraulics,  where  b  b  is  the  gland  of  the  stuffing-box  just  below  it  between  taud  t ;  and  tha 
small  circle  shows  the  cavity  for  the  packing  or  stuffing. 

Sumpt,  or  sump;  a  draining  well  into  which  rain  or  other  water  may  be  led  by  little  ditches  from 
different  parts  of  a  work  to  which  it  would  do  injury. 

Surbase;  the  inside  horizontal  mouldings  just  under  a  window-sill.  Also  those  around  the  top  of  a 
pedestal,  or  of  wainscoting,  &c. 

Swage,  or  swedge;  a  kind  of  hammer,  on  the  face  of  which  is  a  semi-cylindrical,  or  other  shaped 

S:oove  or  indentation  ;  and  which,  beiug  held  upon  a  piece  of  hot  iron  and  struck  by  a  heavy  hammer, 
ave.s  the  shape  of  the  indentation  upon  the  iron. 

Switch, ;  the  movable  tongue  or  rail  by  which  a  train  is  directed  from  one  track  to  another. 

Swivels;  devices  for  permitting  one  piece  to  turn  readily  in  various  directions  upon  another,  with- 
out danger  of  entanglement  or  separation.  At  13,  p  265,  of  Trusses,  is  a  tightening  swivel;  the 
castors  under  the  legs  of  heavy  furniture  are  swivelled  rollers. 

Synclinal  axis ;  in  geology,  a  valley  axis,  or  one  toward  which  the  strata  of  rocks  slope  downward 
from  opposite  directions.  The  line  of  the  gutter  in  a  valley  roof  may  represent  such  an  axis. 

Ts ;  pieces  of  metal  in  that  shape,  whether  to  serve  aa  straps,  or  for  other  purposes.  So  also  with 
L's.  S's,  Ws,  -f-'s,  &c.  See  figs  to  Welded  Iron  Tubes,  p  365. 

Tackle;  a  combination  of  ropes  and  pulleys. 

Talus ;  the  same  as  batter. 

Tamp;  to  fill  up  with  sand  or  earth,  &c,  the  remainder  of  the  hole  in  which  the  powder  has  been 
poured  for  blasting  rock. 

Tap;  a  kind  of  screw  made  of  hard  steel,  and  having  a  square  head  which  may  be  grasped  by  a 
wrench  for  turning  it  around,  and  thus  forcing  it  through  a  hole  around  the  inside  of  which  it  cuts  an 
interior  screw.  To  strike  with  moderate  force.  To  make  an  opening  in  the  side  of  any  vessel. 

Tappet;  a  pin  or  short  arm  projecting  from  a  revolving  shaft;  or  from  an  alternating  bar,  and  in- 
tended to  come  into  contact  with,  or  tap,  something  at  each  revolution  or  stroke. 

Teeth ;  or  cogs  of  wheels. 

Temper ;  to  change  the  hardness  of  metals  by  first  heating,  and  then  plunging  them  into  water,  oil, 
&c.  To  mix  mortar,  or  to  prepare  clay  for  bricks,  &c. 

Templet;  the  outline  of  a  moulding  or  other  article,  cut  out  of  sheet  metal  or  thin  wood,  to  serve 
as  a  pattern  for  stonecutters,  carpenters,  &c. 

Tenon;  a  projecting  tongue  fitting  into  a  corresponding  cavity  called  a  mortise. 

Terracotta;  baked  clay.     Brick  is  a  coarse  kind. 

Thimble ;  an  iron  ring  with  its  outer  face  curved  into  a  continuous  groove.  A  rope  being  doubled 
around  this  and  tied,  the  thimble  acts  as  an  eye  for  it,  and  prevents  that  part  of  the  rope  from  wear- 
ing. Also,  a  short  piece  of  tube  slid  over  another  piece,  or  over  a  rod,  &c,  to  strengthen  a  joint,  &c. 

Thread ;  the  continuous  spiral  projection  or  woi-m  of  a  screw. 

Through- stone;  a  stone  that  extends  entirely  through  a  wall. 

Throw;  the  radius,  or  distance  to  which  a  crank  "  throws  out "  its  arm.  Applies  in  the  same  way 
to  lathes.  Some  use  it  to  express  the  diameter  instead  of  the  radius.  To  avoid  mistakes,  the  terms 
"single"  and  "double"  throw  might  be  used. 

Tides;  those  well-known  rises  and  falls  of  the  surface  of  the  sea  and  of  some  rivers,  caused  by  the 
attraction  of  the  sun  and  moon.  There  are  two  rises,  floods,  or,  high  tides ;  and  two  falls,  ebbs,  or  low 
tides,  every  24  hours  and  50  minutes  ;  a  lunar  day  j  making  the  average  of  6  hours  12V6  minutes  be- 
tween high  and  low  water.  These  intervals  are,  however,  subject  to  great  variations;  as  are  also  the 
heights  of  the  tides ;  and  this  not  only  at  diif^rent  places,  but  at  the  same  place.  These  irregularities 
are  owing  to  the  shape  of  the  coast  line,  the  depth  of  water,  winds,  and  other  causes.  Usually  at  new 
and  full  moon,  or  rather  a  day  or  two  after,  (or  twice  in  each  lunar  month,  at  intervals  of  two  weeks,) 
the  tides  rise  higher,  and  fall  lower  than  at  other  times;  and  these  are  called  spring  tides.  Also,  one 
or  two  days  after  the  moon  is  in  her  quarters,  twice  in  a  lunar  mouth,  they  both  rise  andfall  lass  than 
at  other  times;  and  are  then  called  neap  tides.  From  neap  to  spring  they  rise  and  fall  more  daily; 
and  vice  versa.  The  time  of  high  water  at  any  place,  is  generally  two  or  three  hours  after  the  moon  has 
passed  over  either  the  upper  or  lower  meridian  ;  and  is  called  the  establishment  of  that  place ;  because, 
when  this  time  is  established,  the  time  of  high  water  on  any  other  day  may  be  found  from  it  in  most 
cases.  The  total  height  of  spring  tides  is  generally  from  1^  to  2  times  as  great  as  that  of  neaps.  The 
great  tidal  wave  is  merely  an  undulation,  unattended  by  any  current,  or  progressive  motion  of  the 
particles  of  water.  Each  successive  high  tide  occurs  about  24  minutes  later  thau  the  preceding  one; 
and  so  with  the  low  tides. 

Tie;  any  piece  that  sustains  tension  or  pull. 

Tightning-ring.     See  14,  of  Figs  21^,  of  Trusses,  p  265. 

Tightning- screw.     See  Set-screw. 

Tire ;  the  iron  ring  placed  around  the  outer  circumference  of  the  felloe  of  a  wheel. 

Toggle  joint.  In  Fig  16,  of  Force  in  page  461,  suppose  a  TO  and  a  n  to  be  two  stiff  bars,  hinged 
together  at  a.  It  is  plain  that  if  we  press  downward  at  a,  the  result  will  be  a  great  pushing  force 
against  any  bodies  placed  at  the  ends  m  and  n.  Such  an  arrangement  of  two  bars  for  producing  such 
pressure  is  a  toggle-joint. 

Tongue ;  a  long  slightly  projecting  strip  to  be  inserted  into  aoorresponding  groove,  as  in  tongued 
and  grooved  floors. 

Tooling  ;  dressing  stone  by  means  of  a  tool  and  mallet;  the  tool  being  a  chisel  with  a  cutting  edge 
of  I  to  2  inches  wide.  Tooling  is  generally  done  in  parallel  stripes  across  the  stone. 

Torus;  a  projecting  semi-circular,  or  semi-elliptic  moulding;  often  used  in  the  bases  of  columns, 
It  is  the  reverse  of  a  scotia. 

Trailing -wheels  ;  in  a  locomotive,  those  sometimes  placed  behind  the  driving-wheels. 

Train;  a  number  of  cog-wheels  working  into  each  other. 

Tramway  ;  any  two  smooth  parallel  tracks  upon  which  wheels  without  flanges  may  run.    In  rail 


OP   TERMS.  627 

tramways  the  rails  thamselve^  htfve  flanges  ;  but  ia  wide  stone  ones  for  common  vehicles,  none  are 
required.  S 

Transom  ;  a  beam  acre^s  the  opening  for  a  door,  &c.  Also,  a  horizontal  piece  dividing  a  high  win- 
dow into  two  storiesyfce,  &c. 

Tread',  the  boria<mtal  part  of  a  step. 

Treadle;  a  k.kfa  of  foot-lever,  for  turning  a  lathe,  grindstone,  &c,  by  the  foot. 

Treenail;  a  long  wooden  pin. 

Trimmer;  a  short  cross-timber  framed  into  two  joists  so  as  to  sustain  the  ends  of  intermediate 
Joists,  to  prevent  the  latter  from  entering  a  chimney-flue,  or  interfering  with  a  window,  &c. 

Trip-hammer,  or  tilt-hammer;  a  large  hammer  worked  by  camb  machinery,  and  used  for  heavy 
Iron  work,  especially  for  hammering  irregular  masses  iuto  the  shape  of  bars,  &c. 

Truck;  a  kind  of  small  wagon  consisting  of  a  platform  on  two  or  more  low  wheels.  Also,  those 
frames  and  wheels  usually  placed  under  railroad  cars  and  engines,  and  which,  by  means  of  a  pintle 
tonnecting  the  two,  allows  them  to  vibrate  or  move  laterally  to  some  extent  independently  of  each  other. 

Trundle,  lantern-wheel,  or  u  ullow  er  ;  used  instead  of  a  cog-wheel,  and  consisting  of  two  parallel 
circular  pieces  some  distance  apart,  and  united  by  a  central  axis,  and  by  cylindrical  rods  placed 
around  and  parallel  to  the  axis,  to  serve  instead  of  cogs  or  teeth. 

Trunk  ;  a  long  wooden  boxing  forming  a  water  channel. 

Trunnions  ;  cylindrical  projections,  as  at  the  sides  of  a  cannon,  forming  as  it  were  an  interrupted 
axle  or  shaft  for  supporting  the  cannon  on  its  carriage;  and  allowing  it  to  revolve  vertically  through 
some  distance. 

Tumbler;  a  kind  of  spring  catch,  which  at  the  proper  moment  falls  or  tumbles  into  a  notch  or 
hole  prepared  for  it  in  a  piece  ;  thus  holding  the  piece  in  position  until  the  tumbler  is  lifted  out  of  the 
notch. 

Tumbling-shaft;  in  locomotives,  a  shaft  used  in  the  "link  motion." 

Tunnel  ;  a  passage-way  excavated  beneath  the  surface  of  the  ground.  For  railroads,  they  should, 
if  possible,  be  straight,  especially  when  there  is  but  a  single  track  ;  inasmuch  as  collisions  or  other 
accidents  in  a  tunnel  would  be  peculiarly  disastrous.  A  tunnel  will  rarely  be  expedient  before  the 
depth  of  cutting  exceeds  60  feet.  Firm  rock  of  moderate  hardness,  and  of  a  durable  nature,  is  the 
most  favorable  material  for  a  tunnel  ;  especially  if  free  from  springs,  and  lying  in  horizontal  strata. 
In  soft  rock,  or  in  shales,  (even  if  hard  and  firm  at  first,)  or  in  earth,  a  linikg  of  hard  brick  or  ma- 
sonry in  cement,  is  necessary.  A  tunnel  should  have  a  grade  or  inclination  in  one  direction,  for  ease 
of  future  drainage  and  ventilation.  No  special  arrangement  is  essential  for  ventilation  either  during 
construction,  or  after,  if  the  length  does  not  exceed  about  1COO  ft  ;  but  beyond  that,  generally  during 
«onstruction  either  shafts  are  resorted  to,  or  means  provided  for  forcing  air  into  the  tunnel  through 
pipes  from  its  ends.  But  after  the  work  is  finished,  except  under  peculiar  circumstances,  nothing  of 
the  kind  is  necessary.  Shafts  often  draw  air  downwards  ;  and  frequently,  even  when  aided  by  a  steep, 
uniform  grade,  do  not  sectire  ventilation.  The  Mont  Genis  tunnel  under  the  Alps,  completed  in  1871, 
is  7>i  miles  long,  and  has  no  shafts,  although  it  grades  up  from  each  end,  which  is  the  most  unfavor- 
able  of  all  conditions  for  ventilation  without  shafts.  It  was  made  so  for  facilitating  drainage.  Its  ven- 
tilation is  maintained  by  air  forced  in  from  the  ends.  The  Hoosick  tunnel,  Mass,  4^  miles  long,  has 
shafts  ;  one  of  them  1030  ft  deep  ;  but  they  are  for  expediting  the  work.  Shafts  generally  cost  from 
1%  to  3  times  as  much  per  cub  yd  as  the  main  tunnel,  owing  to  the  greater  difficulty  of  excavating 
and  removing  the  material  ;  and  getting  rid  of  the  water,  all  of  which  must  be  done  by  hoisting, 
When  through  earth,  they  must  be  lined  as  well  as  the  tunnel  ;  and  the  lining  must  usually  be  an 


, 

under-pinning  process.     Or  the  lining  may  first  be  built  over  the  intended  shaft,  and  then  sunk  by 
page  327.     Their  sectional  area  commonly  varies  from  about  40  to  100 


. 
undermining  it  gradually  ;  see  p 


. 

sq  ft.  They  have  the  great  advantage  of  expediting  the  work  by  increasing  the  number  of  points  at 
which  it  can  be  carried  on  ;  but  if  placed  too  close  together,  their  cost  more  than  compensates  for 
this.  The  air  in  some  tunnels,  while  being  constructed,  is  much  more  foul  than  in  others;  so  that 
after  the  work  has  been  commenced,  shafts  with  forced  air  may  be  expedient  where  they  were  not 
anticipated.  In  excavating  the  tunnel  itself,  a  heading  or  passage-way,  5  or  8  ft  high,  and  3  to  12  ft 
wide,  is  driven  and  maintained  a  short  distance  (10  to  100  ft,  or  more,  according  to  the  firmness  of 
the  material)  in  advance  of  the  main  work.  In  rock,  the  heading  is  just  below  the  top  of  the  tunnel, 
so  that  the  men  can  conveniently  drill  holes  in  its  floor  for  blasting;  but  in  earth,  the  heading  ia 
driven  along  the  bottom  of  the  tunnel,  that  being  the  most  convenient  for  enlarging  the  aperture  to 


the  full  tunnel  size,  by  undermining  the  earth,  and  letting  it  fall.  In  earth,  the  top  and  sides  of  the 
heading,  as  well  as  of  the  tunnel,  must  be  carefully  prevented  from  caving  in  before  the  lining  is 
built;  and  this  is  done  by  means  of  rows  of  vertical  rough  timber  props,  and  horizontal  caps  or  over- 


head pieces,  between  which  and  the  earth  rous;h  boards  are  placed  to  form  temporary  supporting  sides 
and  ceiling  to  the  excavation.  The  props  and  caps  are  placed  first :  and  the  boards  are  then  driven 
in  between  them  and  the  earthen  sides  of  the  excavation.  These  are  gradually  removed  as  the  lining 
is  carried  forward.  The  lining,  when  of  brick,  is  usually  from  2  to  3  bricks  thick  (17  to  26  inches)  at 
bottom,  and  from  1^  to  2}<j  bricks  thick  at  top ;  and  when  of  rough  rubble  in  cement,  about  half 
again  as  thick.  It  is  important  that  the  bricks  or  stone  should  be  of  excellent  hard  quality,  and  laid 
in  good  cement.  The  bricks  should  be  moulded  to  the  shape  of  the  arch.  .As  the  tining'is  finished 
in  short  lengths,  and  before  the  centers  are  removed,  any  cavities  or  voids  between  it  and  the  earth 
should  be  carefully  and  compactly  filled  up.  Even  in  rock,  if  much  fissured,  or  if  not  of  durable 
character,  as  common  shale,  lining  is  necessary.  The  cross  section  of  a  single-track  railroad  tunnel, 
in  the  clear  of  everything,  and  for  cars  of  11  ft  extreme  width,  should  not  be  less  than  about  15  feet 
wide,  by  18  feet  high ;  nor  a  double-track  one,  less  than  27  ft  wide,  by  24  ft  high  ;  unless  in  the  last 
case  the  material  is  firm  rock,  in  which  a  high  arch  is  not  necessary  for  lining.  The  roof  may  then 
be  much  flatter,  so  that  a  height  of  20  ft  may  answer.  With  cars  of  10  ft  extreme  width,  the  width 
of  the  tunnel  may  be  reduced  to  25  ft ;  Or  with  9  ft  cars,  to  23  ft.  Many  have  been  made  22  ft. 
The  Mont  Cenis  is  26  ft  wide,  by  25  high.  The  rate  of  daily  progress  from  each  lace  of  a  tunnel 
varies  from  18  inches  to  9  feet  of  length  per  24  hours,  with  three  relays  of  workmen.  On  the  Mont 
Cenis,  for  several  years  past,  the  extremes  have  been  about  4  to  9  ft  daily  for  a  whole  year,  from 
each  face.  On  that  work  greater  expedition  than  usual  was  secured,  by  the  use  of  improved  drills 
worked  by  compressed  air:  although  they  were  employed  only  in  the  headings,  which  were  12  ft  wide 
by  8  ft  high.  Ordinarily,  from  1^£  to  3  ft  may  be  taken  as  averages.  The  difference  of  rate  of  pro- 
gress between  a  single  and  a  double  track  tunnel,  is  not  so  great  as  might  be  supposed  ;  inasmuch  as 
,1  larger  force  can  be  employed  on  the  wider  one.  If  the  tunnel  is  in  earth,  the  construction  of  the 
lining  about  makes  up  for  the  slower  excavation  of  one  in  rock.  In  rock,  with  labor  «.t  $\  per  day, 
the  cost  will  usually  vary  with  the  character  of  the  rock,  from  $2  to  $5  per  cub  yd  for  the  main  tun- 
nel ;  and  from  $3  to  $10  for  the  heading ;  while  shafts  will  average  about  50  per  ct  more  than  beading. 
Tunnel  work,  however,  is  liable  to  serious  contingencies  which  cannot  be  foreseen.  Since  the  «idea 


628 


GLOSSARY   OF   TERMS. 


and  roof  are  rough  as  blasted,  the  width  and  height  should  each  be  estimated  to  the  contractor  as 
about  18  ins  or  2  ft  greater  than  the  established  clear  ones.  At  any  rate,  the  mode  of  measurement 
should  be  clearly  stated  in  the  specifications  for  the  work.  When  a  tunnel  is  made  with  a  uniform 
grade,  the  work  generally  progresses  in  a  more  satisfactory  manner  from  the  lower  end,  because  the 
descent  favors  the  drainage  of  the  spring  water  that  is  usually  met  with  ;  whereas,  at  the  upper  end, 
it  must  be  removed  by  pumps  or  by  bailing.  The  upper  end  has,  however,  the  advantage  of  sooner 
getting  rid  of  the  smoke  in  blasting.  Before  coinmeucing  a  tunnel,  or  even  deciding  upon  one,  trial 
shafts  should  be  sunk  to  ascertain  the  nature  of  the  material.  In  long  ones,  the  greatest  care  and 
accuracy  are  necessary  for  preserving  the  line  of  direction,  so  that  the  work  from  both  ends  shall 
meet  properly  at  the  center.  The  cost  of  a  single-track  tunnel,  when  common  labor  is  $1  per  day, 
will  generally  range  between  $30  and  $75  per  foot  of  length.  Our  limits  prevent  our  saying  more  on 
this  important  subject,  which  of  itself  would  require  a  volume. 

Turnbuckle;  variously  applied,  as  to  the  ordinary  fastenings  at  the  outer  face  of  a  wall,  for  hold 
ing  window-shutters  back  when  opened ;  also,  to  the  tightening  swivel  at  13,  page  265,  of  Trusses,  &c. 

Turnout ;  a  kind  of  siding,  or  short  piece  of  track,  for  permitting  railroad  trains  to  pass  each  other 
•n  a  single-track  road. 

Turntable;  the  well-known  arrangement  for  turning  locomotives  at  rest.    See  page  428. 

Undermine  ;  to  excavate  beneath  anything. 

Underpin;  to  add  to  the  height  of  a  wall  already  constructed,  by  excavating  and  building  beneath 
it.  Also,  to  introduce  additional  support  of  any  kind  beneath  anything  already  completed. 

Upset.    See  Stove- up. 

Valves;  various  devices  for  permitting  or  stopping  at  pleasure  the  flow  of  water,  steam,  gas,  &c. 
A  SAFETY  VALVE  is  one  so  balanced  as  to  open  of  itself  when  the  pressure  becomes  too  great  for 
safety.  A  SLIDE  VALVE  is  one  that  slides  backward  and  forward  over  the  opening  through  which  the 
flow  takes  place.  A  BALL  VALVE,  or  spherical  valve,  is  a  sphere,  which  in  any  position  fits  the  open- 
ing. When  the  pressure  below  it  raises  it  off  from  its  seat,  it  is  prevented  from  rolling  away  by 
means  of  a  kind  of  open  caging  which  surrounds  it.  A  CONICAL  or  PUPPET  VALVE  is  a  horizontal  slice 
of  a  cone,  which  fits  into  a  corresponding  conical  seat  made  in  the  opening.  In  rising  and  falling  it 
is  kept  in  position  by  a  vertical  valve-stem  or  spindle,  which  passes  through  its  center,  and  which 
plays  through  guide-holes  in  bridge-pieces  placed  above  and  below  the  valve.  A  TRAP,  CLACK,  FLAP. 
or  DOOR  VALVE,  is  a  plate  with  hinges  like  a  door.  When  two  such  valves  are  used,  with  their  hinged 
edges  adjacent  to  each  other,  so  that  in  opening  and  shutting  they  flap  like  the  wings  of  a  butterfly, 
they  constitute  a  butterfly  valve.  A  THROTTLE  VALVE  is  one  which  when  closed  forms  a  partition 
across  a  pipe ;  and  opens  by  partially  revolving  upon  an  axis  placed  along  its  diameter.  A  ROTARY 
VALVE  works  like  a  common  stopcock.  A  SHIFTING  VALVE  is  one  which  lets  out  steam  under  water  ;  and 
is  so  called  from  the  snifting  noise  thereby  produced.  The  PORT  VALVE  is  the  sliding  one  which  ad- 
mits steam  from  the  steam-chest  into  the  cylinders.  A  DOUBLE-SEAT,  or  DOUBLE  BEAT  VALVE  is  a  pe- 
culiar one  with  two  seats,  one  above  the  other  :  and  so  arranged  that  the  pressure  of  steam  or  water 
against  it  when  shut,  does  not  oppose  its  being  opened.  A  CUP  VALVE  is  in  shape  of  an  inverted 
cylindrical  cup,  with  a  length  somewhat  greater  than  its  diameter.  Its  lower  or  open  edge  is  ground 
to  fit  the  seat  over  which  it  rests.  As  this  cup  rises  and  falls,  it  is  kept  in  place  by  a  cylindrical 
caging  closed  at  top,  and  having  for  its  sides  four  or  more  vertical  pieces,  against  the  inner  sides  of 
which  the  sides  of  the  cup  play.  A  CHECK  VALVE  is  any  kind  so  placed  as  to  check  or  prevent  the 
return  of  the  fluid  after  its  passage  through  the  valve  into  the  pipe  or  vessel  beyond  it. 

Vault;  an  arch  long  in  comparison  with  its  span.     The  space  covered  by  such  an  arch. 

Veneer;  a  very  thin  sheet  of  ornamental  wood  glued  over  a  more  common  variety. 

Wainscot;  a  wooden  facing  to  walls  in  rooms,  instead  of  plaster,  or  over  a  facing  of  plaster ;  usually 
not  more  than  3  or  4  feet  high  above  the  floor. 

Wales ;  long  longitudinal  timbers  in  the  sides  of  a  ship,  coffer-dam,  caisson,  &c. 

Wallow;  a  water-wheel,  &c,  is  said  to  wallow  when  it  does  not  revolve  evenly  on  its  journals. 

Wallower.    See  Trundle. 

Wall-plate,  or  raising-plate ;  a  timber  laid  along  the  tops  of  walls  for  the  roof  trusses  or  rafters  to 
rest  on,  so  as  to  distribute  their  weight  more  equally  upon  the  wall. 

Warped;  twisted,  as  a  board,  or  the  face  of  a  stone,  &c,  which  is  not  perfectly  flat.  To  warp;  to 
haul  a  vessel  ahead  by  means  of  an  anchor  dropped  some  distance  ahead.  To  flood  an  extent  of 
ground  with  water  for  a  short  time  to  increase  its  fertility. 

Washboards ;  boards  nailed  around  the  walls  of  rooms  at  the  floor,  so  as  to  prevent  injury  to  the 
plaster  when  washing  the  floors. 

Washers ;  broad  pieces  of  metal  surrounding  a  bolt,  and  placed  between  the  faces  of  the  timber 
through  which  the  bolt  passes,  and  the  head  and  n*it  of  the  bolt,  so  as  to  distribute  the  pressure  over 
a  larger  surface,  and  prevent  the  timber  from  being  crushed  when  the  bolt  is  tightly  screwed  up. 

Waste-weir;  an  overfall  provided  along  a  canal,  &c,  at  which  the  water  may  discharge  itself  in 
case  of  becoming  too  high  by  rain.  <fec.  Sometimes  called  a  tumbling-bay. 

Water-shed;  the  sloping  ground  from  which  rain-water  descends  into  a  stream. 

Water-table ;  a  slight  projection  of  the  lower  masonry  or  brickwork  on  the  outside  of  a  wall,  and 
reaching  to  a  few  feet  above  the  ground  surface,  as  a  partial  protection  against  rain,  or  as  ornament. 

Ways ;  the  inclined  timbers  along  which  a  vessel  glides  when  being  launched. 

Wether-boards  ;  boards  used  instead  of  bricks  or  masonry  for  the  outsides  of  a  building,  or  bridge, 
&c.  'Tbey  are  nailed  to  vertical  and  inclined  indoor  timbers ;  and  may  be  either  vertical  or  hor. 
When  hor,  they  are  so  placed  that  the  lower  edge  of  one  overlaps  the  upper  edge  of  the  one  below. 
When  vert,  their  edges  should  be  tongued  and  grooved  ;  and  narrow  slips  be  nailed  over  the  vert  joints, 
to  keep  out  rain,  &c. 

Weir,  or  wier;  a  dam,  or  an  overfall. 

Weld ;  to  join  two  pieces  of  metal  together  by  first  softening  them  by  heat,  and  then  hammering 
them  in  contact  with  each  other.  In  this  operation  fluxes  are  used. 

Wharf;  a  level  space  upon  which  vessels  lying  along  its  sides  can  discharge  their  cargoes ;  or  from 
which  they  can  receive  them. 

Wheel-base;  the  distance  from  center  to  center  from  the  extreme  front  wheels,  to  the  extreme  hind 
ones  in  a  locomotive,  car,  <fec. 

Wicket;  a  small  door  or  gate  made  in  a  larger  one ;  as  the  shuttle  or  valve  in  a  lock-gate,  for  letting 
out  the  water. 

Winch ;  a  handle  bent  at  right  angles,  and  used  for  turning  an  axis ;  that  of  a  common  grindstone. 

Wind.     See  Out  of  wind. 

Winders;  those  steps  (often  triangular)  in  a  staircase  by  which  we  wind,  or  turn  angles. 

Windlass;  tbe  wheel  and  axle,  or  winch  and  drum,  as  often  used  in  common  wells.    Also,  a  hori* 


SARY   OF   TERMS. 


629 


Eontal  shaft  on  shipboardy^y  which  the  anchor  is  raised ;  the  windlass  being  revolved  by  means  of 
wooden  levers  called  hau&spikes. 

Wing-dam;  a  projeouon  carried  out  part  way  across  a  shallow  stream,  so  as  to  force  all  the  water 
to  flow  deeper  through  the  channel  thus  contracted. 

Wings ;  applied  in  many  ways  to  projections.  The  flanges  which  radiate  out  from  a  gudgeon  ;  and 
by  which  it  is  fastened  to  the  shaft.  Small  buildings  projecting  from  a  main  one.  The  wings  or 
flaring  wing-walls  of  a  culvert  or  bridge. 

Wing-walls;  the  retaining- walls  which  flare  out  from  the  ends  of  bridges,  culverts,  &c. 

Wiper.    See  Camb. 

Working-beam,  or  walking  -beam ;  a  beam  vibrating  vertically  on  a  rock-shaft  at  its  center,  as  seen 
in  some  steam-engines  ;  one  end  of  it  having  a  connection  with  the  piston-rod ;  and  the  other  end  with 
a  crank,  or  with  a  pump-rod,  &c. 

Worm;  the  so-called  endless  screw,  which  by  revolving  without  advancing  gives  motion  to  a  cog- 
wheel (worm-wheel),  the  teeth  of  which  catch  in  the  thread  of  the  screw. 

Wrench;  a  long  handle  having  at  one  end  an  eye  or  jaw  which  may  catch  bold  of  anything  to  be 
twisted  or  turned  around,  as  a  screw-nut,  &c.  When  it  has  a  jaw  which  by  means  of  a  screw  is 
adaptable  to  nuts,  &c,  of  different  sizes,  it  is  a  monkey- wrench,  or  screw-wrench. 

SLOPES  IIC  FEET  PER  10O  FT.  HORIZONTAL. 

See  also  pp  388  and  389,  for  other  tables. 
The  fractions  of  minutes  are  given  only  to  34  ft  in  100. 

A  clinometer  graduated  by  the  3d  column,  and  numbered  by  the  first  one, 
will  give  at  sight  the  slopes  in  feet  per  100  ft  horizontal.  No  errors.  Original. 


Rise  in  ft 
per  100 
ft  hor. 

Length  of 
slope  per 
100  ft  hor. 

Angle  of 
slope. 

Si 

Length  of 
slope  per 
100  ft  hor. 

Angle  of 
slope. 

S§S 

|IS 

Length  of 
slope  per 
100  ft  hor. 

Angle  of 
elope. 

Feet. 

Deg.  Min. 

Feet. 

Deg.  Min. 

Feet. 

Deg.-  Min. 

1 

100.005 

0  34.4 

35 

105.948 

19  17 

69 

121.495 

34  36 

2 

100.020 

1   8.7 

36 

106.283 

19  48 

70 

T22.066 

35   0 

3 

100.045 

1  43.1 

37 

106.626 

20  18 

71 

122  642 

35  23 

4 

100.080 

2  17.5 

38 

106.977 

20  48 

72 

123.223 

35  45 

5 

100.125 

2  51.8 

39 

107.336 

21  18 

73 

123.810 

36   8 

6 

100.180 

3  26.0 

40 

107.703 

21  48 

74 

124.403 

36  30 

7 

100.245 

4   0.3 

41 

108.079 

22  18 

75 

125.000 

36  52 

8 

100.319 

4  34.4 

42 

108.462 

22  47 

76 

1-25.603 

37  14 

9 

100.404 

5   8.6 

43 

108.853 

23  16 

77 

126.210 

37  36 

10 

100.499 

5  4-2.6 

44 

109.252 

23  45 

78 

126.823 

37  57 

11 

100.603 

6  16.6 

45 

109.659 

24  14 

79 

127.440 

38  19 

11 

100.717 

6  50.6 

46 

110.073 

24  42 

80 

128062 

38  40 

13 

100.841 

7  24.4 

47 

110.494 

25  10 

81 

128.690 

39   1 

14 

100.975 

7  58.2 

48 

110.923 

25  38 

82 

129321 

39  21 

15 

101.119 

8  31.9 

49 

111.359 

26   6 

HI 

129.958 

39  42 

16 

101.272 

9   5.4 

50 

111  803 

26  34 

84 

130.599 

40   2 

17 

101.435 

9  38.9 

51 

112.254 

27   1 

85 

131.244 

40  22 

18 

101.607 

10  12.2 

52 

112.712 

27  28 

86 

131.894 

40  42 

19 

101.789 

10  45.5 

53 

113.177 

27  55 

87 

132.548 

41   1 

20 

101.980 

11  18.6 

54 

113.649 

28  22 

88 

133.207 

41  21 

21 

102.181 

11  51.6 

55 

114.127 

28  49 

89 

133.869 

41  40 

22 

10-2.391 

12  24.5 

56 

114.612 

29  15 

90 

134.536 

41  59 

M 

lOL'.fill 

12  57.2 

57 

115.104 

29  41 

91 

135.207 

42  18 

24 

10-2.840 

13  29.8 

58 

115.603 

30   7 

92 

135.882 

42  37 

25 

103.078 

4   2.2 

59 

116.108 

30  32 

93 

136.561 

42  55 

26 

103.325 

4  34.5 

60 

116.619 

30  58 

94 

137.244 

43  14 

27 

103.581 

5   6.6 

61 

117.137 

31  23 

95 

137.931 

43  32 

28 

103.846 

a  38.5 

62 

117.661 

31  48 

96 

138.622 

43  50 

29 

104.120 

6  103 

63 

118.191 

32  13 

97 

139  316 

44   8 

30 

104.4C3 

6  42.0 

64 

118.727 

32  37 

98 

140.014 

44  25- 

31 

104.695 

7  13.4 

65 

119.269 

33   1 

99 

140.716 

44  43 

32 

104.995 

7  44.7 

66 

119.817 

33  25 

100 

141.421 

45  00 

33 

105.304 

18  15.8 

67 

120.370 

33  49 

101 

142.130 

45  17 

34 

105.622 

18  46.7 

68 

120.930 

34  13 

102 

142.843 

45  34 

Any  hor  (list  is  =  sloping  dist  X  nat  cosine  ang  of  slope. < 

*'     sloping'  clist  is  =  hor  dist         X  nat  secant    "    "        " 
*'     vert  height    is  =  hor  dist         X  nat  tangent  " 

or  =  sloping  dist  X  uat  sine        "    "        " 


APPENDIX. 


Page  23,  near  top,  after  Rem.  1.     To  find  the  area  of  a  circular 

segment.  If  the  height,  rffe,  exceeds  half  the  diain  of  the  circle,  the  quotient 
will  exceed  .5;  and  will  not  be  found  in  the  table.  In  such  cases,  subtract  the  height  from  the  diam  ; 
and  use  the  remainder  as  the  height.  Then  find  the  area  of  a  segment  having  this  new  height;  and 
subtract  it  from  the  area  of  the  entire  circle.  The  rem  will  plainly  be  the  reqd  area. 

Page  26.  The  following  Rule  for  the  circiimfof  an  ellipse  orig- 

inated with  Ulr.  HI.  Arnold  Pears  of  New  South  Wales,  Australia,  and  was 
by  him  kindly  communicated  to  the  writer. 

Circumf  =  3.1416  d  +  2  (D—  d)  dD—d) 


-- 

V(D-r-d)X(D+2d) 


It  is  not  more  accurate  than  our  own  on  p  26,  but  is  much  neater. 
3V 


P.  32  ;  just  above  Circular  Rings. 
When  am  exceeds  cm,  for  the  solidity 

of  the  ungula  dumb  a,  fiud  the  cube  of  ab,  or  of 
ad,  and  take  %  of  it,  which  call  p.  Mult  the  area 
of  the  base  a  d  m  b i  by  a  c.  Add  the  prod  to  p.  Mult 
the  sum  by  the  height  mn.  Divide  by  am. 

And  for  the  area  of  the  convex 

surface,  mult  the  diain  y  in  by  a  b  or  by 
ad.  Call  the  prod  p.  Take  a  d  from  am.  Mult 
the  rem  by  the  length  of  the  arc  dmb.  Add  the 
prod  to  p.  Call  the  sum  s.  Divide  the  height  mn 
by  a  m.  Mult  the  quot  by  «. 


P.  312.  Cost  of  ashlar  facing:  masonry.  If  the  stone  be  sandstone 

with  good  natural  beds,  the  getting  ouc  may  be  put  at  $3.00  per  cubic  yard.  Face  dressing  at  26  cts 
per  sq  ft :  say  $8.64  per  cubic  yd.  Beds  and  joints  13  cts  per  sq  ft ;  say  $6.76  per  cub  yd.  The  neat 
cost,  laid,  $17.00,  instead  of  the  $21.86  per  cub  yd  for  granite ;  and  the  total  cost  $19.55,'  instead  of  the 

$25.14  for  granite.  And  the  total  cost  of  large  well  scabbled  ranged 
sandstone  masonry  in  mortar,  may  be  taken  at  about  $10  per  cub  yd, 
at  the  wages  on  p  312. 

Embankment,  shrinkage  of. 

Although  earth,  when  first  dug,  und  loosely  thrown  out,  swells  about  i  part,  so  that  a  cub  yd  in 
place  averages  about  li  or  1.2  cub  yds  when  dug;  or  1  cub  yd  dug  is  equal  to  j|,  or  to  .8333  of  a  cub 
yd  in  place:  yet  when  made  into  embankment  it  gradually  subsides,  settles,  or  shrinks,  into  a  less 
bulk  than  it  occupied  before  being  dug. 

The  following  are  approximate  averages  of  the  shrinkage;  or,  in  other  words,  the  earth  measured 
in  place  in  a  cut,  will,  when  made  into  embankment,  occupy  a  bulk  less  than  before  by  about  the 
following  proportions : 

Gravel  or  sand about    8  per  ct ;  or  1  in  1 2^  less. 

Clay "      10  per  ct;  or  1  in  10      less. 

Loam "      12  per  ct;  or  1  in    8>£  less. 

Loose  vegetable  surface  soil  »      "      15  per  ct:  or  1  in    6%  less. 

Puddled  clay "      25  per  <jt;  or  1  in    4     less. 

The  writer  thinks,  from  some  trials  of  his  own,  that  1  cub  yd  of  any  hard  rock  in  place,  will  make 
from  1%  to  \y±  cub  yds  of  embankment;  say  on  an  average  1.7  cub  yds.  Or  that  1  cub  yd  of  rock 
embankment  requires  .5882  of  a  cub  yd  in  place.  He  found  that  a  solid  cub  yd  when  broken  into 
fragments,  made  about  1.9  cub  yds  of  loose  heap;  *  1%  yds  carelessly  piled  ;  t  and  1.6  yds  carefully 
piled. J  Or  \y>  cub  yds  of  very  carelessly  scabbled  rubble;  §  or  1^  yds  of  somewhat  carefully  scab- 
bled.  II  Seep  504. 

*  Solid  part  .526 ;  voids  .474  of  the  entire  bulk. 

t       "       "  .570;      "      .430       "         "          " 

J      "       "  .630;      "      .370       "        "          " 

§       "       "  .670;      "      .320       "         t4 

I      "       "  .800;      "      .200       "        "         " 

630 


APPENDIX.  631 


Approximate  c4tst  of  buildings  per  cubic  foot,  at  Philada  pric 

n  1873  :  iucluerfng  every  cub  ft  of  space  from  roof  to  cellar  floor.     Plaiu  brick  dwellings,  such  ;is  mo 


__m,u _  rices 

In  187*3  rinclu*rfng  every  cub  ft  of  space  from  roof  to  cellar  floor.  Plain  brick  dwellings,  such  as  most 
of  those  in  Poilada.  12  to  15  cts.  Better  class,  highly  finished  throughout,  15  to  18  cts.  First  class, 
•with  cut  stone  fronts,  20  to  30  cts.  Plain  brick  churches,  public  schools,  court-houses,  theaters,  &c, 
12  to  16  cu.  Ornate  Gothic  churches  with  much  cut  stone  facing,  30  to  45  cts,  exclusive  of  spires. 
Large  plain  brick  or  rubble  RR  shops,  depots,  station-houses,  &c,  9  to  12  cts;  or  with  ornamental 
finish  and  best  materials,  15  to  20  cts.  First  class  city  stores,  marble  fronts,  high  stories,  fire-proof, 
(so  called,)  throughout  to  roof;  best  materials  and  workmanship,  18  to  25  cts. 

Small  buildings  cost  more  per  cub  ft  than  large  ones  of  the  same 
finish.  Also  isolated  or  corner  buildings,  cost  more  than  those  which  have  two  party-walls. 

In  Philada  dwellings  of  brick,  the  carpentry  and  lumber  usually  cost 
each  about  one-fourth  of  the  entire  building.  Memorial  Hall  of  the  Centen- 
nial Buildings  cost  68  cts  per  cub  ft,  exclusive  of  iron  dome. 

P  327.  The  plenum  process  as  applied  at  the  South  St  bridge,  Philada, 
by  Mr.  John  W.  Murphy,  contracting  engineer,  differs  materially  from  that  described  on  p  327  ;  and 
moreover  deserves  notice  on  account  of  the  great  simplicity  and  efficacy  of  his  plant.*  This  consisted 
partly  of  two  canal  boats,  decked,  each  100  ft  long,  by  17}^  ft  wide,  and  8  ft  depth  of  hold.  They 
were 'anchored  parallel  to  each  other,  15  ft  apart.  Supported  by  the  boats,  and  over  the  space  between 
them,  was  a  strong  four-legged  shears  about  50  ft  high ;  at  the  lop  of  which  was  attached  tackle  for 
handling  the  cast  iron  cylinders.  In  the  hold  of  one  of  the  boats  was  a  Burleigh 
Compressor  having  two  pistons  of  10  ins  diam,  and  9  ins  stroke;  together  with 
its  boiler.  On  the  deck  of  the  same  boat  stooda  vertical  air-tank  or  regulator, 
22  ft  long,  by  2  ft  diam,  made  of  quarter  inch  boiler  iron.  This  served  to  maintain  a  supply  of  com- 
pressed air  in  the  submerged  cylinder  in  case  of  an  accidental  stopping  of  the  compressor;  which 
otherwise  would  probably  be  fatal  to  the  laborers  in  the  cylinder.  The  condensed  air  flowed  from 
this  air-tank  to  the  air-lock  of  the  cylinder  through  a  hose  4  ins  diam,  made  of  gum  elastic  and  can- 
vas, and  so  long,  and  so  placed,  as  to  extend  itself  as  the  cylinder  went  down,  thus  maintaining  the 
communication  at  all  times.  Entirely  across  both  boats,  and  across  the  interval  between  them,  ex- 
tended two  heavy  wooden  clamps,  each  3  ft  wide  by  18  ins  high;  each  composed 
of  three  pieces  of  12  X  18  inch  timber  strongly  bolted  together.  At  the  centers  of  these  clamps  the 
two  inner  vertical  sides  which  faced  each  other  were  hollowed  out  to  the  depth  of  a  foot  by  concavi- 
ties corresponding  to  the  curve  of  the  cylinders.  The  distance  apart  of  the  clamps  was  regulated  by 
two  strong  iron  rods,  having  screws  and  nuts  at  their  ends  for  that  purpose.  Thus  when  a  section 
of  a  cylinder  was  hoisted  by  means  of  the  shears  into  its  position  over  the  space  between  the  two 
boats,  the  two  concavities  of  the  clamps  w«re  brought  into  contact  with  it,  and  the  nuts  being  then 
screwed  up,  the  cylinder  was  firmly  held  in  place  by  the  clamps.  The  shears  could  then  be  used  to 
raise  another  section  of  the  cylinder  to  its  place  upon  the  first  one.  that  the  two  might  be  bolted  to- 
gether. By  repeating  this  process  the  height  of  the  cylinder  would  soon  become  too  great  to  allow 
the  shears  to  place  another  section  upon  it;  in  which  case  the  nuts  of  the  screws  were  slightly 
loosened,  and  the  cylinder  was  allowed  to  slip  down  slowly  into  the  water  until  its  top  was  but  a 
little  above  the  surface.  The  screws  were  then  again  tightened,  and  the  cylinder  again  held  fast 
until  other  sections  were  added  and  bolted  to  it.  When  there  was  danger  that  the  upward  pressure 
of  the  condensed  air  might  lift  a  cylinder,  the  clamps  were  raised  by  the  shears  clear  of  the  boats; 
then  tightened  to  the  cylinder,  and  a  platform  of  plauks  laid  upon  them,  and  loaded  with  stone. 

The  air-lock  was  so  arranged  as  not  to  require  to  be  removed  when  a  new  sec- 
tion was  to  be  bolted  on.  This  was  effected  as  follows.  .  Sections  of  the  cylinder  were  bolted  together 
in  the  manner  just  described,  until  its  foot  rested  on  the  bottom,  with  its  top  a  few  feet  above  high 
water.  A  heavy  cast  iron  diaphragm  1^  inches  thick,  to  form  the  floor  of  the 
air-lock,  was  then  placed  on  top.  Then  was  added  another  10  ft  high  section  of  the  cvlinder,  to  form 
the  chamber  of  the  air-lock.  These  were  bolted  together  ;  and  then  another  diaphragm  was  added 
at  top  to  form  the  roof  of  the  air-lock.  These  diaphragms  were  furnished  with  openings,  and  with 
doors  and  valves  corresponding  with  those  shown  in  Fig  17.  p  327.  and  remained  permanently  in  the 
cylinders  when  the  work  was  finished.  If  the  depth  of  soil  to  be  passed  through  before  reaching 
rock  is  so  great  as  to  require  other  sections  of  cvlinder  to  be  bolted  on  above  the  top  of  the  air-lock 
this  may  be  done  to  any  extent,  inasmuch  as  it  is  immaterial  whether  the  air-lock  is  under  water  or 

not.    To  keep  the  cylinder  both  air-  and  water-tight  the  faces  of 

the  Hanges  before  being  bolted  together  were  smeared  with  a  mixture  of  red  and  white  lead  and  cot- 
ton fiber. 

At  the  South  St  bridge  the  cylinders  were  4.  6,  and  8  ft  diam  ;  in  lengths 
or  sections  10  ft  long.  They  were  all  \y±  inch  thick.  Inside  flnnges  2%  ins  wide,  1}£  thick,  with  bolt- 
holes  1J4  inch  diam,  by  5  ins  apart  from  center  to  center.  The  bottom  edge  has  no  flange.  A  10  ft 
section  of  an  8  ft  Cylinder  weighs  14600  fts ;  of  a  6  ft  one,  10800 ;  of  a  4  ft  one,  6800.  An  8  ft  dia- 
phragm, 2800  fts  ;  6  ft,  1600;  4  ft,  783.  The  rock  under  the  soil  was  quite  uneven  in  places ;  but  was 
levelled  off  as  the  cylinders  went  down.  These  were  then  bolted  to  it  by  cast  iron  brackets. 
The  work  went  on,  day  and  night,  summer  and  winter:  with  no  inter- 
ruption from  the  tides,  floods,  or  floating  ice;  and  the  thirteen  columns  were  sunk,  filled  with  con- 
crete, and  completed  in  11  months  ;  much  of  which  was  consumed  in  levelling  off  the  rock,  and  bolt- 
ing the  brackets.  The  want  of  guides  caused  much  tilting,  trouble,  and  delay. 

Rise  and  fall  of  tide  about  7  ft.  Greatest  depth  of  soil,  gravel,  Ac.  passed  through.  30  ft :  least,  6 
ft.  Depth  of  water  about  25  ft.  The  work  was  under  charge  of  John  Anderson,  a  very  skillful  and 
energetic  superintendent  of  such  matters.  The  entire  neat  cost  of  the  cylin- 
ders in  place,  and  filled  with  hydraulic  concrete  was  probably  not  far  from  $120  per  ft  of  total  length 
for  the  8  ft  ones ;  $80  for  ttie  6  ft ;  and  $50  for  the  4  ft  diams.  There  were  three  gangs  of  workmen  • 
and  each  gang  worked  4  hours  at  a  time. 

Such  cylinders  have  cracked  through,  around  their  entire  circum- 
ference, in  many  parts  of  the  U.  S.  in  very  cold  weather;  owing  to  the  diff  of  contraction  of  the 
iron,  and  of  the  concrete  filling.  Ignorant  use  of  them  may  be  attended  by  great  danger. 

*  See 'a  full  and  very  instructive  description  with  engraving*,  by  D.  M.  Stauffer,  Superintending 
Engineer  for  the  city,  in  the  Journal  of  the  Franklin  lost,  Nov.  1872.  From  it  the  above  few  item* 


632 


APPENDIX. 


P  177.    Moduliises  of  Elasticity  in  Ibs  per  sq  incb.    For  the 

meaning  of  the  term  see  p  177.  Authors  differ  considerably  in  their  data  on  this  subject.  Where  it 
was  possible  to  adopt  an  average  without  error  of  practical  importance,  we  have  done  so ;  but  where 
the  differences  were  too  great  for  this,  we  have  omitted  the  material  entirely.  Fortunately  it  is  a 
matter  of  but  little  importance  to  the  engineer  ;  for,  with  the  exception  of  iron  and  steel,  he  rarely 
need  care  to  know  the  exact  degree  to  which  his  materials  are  lengthened  or  shortened  by  their  loads. 
It  must  be  carefully  remembered  that  in  practice  the  principle  of  mod  of  elas  applies  only  within 
the  elas  limit  of  materials.  This  limit  may  ordinarily  be  safely  taken  at  about  one-third  of  the 
breaking  tensile  strengths  given  on  pages  177  to  180.  This  is  done  in  our  5th  column.  It  is,  how- 
ever, but  a  tolerable  approximation.  There  is  reason  to  doubt  the  accuracy  of  many  of  the  items  ic 
this  table.  Round  and  square  bare  do  not  stretch  the  same. 


MATERIAL. 

Modulus 
or  Coeff 
of 
Elasticity. 

Stretch  or  C 
in  a  lengtl 
under  a 
1000  Ibs  per 
sq  iu. 

ompression 
i  of  10  ft, 
load  of 
1  ton  per 
sq  in. 

Approx  elas 
limit. 

Ash 

Ibs  per  sq  in. 
1  600  000 

IDS. 
075 

Ins. 
168 

Ibs  per  sqin. 
4500 

Beech 

1  HOO  000 

092 

207 

4000 

Birch 

1  400  000 

086 

192 

5000 

Brass,  cast  

9  200  000 

.013 

.029 

6000 

wire  

14  200  000 

.009 

.019 

16000 

Chestnut.  

1  000  000 

.120 

269 

4500 

18  000  000 

.007 

015 

6300 

•  <      '  wire  

18  000  000 

.007 

015 

10000 

Elm  ...          

1  000  000 

120 

269 

2000 

Glass      .                     

8  000  000 

015 

034 

3200 

12  000  000 

010 

022 

4500 

"      "    average  

to 
23  000  000 
17  500  000 

to 
.005 

.007 

to 
.012 
.015 

to 
8000 
6250 

"    wrought,  in  either                             < 

18  000  000 
to 

.006 
(to 

.015 
to 

13000 

"         "          "    average  

40  000  000 
29  000  000 

.003 
.004 

.007 
.009 

26000 
19500 

•«          «'          "     wire,  hard  

26  000  000 

.005 

.010 

27000 

15  000  000 

.008 

.018 

13000 

Larch 

1  100  000 

.109 

.244 

2300 

I  ead   sheet 

720  000 

.167 

1100 

«<      wire                    

1  000  000 

.120 

1100 

I  400  000 

.086 

192 

2700 

0ak                                                       

1  000  000 

120 

269 

to 
2  000  000 

to 

060 

to 
134 

„    Bveraee 

1  500  000 

.080 

.179 

3300 

Pine  white  or  yellow  

1  600  000 

.075 

.168 

3300 

glate                         

14  500  000 

.008 

.018 

3700 

1  600  000 

.075 

.168 

S.iOO 

oPppi   Kara 

29  000  000 

.004 

.009 

34000 

to 
42  000  000 
35  500  000 

to 
.003 
003 

to 
.006 
007 

to 
44000 
39000 

Sycamore 

1  000  000 

.120 

.269 

4000 

Teak  

2  000  000 

.060 

.134 

5000 

Tin.  cast  

4  600  000 

.026 

1500 

P  603.    Friction  of  Hydraulic  Presses.    Is  not  certainly  known. 

Mr.  John  Hick,  C  E,  England,  says  that  it  is  independent  both  of  the  width  of  the  leather  collar, 
and  of  the  length  of  the  ram  in"  the  cylinder ;  and  that  in  a  press  in  good  order  it  may  be  found 
very  approximately  thus, 

Rule.  Mult  the  diam  of  the  ram  in  inches  by  25.  Divide  the  total  pressure 
against  the  bottom  of  the  ram  by  the  product. 

Example.  In  a  press  of  18  ins  diam  the  total  pressure  is  680000  Ibs;  how 
much  must  be  deducted  from  this  for  friction  ?  Here, 


18  X  25  =  450.     And  - 


=  1511  Tbs,  answer. 


Hence  iu  an  18  inch  press  the  ratio  of  the  friction  to  the  total  pressure  is  ^ftfkvr  —  .00222  ;  or 


s  _  L_  =:  450,  the  fric  = 


of  the  pressure. 


;  15  ins, 


;  12  ins, 


In  a  press  24  ins  diam  it  will  be  ^-^;  18  ins, 
6  ins,  y4-7y  ;  3  ins,  -J-r  part  of  the  entire  pressure. 

But  a  small   press   by  Tang-ey  of  Birmingham,  under  a  pressure  of 

but  360  Ibs  per  sq  inch,  showed  a  friction  of  1  per  ct,  or  4}^  times  greater  than  the  above;  while  other 
experiments  make  it  equal  to  the  pressure  in  fts  per  sq  inch  X  .4  of  the  depth  of  collar  in  ius  X  cir- 
cumf  in  ins.  which  is  much  greater  than  either.  Recent  expts  in  this  country  make  ifr  iu  some 

cases  vastly  greater  still;  while  others  by  Gen  Q,.  A.  Gillmore  con- 
firm Mr.  Hicks. 


BLE   OF   ORDINATES. 


633 


of  Orclinates  5  ft  apart.    Chord  1OO  ft. 

For  Railroad  Curves, 

For  the  radii  corresponding  to  the  angles  of  deflection,  see  page  416. 
Ordinates  for  angles  intermediate  of  those  in  the  table  can  at  once  be  found  by 
simple  proportion.    This  table  was  calculated  by  rule  p  20. 


Distances  of  the  Ordinates  from  the  end  of  the  100  feet  Chord. 


Ang.  of 
Defl. 

Mid. 
50ft. 

45ft. 

40ft. 

35ft. 

30ft. 

25ft.    | 

20ft. 

15ft. 

10  ft. 

5ft. 

0      ' 

i 

4 

.014 

.014 

.014 

.013 

.012 

.010 

.008 

.008 

.005 

.003 

8 

.029 

.029 

.028 

.026 

.024 

.022 

.018 

.015 

.010 

.005 

12 

.043 

.043 

.041 

.038 

.037 

.033 

.028 

.022 

.015 

.008 

16 

.058 

.058 

.056 

.052 

.049 

.044 

.037 

,030 

.020 

.011 

20 

.073 

.072 

.070 

.066 

.061 

.055 

.047 

.037 

.026 

.014 

24 

.087 

.086 

.083 

.077 

.074 

.066 

.056 

.045 

.031 

.017 

28 

.102 

.101 

.098 

.092 

.086 

.077 

.065 

.052 

.036 

.019 

32 

.116 

.115 

.112 

.106 

.098 

.088 

.075 

.058 

.042 

.022 

36 

.181 

.130 

.126 

.119 

.110 

.099 

.084 

.066 

.047 

.024 

40 

.145 

.144 

.140 

.133 

.123 

.110 

.093 

.074 

.052 

.027 

44 

.160 

.158 

.153 

.145 

.135 

.121 

.103 

.081 

.057 

.030 

48 

.174 

.172 

.167 

.158 

.147 

.132 

.112 

.088 

.062 

.033 

52 

.189 

.187 

.181 

.171 

.159 

.143 

.122 

.095 

.068 

.035 

56 

.204 

.202 

.195 

.185 

.171 

.154 

.131 

.103 

.073 

.038 

1 

.218 

.216 

.209 

.198 

.183 

.164 

.140 

.111 

.078 

.041 

4 

.233 

.231 

.223 

.211 

.196 

.175 

.150 

.118   " 

.DBS 

.043 

8 

.247 

.245 

.237 

.224 

.208 

.186 

.159 

.125 

.088 

.046 

12 

.262 

.260 

.252 

.237 

.220 

.196 

.168 

.133 

.094 

.049 

16 

.276 

.274 

.265 

.251 

.232 

.207 

.177 

.140 

.099 

.052 

20 

.291 

.288 

.279 

.264 

.244 

.218 

.187 

.148 

.104 

.055 

24 

.306 

.303 

.293 

.277 

.256 

.229 

.197 

.155 

.109 

.057 

28 

.320 

.317 

.307 

.291 

.269 

.240 

.206 

.163 

.114 

.060 

32 

.334 

.331 

.321 

.304 

.281 

.251 

.215 

.171 

.120 

.063 

86 

.349 

.345 

.335 

.317 

.293 

.262 

.224 

.178 

.125 

.066 

40 

.364 

.360 

.349 

.330 

.305 

.273 

.233 

.185 

.130 

.069 

44 

.378 

.374 

.363 

.343 

.318 

.284 

.242 

.192 

.135 

.072 

48 

.393 

.389 

.377 

.356 

.330 

.295 

.251 

.200 

.141 

.075 

52 

.407 

.403 

.391 

.370 

.342 

.305 

.261 

.208 

.147 

.077 

56 

.422 

.418 

.405 

.383 

.354 

.316 

.270 

.215 

.152 

.080 

2 

.436 

.432 

.419 

.397 

.366 

.327 

.280 

.222 

.157 

.083 

4 

.451 

.446 

.433 

.409 

.379 

.338 

.289 

.230 

.162 

.086 

8 

.465 

.461 

.447 

.425 

.391 

.349 

.298 

.237 

.167 

.088 

12 

.480 

.475 

.461 

.437 

.403 

.360 

.308 

.245 

.173 

.090 

16 

.495 

.490 

.475 

.450 

.415 

.371 

.317 

.252 

.178 

.093 

20 

.509 

.504 

.489 

.463 

.428 

.382 

.326 

.260 

.183 

.096 

24 

.523 

.518 

.503 

.476 

.440 

.393 

.334 

.267 

.188 

.099 

28 

.538 

.533 

.517 

.489 

.452 

.404 

.346 

.275 

.194 

.102 

32 

.552 

.547 

.531 

.503 

.465 

.415 

.355 

.282 

.199 

.104 

36 

.567 

.562 

.545 

.516 

.477 

.425 

.364 

.289 

.204 

.107 

40 

.582 

.576 

.559 

.529 

.489 

.436 

.373 

.297 

.209 

.110 

44 

.596 

.590 

.573 

.542 

.501 

.447 

.382 

.304 

.214 

.113 

48 

.611 

.605 

.587 

.555 

.513 

.458 

.391 

.312 

.219 

.116 

52 

.625 

.619 

.601 

.569 

.526 

.469 

.401 

.319 

.225 

.118 

56 

.640 

.634 

.615 

.582 

.538 

.480 

.410 

.326 

.230 

.121 

3 

.654 

.648 

.629 

.595 

.550 

.491 

.419 

.334 

.235 

.124 

4 

.669 

.662 

.643 

.608 

.562 

.502 

.428 

.341 

.240 

.127 

8 

.683 

.677 

.657 

.621 

.574 

.512 

.438 

.349 

.246 

.130 

12 

.698 

.691 

.671 

.635 

.587 

.523 

.448 

.357 

.251 

.132 

16 

.713 

.705 

.685 

.649 

.599 

.534 

.457 

.364 

.257 

.135 

20 

.727 

.720 

.699 

.662 

.611 

.545 

.466 

.371 

.262 

.138 

24 

.742 

.734 

.713 

.675 

.623 

.556 

.475 

.378 

.267 

.141 

28 

.756 

.749 

.727 

.688 

.635 

.567 

.485 

.386 

.272 

.144 

32 

.771 

-763 

.741 

.702 

.648 

.578 

.494 

.394 

.278 

.146 

86 

.786 

.777 

.755 

.715 

660 

589 

503 

401 

.283 

149 

40 

.800 

.792 

.769 

.728 

.673 

.600 

.512 

.408 

.288 

.152 

44 

.814 

.806 

.783 

.741 

.685 

.611 

•521 

.415 

.293 

.155 

48 

.829 

.821 

.797 

.754 

.697 

.621 

.531 

.423 

.298 

.158 

52 

.843 

.835 

.811 

.768 

.709 

.632 

.541 

.431 

.304 

.160 

56 

.858 

.850 

.825 

.781 

.721 

.643 

.550 

.438 

.309 

.163 

4 

.873 

.864 

.839 

.794 

.734 

.655 

.559 

.445 

.314 

.166 

10 

.909 

.900 

.874 

.827 

.764 

.682 

.582 

.464 

.327 

.173 

20 

.945 

.936 

.909 

.860 

.795 

.709 

.606 

.482 

.340 

.179 

30 

.981 

.972 

.944 

.893 

.825 

.736 

.629 

.501 

.354 

.186 

40 

1.017 

1.008 

.979 

.926 

.a55 

.764 

.652 

.519 

.367 

.193 

50 

1.054 

1.044 

1.014 

.959 

.886 

.791 

.676 

-.538 

.380 

.199 

5 

1.091 

1.080 

1.048 

.993 

.917 

.818 

.699 

.557 

.393 

.207 

10 

1.127 

1.116 

1.083 

1.026 

.947 

.845 

.722 

.576 

.406 

.214 

20 

1.164 

1.152 

1.118 

1.058 

.978 

.872 

.746 

.594 

.419 

.220 

30 

1.200 

1.188 

1.153 

1.092 

1.009 

.900 

.769 

.613 

.432 

.228 

634 


TABLE   OF   ORDINATES. 


Table  of  Ordinates  5  ft  apart. —(Continued.) 

Distances  of  the  Ordinates  from  the  end  of  the  100  feet  Chord. 


Ang.  of 
Deft. 

Mid. 
50ft. 

45  ft. 

40ft. 

35ft. 

30ft. 

25ft. 

20ft. 

15ft. 

10ft. 

5ft. 

540 

1.236 

1.224 

1.188 

1.124 

1.039 

.927 

.792 

.631 

.445 

.235 

50 

1.273 

1.2HO 

1.223 

1.157 

1.070 

.954 

.816 

.649 

.458 

.241 

6 

1.309 

1.296 

1.258 

1.191 

1.100 

.982 

.839 

.668 

.472 

.243 

10 

1.345 

1.332 

1.293 

1.224 

1.130 

1.009 

.862 

%    .686 

.485 

.255 

20 

1.382 

1.368 

1.328 

1.256 

1.161 

1.036 

.886 

.705 

.498 

.262 

30 

1.419 

1.404 

1.362 

1.290 

1.192 

1.064 

.909 

.724 

.511 

.269 

40 

1.455 

1-440 

1.397 

1.323 

1.222 

1.091 

.932 

.742 

.524 

.276 

50 

1.491 

1.476 

1.432 

1  .355 

1.253 

1.118 

.956 

.761 

.537 

.283 

7 

1.528 

1.512 

1.467 

1.389 

1.284 

1.146 

.979 

.779 

.551 

.290 

10 

1.564 

1.548 

1.502 

1.422 

1.314 

1.173 

1.002 

.798 

.564 

.297 

20 

1.600 

1.584 

1.537 

1.454 

1.345 

1.200 

1.026 

.816 

.576 

.304 

30 

1.637 

1.620 

1.572 

1.488 

1.375 

1.228 

1.048 

.835 

.590 

.311 

40 

1.073 

1.656 

1.607 

1.521 

1.405 

1.255 

1.071 

.854 

.603 

.318 

50 

1.710 

1.692 

1.641 

1.553 

1.436 

1.282 

1.095 

.872 

.616 

.324 

8 

1.746 

1.728 

1.677 

1.587 

1.467 

1.810 

1.118 

.891 

.629 

.332 

80 

1.855 

1.836 

1.782 

1.687 

1.559 

1.392. 

1.188 

.946 

.669 

.353 

e 

1.965 

1.944 

1.886 

1.787 

1.651 

1.474 

1.258 

1.002 

.708 

.873 

30 

2.074 

2.052 

1.991 

1.887 

1.742 

1.556 

1.328 

1.057 

.748 

.394 

10 

2.183 

2.161 

2.096 

1.987 

1.834 

1.637 

1.398 

1.114 

.787 

.415 

30 

2.292 

2.269 

2.201 

2.087 

1.926 

1.719 

1.468 

1.170 

.827 

.436 

11 

2.401 

2.377 

2.306 

2.186 

2.018 

1.802 

1.538 

1.226 

.866 

.457 

30 

2.511 

2.486 

2.411 

2.286 

2.110 

1.884 

1.609 

1.282 

.906 

.478 

12 

2.620 

2.594 

2.516 

2.386 

2.203 

1.967 

1.680 

1.339 

.946 

.499 

30 

2.730 

2.703 

2.621 

2.485 

2.295 

2.049 

1.750 

1.395 

.985 

.520 

13 

2.889 

2.811 

2.726 

2.585 

2.387 

2.132 

1.820 

1.451 

1  .025 

.541 

30 

2.949 

2.920 

2.832 

2.685 

2.479 

2.214 

1.891 

1.507 

1.065 

.562 

14 

3.058 

3.028 

2.937 

2.785 

2.571 

2.297 

1.961 

1.564 

1-105 

.583 

30 

3.168 

3.136 

3.042 

2.884 

2664 

2.379 

2.031 

1.620 

1.144 

.604 

15 

3.277 

3.245 

3.147 

2.984 

2.756 

2.462 

2.102 

1.676 

1.184 

.625 

30 

3.387 

3.354 

3.252 

3.084 

2.848 

2.544 

2.172 

1.732 

1.224 

.646 

16 

3.496 

3.462 

3.358 

3.184 

2.941 

2.627 

2.243 

1.789 

1.264 

.667 

17 

3.716 

3.680 

3.569 

3.384 

3.125 

2.792 

2.384 

1.902 

1.344 

.709 

18 

3.935 

3.897 

3.779 

3.584 

8.310 

2.958 

2.525 

2.014 

1.424 

.751 

19 

4.155 

4.115 

3.990 

3.784 

3.495 

3.123 

2.666 

2.127 

1.504 

.793 

20 

4.375 

4.332 

4.201 

3.984 

3.680 

3.288 

2.808 

2.240 

1.583 

.836 

22 

4.815 

4.768 

4.624 

4.386 

4.050 

3.620 

3.093 

2.467 

1.744 

.922 

24 

5.255 

5.204 

5.048 

4.789 

4.423 

3.952 

3.379 

2.695 

1.905 

1.008 

26 

5.697 

5.642 

5.473 

5.192 

4.798 

4.286 

3.665 

2.924 

2.068 

1.094 

28 

6.139 

6079 

5.898 

5.595 

5-171 

4.622 

3.952 

3.154 

2.232 

1.181 

30 

6.582 

6.517 

6.323 

5.999 

5.544 

4.958 

4.239 

3.385 

2.396 

1.268 

32 

7-027 

6.957 

6.751 

6.406 

5.922 

5.297 

4.530 

3.619 

2.565 

1.356 

34 

7.472 

7.398 

7.179 

6.813 

6.300 

5.637 

4.822 

3.854 

2.733 

1.445 

36 

7.918 

7.841 

7.609 

7.2'22 

6.679 

5.978 

5.115 

4.090 

2.901 

1.535 

38 

8.367 

8.286 

8.041 

7.633 

7.060 

6.320 

5.410 

4.327 

3.069 

1.626 

40 

8.816 

8.731 

8.474 

8.044 

7.442 

6.663 

5.705 

4.565 

3.238 

1.718 

GEILLAGE, 


After  piles  have  been  driven,  and  their  heads  carefully  sawed  off  to 

a  level,  if  not  under  water,  the  spaces  between  them  are  in  important  cases  filled  up  level  with  their 
tops  with  well  rammed  gravel,  stone  spawls,  or  concrete,  in  order 
to  impart  some  sustaining  power  to  the  soil  between  the  piles.  Two 
courses  of  stout  timbers,  ("from  8  to  12  ins  square,  according  to  the 
weight  to  be  carried)  are  then  bolted  or  treeuailed  to  the  tops  of  the 
piles  and  to  each  other,  as  shown  in  the  Fig,  forming  what  is  called 
a  grillage.  On  top  of  these  is  bolted  a  floor  or  plat- 
form of  thick  plank  for  the  support  of  the  masonry  ;  or  the  timbers 
of  the  upper  course  of  the  grillage  may  be  laid  close  together  to  form  the  floor.  The  space  below  the 
floor  should  also,  in  important  cases,  be  well  packed  with  gravel,  spawls,  or  concrete. 
If  under  water,  the  piles  are  sawed  off  by  a  diver,  or  by  a  circular  saw  driven 
by  the  engine  of  the  pile-driver,  and  the  grillage  is  omitted.  Instead  of  it  the  masonry  or  concrete 
may  be  built  in  the  open  air  in  a  caiasoti,  p  316  ;  which  gradually  sinks  as  it  becomes  filled  ;  or  on  a 
strong  platform  which  is  lowered  upon  the  piles  by  screws  as  the  work  progresses,  p  328.  Or  a  strong 
caisson  may  first  be  sunk  entirely  under  water,  and  then  be  filled  with  concrete,  p  507,  up  to  near 
low  water;  the  caisson  being  allowed  to  remain.  Or  the  caisson  may  form  a  cofferdam,  to  be  first 
sunk,  and  then  pumped  dut.  If  the  ground  is  liable  to  wash  away  from  around  the  piles,  as  in  the 
case  of  bridge  piers,  &c,  defend  it  toy  sheet-piles,  or  Tip- rap,  or  both  ;  p  314. 


BUOYANCY, 


ATION,    METACENTER,    ETC.          635 


or  O,  either  solid  or  a  vessel  of  any  shape,  at  rest,  as  L, 

*"•       buoyancy  and  flotation)  be  considered  as  acted  upon  only  by  two 


A  floating  bodv 

may  (so  far  as 
equal  vert  for 
shown  by  Jtf, 
equal  arrows.*  One 
of  them  is  the  wt  of 
the  body  itself,  al- 
ways acting  vert 
downwards,  and  as 
if  concentrated  at 
the  oen  of  grav  G  of 
the  body.  The  other 
is  the  "buoyancy  or 
upward  pres  of  the 
water;  and  is  equal 
to  the  wtof  the  water 
displaced  by  the 
body  :  or  to  the  wt  of 
the  "floating  body  it- 
self. It  always  acts  vert  upwards,  and  as  If  concentrated  at  the  cen  of  grav  W  of  the  displaced  water.f 

W  is  also  called  the  center  of  pressure,  or  of  buoyancy  of  the  water; 
and  a  vert  line  drawn  through  it  is  called  the  axis  or  vertical  of  buoyancy, 
or  of  flotation.  This  W  of  course  shifts  its  position  with  every  change  in  that 
of  the  body.  Thus  in  L  it  is  at  the  cen  of  grav  of  the  rectangle  o  o  b  b  :  and  in  N,  at  that  of  the  tri- 
angle a  a  v.  When  a  floating  body  L  or  P  is  at  rest  or  undisturbed  by  any  third  force,  then  G  and  W 
will  be  in  the  same  vert  line  t  t&g  L;  or  e  e  fig  P  ;  which  line  is  called  the  axis, 
or  vertical  of  equilibrium.  If  (T  fig  L  is  then  above  W,  the  body  will 
be  in  unstable  equilibrium ;  that  is,  if  any  third  force,  as  the  wind,  F  fig  N, 

causes  the  axis  of  equilibrium  to  lean,  the  body  will  upset;  for  the  forces  G  and  W  then  no  longer  act 
in  the  same  straight  line,  but  in  two  parallel  lines;  thus  forming  a  coil  pie  (Case 
3,  p  483):  and  instead  of  holding  each  other,  and  the  body,  in  equilibrium,  they  cause  it  to  rotate 
around  a  point  half  way  between  them.  Thus  the  force  G  fig  N  impels  the  upper  part  of  the  body 
downwards,  while  W  impels  its  lower  part  upwards.  Hence  the  body  must  upset,  even  if  F  ceases. J 

But  if  the  upright  vessel  L  be  so  loaded  that  G  shall  be  vert  below  W,  then  it 
will  be  in  stable  equilib;  so  that  if  a  third  force  F,  causes  the  axis  of  equilib  tt 

to  lean  as  in  fig  O,  the  couple  or  forces  G  and  W  will  tend  to  prevent  it  from  upsetting ;  since  G  impels 
the  lower  part  downward,  while  W  impels  the  upper  part  upward ;  so  that  if  F  ceases  to  act,  the  vessel 
•will  right  itself.  It  is  true  however  that  even  in  this  case  the  thiid  force  F  may  be  so  great  as  to  en- 
tirely overpower  the  combined  forces  G  and  W,  so  that  a  vessel  may  upset  in  a  hurricane,  although 
judiciously  loaded  and  ballasted  for  ordinary  winds.  The  tendency  of  the  leaning  bouy  either  to 
upset  or  to  right  itself  increases  with  its  inclination,  and  is  readily  found  in  foot-pounds  as  stated 
in  Case  3,  p  483.  by  mult  one  of  the  forces  by  the  perp  dist  in  ft  existing  between  the  two  forces  at 
any  given  inclination. 

Uneven  loading1,  instead  of  a  third  force,  may  cause  a  vessel  at  rest  to 

incline  as  in  fig  F ;  and  yet  the  vessel  so  leaning  will  be  in  stable  equilibrium,  G  being  below  W  ;  for 
its  axis  e  e  of  equilibrium  is  vert,  although  not  coinciding  with  the  axis  of 
symmetry  of  the  vessel,  as  it  does  at  t  tin  L.  Hence  if  a  third 

force,  as  wind,  should  cause  e  e  to  lean,  G  and  W  will  both  tend  to  bring  it  back  to  a  vert  position. 
It  is  plain  however  that  a  vessel  thus  unevenly  loaded  would  more  easily  upset  towards  the  right  hand 
than  towards  the  left;  hence  it  should  be  so  loaded  as  to  float  upright  when  at  rest.  Also  the  heaviest 
articles  of  the  cargo  should  be  placed  lowest  in  the  hold,  in  order  to  keep  G  as  far  below  W  as  pos- 
sible. Persons  in  a  boat  in  danger  of  upsetting  should  squat  down  instead  of  standing  up. 

When  a  third  force  causes  the  axis  of  equilibrium  1 1  fig  L  of  a  floating  body  to  lean  as  in  figs  N  and 
O,  then  if  a  vert  line  be  drawn  upwards  from  the  center  W  of  flotation,  the  point  M  at  which  said  line 

cuts  said  axis  is  called  the  metaceiiter  of  the  body.    This  metacenter  shifts  its 

position  according  to  the  inclination  of  the  axis  of  equilib ;  but  so  long  as  it  is  higher  than  the  cen  of 
nv  G  of  the  body,  as  in  fig  O,  the  body  will  remain  in  stable  equilib,  and  will  restore  itself  to  its 
iginal  position  as  soon  as  the  disturbing  force  ceases  to  act.  But  if  M  is  below  G  as  in  fig  N,  the 

equilib  will  be  unstable,  and  the  body  must  upset  even  if  the  third  force  ceases  to  act. 

A  hor  section  of  a  body  at  water-line  is  called  its  plane  of  flotation. 

*  The  body  is  in  fact  acted  upon  by  other  forces,  such  as  the  hor 

pressures  of  the  water  against  its  immersed  portions  ;  but  as  all  of  these  in  any  one  given  direction 
are  balanced  by  equal  ones  in  the  opposite  direction,  they  have  no  effect  upon  the  forces  G  and  W. 
It  is  also  acted  upon  by  the  air,  which  presses  it  downwards  with  a  force  of  14.75  Ibs  per  sq  inch  ;  but 
this  is  balanced  by  an  equal  pres  of  the  surrounding  air  upon  the  surface  of  the  water,  and  which  is 
nsnaitted  'art  7,  p  526)  vert  upwards  against  the  immersed  bottom  of  the  floating  body. 

t  This  buoyancy  is  made  lip  of  the  parallel  upward  pressures  of  the 
innumerable  vert  filaments  of  the  displaced  water  as  shown  by  Fig  26,  p  533 ;  and 
;he  axis  of  flotation  is  their  resultant,  as  in  the  case  of  parallel  forces  Fig  55,  p  481. 

t  When  a  body  is  held  in  eqiiilib  by  two  forces,  one  of  which  is  its 

>wn  wt,  it  follows,  that  inasmuch  as  gravity  or  wt  acts  vert  downwards  only,  the  two  forces  must  be 

in  the  same  vert  line.     If  a  disturbing  force  should  raise  the  cen  of  grav  of  the 

body,  then  on  the  removal  of  said  force  this  cen  of  grav  will  naturally  descend  to  its  original  position, 

and  thus  restore  the  equilibrium,  which  in  such  cases  is  called  stable. 
But  if  the  disturbing  force  lowers  the  cen  of  grav  of  the  body,  said  cen  will  not 

return  to  its  original  position  ;  but  will  continue  to  descend,  and  the  body  will" fall ;  its  equilib  having 

been  unstable.  If  the  cen  of  j;  rav  of  the  body  neither  rises  nor 
falls,  its  equilibrium  is  called  indifferent;  and  the  body  will  remain  in  any 
position  in  which  a  third  force  may  place  it,  as  a  ball  on  a  level  table,  or  a  grindstone  on  its  axis. 


636 


TEST   BORINGS. 


Pierce's  well  borer  is  an  excellent  tool  for  boring  into  soils,  clay,  sand  or 

gravel,  even  when  quite  indurated.  It  removes  pebbles  and  stones  smaller  than  the  bore.  The 
augers  are  made  from  6  to  18  ins  diain,  according  to  the  required  purpose;  and  if  the  hole  should  re- 
quire to  be  enlarged  for  the  insertion  of  tubing  or  curbing,  a  rimmer  is  at- 
tached. If  loose  running  sand,  slush,  &c,  are  met  with,  the  sand-sides  and 
valves  are  put  on  ;  but  for  these  materials,  (which  require  tubing  even  for  test 
holes)  Pierce's  sand-auger  is  better.  A  light  derrick  about  25  or  30  feet 

high,  with  winch,  drum,  simple  cog-gear,  and  two-block  tackle  are  required  for  raising  the  auger  iiud 

its  rods  at  short  intervals  for  emptying.    The  square  socket-jointed  rods 

are  10  to  14  ft  long,  of  1.5  inch  sq  iron.  Where  no  tubing  is  needed  the  auger  is  screwed  down  at  the 
rate  of  from  3  to  20  ft  per  hour  by  hook- wrench  levers  6  ft  long,  worked  by  '2  to  4  men,  or  by  a  horse, 
according  to  depth,  hardness,  diam,  &c.  Another  man  attends  the  winch  by  which  the  auger  is  fed 
and  raised.  Where  tubing  is  required  the  progress  is  of  course  much  slower.  In  dry  soils  a  bucket 
of  water  is  thrown  down  the  hole  whenever  the  auger  is  raised.  In  boring  many  shallow  holes  near 
each  other  the  moving  of  the  derrick  consumes  much  of  the  time.  The  Pierce  borer  may  be  ad- 

vantageously  used  for  sand-piles,  p  328;  and  at  times  instead  of  driving 
wooden  piles,  it  may  be  better  to  plant  them  (perhaps  butt  down)  in  holes 

bored  by  this  auger,  ramming  the  earth  well  around  them  afterwards.  This  will  save  adjacent  build- 
ings from  the  jarring  and  injury  done  by  a  pile-driver. -X- 

In  testing;  through  hard  gravel  mixed  with  cobbles,  or  even  through 

rock,  a  welded  iron  tube  about  4  to  6  ins  diam,  in  screw  jointed  lengths  of  about  8  ft,  and  with  a  steel 
cutting-edge  ring  screwed  around  its  foot,  may  be  held  over  the  spot,  and  if  possible  be  driven  down 
so  far  as  to  stand  by  itself;  or  if  this  cannot  be  done  it  must  be  held  in  place  at  first  by  other  means. 
Inside  of  this  tube  the  boring  is  done  by  a  boring-bit  weighing  about  150  Ibs, 
of  about  3  inch  diam  iron,  3  ft  long~  with  a  steel  chisel-edge  foot  about  4  ins  wide  ,  and  with  an  eye 
at  top  for  a  derrick  rope.  It  is  expedient  to  have  at  bund  an  extra  boring  bit  of  about  400  Ibs,  in  case 
the  hardness  of  the  material  should  require  it.  To  vork  the  bit,  a  few  turns  of  the  i  ope  are  taken 
around  the  drum,  and  a  man  pulls  at  the  slack  end  of  the  rcpe,  while  the  bit  is  being  raised  a  foot  or 
two  by  the  crab.  When  so  raised  this  oian  lets  go  his  end  of  the  rope,  thus  loosening  the  turns 
around  the  drum,  and  the  bit  falls.  Previous  to  each  stroke  of  the  bit  the  rope  must  be  slightly 
twisted,  in  order  to  change  the  position  of  the  chisel  at  each  stroke,  so  that  the  hole  may  be  round. 

At  intervals  of  about  a  foot  in  depth,  as  the  boring  goes  on,  the  bit  must  (as  in  other  drilling  opera- 
tions) be  lifted  out  by  the  crab  to  allow  the  debris  which  has  accumulated  in  the  well  tube  to  be  re- 
moved. Except  for  very  shallow  holes  this  is  best  done  by  a  sand-pump,  a 
simple  form  of  which  is  a  welded  iron  tube  for  the  pump  barrel,  about  4  ft  long,  and  with  a  diam  say 
about  an  inch  less  than  that  of  the  well  tube  into  which  it  is  to  be  lowered.  At  its  foot  is  a  leathe'r 
up- valve  ;  and  at  top  is  a  falling  or  bucket  handle  for  lowering  it  into,  or  lifting  it  out  of  the  well-tube 
by  the  derrick  rope  and  crab.  In  this  pump-barrel  works  a  close-fitting  sucker 
or  piston  of  wood  and  leather,  with  a  handle  or  pump-rod  about  6  ft  long,  with  an  eye  at  top  for  a 
rope.  By  this  rope  the  sucker-rod  is  jerked  suddenly  upwards  a  foot  or  two.  by  hand  a  few  times,  de 
scending  each  time  by  its  own  wt.  Water  having  first  been  poured  into  the  well,  this  process  pumps 
the  debris  (including  fragments  of  egg  size)  from  the  bottom  of  the  well  tube  into  the  pump  barrel, 
which  is  then  lifted  out  of  the  well  and  emptied.  The  well  tube  is  then  driven  down  a  little  farther, 
the  boring  bit  is  again  lowered  into  it,  and  the  boring  is  continued  until  enough  more  debris  accumu- 
lates to  again  require  the  pump;  and  so  on  alternately.  Frequently  something  will  stick  in  the 
sucker,  and  keep  it  open  so  as  to  prevent  the  pump  from  working.  The  pump  must  then  be  lifted  out 
from  the  well,  and  the  obstacle  removed. 

An  ordinary  clay's  work  with  this  tool  in  hard  gravel  mixed  with  cobbles, 

•will  be  but  from  2  to  4  ft  of  depth.     Pierce's  hand-drill  is  much  more  rapid. 

To  avoid  bruising  the  top  of  the  well-tube  by  hard  driving,  and 

thus  destroying  the  fit  of  the  screw  joints  of  its  separate  lengths,  there  must,  while  driving,  be  placed 
upon  it  some  attachment  of  hard  wood  or  iron  to  receive  the  blows  of  the  heavy  maul. 

For  boring:  under  water  the  derrick  may  be  placed  at  the  center  of  a 

raft  or  scow  of  about  15  by  30  ft,  with  an  opening  about  a  foot  square  for  working  the  tools.  If  in  a 
tideway  there  will  be  trouble  in  keeping  a  raft  or  scow  constantly  in  position  ;  and  some  arrangement 
must  be  made  for  a  fixed  platform  to  work  from. 

*  The  Pierce  Well  Excavator  Co,  No  29  Rose  St,  New  York,  furnish 

these  tools  complete,  as  well  as  all  those  for  artesian  and  other  boring,  together  with  earth  screw- 
augers,  windmills,  pumps,  tubings,  &c.  Or  they  will  contract  to  do  the  work  itself,  using  their  own 

tools.    Their  charges,  in  1878,  were  about  $1  per  ft  of  depth  for  a  well  17  ins 

diam,  lined  with  wood;  and  from  $3  to  $5  per  foot  for  holes  5  ins  diam  in  solid  rock,  to  a  depth  of  1000 

ft  or  more.    Their  price  for  the  above  well-borer  of  any  one  given  diam  from 

6  to  18  ins,  and  with  rods  for  a  depth  of  100  ft,  including  derrick,  tackle,  levers,  Ac,  complete,  is 
about  $150,  including  rimmer,  $5,  and  sand-sides  and  valves,  $10.  The  sand-auger  $35  extra.  They 
furnish  well-tubing  of  No  16  to  18  galvanized  iron,  1  foot  diam,  at  50  to  75  cts  per  ft  run. 

A  lighter  tool  for  holes  about  6  ins  diam,  and  10  to  20  ft  deep,  and  which 

would  require  no  derrick,  but  would  be  raised  by  hand,  can  be  furnished  for  about  $40,  exclusive  of 
rimmer,  and  sand-sides.  The  same  Co  furnish  their  Portable  Hand  Rock 
]>rill  including  drill  sharpener,  6  drills,  &c,  at  $230.  With  it  one  man  can  drill 
3  inch  holes  to  a  depth  of  50  or  more  feet,  at  the  rate  of  from  1  to  5  ft  per  hour,  according  to  hard 
ness,  depth,  &c.  It  requires,  of  course,  a  sand-pump. 

A  screw-auger  of  steel  about  15  ins  long,  and  1.75  ins  diarn,  either  single 

or  (far  better)  double  twist,  with  jointed  boring  rods  of  1.5  inch  square  iron  or  (much  better)  steel, 
each  joint  about  10  ft  long,  may  be  u.sed  for  test  boring  in  clay,  sand,  or  fine  gravel,  of  all  which  it 
will  bring  up  samples.  It  will  not  bore  through  pebbles ;  but  at  moderate  depths  these  may  be  broken 
up,  or  penetrated  by  a  bar  with  a  cutting  edge.  Loose  sand  or  slush  will  of  course  require  gas-pipe  tub 
ing.  It  may  be  worked  to  a  depth  of  100  ft  in  a  day  or  two,  by  '2  to  4  men  according  to  hardness  and 
depth  ,  levers  3  to  4  ft  long.  It  must  have  a  derrick,  &c,  like  the  well-borer.  The  above  Co  can  fur- 


AND  +   PILLARS. 


637 


Table  A.  Br<*lLRing:  loads  in  tons  (224O  Ibs)  per  square  inch 
of  metal  area,  of  pillars  or  struts  of  L,  T,  or  +  section  of 
equal  arms,  and  of  uniform  thickness;  by  formulas  on  p  235. 

The  heights  or  lengths  of  the  pillars  are  in  out  to  out  arms  of  the  sec- 
tion. See  "  Remarks  "  below.  For  table  of  pillars  of  H  section  see  pp  236  and  638.  (Original.) 


Hts 

Cast. 

Wrt. 

Hts 

Cast. 

Wrt. 

Hts 

Cast. 

Wrt. 

Hts 

Cast. 

Wrt. 

Arras. 

Tons. 

Tons. 

in 
Arms. 

Tons. 

Tons. 

in 
Arms. 

Tons. 

Tons. 

in 
Arms. 

Tons. 

Tons. 

1 

35.5 

16.1 

12 

17.2 

14.7 

23 

7.20 

11.9 

38 

3.02 

8.19 

2 

34.7 

16.0 

13 

15.8 

14.4 

24 

6.72 

11.6 

40 

2.75 

7.78 

3 

33.5 

16.0 

14 

14.5 

14.2 

25 

6.28 

11.3 

42 

2  52 

7.39 

4 

32.0 

15.9 

15 

13.3 

14.0 

26 

5.88 

11.0 

45 

2.21 

6.84 

5 

30.1 

15.8 

16 

12.2 

13.7 

27 

5.52 

10.8 

50 

1.81 

6.03 

6 

28.1 

15.7 

17 

11.3 

13.5 

28 

5.19 

106 

55 

1.51 

5.34 

7 

26.1 

15.5 

18 

10.4 

13.2 

29 

4.88 

10.3 

60 

1.28 

4.73 

8 

24.1 

15.4 

19 

9.6 

13.0 

30 

4.61 

10.0 

70 

.95 

3.77 

9 

22.2 

15.2 

20 

8.9 

12.7 

32 

4.11 

9.6 

80 

.73 

3.05 

10 

20.4 

15.1 

21 

8.3 

12.4 

34 

3.69 

9.1 

90 

.58 

2.51 

11 

18.7 

14.9 

22 

7.7 

12.1 

36 

3.33 

8.6 

100 

.47 

2.10 

Remarks.    If  the  arms  or  members  taper  towards  their  ends, 

as  they  nearly  always  do  in  practice,  instead  of  being  of  uniform  thickness,  the  strength  is  thereby 
lessened  to  an  extent  that  varies  with  the  proportions  of  the  section,  the  height,  and  whether  of  cast 
or  of  wrought  iron  ;  and  although  in  common  practice  this  taper  is  not  great,  still  for  safety  it  is  well 
to  assume  the  metal  area  at  only  what  it  would  be  if  the  tapering  members  had  a  uniform  thickness 
about  equal  to  that  at  their  ends. 

If  one  of  the  two  members  is  shorter  than  the  other  the 

height  must  be  measured  by  the  short  one  ;  and  we  shall  in  that  case  err  on  the  safe  side  by  using  the 
loads  in  the  table  as  the  breaking  ones,  for  these  last  will  in  fact  be  greater  at  the  reduced  height. 
Still  if  lengths  of  arms  differ  more  than  one-sixth  part  use  the  formula. 

The  above  table  may  also  be  used  for  this  channel  form  LJ  , 

thus,  If  the  out  to  out  length  of  a  flange  is  just  one-half  the  out  to  out  length  of  web,  and  if  the  uni- 
form thickness  is  not  greater  than  one-sixth,  nor  less  than  one-eighteenth  of  the  out  to  out  length 
of  web,  th  ;n  the  breaking  load  per  sq  inch  for  any  length  or  height  measured  in  flanges  may  be  taken 

at  once  from  the  table,  sufficiently  correct  for  practice.  But  if  the  out  to 
out  length  of  a  flange  is  either  greater  or  less  than  half  that  of 

the  web  of  the  uniformly  thick  channel  iron,  but  not  shorter  than  one-fifth  of  the  out  to  out  length 
of  web,  (or  than  two  thicknesses  of  web,  if  they  amount  to  more  than  the  other)  tiien  multiply  the 
load  in  the  above  table  by  the  corresponding  multiplier  in  the  table  below.  See  another  table  of 
channel-bar  pillars,  p  640.  (Original.) 


a  • 

la  « 

Length  of  flange  in  parts  of  length  of  web  ;  both  from  oat  to  out. 

si 

.2     |  .25 

.3 

.35 

.4 

.45 

.5 

.6 

.7 

.8 

.9 

l.O 

— 

CAST.    Multipliers  for  the  co  umns  of  cast  iron  in  table  A. 

5 

.85 

.90 

.93 

.95 

.97 

.99 

i. 

1.01 

1.02 

1.03 

1.03 

1.04 

10 

.67 

.77 

.85 

.90 

.94 

.97 

I. 

1.04 

1.07 

1.09 

1.11 

1.13 

15 

.58 

.69 

.78 

.85 

.91 

.96 

1.06 

1.10 

1.14 

1.17 

1.20 

20 

.54 

.66 

.75 

.83 

.89 

.95 

1.07 

1.13 

1.17 

1.21 

1.24 

30 

.50 

.62 

.72 

.81 

.88 

.95 

1.08 

1.15 

1.20 

1.25 

1.29 

40 

.49 

.61 

.71 

.80 

.87 

.94 

1.09 

1.16 

1.21 

1.27 

1.32 

50 

.48 

.60 

.70 

.79 

.87 

.94 

1.09 

1.16 

1.22 

1.28 

1.33 

100 

.47 

.59 

.70 

.79 

.86 

.94 

1.09 

1.18 

1.24 

1.30 

1.35 

ROLLED.    Multipliers  for  the  co  umns  of  wrought  iron  in  table  A. 

5 

.98 

.99 

.99 

.99 

.99 

.99 

1. 

I, 

1. 

1. 

j 

10 

.93 

.96 

.97 

.98 

.99 

.99 

1. 

1. 

1.01 

1.01 

L01 

15 

.87 

.92 

.94 

.96 

.98 

.99 

1.01 

1.01 

1.02 

1.03 

1.04 

20 

.81 

.87 

.91 

.94 

.97 

.99 

1.02 

1.02 

1.04 

1.05 

1.06 

30 

.70 

.80 

.86 

.90 

.95 

.98 

1.03 

1.06 

1.09 

1.11 

1.12 

40 

.63 

.74 

.82 

.87 

.92 

.97 

1.04 

1.08 

1.11 

1.14 

1.16 

50 

.59 

.70 

.78 

.85 

.91 

.96 

i! 

1.05 

1.10 

1.14 

1.17 

1.19 

100 

.50 

.62 

.72 

.81 

.88 

.95 

i. 

1.08 

1.15 

1.20 

1.25 

1.29 

If  the  flanges  are  thinner  than  the  web  the  strength  per  square 

inch  of  the  whole  metal  area  becomes  less  ;. and  if  thicker  than  the  web  it  becomes  greater;  both  to 
an  extent  varying  with  every  different  proportion  of  the  parts,  and  with  the  height  or  length  of 
strut  or  pillar  as  measured  by  flanges,  and  also  with  whether  it  is  of  cast  or  of  wrought  iron. 

If  the  difference  in  thickness  is  not  more  than  one-eighth  part  either 

way  it  may  be  disregarded  without  serious  error  ;  but  if  more,  use  the  formula  (Raukine's)  p  236. 

nish  it  complete  for  about  $125.  In  non-running  earths  and  sands  considerable  depths  may  be  reached 
without  tubing,  as  the  sides  will  not  cave. 

For  work  in  loam,  clay,  or  non-running  sand,  an  effective  screw-auger 

can  be  made  by  any  good  blacksmith,  by  merely  forming  a  one  inch  sq  bar  of  iron  or  steel  into  cork- 
screw shape  about  2  ft  long,  with  6  complete  turns  6  ins  iu  diam ;  its  lower  end  sharpened  to  form  a 
vertical  cutting  edge,  which  should  project  say  .5  of  an  inch  beyond  the  spiral  of  the  screw,  in  order 
to  diminish  friction.  It  will  hring  up  full  samples.  Requires  a  derrick,  or  some  other  simple  mode 
of  lifting,  when  the  screw  is  full. 

41 


638 


ROLLED   I    BEAMS   AS   PILLARS. 


2  .a  .23 


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LLED    I    BEAMS   AS    PILLARS. 


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640 


ROLLED   CHANNEL-BAR    PILLARS. 


Table  of  quiescent  breaking:  loads  in  short  tons  (2OOO  Ibs) 
of  rolled  channel  iron  LJ  *  as  pillars  or  struts.    From  Carnegie 

Bros  &  Cos.  (Union  Iron  Milla,  Pittsburgh,  Penn)  "  Tables  and  Information  on  Wrought  Iron." 
For  manner  of  use  see  the  paragraph  above  the  preceding  table.     Hgd,  fxd,  mean  hinged,  fixed. 
The  headings  12  Hy,  12  Mm,  &c,  mean  12  inch  heavy,  and  12  inch  medium 
channels.    For  the  weights  and  dimensions  of  these  see  Table  p  211.   Sideway 
means  across  the  web;  Edgeway  means  along:  the  web. 


Length  of  Strut. 

12  Hy. 

12  Mm. 

10  Hy. 

10  Mm. 

Both 
ends 
flxd. 

1  fxd. 
1  hgd. 

Both 
ends 
hgd. 

Side 
Way. 
Sht 

Edge 
Way. 

Sht 

Side 
Way. 
Sht 

Edge 
Way. 
Sht 

Side 
Way. 

Sht 

Isr. 

Sht 

Side 
Way. 

Sht 

Edge 
Way. 
Sbt 

Ft. 

Ft. 

Ft. 

268 

160 

160 

121 

124 

1 

270 

183 

189 

2 

1.5 

1 

260 

268 

154 

160 

181 

189 

118 

123 

4 

3. 

2 

235 

266 

135 

160 

J*51 

188 

104 

123 

6 

4.5 

3 

200 

264 

112 

160 

135 

187 

87 

122 

8 

6. 

4 

168 

262 

90 

159 

111 

86 

71 

122 

10 

7.5 

5 

138 

260 

72 

159 

90 

85 

57 

121 

12 

9. 

6 

114 

258 

60 

159 

75 

83 

46 

121 

14 

10.5 

7 

95 

256 

47 

158 

60 

81 

38 

120 

16 

18 

12. 
13.5 

8 
9 

79 
66 

g 

38 
32 

156 
155 

50 
41 

78 
76 

31 

26 

118 
116 

20 

15. 

10 

56 

252 

27 

153 

35 

74 

22 

115 

22 

16.5 

11 

48 

249 

23 

51 

31 

71 

19 

113 

24 

18. 

12 

41 

246 

20 

49 

27 

67 

16 

111 

26 

19.5 

13 

36 

243 

17 

47 

24 

63 

14 

10t> 

28 

21. 

14 

32 

239 

15 

45 

20 

159 

12 

107 

30 

22.5 

15 

28 

235 

13 

43 

17 

156 

11 

105 

32 

24. 

16 

25 

230 

11 

40 

15 

153 

10 

103 

34 

25.5 

17 

23 

226 

10 

38 

14 

149 

9 

100 

36 

27. 

18 

21 

222 

9 

36 

13 

146 

8 

98 

38 

28.5 

19 

19 

218 

8 

34 

11 

143 

7 

96 

40 

30. 

20 

17 

213 

7 

32 

10 

140 

6 

94 

9  Hy. 

9  Mm. 

8  Hy. 

8  Mm. 

1 

160 

162 

96 

97 

134 

135 

85 

86 

2 

1.5 

1 

154 

161 

93 

97 

130 

134 

83 

85 

4 

3. 

134 

160 

80 

96 

112 

133 

70 

85 

6 

45 

3 

110 

159 

65 

96 

91 

132 

57 

84 

8 

6. 

4 

90 

158 

53 

95 

72 

131 

45 

84 

10 

7.5 

5 

72 

157 

42 

95 

58 

130 

36 

83 

12 

9. 

6 

58 

154 

33 

94 

46 

127 

28 

83 

14 

10.5 

7 

47 

150 

27 

93 

30 

125 

23 

82 

16 

12. 

8 

38 

147 

22 

92 

25 

123 

19 

81 

18 

13.5 

9 

32 

143 

18 

91 

21 

120 

15 

80 

20 

15. 

10 

27 

140 

15 

90 

19 

118 

13 

78 

22 

16.5 

11 

23 

138 

13 

88 

17 

115 

11 

76 

24 

18. 

12 

20 

136 

11 

86 

15 

112 

10 

74 

26 

19.5 

13 

17 

134 

9 

84 

13 

109 

8 

72 

28 

21. 

14 

15 

131 

8 

82 

12 

105 

7 

70 

30 

22.5 

15 

13 

129 

7.3 

80 

10 

102 

6 

68 

32 

24. 

16 

12 

126 

6.6 

78 

9 

99 

5.5 

66 

34 

25.5 

17 

11 

122 

5.9 

76 

8 

96 

5 

64 

36 

27. 

18 

10 

118 

5.2 

74 

7.4 

93 

4.5 

62 

38 

28.5 

19 

9 

115 

4.5 

72 

6.7 

89 

4 

60 

40 

30. 

20 

8 

111 

3.8 

70 

6.0 

86 

3.5 

58 

7Hy. 

7  Mm. 

6  Hy. 

5  Hy. 

1 

105 

106 

75 

76 

54 

55 

53 

54 

2 

1.5 

1 

103 

106 

72 

75 

51 

54 

50 

54 

4 

3. 

2 

87 

105 

61 

75 

43 

53 

41 

53 

6 

4.5 

3 

70 

104 

49 

74 

33 

52 

31 

52 

8 

6. 

4 

55 

104 

39 

74 

25 

51 

24 

51 

10 

7.5 

5 

43 

103 

30 

73 

20 

50 

18 

50 

12 

9. 

6 

34 

100 

24 

72 

16 

49 

14 

49 

14 

10.5 

7 

27 

98 

19 

71 

13 

48 

11 

47 

16 

12. 

8 

23 

96 

16 

70 

10 

47 

8.8 

45 

18 

13.5 

9 

18.2 

94 

13 

68 

8 

46 

7.2 

43 

20 

15. 

10 

15.5 

"  92 

11 

66 

7 

45 

6 

41 

22 

16.5 

11 

13.5 

89 

9 

64 

6 

44 

5 

40 

24 

18. 

12 

12 

86 

8 

62 

5 

42 

4.3 

38 

26 

19.5 

13 

10.5 

83 

7 

60 

4.5 

40 

3.7 

36 

28 

21. 

14 

9 

80 

6 

58 

4. 

38 

3.2 

34 

30 

22.5 

15 

7.5 

77 

5 

56 

3.6 

37 

2.8 

32 

32 

24. 

16 

6.9 

75 

4.6 

54 

3. 

36 

2.5 

31 

34 

25.5 

17 

6.2 

72 

4.2 

52 

2.6 

35 

2.2 

30 

36 

27. 

18 

5.5 

69 

3.7 

50 

2.3 

34 

2.0 

28 

38 

28.5 

19 

5.0 

66 

3.3 

48 

2.1 

33 

1.8 

26 

40 

30. 

20 

4.5 

63 

3.1 

46 

1.9 

30 

1.6 

24 

3NTINUOUS   BEAMS.      MODELS. 


641 


Copffnuous  beams.  When  a  single  beam,  as  a  b,  Fig  40,  is  supported  not 
only-lit  its  two  ends,  but  at  one  or  more  intermediate  points,  it  is  said  to  be  con- 
tinuous. It  is  stronger  than  if  it  were  cut  into  two  parts,  a  c,  b  c,  each  supported 

at  both  ends ;  because  the  tensile 
strength  of  the  particles  at  o  (lower 
Pig)  assists  in  counteracting  the  bendg 
or  breakg  tendency  of  loads  on  the  in- 
termediate parts  o  ra,  on,  of  the  lower 
Fig.  These  particles  at  o  must  be  torn 
asunder  before  the  beam  (if  properly 
proportioned)  can  fail.  Such  a  beam, 


rara,  if  very  long  and  flexible,  will, 
under  its  own  wt,  assume  the  shape  of 
the  reversed  curve  m  so  s  n ;  or  if  it  be 
stiff,  and  heavily  loaded,  the  same 
effect  will  follow.  The  points  ss,  at  which  the  curves  reverse,  are  called  tlie 
points  of  contrary  flexure;  and  the  spans  are  virtually  reduced  from  mo 
and  n  o,  to  ms  and  ns.  When  the  beam  is  supported  at  only  3  points,  as  in  the  Fig, 
and  uniformly  loaded,  the  point  of  contrary  flexure  is  dist  from  the  central  support 
%  of  the  span ;  so  that  each  span,  om,  on,  becomes  virtually  reduced  about ^  part; 
and  the  defs  will  be  but  about  y%  as  great  as  if  there  were  two  separate  beams.  The 
sections  of  the  beam  at  s  and  s  will  then  experience  no  hor  strain  ;  but  merely  the 
vert  one  arising  from  half  the  wt  between  m  and  s,  and  n  and  s.  The  position 
of  the  point  of  contrary  flexure  varies  with  the  number  of  interme- 
diate supports,  and  with  the  manner  of  loading ;  and  in  bridges,  &c,  where  the  load 
moves  along  the  beam,  it  changes  its  place  during  the  transit,  so  as  to  bring  the  points 
s  s  considerably  nearer  to  the  central  support  o;  thus  reducing  materially  the  ad- 
vantage commonly  supposed  to  arise  from  connecting  together  the  ends  of  adjacent 
bridge-trusses ;  if  indeed  there  is  any  advantage  in  so  doing,  which  is  doubtful.  The 
principle,  however,  becomes  very  useful  in  the  case  of  long  rafters  or  girders,  stretch- 
ing over  several  points  of  support,  especially  when  uniformly  loaded.  Each  interval, 
except  the  two  end  ones,  will  have  two  points  of  contrary  flexure ;  and  will  then  have 
nearly  twice  as  much  strength,  under  an  equally  distributed  load,  as  a  single  beam 
no  longer  than  said  interval. 

Comparison  between  models  and  actual  structures.  Many 
practical  men  imagine  that  if  a  model  is  strong,  an  actual  bridge,  roof,  Ac,  con- 
structed with  precisely  the  same  proportions,  must  be  equally  strong  in  propor- 
tion to  its  size.  This  arises  from  their  ignorance  of  the  fact  that  the  strength 
of  similar  beams,  trusses,  &c,  increases  only  in  proportion  to  the  squares  of  their 
spans ;  while  their  weight  increases  as  the  cubes  of  the  spans ;  so  that  a  model 
5  or  10  ft  long  may  show  a  great  surplus  of  strength;  while  the  roof  or  bridge  of  50 
or  100  ft  span,  constructed  like  it  in  every  respect,  may  break  down  under  its  own 
weight. 

We  may  compare  the  two  in  the  following  manner:  Let  us  suppose  a  model  4  feet 
long  of  a  bridge  truss,  its  wt  6  ft>s,  and  the  extraneous  center  load  reqd  to  break  it 
120  fbs,  or  20  times  its  own  wt.  Then  its  entire  center  breakg  load,  including  half 
its  own  wt,  is  120  +  3  =  12-i  ft>s.  Now  suppose  we  are  going  to  build  a  bridge  truss 
of  200  ft,  or  50  times  the  span  of  the  model.  The  strength  of  the  truss  will  be  50a, 
or  2500  times  that  of  the  model;  that  is,  it  will  require  f<»r  its  entire  center  breakg 
load,  50*  X  123  =  2500  X  I'M  =  307500  fbs.  Its  wt,  however,  will  be  503,  or  12-;>000 
times  that  of  the  model,  or  125000  X  6  =  750000  ft>s  ;  and  one-half  of  this  weight,  or 
375000  fbs,  must  be  deducted  from  its  entire  center  strength,  in  order  to  find  its  ex- 
traneous center  load.  But  in  this  case  the  half  weight  is  greater  than  the  entire 
center  strength ;  consequently  the  truss  would  break  under  its  own  wt.  If,  instead 
of  a  center  load  in  the  model,  we  had  broken  it  by  an  equally  distributed  one,  the 
calculation  would  plainly  be  the  same,  except  that  in  the  model  the  entire  weight, 
instead  of  %  of  it,  would  be  added  to  the  extraneous  load  for  the  entire  distributed 
breakg  load;  and  in  the  truss,  its  whole  weight  must  be  deducted  from  its  breaking 
strength,  to  get  the  extraneous  distributed  load. 

If  the  breaking  load  of  a  model  is  2,  3  or  4,  &c,  times  as  great  as  its 
weight,  then  a  similar  structure  2, 3  or  4,  <fec,  times  as  large  in  every  particular  will 
break  under  its  own  weight. 


642 


SHEARING   OF    BEAMS. 


— . 

V\\ 

..  .  J. 


SHEARING  OF  BEAMS. 

Art.  1.    The  Rules  here  given  for  shearing  apply  only  to  hor  beams  with  hor 

flanges  if  any.    If  the  flanges  are  inclined   or  curved,  see  Art  12,  p  649.     If  a 

beam  a  v,  Figs  1,  is  placed  upon  two  supports  n  and  s  its 

Pf»    r|  own  wt  and  that  of  any  load  upon  it  between  the  sup- 

ports, tend  not  only  to  bend  it  by  bringing  into  action 
longitudinal  forces  which  compress  lengthwise  some  of  its 
fibres,  and  stretch  others,  thereby  causing  the  beam  to 
sag  as  in  Fig  3,  p  246,  but  also  to  cut  it. across  vertically  at 
two  sections,  inclining  one  part  to  slide  down  past  the 
others  as  in  the  lower  fig,  leaving  its  ends  standing  on 
the  supports.  This  kind  of  vertical  yielding  is  called 
shearing-.  In  practice  it  rarely  if  ever  happens  that 
common  rectangular  building  beams  of  sound  timber  fail 
thus.  In  many  experiments  by  the  writer  with  reference 

to  this  point  they  invariably  failed  either  as  in  Fig  3,  p  246,  even  when  very  short 
and  deep,  or  by  warping,  or  by  combined  splitting  and  crushing  at  their  ends  on 
the  supports.  Still,  shearing  is  assumed  to  require  attention  especially  at  the 
ends  of  beams  in  heavily  loaded  warehouses,  and  at  the  ends  of  tie-beams  of  heavy 
roofs,  or  where  the  chords  of  bridges  rest  on  their  abutments,  &c.  In  trusses  in 
which  the  loads  are  assumed  to  be  concentrated  at  panel-points,  and  therefore  to 
be  borne  as  end  loads  by  the  web  members,  this  shearing  tendency  of  the  load 
does  not  exist,  or  but  to  a  trifling  extent  (which  may  usually  be  neglected)  between 
the  panel-points,  where  any  partial  loads  resting  on  those  intervals  tend  to  shear 
the  chords.  But  in  rectangular  or  in  I  beams,  or  others  with  continuous  webs,  it 
exists  throughout  every  part,  being  the  greatest  at  the  supports,  inasmuch  as 
they  bear  all  the  wt  of  the  beam  and  its  load.  Thus  the  wt  of  the  central  part  c 
and  its  load  tends  to  shear  the  beam  across  as  shown  by  the  dotted  lines,  and  that 
of  the  part  e  o  tends  to  do  the  same  at  e  and  o ;  and  so  at  every  vert  section  of  the 
beam,  but  evidently  most  so  at  the  supports.  If  the  load  is  either  central  or  uni- 
form, the  shearing  force  at  each  support  is  equal  to  half  the  wt  of  the  beam  and 
load ;  and  in  all  cases  it  is  there  equal  to  that  portion  of  the  entire  wt  that  is 
borne  by  each  support,  (as  found  by  Art.  47,  p  219.) 

Art.  2.  In  order  to  have  a  safety  of  3,  4  or  6,  &c,  against  the  shearing  force  at 
any  cross  section  of  its  length,  the  sheared  area  in  sq  ins  of  that  section  when 
mult  by  the  ultimate  coefficient  or  constant  of  shearing  per  sq  inch  must  give  a 
product  3, 4  or  6,  &c,  times  as  great  as  that  shearing  force ;  or  its  area  mult  by  one- 
third,  one-fourth,  Ac,  of  the  ult  coef,  or  in  other  words  by  the  safe  coef,  must  be 
equal  to  the  shearing  force.  The  lilt  coef  for  shearing*  for  cast  iron  of 
average  quality  is  about  22000  Ibs  or  10  tons  per  sq  inch ;  of  wrought  iron  about 
45000  Ibs  or  20.1  tons ;  of  average  steel  about  67000  Ibs  or  30  tons.  Of  woo«ls, 
across  the  fibres  as  in  beams,  by  the  writer's  own  experiments,  with 
pins  %  inch  diam,  of  fairly  seasoned  timber,  as  below.  Differences  of  10  to  20  per 
ct  either  way  will  of  course  frequently  occur  even  in  good  sound  timber,  owing  to 
whether  it  is  heart  wood  or  sap,  age,  degree  of  seasoning,  &c. 

Ibs  pr  sq  inch. 

Pine,  white 2480 

"  yellow,  Northern...  4340 
"  "  Southern...  5735 
"  yel,  S'n,  very  resin's  5053 

Poplar 4418 

Spruce 3255 

Walnut,  black 4728 

"         common 2830 

Example.  Let  a  v,  Fig  1,  be  a  hor  white  pine  beam  16  ft  clear  span,  12  ins 
deep,  and  4  ins  wide,  loaded  uniformly  from  end  to  end,  and  weighing  with  its 
load  5400  Ibs.  Then  each  support  upholds  2700  Ibs,  which  is  therefore  the  shear- 
ing force  at  each  support.  Since  the  average  ult  shearing  coef  of  white  pine  is 
about  2400  Ibs  per  sq  inch,  the  cross  section  at  each  support  to  sustain  a  shear  of 
2700  Ibs  with  a  safety  of  say  6,  must  have  an  area  =  (2700  -f-  2400)  X  6  =  6.75  sq 
ins.  It  actually  has  (4  X  12)  =  48  sq  ins,  or  a  safety  of  about  43  against  shearing 
at  its  two  most  dangerous  points  on  that  score.  This  same  beam  has  (see 
Table,  p  191)  a  safety  of  just  6  against  breaking  transversely  under  its  distributed 
load  of  5400  Ibs. 

Again  white  pine  crushes  across  the  grain  with  about  800  Ibs  per  sq  inch,  and 
has  a  safety  of  6  only  under  133  Ibs.  Therefore  each  end  of  the  beam  should  have 


Ash?  

Ibs  per  sq  inch. 
6280 

Ibs  pr  sq  inch. 

Ebony  7750 

Beech  

5223 

Gum                 ..  5890 

Birch  

5595 

Hemlock  2750 
Hickory  6045  to  7285 
Locust  .  ..           7176 

Cedar  white 

1445 

"     Central  America  3410 
Cherry                              9Q45! 

Maple                  6355 

Chestnut  ..  .. 

1535 

Oak,  white  4425 
"     live....  8480 

Dog;  wood  .... 

....  6510 

SHEARING   OF    BEAMS. 


643 


s 


a  resting  area  of  2700  -f-  133  =  say  20  sq  ins;  which  since  it  is  4  ins  wide  will 
require  its  length  of  bearing  on  each  support  to  be  5  ins. 

Art.  3.  The  shearing-  force  at  any  cross  section  of  a  nor 
cantilever  (a  projecting  beam  fixed  at  one  end  and  free  at  the  other),  no  mat- 
ter how  the  load  is  disposed,  is  always  equal  to  the  wt  of  that  part  of  the  beam 
and  its  load  which  is  between  said  section  and  the  free  end. 

For  the  shearing  force  at  any  cross  section  of  a  nor  beam 
supported  at  both  ends  no  matter  how  the  load  may  be  disposed,  Rule 
1,  find  the  common  cen  of  grav  of  the  beam  and  its  load,  and  by  Art.  47,  p  219, 
find  what  portion  of  the  entire  wt  is  borne  by  each  support.  Then  for  the  shear- 
ing force  at  any  section  find  the  difference  between  the  portion  that  rests  on  one 
of  the  supports,  and  the  wt  of  the  beam  and  load  between  that  support  and  the 
given  section.  Do  the  same  with  the  portion  on  the  other  support,  and  the  wt  of 
beam  and  load  between  it  and  the  given  section.  If  these  two  diffs  are  equal, 
either  one  of  them  is  the  required  shear.  If  they  are  unequal  (which  will  occur 
only  with  a  concentrated  load  not  at  the  center)  use  the  greater  one.  At  each 
support  the  shear  is  equal  to  that  portion  of  the  load  that  goes  to  it. 

Art.  4.  Rule  2.  Or  the  shear  at  any  section  of  a  hor  beam  supported  at 
each  end  may  be  found  thus:  Let  all.  the  wt  of  beam  and  load  resting  on  it  at  the 
right  hand  of  the  section  be  called  R,  and  all  that  on  the  left  hand,  L.  Then  the 
shear  at  that  section  will  be  equal  to  the  diff  between  that  portion  of  R  that  goes 
to  the  left  hand  support,  and  that  portion  of  L  that  goes  to  the  right  hand  sup- 
port. But  the  first  rule  is  the  most  easily  applied.  In  common  building  operations 
the  wt  of  the  beam  is  usually  so  trifling  in  comparison  with  its  load  that  it  may 
be  entirely  disregarded,  thereby  greatly  simplifying  the  calculations.  In  roofs, 
bridges,  &c,  this  cannot  be  done. 

Example.  If  the  concentrated  load  on  the  hor  beam  s  s  in  Fig  2  is  12  tons, 
and  the  clear  span  s  s  of  the  beam  3  ft,  the  portion  on  the  support  a  will  plainly 
be  8  tons,  and  that  on  c  4  tons.  Now  (omitting  the 
wt  of  the  beam  itself)  inasmuch  as  a  concentrated 
load  is  one  assumed  to  rest  on  a  mathematical  point, 
there  can  be  no  load  between  a  and  o,  although 
there  is  one  at  O ;  and  therefore  by  Rule  1  the  diff 
between  the  8  ton  portion  on  a,  and  the  load  between 
a  and  o  is  8  tons.  In  like  manner  there  is  no  load 
between  the  4  tons  on  c  and  the  section  o ;  and  there- 
fore the  diff  between  4  tons  and  the  load  on  any 
point  between  c  and  o  is  4  tons.  Now  of  these  two  diffs  the  8  tons  is  the  greatest, 
and  therefore  the  shear  at  o  is  8  tons.  The  shear  at  any  section  between  a  and  o 
is  8  tons ;  and  at  any  section  between  c  and  o  it  is  4  tons,  as  shown  by  the  vert 
lines  below  the  beam  s  s.  Let  us  take  the  section  at  e.  Here  the  portion  on  a  is 
8  tons,  aud  the  load  between  a  and  e  is  12  tons,  diff  4  tons.  Also  the  portion  on  c 
is  4  tons,  and  the  load  between  c  and  e  is  O,  diff  4  tons.  Hence  here  the  two  diffs 
are  equal,  and  therefore  the  shear  at  e  is  4  tons  as  before.  Taking  the  section  at 
t  in  the  same  way  we  shall  find  the  shear  there  to  be  8  tons  as  before. 

Art.  5.  Fig  3  shows  the  shearing  forces  (10,  6,  2  and  14  tons),  resulting  from 
the  three  concentrated  loads  4,  8,  and  12 
tons,  supported  at  intervals  of  3  ft  by  a 
beam  a  a  12  ft  long.  Now  a  square 
wrought  iron  beam  to  have  a  safety  of 
4  against  14  tons  shear  need  be  only 
about  1.75  ins  square  from  a  to  o;  and 
for  the  2  tons  shear  only  .65  of  an  inch 
square  from  o  to  e.  The  absurdity  of  so 
small  dimensions  for  such  a  beam  and 
load  shows  how  trifling  a  part  the  shear- 
ing force  at  times  performs  in  such 
cases;  for  with  the  foregoing  load  a 
uniform  beam  of  wrought  iron  3  ins 
thick  and  15  ins  deep  would  barely  have  a  safety  of  4  against  yielding  by  the  hor 
stretching  and  crushing  of  its  fibres  as  shown  by  Fig  3,  p  246 ;  but  would  evidently 
have  a  very  superabundant  safety  against  the  vertical  shear.  As  said  in  Art  1 
the  shear  may  usually  be  neglected  entirely  in  calculating  the  strength  of  ordi- 
nary rectangular  building-beams,  taking  care  only  that  the  ends  rest  far  enough 
on  the  supports  to  be  safe  against  crushing. 

Rein.  1.  The  vertical  shearing  force  in  hor  beams  with  closed  or  continuous 
webs  like  a  rolled  I  beam,  or  Figs  22  to  24,  p  214,  is  in  practice  usually  assumed  to 


a 


644  OPEN  AND  CLOSED  BEAMS. 

be  borne  by  the  web  only,  although  it  is  manifest  that  the  flanges  must 
assist,  notwithstanding  that  they  (except  when  inclined  or  curved,  see  Art  12,  p 
649)  are  as  generally  assumed  to  resist  only  the  hor  crushing  and  pulling  forces 
of  the  load.  The  assumption  however  simplifies  calculations,  and  is  safe. 

Rem.  2.  But  the  shearing  force  in  open  beams,  as  in  bridge  and 
roof  trusses,  &c,  is  converted  into  strains  of  tension  and  compression  acting  as  end 
loads  lengthwise  of  the  vertical  and  inclined  members  which  constitute  the  webs 
of  such  trusses,  and  the  duty  of  which  is  thus  to  convert  it  into  end  loads,  and  to 
convey  it  in  that  shape  from  its  places  at  the  panel-points  along  the  chords,  to  the 
points  of  final  support  at  the  abutments,  as  shown  under  the  head  "Trusses,"  p 
243,  <fcc. 

Art.  6.  Although  the  two  general  rules  in  Art  3  will  answer  for  any  case  of 
hor  beams,  the  following?  will  at  times  be  found  convenient.  In 
a  cantilever  the  shear  is  greatest  at  the  fixed  end;  and  there,  or  at  any  other 
section,  it  is  equal  to  all  the  wt  of  beam  and  load  between  said  section  and  the  free 
end.  In  a  beam  supported  at  both  ends  the  shear  at  either  support  is 
equal  to  the  wt  of  that  portion  of  the  beam  and  load  (as  found  by  Art  47,  p  219) 
that  rests  on  said  support.  When  uniformly  loaded  alone  the  entire 
span  the  greatest  shear  is  at  each  support,  and  is  =  half  wt  of  beam  and  load. 
From  each  support  it  diminishes  regularly  to  the  center  where  it  is  O.  When 
uniformly  loaded  from  one  support  (which  call  s)  to  only  part 
way  across,  say  to  n,  (as  when  a  train  as  long  as  the  span  comes  upon  a  bridge) 
the  greater  shear  is  at  s,  and  is  =  the  portion  resting  on  s.  The  least  is  zero,  and 
is  always  within,  the  load,  that  is,  between  s  and  n,  say  at  I.  To  find  the  dist  ol  I 
from  n  (omitting  wt  of  beam  itself )  say  as  Twice  the  span  :  whole  uniform  load  :  : 
said  load  :  n  1.  This  I  is  also  the  section  of  greatest  moment  of  rupture  of  the  load. 

Rem.  1.  At  the  center  of  the  span,  or  at  any  point  beyond  the  center,  the  shear 
increases  as  the  uniform  load  advances,  and  is  greatest  when  the  load  just  reaches 
it;  and  afterwards  diminishes  as  the  load  becomes  longer. 

Rem.  2.  But  at  any  point  before  reaching  the  center  the  shear  increases  from 
the  time  the  load  first  touches  the  span  until  it  covers  the  entire  span. 

Rein.  3.  The  greatest  shear  at  any  given  point  is  when  the  longest  segment  of 
the  span  is  loaded  to  that  point.  It  is  then  greater  at  that  point  than  when  the 
whole  span  is  loaded. 

Rem.  4.  These  shears  at  any  given  point  must  not  be  confounded  with  the 
greatest  one  of  all  which  the  partial  load  is  at  the  same  instant  causing  at  one  of 
the  supports. 

Concentrated  load  at  center,  (omitting  the  beam)  the  shear  at  any  sec- 
tion is  =  half  the  load.  A  concentrated  load  moving  along-  the 
hor  beams  from  end  to  end,  (omitting  the  beam)  produces  its  greatest 
shear  when  at  an  end  support,  at  either  of  which  it  then  is  =  whole  load.  Thence 
it  diminishes  uniformly  until  the  load  reaches  the  center,  where  it  is  least  and 
=  half  load. 

Rem.  5.  If  both  ends  of  the  hor  beam  are  fixed,  or  one  end 
fixed  and  the  other  merely  supported,  the  results  will  generally  be  different. 


OPEN  AND  CLOSED  BEAMS. 

General  Remarks. 
Art.  1.    An  open  beam,  Fig  6,  as  distinguished  from  a  closed  one  is 

one  which,  like  an  ordinary  bridge  truss,  has  openings  between  its  top  and  bottom 
flanges  or  chords;  or  in  other  words,  one  whose  web  is  not  solid  or  continuous  as 
it  is  in  a  common  rolled  I  beam,  or  in  Figs  22  to  24,  p  214.  In  the  term  open  beam 
we  here  include  trusses.  There  is  an  essential  difference  in  the  modes  of  resist- 
ance of  the  two  kinds  of  beams  against  the  action  of  their  loads.  While  pointing 
this  out  we  will  repeat  a  few  facts  alluded  to  in  different  parts  of  this  volume,  but 
which  it  is  not  amiss  to  bring  together  in  this  connection. 

When  either  a  load  I,  Fig  1,  or  any  other  vert  force  acts  upon  a  hor  beam  /  o 
fixed  at  one  end  the  beam  becomes  a  lever,  and  with  the  load  has  a  tendency  to 
move  or  revolve  about  the  fixed  end, /as  a  supporting  fulcrum,  and  in  so  doing 
to  strain  or  break  the  beam  at  said  fulcrum,  by  forces  of  tension  and  compres- 
sion, acting  hor  or  lengthwise  of  the  beam,  pulling  apart  the  upper  fibres  at/,  and 
crushing  together  the  lower  ones.  The  load  or  other  force  together  with  the  wt 
of  the  beam  also  tends  to  break  the  beam  by  shearing  or  cutting  it  across  verti- 


PEN   AND   CLOSED   BEAMS. 


645 


a 


o 


frn  at  Figs  12  p  642,  see  Shearing,  p  642,  &c.  But  it  is  only  the  first  of 
these  telidencies  of  which  we  speak  now.  It  is  called  the  load's  Moment  of 
Rupture,  or  Breaking  Moment,  or  merely  its  mo- 
ment about  the  fulcrum/;  and  is  measured  in  foot-tons, 
or  foot-ft>s,  inch-lbs,  &c,  by  mult  the  load  in  tons  or  fts, 
&c,  by  its  leverage,  or  the  shortest  or  perp  dist  h  e  or  / 
s  of  its  line  of  direction  a  m  from  the  fulcrum  in  ft  or 
ins.  See  "  Moments,"  p  217  and  473. 

If  the  load  instead  of  being*  concen- 
trated like  I  is  distributed  in  any  way  along  the  whole 
or  a  part  of  the  beam,  its  leverage  is  measured  from  the 
fulcrum  perp  to  the  line  of  direction  of  its  cen  of  grav ; 
which  is  plainly  the  case  also  with  a  concentrated  load, 
because  its  line  of  direction  also  passes  through  its  cen  of  grav.  Before  the  beam 
bends,  its  leverage  is  evidently  greater  than  afterwards,  and  it  becomes  less  as  the 
bending  increases;  but  as  very  little  bending  is  allowed  in  practical  cases  the 
leverage  may  generally  be  assumed  not  to  change,  but  to  remain  as  when  the 
beam  is  hor. 

The  load  evidently  tends  also  to  strain,  or  break  the  beam  at  any  point  what- 
ever as  t,  Fig  1,  between  itself  and  the  fulcrum  /,  and  is  assisted  in  so  doing  by  the 
wt  of  beam  between  t  and  o.  Therefore  any  such  point  t  may  also  be  assumed  to 
be  a  fulcrum.  The  moment  of  the  load  will  of  course  be  less  at  such  point  than 
at /because  its  leverage  t  s  will  be  shorter. 

Art.  2.  In  the  closed  beam  i  a  e  o,  Fig  2,  the  load  tends  to  revolve  about 
the  neutral  axis  n  as  the  supporting  fulcrum  of 
its  lever  the  beam,  as  shown  by  the  dotted  lines,  and 
thereby  to  strain  all  the  fibres  from  top  to  bottom  of 
the  beam  at  the  section  i  n  e,  by  stretching  lengthwise 
those  above  w,  and  compressing  lengthwise  those  below 
n.  The  greatest  strain  is  at  the  top  and  bottom  fibres ; 
and  from  them  both  ways  it  diminishes  until  at  n  it  is 
nothing.  The  load  also  stretches  or  compresses  the 
fibres  length  wise  at  every  vert  section  along  the  entire 
length  of  the  beam,  more  or  less  according  to  its  lever- 
age and  moment  at  said  section  ;  most  near  the  fixed 
end  and  least  near  the  free  end;  so  that  the  extent  of 
stretch  indicated  by  s  i  is  the  total  accumulated 
stretchings  that  have  taken  place  in  the  top  fibres  at  every  point  from  i  to  a.  The 
same  is  the  case  with  the  stretches  and  compressions  of  the  fibres  anywhere  be- 
tween i  e  and  a  o,  as  indicated  by  the  varying  hor  dists  between  n  i  and  n  s,  or 
between  n  v  and  n  e.  The  compressed  fibres  below  n,  and  comprised  between  n  v 
and  n  e  would  as  it  were  vanish,  being  crushed  or  mashed  flat  against  the  face  of 
the  wall. 

Art.  3.  A  closed  beam  a  a.  Fig-  3,  supported  at  only  one 
point  whether  at  the  center  or  not,  and  balanced  by  two  either  equal  or  un- 
equal loads,  may  plainly  be  regarded  as  two 
levers  each  of  which  is  essentially  in  the  same 
condition  as  Fig  2.  Whether  the  loads  are  con- 
centrated or  distributed  their  leverages  n  e,  n  e 
are  as  before  to  be  measured  from  n  and  perp  to 
the  lines  of  direction  v  o,  v  o  of  their  centers  of 
grav  as  in  Fig  2.  Both  the  5  ton  loads  are  mani- 
festly upheld  by  the  support,  which  of  course 
reacts  vert  upwards  against  them  in  a  vert  line 
with  their  common  cen  of  grav  n,  with  a  force 
of  10  tons  as  per  the  central  arrow. 

Item.  1.  Each  end  load  in  Fig  3  being  5 
tons,  suppose  each  lever  n  e  to  be  4  ft.  Then  the  moment  of  each  load  about  the 
fulcrum  n  would  be  =  5  X  4  =  20  ft-tons.  Hence  it  might  seem  that  over  the 
support  the  fibres  of  the  beam  near  n  would  have  to  resist  a  combined  moment 
of  40  ft-tons.  Hut  they  have  actually  to  present  a  resistance  of  but  20  ft-tons,  on 
the  same  principle  that  if  two  men  pull  against  each  other  at  two  ends  of  a  rope, 
each  with  a  force  of  say  30  fbs,  the  strain  or  pull  on  the  rope  is  not  60  but  only  30 
fts,  because  strain  is  the  reaction,  (pressure  or  pull)  against  each  other  of  two  equal 
opposing  forces,  and  is  equal  to  only  one  of  them.  *The  two  above  equal  moments 
are  merely  two  forces  acting  through  leverages.  See  "Strain,"  Art  2,  p  444. 

Art.  4.  A  closed  beam,  Fig1  4,  supported  at  both  ends,  and 
loaded  at  only  one  point,  whether  at  the  center  or  not,  with  a  concentrated  load, 


:o 


646 


OPEN  AND  CLOSED  BEAMS. 


may  also  like  Fig  3  be  regarded  as  two  levers  with  their  common  fulcrum  at  n 
in  a  vert  line  with  the  cen  of  grav  of  the  load.  This  however  is  by  no  means 
so  manifest  at  first  sight  as  in  Fig  3,  but  needs  a  little  explanation.  Let  the 
beam  bear  10  tons  concentrated  at  its  center,  then 
evidently  5  tons  of  it  will  rest  pressing  down- 
wards on  each  end  support;  and  each  support 
will  therefore  press  upward  or  react  against  an 
end  of  the  beam  with  a  force  of  5  tons  as  per  the 
arrows.  Now  these  two  5-ton  reactions  of  the  sup- 
ports in  Fig  4  are  to  be  considered  as  taking  the 
place  of  the  two  5-ton  end  loads  in  Fig  3 ;  while 
the  10-ton  load  in  Fig  4  takes  the  place  of  the 
10-ton  reaction  of  the  support  in  Fig  3,  and  hence  in  this  view  of  the  case  is  no  longer 
to  be  considered  at  all  as  load,  but  merely  as  a  fixture  for  holding  the  common  ful- 
crum n  of  the  two  levers  in  place,  or  in  equilibrium  with  the  upward  end  reactions. 
Being  no  longer  regarded  as  load,  it  of  course  cannot  in  such  cases  be  assumed  to 
have  any  moment  of  rupture;  that  property  being  now  transferred,  to  the  end 
reactions.  Still,  to  avoid  awkwardness  of  expression  we  always  speak  of  the  mo- 
ment of  the  load  even  in  such  cases,  rather  than  of  the  moments  of  the  reactions 
of  the  load.  In  both  Figs  3  and  4  the  forces  at  work  are  the  same  in  amount,  but 
plainly  reversed  in  direction. 

Rent.  If  the  load  Is  diatribnted  as  the  6  tons  in  Fig  5,  instead  of 
concentrated  as  in  Fig  4,  we  still  consider  the  beam  as  consisting  of  two  levers 
with  their  common  fulcrum  n  in  a  vert  line  with 
the  cen  of  grav  c  of  the  load.  But  to  find  the  mo- 
ment of  the  load  (or  more  correctly,  the  moment 
of  the  reactions  of  the  supports)  about  n  we  must 
proceed  a  little  differently.  Thus  let  the  beam  be 
3  ft  span,  and  the  load  uniform,  weighing  6  tons, 


6tons 


c 

4.  X 


Rq5. 


and  being  1  ft  long.    Find  by  Art  47,  p  219,  how 
much  of  this  load  rests  on  each  support,  (4  tons  on 


*  a,  and  2  tons  on  o.)  The  upward  reactions  of  the 
£  supports  will  therefore  also  be  4  and  2  tons.  Then 
first  find  the  moment  about  n  of  either  one  of  the  reac- 
tions, say  of  the  4-ton  one  at  a.  This  moment  will  plainly  be  (4  tons  X  1  ft)  =  4 
ft-tons.  Then  find  the  moment  about  n  of  that  part  (3  tons)  of  the  load  that  is  between 
n  and  a,  by  mult  the  wt  (3  tons)  of  that  part  by  the  hor  dist  (en  =  .25  of  a  ft) 
between  its  cen  of  grav  and  n.  This  last  moment  (3  tons  X  -25  of  a  ft)  =  .75  of  a 
ft-ton,  being  downward,  evidently  diminishes  or  counteracts  the  upward  moment 
of  the  4-ton  reaction  at  a  about  the  same  fulcrum  n  to  the  same  extent,  and  is 
therefore  to  be  subtracted  from  it,  thus  leaving  4  —  .75  =  3.25  ft-ton  for  the  mo- 
ment of  the  6-ton  load  about  n. 

The  same  result  will  follow  if  we  use  the  2-ton  reaction  of  o,  with  the  hor  lever- 
age o  w,  and  the  part  of  the  load  between  o  and  n.  To  find  the  moment  for  any 
other  point  than  n  see  Moments,  Case  11,  p  220^ 

Art.  5.  In  a  closed  beam,  say  Fig  2,  each  of  the  fibres  throughout  the 
entire  depth  of  the  yielding  section  i  n  e  opposes  the  breaking  moment  of  the 
load  by  a  Resisting  Moment  or  Moment  of  Resistance  of  its  own.  As  the 
breaking  moment  about  n  of  the  load  is  made  up  of  its  gravity-force  or  weight 
mult  by  its  leverage  or  perp  distance  n  c  from  the  fulcrum  or  neutral  axis  n,  so 
the  resisting  moment  about  n  of  each  separate  fibre,  say  for  instance  the  one  at  »*, 
is  made  up  of  its  natural  longitudinal  tensile  or  compressive  force  or  strength 
mult  by  its  leverage  or  perp  dist  n  i  above  or  below  the  same  fulcrum  n.  We  have 
already  said  (and  it  is  self-evident  from  the  fig)  that  the  extent  to  which  any 
fibre  is  stretched  or  crushed  lengthwise  is  in  proportion  to  its  dist  above  or  below 
n;  hence  on  the  principle  that  action  and  reaction  are  equal  and  in  opposite 
directions,  (Arts  13  and  14,  p  449)  and  that  within  the  elastic  limit  •'«/  tensio  sic 
vis"  (as  is  the  stretch  so  is  the  force)  the  length  wise  force  which  stretches  or  com- 
presses any  fibre,  and  with  which  that  fibre  reacts  to  resist  being  stretched  or 
compressed  is  also  as  its  dist  or  leverage  from  n.  Now  suppose  three  fibres  to  be 
at  the  dists  1,  2,  3  iris  from  n,  or  in  other  words  let  their  leverages  about  n  be  1,  2 
and  3  ins.  Then  as  just  said  their  lengthwise  resisting  forces  must  also  be  as  1,  2, 
and  3 ;  and  hence  (since  the  moment  of  a  force  is  the  force  mult  by  its  leverage) 
the  resisting  moments  about  n  of  these  three  fibres  are  as  1  X  lj  2  X  2,  and  3X3, 
or  as  1,  4  and  9  ;  that  is  as  the  square.?  of  their  dists  from  the  fulcrum  n.  Thus  we 
see  that  in  a  closed  beam  the  lengthwise  resistance  in  ft>s  or  tons,  &c,  of  each  fibre 
is  as  its  dist  from  the  fulcrum ;  while  its  resisting  moment  about  that  fulcrum  is 
as  the  square  of  said  dist. 


CLOSED   BEAMS. 


647 


Art.  6.  Thte^ft  fact  as  will  be  shown  in  Art  8  constitutes  the  great  differ- 
ence betweeni^sed  beams  and  open  ones.  It  also  explains  why  the  strengths 
of  closed^beams  are  as  the  squares  of  their  depths,  while  in  open 
ones  such  as  the  trusses  of  bridges,  roofs,  &c,  they  are  simply  as  their  depths.  It 
also  shows  that  the  strongest  form  of  beam  is  that  in  which  as  much  of 
the  material  as  possible  is  taken  from  near  the  neutral  axis  where  it  has  but  little 
resistance,  and  placed  at  the  top  and  bottom  of  the  beam  where  it  may  exert  great 
resistance,  as  in  the  common  rolled  I  beam,  and  the  Hodgkinson  cast  ones,  p  208. 

Art.  7.  The  Moment  of  Inertia  (which  may  be  found  by  the  ap- 
proximate method  on  p  195;  at  any  section  of  a  closed  beam,  when  multiplied  by 
the  Constant  of  Rupture  for  the  material  of  which  the  beam  is  made  (the 
strain  in  Ibs  per  sq  inch  borne  by  the  extreme  fibres  when  on  the  point  of  yield- 
ing, see  p  195)  and  the  product  divided  by  the  dist  in  ins  of  the  farthest  fibre  of 
the  section  (or  according  to  Prof  De  Volson  Wood  the  farthest  fibre  on  the  side  thai, 
will  yield  first,  see  Art  25V£,  p  194)  from  the  neutral  axis  n,  gives  the  Moment  ol 
Resistance  of  the  closed  beam  at  that  section.  This  is  usually  expressed  by  the 

formula  R  =  i-±. 

t 

In  order  that  the  beam  shall  not  fail  at  that  section,  its  moment  of  resistance  OT 
its  1  C  -±t  must  there  be  at  least  equal  to  the  load's  Moment  of  Rupture;  and  for 
a  safety  of  3, 4  or  6,  &c,  it  must  be  3,  4  or  6,  &c,  times  as  great.  The  modes  of  finding- 
the  moments  of  rupture  and  of  resistance  in  many  cases  are  given  under  that 
head,  p  217,  &c. 

Art.  8.  But  in  an  open  beam  i  x  e  a,  Fig  6,  there  is  no  neutral  axis  to 
act  as  a  supporting  fulcrum  for  the  beam  and  its  load 
at  n,  but  the  inner  end  e  of  the  lower  chord  e  a  now 
becomes  the  fulcrum ;  and  the  beam  with  its  load 
now  tends  to  revolve  about  it  as  per  the  dotted  lines, 
stretching  lengthwise  the  fibres  of  the  upper  chord 
or  flange  i  z,  and  crushing  in  like  manner  those  of 
the  lower  one  e  a.  It  is  plain  that  the  breaking 
moment  of  the  load  is  the  same  as  in  the  closed  beam, 
Fig  2,  being  still  the  load  X  its  leverage  ea  orix', 
but  the  resisting  moment  of  the  beam  consists  now 
of  the  longitudinal  tensile  and  crushing  strengths 
of  the  fibres  of  the  two  chords  or  flanges  only,  X  by 
their  leverage,  the  depth  of  the  beam  ;  while  the  web 
members  resist  only  the  vert  or  shearing  force  of 
the  load,  which  in  an  open  beam  does  not  tend  to 
shear  the  web  members,  but  as  end  loads  to  press  or  pull  them  in  the  direction  of  their 
lengths.  In  practice  the  depth  of  any  open  beam  or  truss  is  measured 
from  i  to  e,  the  centers  of  grav  of  the  cross  sections  of  the  chords,  which  are  as- 
sumed to  be  so  far  apart  and  so  thin  that  we  may  do  so  without  sensible  error. 
In  doing  this  we  of  course  thereby  assume  what  is  impossible,  namely,  that  all  the 
fibres  of  one  chord  at  any  vert  section  of  the  truss  are  equally  distant  from  all 
those  of  the  other  chord,  and  hence  that  all  of  them  have  the  same  leverage, 
namely  the  depth  of  truss  measured  between  the  centers  of  grav  of  the  chords. 
Although  this  cannot  be  strictly  correct  under  any  circumstances,  still  in  beams 
whose  chords  or  flanges  are  thin  in  proportion  to  the  aforesaid  depth  it  is  suffi- 
ciently so  for  practice. 

This  is  much  more  simple  than  the  case  of  closed  beams,  and  greatly  facilitates 
the  finding  of  the  moments  and  hor  strains  in  the  chords  of  open  ones,  as  follows. 

Art.  9.  Let  Fig  6  be  a  hor  open  beam  1.5  ft  deep  from  i  to  e,  and  projecting  6 
ft  from  a  wall  into  which  it  is  firmly  fixed  by  its  flanges  or  chords  i  and  e;  and 
let  the  concentrated  load  L<  at  its  outer  end  be  1  ton.  This  load  tends  to  pull  the 
beam  into  the  dotted  position  by  stretching  or  tearing  apart  the  fibres  of  the 
upper  chord  at  i.  Now  with  how  great  a  moment  of  rupture  does  it  tend  to  do 
this,  and  how  strong  must  the  fibres  of  the  chord  be  at  i  in  order  that  their  mo- 
ment of  resistance  may  oppose  it  safely  ?  We  shall  here  leave  the  wt  of  the  beam 
itself  out  of  consideration.  When  required  to  be  included  see  Case  10,  p  220^- 

Regard  the  lines  i  e  and  e  a  as  the  two  arms  of  a  bent  lever  resting  on  its  ful- 
crum e.  This  lever  is  plainly  acted  upon  and  balanced  by  two  equal  moments, 
one  at  each  end  a  and  i\  namely  at  a  the  moment  of  rupture  of  the  load,  equal  to 
(1  ton  X  6  ft  leverage  a  e}  =  6  ft  tons ;  and  at  i  the  resisting  moment  of  the  beam, 
equal  to  the  hor  pull  or  strain  on  the  fibres  at  the  chord  i  X  1-5  ft,  leverage  *'  e. 
But  we  do  not  yet  know  what  amount  of  hor  pull  by  the  fibres  at  i  is  required  to 


648  OPEN   AND    CLOSED   BEAMS. 

balance  the  moment  of  the  load.  It  is  however  very  easily  found  by  merely  divid- 
ing the  6  ft  tons  moment  of  the  load  by  the  1.5  ft  leverage  of  the  fibres,  that  is,  by 
the  depth  of  the  beam.  Thus  we  get  (6  -f-  1.5)  =  4  tons  pall  at  ?'  ;  and  we  then  have 
the  6  X  1  =  6  ft  tons  moment  of  the  load,  balanced  by  the  1.5  X  4  =  6  ft  tons  mo- 
ment of  the  fibres.  Therefore  in  order  just  to  balance  the  moment  of  the  load, 
thecho  ' 
of  3,4 

lower  one  upon  which  it  acts  as  an  equal  hor  compressing  one. 
In  shape  of  a  formula  the  above  stands  thus. 

Hor  strain  at  any       Moment  of  load      Load  X  its  leverage 
point  in  a  hor  flange  =     at  that  point     _        at  that  point 
of  an  open  cantilever         Depth  of  beam  Depth  of  beam 

Hence  if  we  know  the  size  and  of  course  the  ultimate  longitudinal  tensile  and 
corupressive  strength  of  the  flange  or  chord,  we  have  by  transposition  the  ulti- 
mate or  breaking  load  of  the  hor  open  beam,  thus, 

Breaking  load  at  any  _  Ultimate  strength  of  flange  X  Depth  of  beam. 
point  of  a  hor  open  cantilever  Leverage  of  load  at  that  point. 

And  for  a  safety  of  3,  4  or  6,  &c,  we  have 


Safe  load  =  ^»  &c*  the  ult  strength  of  flange  X  Depth  of  beam. 

Levefage  of  load  at  that  point. 

Art.  1O.  Also  in  a  hor  open  beam  or  truss  supported  at  both 
ends,  after  having  found  the  moment  of  the  load  at  any  point  (by  "moments," 
p  217,  &c)  the  strain  on  the  beam  as  also  its  load  in  Ibs  or  tons  are  found  in  the 
same  way  or  by  the  same  formulas. 

Rein.  1.  The  longitudinal  strains  on  the  flanges  of  hor  elosed  beams 
with  thin  webs  such  as  common  rolled  I  beams,  or  Figs  22  to  24,  p  214,  as  well  as 
their  loads,  are  also  frequently  computed  in  this  same  ready  way,  instead  of  the 
more  troublesome  one  at  foot  of  p  194,  or  in  "  Moments,"  p  217,  &c.  The  webs  are 
then  left  entirely  out  of  consideration  as  regards  the  hor  strains.  Although  not 
strictly  correct,  it  is  sufficiently  so  for  ordinary  practice,  and  is  safe.  With  these 
assumptions  the  dimensions  or  sectional  areas  of  the  top  and  bottom  flanges  are 
proportioned  to  the  safe  unit  strains  of  the  material.  Thus  Hodgkinson  having 
found  that  the  ultimate  coinpressive  strength  of  cast-iron  averaged  about  6  times 
as  great  as  its  tensile  one,  gave  his  upper  flange  only  one-sixth  the  area  of  the 
lower  one,  in  order  that  both  should  be  equally  strong.  In  wrought-iron  the  ten- 
sile strength  is  somewhat  the  greatest,  which  would  lead  to  making  the  lower 
flange  the  smallest,  but  here  this  consideration  is  outweighed  by  the  practical 
ones  of  greater  ease  of  manufacture  and  of  handling  or  placing  which  require 
equal  flanges. 

Item.  2.  If  the  flanges  are  not  horizontal,  although  the  beam  or 
truss  itself  may  be  so,  the  longitudinal  strains  on  the  flanges  will  be  increased; 
and  the  transverse  or  shearing  strains  on  the  webs  will  also  be  changed  as  stated 
in  Art  12.  If  the  beams  are  inclined,  modifications  arise  which  we  shall 
not  treat  of.  Strangely,  most  of  our  standard  authorities  on  bridge  building  do 
not  even  allude  to  them. 

Rein.  3.  The  principle  of  the  bent  lever  in  open  beams  explains  why  the 
strength  of  a  truss  is  as  its  depth,  (the  length  of  its  vert  lever-arm) 
instead  of  as  the  square  of  its  depth  as  in  closed  beams.  The  strength  however  is 
inversely  as  the  length  in  both  kinds. 

Art.  11.  The  web  members  of  an  open  beam  or  common  truss  like  Figs 
10  and  11,  p  254,  uniformly  loaded,  carry  the  vert  or  shearing  forces  of  the  load 
and  beam  from  the  center  each  way,  up  and  down  alternately  from  one  chord  to 
the  other,  until  finally  the  end  ones  deposit  it  as  load  on  the  supports  or  abut- 
ments. For  each  member  receives  and  carries  its  share  of  the  shearing  force  in 
the  shape  of  an  end  load,  thus  changing  the  shearing  tendency  into  an  alternately 
pulling  and  compressing  one  according  as  the  alternate  members  are  ties  or  struts. 
In  doing  this  any  web  member  that  is  oblique  is  (on  account  of  its  obliquity) 
strained  to  an  extent  that  exceeds  its  load  in  the  same  proportion  that  the  oblique 
length  of  the  member  exceeds  the  length  it  would  have  had  if  it  had  been  vert,  as 
explained  in  Art  11,  p  253,  &c.  This  excess  of  strain  over  the  load  on  the  obliques 


f  I 

exhibits  itself  at  th^fr  ends  as  hor  pull  along  one  chord,  and  hor  compression 
along  the  other j^trnd  th/se  hor  strains  on  the  chords  are  the  same  as  those  found 


e  Case  10,  p  220%.    Thus  it  is  seen  in  Figs  10  and  11  that  the  hor 


chord  (as  there  found  by  tracing  up  the  dift'oreri 


incuts,       v  ertiua.1  mciiiucia  incicij'   ^uuvc^    HJCJLJ   lutiuo  veil*    uu   ui    vtuvvn   iiuiii  unc 

chord  to  the  other,  at  which  last  they  transfer  them  to  oblique  members  which 


cnora  to  me  ouier,  ai  wiucii  iasi  iney  uaiisier  mem  10  oouque  memuers  wnica 
can  convey  them  laterally.  If  both  a  pull  and  a  push  act  at  once  in  opposite 
directions  on  a  web  member,  their  din0  is  the  actual  strain. 

Rem.  As  a  matter  of  economy  in  small  spans  it  is  often  better  not  to  , 
proportion  the  sizes  of  the  individual  members  to  the  strains  they  have  to  bear  ; 
but  to  give  to  the  flanges  throughout  their  entire  length  the  same  dimensions  as 
are  required  at  their  most  strained  part,  namely,  at  the  center  ;  and  to  make  all 
the  web  members  as  strong  as  the  most  strained  or  end  ones.  This  avoids  the 
extra  trouble  and  expense  of  getting  out  and  fitting  together  many  pieces  of 
yarious  sizes. 

Art.  12.  Oblique  or  curved  flanges.  We  have  hitherto  supposed 
the  beams  and  their  flanges  to  be  horizontal;  but  a  beam  may  be  hor,  and  yet 
have  one  or  both  of  its  flanges  oblique  or  curved  as 
at  A  and  B.  In  such  cases  the  longitudinal  strains 
along  the  flanges  become  greater;  and  the  vert  or 
shearing  strains  across  the  web  in  most  cases  less. 
See  Bern  at  end  of  Art.  It  is  plain  that  such  flanges 
must  as  it  were  intercept  to  some  extent  (depending 
on  their  inclination)  the  vert  force  at  any  point, 
and  convert  it  into  an  oblique  one  along  the  flanges, 
somewhat  as  the  oblique  web  members  of  an  open 
beam  do. 

To  find  these  new  strains  at  any  point  o, 
Figs  A,  B,  of  either  an  upper  or  lower  oblique  or 
curved  flange,  first  ascertain  by  "Moments,"  the 
hor  strain  at  that  point  for  a  beam  with  the  depth 
o  e;  and  by  "Shearing,"  p  642,  &c,  the  shear  also. 
Then  from  that  point  o  draw  a  hor  line  h  equal  by 
scale  to  the  hor  strain  ;  and  from  its  end  draw  v 
vert  and  ending  either  at  the  flange  (produced  if  necessary)  if  straight  as  in  A  ; 
or  at  a  tangent  /  from  o  if  the  flange  is  curved  as  at  B.  Then  will  /  in  either  fig 
give  by  the  same  scale  the  longitudinal  strain  along  the  flange  at  o  ;  and  h  and  v 
are  the  components  of  that  strain.  As  a  formula,  the  Rule  reads  thus,  o  being  the 
angle  formed  by  h  and  I  at  o. 


web  at  o.     For  exceptions,  see  Rein.   The  foregoing  applies  also  to  oblique  flanges 
of  open  hor  beams. 

In  the  hor  triangular  flanged  beam  D  with  a  concentrated  load  at 
its  free  end,  draw  a  o  vert  and  equal  by  scale  to  the  load, 
and  draw  o  c  hor.  Now  here  the  whole  load  rests  upon  the 
upper  end  a  of  the  oblique  flange  a  n,  which  therefore  sus- 
tains all  of  it  as  an  end  load,  which  it  deposits  as  vert  press- 
shear: 


, 

at  ?i,  and  thus  entirely  prevents  it  from  exerting  any 
aring  force  whatever  upon  any  part  of  the  beam.    The 


searng  orce  waever  upon  any  par  o  e  eam.  e 
shaded  web  is  therefore  of  no  use  here.  The  line  a  c  meas- 
ures the  strain  along  the  oblique  flange  ;  a  o  the  vert  pressure 
at  n  ;  and  o  c  the  hor  pull  of  the  load  all  along  the  upper 
flange  a  e.  Also  a  o  and  o  c  are  the  components  of  a  c. 


650 


KUTTER  S    FORMULA. 


a 


the  abutments  or  supports  these  pulls  along  ca  and  cb  become  converted  into  ver- 
tical pressures,  together  equal  to  the  load  I ;  and  into  hor  pressures  compressing  a  b. 
Here  also  the  shaded  web  is  unnecessary ;  as  would 
likewise  be  the  case  if  the  load  were  transferred  to  e, 
and  a  single  vert  post  (shown  by  the  dark  line)  provided 
to  carry  it  down  to  c,  as  the  string  before  carried  it  up 
to  c-  If  there  is  no  such  post  the  web  acts,  and  the 
strain  on  either  oblique  flange  is  found  as  for  A  and  B. 
But  it  is  only  in  a  few  similar  cases  that  the  oblique 
flange  entirely  supplants  the  continuous  web. 

If  umber  give*  for  finding  the  strain  at 
any  point  of  an  oblique  or  curved  web  as  follows. 
First  find  the  shear  as  before   as   for    a  horizontal 
flange.    Then 

If  the  coiiipressed./frm<7e  is  inclined  down  to  the  nearest  suppwt,  or 
Jfthe  stretched  flange  is  inclined  down  from          " 
take  the  diff  between  the  vert  component  v  and  the  shearing  force.    But 
Jfthe  compressed  flange  is  inclined  down  from  the  nearest  support,  or 
Jfthe  stretched  flange  is  inclined  down  to 
take  the  sum  of  the  vert  component  v  and  the  shearing  force. 

Rein.  Hence  in  these  last  two  cases  (which  do  not  include  any  of  our  above 
figs)  the  vertical  force  on  the  web  is  increased. 

As  Humber  remarks,  in  girders  or  beams  with  curved  or  oblique  flanges  the 
greatest  strain  in  the  web  is  not  always  where  the  greatest  shearing  strain  is 
produced. 

Art.  13.  The  moment  of  rupture  at  any  point  t  in  a  bent  piece  R  with  a 
load  upon  or  suspended  from  c,  is  equal  to  the  load  X  its  leverage  1 1,  perp  to  c  w. 

This  moment  tends  to  break 
the  piece  R  at  its  cross  section 
at  t  by  tearing  apart  the  fibres 
to  the  right  of  its  neutral  axis, 
and  by  compressing  those  to 
the  left  of  it ;  and  to  this  mo- 
ment the  piece  R  opposes  the 
moment  of  resistance  of  that 
section  as  in  the  case  of  a  beam. 
In  an  arelieil  piece  as 
S  loaded  at  any  one  point  o, 

draw  on,om,  also  o  w  vertical  and  equal  by  scale  to  the  load,  and  complete  the 
parallelogram  o  e  w  c  of  forces.  Then  will  o  e  and  o  c  by  the  same  scale  give  two 
forces  into  which  the  load  is  resolved,  and  acting  in  the  directions  o  TO,  on,  much 
as  the  two  strings  of  two  bows  o  a  w,  o  u  m.  The  force  o  e  tends  to  break  the  bow 
o  u  m  at  any  section  u  with  a  moment  =  the  force  X  its  leverage  u  v  drawn  from 
the  point,  and  perp  to  o  ra;  and  the  force  o  c  tends  to  break  o  a  n  at  any  point  in 
the  same  way  with  its  leverage.  The  section  at  u  or  elsewhere  resists  as  in  R. 
The  weights  of  the  pieces  R  and  S  themselves  have  not  been  taken  into  con- 
sideration. 


KUTTER'S  FOEMULA, 

Kiit  tor's  general  formula  for  the  mean  velocity  in  feet  per 
second  in  pipes,  aqueducts,  canals,  rivers,  &c.  This  formula  is  the  joint  produc- 
tion of  two  eminent  German  engineers,  Ganguillet  and  Kutter,  but  for  conven- 
ience is  usually  called  by  the  name  of  the  last.  Here  11  is  the  coef  for  roughness 
of  sides  of  pipe  or  channel  as  given  by  the  table  below.  The  slope  is  the  quo- 
tient arising  from  dividing  the  fall  in  any  portion  of  the  length  by  the  length  of 
that  portion.  The  wet  perimeter'is  the  length  in  ft  found  by  measuring 
across  the  channel  such  parts  of  its  sides  and  bottom  as  are  in  contact  with  the 
water;  see  Art  21,  p  564.  The  Mean  Radius  is  the  quotient  arising  from 
the  area  of  cross  section  of  the  water  divided  by  the  wet  perimeter.  In  pipes 
running  full  it  is  always  equal  to  one-fourth  of  the  bore.  All  the  dimen- 
sions must  be  in  ft.  Then  by  Kutter's  formula 


FORMULA.  651 

V/Mean  Had  X  Slope. 
I/  Mean  Rad 

Table  of  n,  or  eoeffs  of  Roughness  of  Wet  Perimeter. 

.009  for  well  planed  timber. 

.010   "  plastered  with  neat  cement;  also  for  glazed  pipes. 

.011    "  plastered  with  1  measure  of  sand  to  3  of  cement. 

.012   "  unplaned  timber,  or  unlined  cast-iron  pipes.    If  the  pipes  are  very  smooth, 

.011  may  be  used. 
.013    "  Ashlar  or  Brickwork. 
.017    "  Rubble. 

.020   u  Canals  in  very  firm  gravel. 
.025    "  Rivers  and  Canals  in  moderately  good  order  and  regimen,  and  perfectly 

free  from  stones  and  weeds. 
.030    "  Rivers  and  Canals  in  moderately  good  order  and  regimen,  having  stones 

and  weeds  occasionally. 
.035    "  Rivers  and  Canals  in  bad  order  and  regimen,  overgrown  with  vegetation, 

and  strewn  with  stones  or  detritus  of  any  sort. 

Rem.  To  avoid  t  lie  use  of  this  troublesome  formula,  at  least  in 
the  case  of  clean  cast-iron  pipes,  the  writer  would  remark  that  he  has  found  that 
for  any  head  not  less  than  at  the  rate  of  4  ft  per  mile,  or  say  .9  inch  in  100  ft, 
(which  is  of  very  rare  occurrence  for  pipes)  it  gives  results  for  a  pipe  of  1  ft 
diam  so  nearly  identical  with  our  Table  on  p  539,  that  said  Table  (alter  4  ft  per 
mile)  may  be  considered  as  drawn  up  from  Kutter's  formula,  using  .012  for  n.  For 
diams  greater  than  1  ft  his  formula  gives  greater  vels,  and  for  smaller  diams 
smaller  vels  than  that  by  Poncelet,  near  top  of  p  538,  from  which  our  Table  p  539 
was  prepared.  But  the  writer  fortunately  also  finds  that  when  the  head  is  not 
less  than  at  the  rate  of  4  ft  per  mile,  we  may  use  the  simple  formula  near  top  of 
p  538,  and  multiply  the  vel  thus  found  by  the  corresponding  number  in  the  Table 
below  the  Rule  for  vels  on  said  page.  The  product  will  be  Kutter's  vel. 

For  the  same  reason  we  add  the  following  table  of  vels  in  sewers. 

Remarks  on  Kutter's  Formula. 


.  - 

also shows  that  the  foundations  of  the 


clusive,  but  cannot  be  given  here.     H 


ves  w      proay  seom        er  more      an    rom 

whereas  in  quite  possible  cases  the  old  formulas  would  err  50  or  even  100  per  cent. 
As  we  have  before  remarked,  extreme  accuracy  is  not  to  be  expected  in  such  mat- 
ters ;  but  we  almost  always  may  and  should  ensure  that  the  error  shall  at  least  be 

be  necessar. 


ters ;  but  we  almost  always  may  and  should  ensure  tnat  tne  error  snail  at  jeasi  oe 
on  the  safe  side  by  making  n  a  little  larger  than  may  be  supposed  to  be  necessary. 
If  in  that  case  the  roughness  of  the  channel  afterwards  proves  to  be  less  than  we 
had  supposed  it  might  be,  and  the  vel  and  discharge  therefore  greater  than  we 
need,  the  supply  may  be  regulated  by  stop-gates.  Inasmuch  as  the  same  degree 
of  roughness,  or  n,  diminishes  the  vel  and  disch  to  a  greater  proportional  extent 
in  small  pipes  or  channels,  especially  in  high  vels,  than  in  large  ones,  care  should 
be  taken  that  their  sides  be  made  as  smooth  as  possible. 


652 


VELOCITIES    IN   SEWERS. 


VELOCITIES  IN  SEWEES, 


Table  of  vels  in  Circular  Brick  Sewers  when  running  full,  by 
Kutter's  formula,  p  650,  but  taking  n  at  .015  instead  of  his  .013,  in  consideration 
of  the  rough  character  of  sewer  brickwork  generally. 

When  running:  only  half  full  the  vel  will  be  the  same  as  when  full, 
but  this  is  not  the  case  at,  any  other  depth  whether  greater  or  less.  At  greater 
ones  it  increases  until  the  depth  equals  very  nearly  .9  of  the  diam,  when  it  is 
about  10  per  cent  greater  than  when  either  full  or  half  full.  From  depth  of  .9  of 
the  diatn  the  vel  decreases  whether  the  depth  becomes  greater  or  less.  At  depth 
of  .25  diam  the  vel  is  about  .78  of  that  when  full ;  and  then  diminishes  much 
more  rapidly  for  less  depths.  All  this  applies  also  to  pipes. 

The  vel  for  any  fall  or  diam  intermediate  of  those  in  the  table  can  be  found  by 
simple  proportion.  Original. 


Fall 

in  ft 
per 
mile. 

2 

3 

] 

Mametei 
6 

rg  in  feel 

8 

12 

16 

20 

Fall 

in  ft 
per 
100  ft. 

Velocities  in  feet  per  second. 

.1 

.19 

.27 

.35 

.50 

.64 

.89 

1.10 

1.34 

.0019 

.2 

.30 

.42 

.53 

.74 

.93 

1.26 

1.56 

1.84 

.0038 

.4 

.46 

.65 

.80 

1.08 

1.39 

1.81 

2.20 

2.60 

.0076 

.6 

.59 

.81 

1.00 

1.35 

1.70 

2.22 

2.70 

3.18 

.0114 

.8 

.69 

.95 

1.17 

1.57 

1.94 

2.56 

3.08 

3.60 

.0151 

1.0 

.79 

1.07 

1.32 

1.77 

2.16 

2.84 

3.43 

3.96 

.0189 

1.25 

.89 

1.21 

1.49 

1.98 

2.42 

3.17 

3.8 

4.5 

.0237 

1.50 

.98 

1.33 

1.64 

2.18 

2.64 

3.5 

4.2 

4.9 

.0284 

1.75 

1.06 

1.44 

1.78 

2.34 

2.85 

3.8 

4.5 

5.3 

.0331 

2.0 

1.15 

1.55 

1.91 

2.53 

3.1 

4.0 

4.8 

5.6 

.0379 

2.5 

1.32 

1.78 

2.18 

2.85 

3.5 

4.5 

5.4 

6.3 

.0473 

3.0 

1.44 

1.94 

2.38 

3.2 

3.8 

5.0 

6.0 

6.9 

.0568 

3.5 

1.58 

2.10 

2.58 

3.4 

4.1 

5.3 

6.5 

7.4 

.0662 

4. 

1.68 

2.2 

2.7 

3.6 

4.4 

5.7 

6.9 

7.9 

.0758 

5. 

1.90 

2.5 

3.1 

4.1 

4.9 

6.3 

7.6 

8.7 

.0947 

6. 

2.06 

2.7 

3.3 

4.4 

5.4 

6.9 

8.3 

9.6 

.1136 

7. 

2.2 

3.0 

3.6 

4.8 

5.8 

7.5 

9.0 

10.4 

.1325 

8. 

2.4 

3.2 

3.8 

5.1 

6.2 

8.0 

9.7 

11.1 

.1514 

9. 

2.5 

3.4 

4.1 

5.4 

6.6 

8.5 

10.3 

11.8 

.1703 

10. 

2.7 

3.5 

4.3 

5.7 

6.9 

9.0 

10.8 

12.5 

.1894 

12. 

2.9 

3.9 

4.8 

6.3 

7.6 

9.9 

11.9 

13.6 

.2273 

15. 

3.3 

4.4 

5.4 

7.1 

8.5 

11.0 

13.3 

15.3 

.2841 

18. 

3.6 

4.8 

5.9 

7.7 

9.3 

12.1 

14.5 

16.7 

.3409 

21. 

3.9 

5.1 

6.3 

8.4 

10.0 

13.0 

15.7 

17.9 

.3975 

24. 

4.2 

5.5 

6.8 

8.9 

10.8 

13.9 

16.8 

19.2 

.4546 

27. 

4.5 

5.9 

7.2 

9.5 

11.4 

14.8 

17.9 

20.4 

.5109 

30. 

4.7 

6.2 

7.5 

9.9 

12.0 

15.6 

18.8 

21.5 

.5682 

35. 

5.0 

6.7 

8.2 

10.8 

13.0 

16.8 

204 

23.2 

.6629 

40. 

5.4 

7.1 

8.7 

11.5 

13.9 

18.0 

21.7 

24.8 

.7576 

45. 

5.6 

7.5 

9.2 

12.2 

14.8 

19.1 

23.0 

26.3 

.8523 

50. 

5.9 

8.0 

9.7 

12.8 

15.5 

20.1 

24.2 

27.7 

.9470 

60. 

6.5 

8.7 

10.7 

14.1 

17.0 

22.1 

26.5 

30.3 

1.136 

70. 

7.0 

9.4 

11.5 

15.2 

18.4 

23.9 

28.5 

32.8 

1.326 

80. 

7.4 

10.1 

12.3 

16.2 

19.7 

25.5 

31.0 

35.0 

1.515 

90. 

7.9 

10.7 

13.1 

17.2 

20.9 

27.0 

32.3 

37.1 

1.705 

100. 

8.4 

11.3 

13.8 

18.2 

22.0 

28.5 

34.1 

39.1 

1.894 

A  vel  of  1O  ft  per  sec  =  600  ft  per  minute  =  36000  ft,  or  6.818  miles  per 
hour.  About  5  ft  per  sec  is  as  great  as  can  be  adopted  in  practice  to  prevent  the 
lower  parts  of  the  sewers  from  wearing  away  too  rapidly  by  the  debris  carried 
along  by  the  water. 


RIV, 


653 


R,  Figs  3  p  654,  shows  the  /urual  shapes  of  rivets  as  sold.* 

The  weights  in  the  following  table  of  course  include  the  head ;  hut  the  lengths,  as  usual, 
are  taken  "  under  the  head/'  or  are  those  of  the  shanks  only.  In  practice,  discrepancies  of  5  or  6 
per  ct  in  wt  may  be  expected. 

From  Carnegie  Bros.  &  Go's  "Useful  Information,"  by  C.  L.  Strobel,  C  E. 


E 


AND    RIVETING. 


AND  RIVETING, 


Length 
of  Shank. 
Ins. 

/ 

%  |  K 

IHnmct 

% 

erg  of  Rl 

M 

vets  in  ii 

% 

iches. 

1 

IK 

Vi 

, 

3.0 

8.5 

Weight  of  100  B 

[vets,  in 

pounds. 

i/ 

3.8 

9.9 

17.3 

i  4 

4.6 

11.2 

19  4 

25.6 

389 

\/ 

5.4 

12.6 

21.5 

28.7 

43.1 

65.3 

91.5 

123 

iz 

6.2 

13.9 

23.7 

31.8 

47.3 

70.7 

98.4 

133 

37 

6.9 

15.3 

25.8 

34.9 

51.4 

76.2 

105 

142 

2 

1.1 

16.6 

27.9 

37.9 

55.6 

81.6 

112 

150 

K 

8.5 

18.0 

30.0 

41.0 

59.8 

87.1 

119 

159 

<A 

9.2 

19.4 

32.2 

44.1 

64.0 

92.5 

126 

167 

% 

10.0 

20.7 

34.3 

47.1 

68.1 

98.0 

133 

176 

3 

10.8 

22.1 

36.4 

50.2 

72.3 

103 

140 

184 

1A 

11.5 

23.5 

38.6 

53.3 

76.5 

109 

147 

193 

\/ 

12.3 

24.8 

40.7 

56.4 

80.7 

114 

154 

201 

y. 

13.1 

26.2 

42.8 

59.4 

84.8 

120 

161 

210 

4 

13.8 

27.5 

45.0 

62.5 

89.0 

125 

167 

218 

K 

14.6 

28.9 

47.1 

65.6 

93.2 

131 

174 

227 

% 

15.4 

30.3 

49.2 

68.6 

97.4 

136 

181 

236 

M 

16.2 

31.6 

51.4 

71.7 

102 

142 

188 

244 

5 

16.9 

33.0 

53.5 

74.8 

106 

147 

195 

253 

K 

17.7 

34.4 

55.6 

77.8 

110 

153 

202 

261 

<A 

18.4 

35.7 

57.7 

80.9 

114 

158 

209 

270 

% 

19.2 

37.1 

59.9 

84.0 

118 

163 

216 

278 

6 

20.0 

38.5 

62.0 

87.0 

122 

169 

223 

287 

1A 

21.5 

41.2 

66.3 

93.2 

131 

180 

236 

304 

1 

23.0 

43.9 

70.5 

99.3 

139 

191 

250 

321 

1A 

24.6 

46.6 

74.8 

106 

147 

202 

264 

338 

8 

26.1 

49.4 

79.0 

112 

156 

213 

278 

355 

9 

29.2 

54.8 

87.6 

124 

173 

234 

306 

389 

10 

32.2 

60.3 

96.1 

136 

189 

256 

333 

423 

11 

35.3 

65.7 

105 

148 

206 

278 

361 

457 

12 

38.4 

71.2 

113 

161 

223 

300 

388 

491 

The  diam  of  rivets  for  bridge  work  is  from  %  to  1  inch;  usually  %  to 
%;  and  for  plates  more  than  .5  inch  thick,  it  is  about  1.5  times  the  thickness; 
and  for  thinner  ones  about  twice ;  but  these  proportions  are  not  closely  adhered 
to.  The  common  form  of  rivets  as  sold  is  shown  at  R,  Figs  3,  a  head 
and  the  shank  in  one  piece ;  and  S  shows  the  same  when  after  being  heated 
white  hot  it  is  inserted  into  its  hole,  and  a  second  head  (conical)  formed  on  it  by 
rapid  hand-riveting  as  it  cools.  When  longer  than  about  6  ins  they 
are  cooled  near  the  middle  before  being  inserted,  les^their  contraction  in  cooling 
should  split  off  their  heads.  Why  this  should  occur  in  long,  and  not  in  short 
ones,  is  not  very  evident.  The  hemispherical  heads  often  seen,  and  called  snap 
heads,  are  formed  by  a  machine.  The  two  heads  alone  req  u  i  re  about 
as  much  iron  as  3  diams  length  of  shank.  Length  of  a  head  =  about  1 
diam  of  shank ;  and  its  width  about  2  diams  of  shank. 

Riveting  of  Steam  and  Water  Tight  Joints. 

Joints  for  boilers  and  water-tight  cisterns  are  usually  proportioned  about 
as  per  the  following  table  by  Fairbairn  ;  and  are  made  as  shown  either  by  Fig  1, 
or  Fig  2.  Fig  1  is  called  a  single-riveted,  and  Fig  2  a  double-riveted 

lap-joint.    The  dist  a  a,  or  c  c,  is  the  lap. 

Mr  Fairbairn  considers  the  strength  of  the  single-riveted  lap-joint  to  be  about 
.56 ;  and  that  of  the  double-riveted,  about  .7  that  of  one  of  the  full  unholed 

*  Price  in  Philada,  1880,  about  6  cts  per  fb. 
42 


654 


RIVETS   AND    RIVETING. 


plates,  when  both  joints  are  proportioned  as  in  his  following  table.    But  some 

later  experimenters  consider  about 
,**  r~\    **••  _  „          a  .    ^       ^ 


O 
O 
O 


c 

0 
O 
O 

0 
0 

H 

Fig  1.  Fig  2. 

proportions  include  friction  (Art  4),  without  which  they  would  6e  about  A  and  .5. 


.5  and  .6  as  nearer  the  correct  aver- 
age. Experiments  on  the  subject 
are  quite  conflicting;  and  it  is 
plain  that  no  one  set  of  propor- 
tions can  precisely  suit  all  the  dif- 
ferent qualities  of  plate  and  rivet 
iron.  With  fair  qualities  of  both, 
there  is  every  reason  to  rely  upon 
.5  and  .6  (or-  about  one-seventh 
part  less  than  Fairbairn's  assump- 
tion) as  safe  for  practice.  These 


Fairbairn's  table  for  proportioning*  the  riveting  for  steam 
and  water-tight  lap-joints. 


Thickness  of 
each  plate. 

Diameter  of 
ri  vets. 

Length  of  shank 
before  driving. 

From  center  to 
center  of  rivets. 

Lap  in  single 
riveting. 

Lap  in  double 
riveting. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

3-16 

% 

% 

jl/ 

1/4 

2J 

\/ 

\/ 

% 

\\/ 

11A 

2V 

6-16 

% 

iM 

1% 

m 

31 

% 

3/ 

J5/C 

J3/ 

2 

33^ 

\£ 

13-16 

2^£ 

2 

214 

3% 

$ 

15-16 

if 

g 

2% 

1 

Riveting  of  iron  girders,  bridges, 


Art.  1.  The  subject  of  riveting  is  abstruse,  and  involved  in 
much  uncertainty ;  and*  expf  rimental  results  are  very  discrepant.  We  here  pro- 
pose merely  to  confine  ourselves  to  what  is  considered  the  best  joint;  and  for 
safety  we  shall  omit  friction;  see  Art  4.  In  girder  and  bridge  work  the  lap- 
joints  above  described  are  seldom  used.  Instead  of  them,  the  plates  p,  Figs  3,  to 
be  joined,  are  butted  up  square  against  each  other,  thus  forming  a  butt-joint, 
i  i,  Fig  D;  and  are  united  by  either  a  single  covering-plate,  cover, 
wrapper,  fish-plate,  or  welt  e  e,  Fig  K;  or  the  best  of  all  by  two  of  them, 
as  at  A,  or  oo,  o  o,  Fig  B.  In  what  follows,  the  term  plate  never  includes  the 
covers.  The  single  cover,  like  the  lap-joint,  allows  both  plates  and  cover  to  bend 
under  a  strong  pull,  somewhat  as  at  W,  thus  weakening  them  materially ;  whereas 
the  double  cover  00,0  o.  Fig  B,  keeps  the  pull  directly  along  the  axis  of  the  plates, 
thus  avoiding  this  bending  tendency.  It  also  brings  the  rivets  into  double  shear, 
thus  doubling  their  strength.  When  there  is  but  one  cover,  it  should  be  at  least 
as  thick  as  a  plate ;  and  when  there  are  two,  experience  shows  that  each  had  bet- 
ter be  about  two-thirds  as  thick  as  a  plate,  although  theory  requires  each  to  be 
but  half  as  thick  as  a  plate. 


RIVETS   AND   RIVETING.  655 

w  w  of  covers  across  the  joint  is  equal  to  that  of  the  joint. 

Butts  require  twice  as  many  rivets  as  laps,  because  in  the  lap  each 
rivet  passes  through  both  the  joined  plates;  and  in  the  butt  through  only  one. 

The  rivets  and  plate  on  one  side  only  (right  or  left)  of  the  joint- 
line  I  i  of  any  properly  proportioned  butt-joint  D,  represent  the  full  strength 
of  the  joint,  inasmuch  as  those  on  one  side  pull  in  one  direction,  against  those  on 
the  other  side,  which  pull  in  the  opposite  direction.  Therefore  in  designing  such 
joints  we  need  keep  in  mind  only  those  on  one  side,  as  is  done  in  what  follows. 
Thus  a  single,  double,  or  triple-riveted  butt-joint  D  implies  one,  two,  or  three 
rows  of  rivets  oil  eacli  side  of  the  joint-line  i  i,  and  parallel  to  it.  In  a  prop- 
erly proportioned  lap  the  strength  is  as  all  the  rivets,  because  one-half  of  them 
do  not  pull  against  the  other  half,  but  one  end  of  every  rivet  pulls  in  one  direc- 
tion, and  its  other  end  in  the  opposite  direction. 

The  net  iron,  net  plate,  or  net  joint,  is  that  which  is  left  between 
the  rivet  holes,  and  outside  of  the  two  outer  ones,  all  on  a  straight  line  drawn 
through  the  centers  of  the  holes  of  one  row.  Its  width  and  area  are  called  the  net 
ones  of  the  joint.  That  between  other  rows  does  not  increase  the  strength. 

In  Figs  1,  2,  N,  and  K,  the  rivets  are  plainly  exposed  only  to  single  shear; 
that  is,  the  opposing  pulls  of  the  two  plates  tend  to  shear  each  rivet  across  only 
one  circular  section  ;  whereas  in  Fig  B,  with  two  covers,  or  in  Fig  A,  each  rivet 
is  exposed  to  double  shear,  one  just  above  and  one  just  below  the  joined 
plates. 

Art.  2.  Bridge-joints  are  not  required  to  be  steam  or  water- 
tight like  those  of  boilers  or  cisterns;  and,  therefore,  by  increasing  the  breadth 
of  the  overlap,  or  the  length  of  the  covers,  the  rivets  may  be  placed  in  several 
rows  behind  each  other,  as  the  3  rows  of  3  rivets  each  in  M  and  D,  instead  of  only 
one  row  of  9  rivets,  as  in  L.  By  this  means,  without  losing  any  of  the  strength  of 
the  9  rivets,  or  of  the  net  iron,  we  may  narrow  the  width  of  the  plate  to  an  ex- 
tent equal  to  the  combined  diams  (6  in  this  case)  of  the  holes  thus  dispensed  with 


in  the  one  row.  Moreover,  by  using  more  than  one  row  we  lessen  the  weakening 
effect  shown  at  W.  This  mode  of  placing  the  rivets  directly  behind  each  other  in 
several  rows,  as  at  M,  and  at  the  left-hand  half  of  Fig  D,  constitutes  Mr  Fair- 


q 
B 


,  , 

bairn's  chain  riveting1  ;  but  the  joint  will  be  somewhat  stronger  if  the  rivets 
are  placed  in  zigzaging  order,  as  in  the  right-hand  half  of  Fig  D. 

The  dist  apart  of  the  rows  from  cen  to  ceil  should  not  be  less 
than  2  diams.    It  is  questionable  to  what  extent  this  increase  in  the  number  of 
rows  may  be  carried  without  an  appreciable  loss  of  strength  in  the  rivets  conse- 
uent upon  the  impossibility  of  quite  equalizing  the  strains  on  the  separate  rows. 

ut  it  is  probable  that  if  we  do  not  exceed  2  or  3  rows  in  laps,  or  the  same  num- 
ber on  each  side  of  the  joint-line  in  butts,  we  may  in  practice  assume  that  each 
row,  and  each  rivet,  is  nearly  equally  strained. 

Rivet-holes  are  usually  of  about  one-sixteenth  inch  greater  diam  than  the 
original  rivet,  so  as  to  allow  the  hot  rivet  to  be  easily  inserted.  The  subsequent 
hammering  swells  the  diam  of  the  rivet  until  it  fills  the  hole.  We  may  either 
take  this  increased  diam  of  rivet  into  consideration,  as  we  have  done,  in  calcula- 
ting its  shearing  and  crippling  strength,  as  explained  farther  on,  or  with  reference 
to  increased  safety  we  may  omit  it.  Drilled  rivet-holes  are  said  to  be  better 
than  punched  ones,  as  the  drilling  does  not  injure  the  iron  around  them;  but  on 
the  other  hand  their  sharper  edges  are  said  to  shear  the  rivets  more  readily. 
Hence,  such  edges  are  sometimes  reamed  off.  Both  these  points  are,  however, 
disputed  ;  and  both  modes  are  in  common  use. 

The  dist  from  the  edge  of  a  hole  to  the  end  of  a  plate  or  cover  should 
not  be  less  than  about  1.2  diams,  to  prevent  the  rivets  from  tearing  out  the  end 
of  the  plate;  nor  nearer  the  side  edge  of  a  plate  than  half  the  clear  dist  between 
two  holes  as  given  by  the  Rule  in  Art  5.  The  first  is  rather  more  than  Fairbairn 
directs. 

Rivet  holes  weaken  the  net  iron  left  between  them,  not  only  by  the 
loss  of  the  part  cut  out,  but  either  by  disturbing  the  iron  around  them,  or  perhaps 
by  changing  the  shape  of  the  net  line  of  fracture,  which  may  not  then  resist 
tension  as  well  as  while  it  was  a  continuous  straight  line.  Some  deny  both  cause 
and  effect  entirely,  each  party  basing  its  opinion  on  experiments.  But  the  mass 
of  evidence  seems  to  the  writer  to  show  that  the  net  iron  loses  on  an  average 
about  one-seventh  of  the  strength  due  to  the  net  width.  With  a  view  to  safety, 
which  we  consider  to  be  of  paramount  importance,  we  shall  in  what  follows 
assume  (until  the  question  is  definitely  settled)  that  there  is  such  a  loss  of 
strength  in  the  net  iron. 

Riveted  joints  for  resisting  compression  should  depend,  not  as 
might  be  supposed  upon  their  butting  ends,  but  upon  either  the  shearing  or  the 
crippling  strength  of  the  rivets;  for  contraction  or  bad  work  may  throw  the 


656 


RIVETS   AND   RIVETING. 


pressure  on  the  rivets.  Machine  riveting-  is  somewhat  stronger  than  that 
done  (as  is  assumed  in  our  examples)  by  hand.  The  thickness  of  plates 

used  in  girders,  tubular  bridges,  <fcc,  is  usually  .25  to  .5  inch ;  with  thicker  ones 
up  to  1  inch  sparingly  in  large  ones.  A  packing  piece,  as  the  shaded  piece 
in  P,  is  one  inserted  between  two  plates  to  prevent  their  being  bent  or  drawn 
together  by  the  rivets. 

Art.  3.  A  riveted  joint  may  yield  in  three  ways  after  being 
properly  proportioned,  namely,  by  the  shearing  of  its  rivets;  or  by  the  pulling 
apart  of  the  net  plate  between  the  rivet  holes ;  or  by  the  crippling-  (a  kind  of 
compression,  mashing,  or  crumpling)  of  the  plates  by  the  rivets,  when  the  two  are 
too  forcibly  pulled  against  each  other.  It  also  compresses  the  rivets  themselves 
transversely,  at  a  less  strain  than  the  shearing  one;  and  this  partial 
yielding  of  both  plates  and  rivets  allows  the  joint  to  stretch,  and  may  thus 
produce  injurious  unlooked-for  strains  in  other  parts  of  a  structure,  considerably 
before  there  is  any  danger  of  actual  fracture.  Or  in  steam  and  water  joints  it  may 
cause  leaks,  without  farther  inconvenience,  or  danger.  For  a  long  time  this 
crippling  had  entirely  escaped  notice,  and  it  was  supposed  that  the  only  important 
point  in  designing  a  riveted  joint  was  that  the  tensile  strength  of  the  net  plate, 
and  the  shearing  strength  of  the  rivets  should  be  equal  to  each  other. 

The  crippling-  strength  of  a. joint  is  as  the  number  of  rivets,  in  a  lap. 
or  the  number  on  one  side  of  the  joint-line  in  a  butt  X  diam  X  thickness  of  joined 

Slate.  This  product  gives  the  crippled  area  of  the  joint.  We  shall  here  call  the 
iam  X  thickness  of  plate,  the  crippling-  area  of  a  rivet.  If  there  are  2  or 
more  plates  (hot  covers)  on  top  of  each  other  at  one  joint,  their  united  thickness 
is  used  for  finding  the  crippling  area.  The  ultimate  crippling-  unit, 
by  which  the  above  product  is  to  be  multiplied  for  the  actual  ultimate  crippling 
strength  of  the  joint,  may  be  safely  taken  at  about  60000  fos,  or  26.8  tons  per  sq 
inch. 

The  diam  of  a  rivet  in  ins  to  resist  safely  a  given  single-shearing 
force  is  found  thus :  Mult  the  shearing  force  by  the  coef  of  safety,  that  is  by  the 
number,  3,  4,  or  6,  &c,  denoting  the  required  degree  of  safety.  Call  the  product  g. 
Mult  the  ultimate  shearing  strength  per  sq  inch  of  the  rivet-iron,  by  the  decimal 
.7854.  Call  the  product  6.  Divide  g  by  b.  Take  the  sq  rt  of  the  quotient.  The 
shearing  force  and  the  shearing  strength  must  both  be  in  either  ft>s  or  tons. 
Or  by  a  formula, 

l>iaiu  ill  ins  =\  /~        Shearing  force  >Qcoef  of  safety 

\/    Ult  shearing  strength  per  sq  inch  X  .7854 

If  the  rivet  is  to  be  double-sheared,  first  mult  only  half  the  shearing 
force  by  the  coef  of  safety.  Then  proceed  as  before. 

Or,  near  enough  for  practice,  mult  the  diam  in  single  shear  by  the  decimal  .7. 

The  ultimate  shearing-  mi  it  for  average  rivet-iron  may  be  taken  at 
about  45000  tt>s,  or  20.1  tons  per  sq  inch  of  circular  sheared  section. 

Table  of  ultimate  single  shearing1  strength  of  rivets. 

(market  sizes),  in  single  shear ;  at  45000  Bbs  or  20.1  tons  per  sq  inch. 

This  table  is  not  to  be  used  when  as  in  our  "  Example,"  Art  5,  the 
crippling  strength  of  the  rivet  governs  the  strength  of  the  joint. 

If  the  rivet  is  in  double  shear  it  will  have  twice  the  strength  in  the 
table. 

For  the  cliam  in  double  shear  to  equal  the  strength  in  the  table,  mult 
the  diam  in  the  table  by  the  decimal  .7 ;  near  enough  for  practice  ;  strictly,  .707. 


Diam. 
Ins. 

Diam. 
ins. 

fts. 

Tons. 

Diam. 
Ins. 

Diam. 
Ins. 

as. 

Tons. 

Diam. 
Ins. 

Diam. 
Ins. 

fts. 

Tons. 

H 

.125 

552 

.246 

.562 

11183 

4.99 

1 

1.000 

35343 

15.8 

.187 

1242 

.554 

% 

.625 

13806 

6.16 

1.062 

39899 

17.8 

K 

.250 

2209 

.986 

.687 

16705 

7.46 

1V& 

1.125 

44731 

20.0 

.312 

3452 

1.54 

'A4 

.750 

19880 

8.88 

1.187 

49838 

22.2 

*4 

.375 

4970 

2.22 

.812 

23332 

10.4 

IK 

1.250 

55224 

24.6 

.437 

6765 

3.02 

% 

.875 

27060 

12.1 

1.312 

60885 

27.2 

1A 

.500 

8836 

3.94 

.937 

31064 

13.9 

m 

1.375 

66820 

29.8 

RIVETS/AND    RIVETING.  657 

The  tensile  stren&m.  of  a  properly  proportioned  joint  is 

equally  as  either  the  sessional  area  of  the  net  plate  (not  covers)  across  the  cen- 
ters of  only  one  rowjpi  rivets  j  or  as  the  shearing  or  the  crippling  (as  the  case 
may  be)  areas  of  alL-die  rivets  in  a  lap,  or  of  all  the  rivets  on  one  side  of  the 
joint-line  in  a  DU£K  The  tensile  strength  of  fair  quality  of  plate  iron,  before  the 
rivet  holes  are  s^nlade,  averages  about  45000  fts,  or  20.1  tons  per  sq  inch  ;  but  we 
shall  for  safety  assume,  as  stated  in  Art  2,  that  the  making  of  the  holes  reduces 
the  strength  of  the  net  iron  that  is  left  about  one-seventh  part,  or  to  38500  ft>s, 
or  17.2  tons  per  sq  inch. 

Rent.  Even  this  is  considerably  too  great  for  laps,  or  for  butts 
with  one  cover,  owing  to  the  weakening  of  the  iron  in  such  by  the  bending  shown 
at  W,  Figs  3.  But  we  are  not  speaking  of  such.  See  Art  7. 

Art.  4.  The  friction  between  the  plates  in  a  lap,  or  between  the 
plates  and  the  covers  in  a  butt,  produced  by  their  being  pressed  tightly  together 
by  the  contraction  of  the  rivets  in  cooling,  adds  much  to  the  strength  of  a  joint 
while  new,  perhaps  as  much  as  1.5  to  3  tons  per  sq  inch  of  circ  section  of  all  the 
rivets  in  a  lap,  or  of  all  on  one  side  of  a  single-cover  butt  ;  or  3  to  6  tons  of  all  on 
one  side  of  a  double-cover  butt.  In  quiet  structures,  this  friction  might  continue 
to  exist,  either  wholly  or  in  part,  for  an  indefinite  period  ;  but  in  bridges,  &c,  sub- 
ject to  incessant  and  violent  jarring  and  tremor,  it  is  probably  soon  diminished, 
or  entirely  dissipated.  Hence  good  authorities  recommend  not  to  rely  on  it,  and 
it  is,  therefore,  omitted  in  what  follows. 

Art.  5.  We  now  give  rules  for  finding  the  number  of  rivets  required  for  a 
double  coyer  butt-joint  (the  only  kind  of  which  we  shall  treat),  and  their 
clear  or  net  distance  apart.  This  dist  -f  one  diam  is  the  pitch  of  the  rivets,  or 
their  dist  from  center  to  center.  The  principle  of  the  rule  will  be  explained 
further  on,  at  Art  7. 

First,  select  a  diam  of  rivet  either  equal  to  or  greater  than  .85  times  the 
thickness  of  the  plate.  In  practice  they  are  generally  1.5  times  for  plates  %  inch 
or  more  thick  ;  and  2  for  thinner  than  %  in. 

Second,  mult  the  greatest  pull  that  can  come  upon  the  joint  by  the  coef  (3,  4, 
or  6,  &c)  of  safety,  and  call  the  product  p. 

Third,  multiply  the  crippling  area  of  the  rivet  (that  is,  its  diam  X  the  thick- 
ness of  plate)  by  60000.  The  prod  is  the  ult  crippling  strength  of  a  rivet.  Call  it  m. 

Fourth,  divide  p  by  m.  The  quotient  will  be  the  number  of  rivets  to  sustain 
the  given  pull  with  the  reqd  degree  of  safety. 

Then,  the  clear  distance  apart  will  be 

Diam  X  thickness  of  plate  X  60000 
Thickness  of  plate  X  38500 

Example.  A  double-cover  butt-joint  in  .5  inch  thick  plate  is  to  bear  an 
actual  pull  of  33750  ft>s,  with  a  safety  of  4  ;  or  not  to  break  with  less  than  33750  X 
4  =  135000  ft)s.  How  many  rivets  must  it  have  ;  and  how  far  apart  must  they  be? 

First,  Here  .85  times  the  thickness  of  the  plate  is  .5  X  .85  =  .425  inch  ;  there- 
fore, our  rivets  must  not  be  less  than  .425  inch  in  diam  ;  but  we  will  take  .75  inch 
diam. 

Second,  The  greatest  pull  X  coef  of  safety  =  33750  X  4  =  135000  Ibs  =  p. 

Third,  The  crippling  area  of  a  rivet  X  60000  =  .75  X  -5  X  60000  =  22500  =  m. 

Fourth,  -*--  =  TTo^T^r  =  6  rivets  required  on  each  side  of  the  joint-line. 


And  the  clear  space  or  net  width  between  them  will  be 
Diam  X  thickness  of  plate  X  60000  _    22500  _ 
Thickness  of  plate  X  38500          =    19250     .    ' 

And  the  pitch  =  net  space  +  diam  =»  1.1688  +  .75  =  1.9188  ins, 

=  2.56  diams.  ,  ** 

In  practice,  to  avoid  troublesome  decimals,  we  might  make  the  net  space  1.2 
ins;  and  the  pitch  1.95;  but  to  show  farther  on  the  working  of  the  rule,  we  ad- 
here to  the  more  exact  ones. 

The  entire  width  of  net  iron,  if  all  6  rivets  are  in  one  row,  is  equal  to 
one  clear  space  X  number  of  rivets,  =  1.1688  X  6=  7.0128  ins;  and  the  entire 
width  of  plate  is  equal  to  one  pitch  X  number  of  rivets,  =  1.9188  X  6  =  11.5128 
ins.  The  net  width  X  by  the  thickness  of  plate  (7.0128  X  .5)  gives  3.5064  sq  ins 
area  of  net  iron  ;  and  this  area  X  by  our  tensile  unit,  38500  fas,  gives  135000  fts  as 
the  ultimate  pull  required  to  break  it,  as  in  the  beginning  of  the  example. 


658  RIVET8    AND    RIVETING. 

The  area  of  the  entire  unholed  plate  is  11.5128  X  .5  =  5.7564  sq  ins;  and  its 
tensile  strength  before  the  Holes  are  made  is  5.7564  X  45000  =  259038  fts. 

1  85000 

The  strength  of  our  joint,  omitting  friction,  is  therefore  ^TTTTT:  =  .52  of  that  of 
the  original  unholed  plate. 

If  we  place  the  6  rivets  ill  2  rows  of  3  each,  we  shall  have  hut  3 
rivet-holes  in  a  straight  line  across  the  plate  ;  and  as  the  united  diams  of  the  3 
holes  thus  dispensed  with  is  equal  to  (3  X  '75)  =  2.25  ins,  the  width  of  net  iron 
in  a  row  is  increased  that  much.  This  of  course  increases  the  strength  of  the 
net  iron,  but  not  that  of  the  joint;  because  the  latter  (as  proportioned  by 
the  rule)  depends  precisely  as  much  on  the  crippling  strength"  of  the  rivets  as  on 
the  tensile  one  of  the  net  iron  ;  and  inasmuch  as  the  6  rivets  will  now  yield  under 
the  same  pull  as  before,  it  is  plain  that  the  joint  is  no  stronger  than  before. 

But  the  2  rows  are  the  most  economical,  because  we  may  reduce  the  width 
of  the  joined  plates  2.25  ins,  or  from  11.5128  to  9.2628  ins,  without  diminishing 
either  the  number  of  rivets,  or  the  calculated  necessary  net  iron  in  one  row.  The 
ult  strength  of  this  net  iron  in  one  row  is  still  135000  ;  but  that  of  the  unholed 
plate  of  reduced  width  is  now  (9.2628  X  .5  X  45000;  =  208413  Ibs;  so  that  the 

strength  of  the  joint  is  now    •      --5  =  .65  of  that  of  the  reduced  unholed  plate. 


If  we  place  the  6  rivets  in  3  rows  of  2  each,  we  may  reduce  the  entire 
width  of  joint  3  ins,  by  saving  4  holes  in  the  first  row  ;  thus  making  it  only 
11.5128  —  3  =  8.5128  ins,  still  retaining  the  first  required  width  (7.0128  ins)  of  net 
iron  in  that  row. 

The  ult  strength  of  the  entire  unholed  twice-reduced  plate  is  then  8.5128  X  -5 

X  45000  =  191538  fts  ;  and  that  of  our  joint  without  friction  is  ~    —  =  .7  of  it. 


Art.  6.  The  distance  apart  of  the  rows,  from  center  to  center  of 
rivets,  should  not  be  less  than  two  diameters  of  a  rivet-hole. 

Rem.  1.  With  our  constants  for  tension,  shearing,  and  compression,  the 
rivets  will  not  yield  first  by  shearing  in  a  double-cover  butt  (and 
of  course  in  double  shear),  except  when  the  diam  is  either  equal  to  or  less  than 
.85  of  the  thickness  of  the  plate,  which  will  rarely  happen.  At  .85  the  crippling 
and  shearing  strength  of  a  rivet  are  equal  when  using  our  assumed  coeffs  of  crip- 
pling, shearing,  and  tension. 

Item.  2.  Our  example  was  chosen  to  illustrate  the  rule.  It  will  rarely  hap- 
pen in  practice  that  the  rule  will  give  a  number  of  rivets  without  a  fraction  ;  or 
that  may  be  divided  by  2  and  by  3  without  a  remainder.  In  case  of  a  fraction,  it 
is  plainly  best  to  call  it  a  whole*  rivet  ;  although  the  joint  thereby  becomes  a  trifle 
stronger  than  necessary.  Or  rivets  of  a  slightly  dirf  diam  may  be  used.  If  the 
number  of  rivets  comes  out  say  7  or  9,  we  may  make  2  rows  of  3  and  4,  or  of  4  and 
5,  &c.  Moreover,  the  width  of  the  plate  is  frequently  fixed  beforehand  by  some 
requirement  of  the  structure,  and  we  must  arrange  the  rivets  to  suit,  taking  care 
in  all  cases  to  maintain  the  calculated  area  of  net  iron  in  one  row,  &c. 
Rem.  3.  We  have  (as  we  at  first  said  we  should  do)  confined  ourselves  to  the 
simple  butt-joint  with  2  covers,  and  with  the 

.JC  _  5  _  .      rivets  in  either  1,  or  in  2  or  more  parallel  rows 

on  each  side  of  the  joint-line  ;  this  being  the 
strongest  and  the  one  in  most  common  use  in 
engineering  structures.  Necessity  at  times 
calls  for  less  simple  arrangements,  for  which 
we  cannot  afford  space,  and  the  strength  of 
which  is  not  so  readily  calculated.  These 
sometimes  yield  results  which  appear  strange 
to  the  uninitiated  ;  thus,  this  lap-joint  breaks 
across  the  net  iron  of  one  plate,  along  either  c  c  or  o  o,  where  there  is  most  of  it,  and 
where,  therefore,  it  might  be  supposed  to  be  the  strongest. 

Rem.  4.  The  following1  table  shows  approximately  the  comparative 
strengths  of  the  common  forms  of  joints  when  properly  proportioned  ;  varying 
with  quality  of  sheets,  and  of  rivets  : 


With 
friction. 

The  original  uuholed  plate 1.00 

Double-riveted  butt  with  two  covers 80 

Double-riveted  butt  with  one  cover 65 

Single-riveted  butt  with  one  cover 50 

Double-riveted  lap 65 

Single-riveted  lap 50 


Without 
friction. 
1.00 
.64 
.52 
.40 
.52 
.40 


RIVI 


RIVETING. 


659 


Rein.  5.    The 

irnately  attain* 
double- 


abular  strengths  for  the  lap-joints  will  be  approx- 
ng  the  following  proportions,  according  as  the  joint  is 


^ 

Calling  thickness  of  plate  

Double  rlv 

In  thicknesses. 

zigzag. 

In  diams. 

Single 

In  thicknesses. 

rlv. 

In  diams. 

1. 

1.67 
9.0 
7.0 

2.0 
3.33 

.6 
1.0 
5.4 
4.2 

1.2 

2.0 

1. 
1.67 
5.67 
4.5 

2.0 

.6 
1.0 
3.4 

2.7 

1.2 

Then  make  diani  of  rivet 

"         "      breadth  of  lap  

"          "      pitch  from  cen  to  cen  
"          "      dist  from  end  of  plate  to 
edge  of  holes      . 

"         "      dist  apart  of  rows  from 
cen  to  cen  — 

Item.  6.    If  two  or  more  plates  on  top  of  each  other,  as  the 

four  in  A  B  or  M  H,  are  to  be  jointed  together  so  as  to  act  as  one  plate  of  the 
thickness  c  e,  the  diams  of  the  rivets,  and  the  thickness  of  the  covers  cc.ee  will 
depend  upon  whether  the  junctions  of  the  plates  are  all  in  one  line  with  each 
other  as  at  c  c,  in  A  B,  or  whether  they  break  joint  with  each  other  as  at  0,  1,  2,  3 
in  M  H. 


It  is  plain  that  the  two  covers  c  c  by  means  of  their  connecting  rivets  convey 
from  A  to  B,  across  the  joint  c  c,  all  the  strength  that  partly  compensates  for  the 
severance  of  the  four  plates  at  that  joint  ;  whereas  the  two  covers  e  e,  e  e,  and 
their  rivets  in  like  manner  convey  from  n  of  one  single  plate,  to  o  of  the  adjoining 
one,  across  the  joint  between  those  two  letters,  only  the  strength  that  partly  com- 
pensates for  the  severance  of  that  single  plate  ;  and  so  with  the  joints  at  1,  2,  and 
3.  Therefore  the  covers  c  c,  and  their  rivets,  must  be  four  times  as  strong  as  those 
at  any  one  of  the  four  joints  0,  1,  2,  3.  The  first,  c  c,  are  to  be  regarded  as  joining 
two  solid  plates  A  and  B,  each  of  the  fourfold  thickness  c  c  ;  and  the  others  as 
joining  two  of  the  single  thickness.  The  covers  c  c  will,  therefore,  each  be  about 
two-thirds  of  the  thickness  c  c  ;  and  the  others  each  about  two-thirds  as  thick  as 
a  single  plate.  Hence,  when  any  number  of  superimposed  plates  break  joint,  the 
covers  and  diams  of  rivets  may  be  of  the  same  as  when  only  two  plates  of  single 
thickness  are  jointed.  Since  the  weakest  point  of  a  structure  is  the  measure  of  its 
effective  strength,  the  tie  A  B  is  weaker  than  M  H  ;  for  although  M  H  has  four 
weak  cross-sections,  namely,  through  all  four  plates  at  each  of  the  joints  0,  1,  2,  3, 
yet  each  section  is  weakened  only  by  the  loss  of  part  of  the  strength  of  a  single 
plate;  whereas  the  section  c  c  is  weakened  by  the  loss  of  an  equal  part  of  the 
strength  of  each  of  the  four  plates;  and  is,  therefore,  weaker  than  any  other  sec- 
tion entirely  across  either  A  B  or  M  H  ;  and  consequently  A  B  is  weaker  than  M  H. 

Art.  7.  Principle  of  the  Rule  in  Art  5.  Omitting  friction,  and 
having  the  proper  coefts  of  tension,  shearing,  and  crippling,  it  is  plain  that  when 
we  equalize  the  tensile  strength  of  the  net  plate  across  one  row  of  rivets,  with 
the  single  or  double  shearing1  strength  of  all  the  rivets  in  a  lap,  or  of  all  on 
one  side  of  the  joint-line  in  a  butt,  we  have  for  the  net  width  or  space  between 
twe  rivets, 

Net  space  againstvy'  Thicknessv/  Tension  _  Once  or  twice  the  cir-vy  Shearing 
shearing         /\    of  plate  S\     unit    —  cular  area  of  a  rivet  /\     unit. 

From  this  follows,  by  transposing, 

_  Once  or  twice  the  circ  area  of  a  rivet  X  Shearing  unit 


R    1     1 


shearing- 


Thickness  of  plate  X  Tension  unit. 
And,  also,  that  when  we  equalize  the  tensile  strength  of  net  plate  with  the 


660  RIVETS  AND   RIVETING. 

crippling  strength  of  the  rivets,  we  have,  calling  the  diain  of  a  rivet  X  thickness 
of  plate,  its  crippling*  area. 

Net  space  against  \s  Thickness  \/  Tension Crippling  area  \s  Crippling 

crippling         S\    of  plate    s\     unit     of  a  rivet      /\      unit. 

From  this  follows,  by  transposing, 

Rule  2.    yeag^inc8tCeii:Cripplipg  area  of  a  rivet  X  CriPPling  imit 
crippling*  Thickness  of  plate  X  Tension  unit. 

Now  take  any  diam  of  rivet,  and  find  its  circular  area.  Mult  this  circ  area  by 
the  shearing  unit;  divide  the  prod  by  the  prod  of  diam  X  crippling  unit.  The 
quotient  will  evidently  be  the  thickness  or  plate  (not  cover)  which,  with  the 
assumed  units,  will  make  the  shearing  and  crippling  strengths  equal  to  each  other 

Assumed  diameter 

in  single  shear.    And  then  -    .-. —    — - — r— -  will  give  the  fixed  proportion  in 
Thickness  of  plate 

single  shear  which  (with  said  units)  any  diam  must  bear  to  any  given  thickness 
of  plate  whatever,  to  ensure  equality  between  the  crippling  and  shearing  strengths. 
With  our  assumed  units  of  45000  Ibs  per  sq  inch  for  shearing,  and  60000  fts  per  sq 
inch  for  crippling,  the  diameters  for  equality  will  be  found  to  be  1.7  times  the 
thickness  of  the  plate  when  in  single  shear ;  and  .85  when  in  double  shear.  If 
the  diam  is  less  than  1.7  or  .85  of  the  thickness  of  the  plate,  as  the  case  may  be, 
the  foregoing  calculation  will  show  that  the  shearing  strength  of  the  rivet  is  less 
than  its  crippling  strength,  and  that,  therefore,  it  will  give  way  by  shearing. 
Hence,  we  equalize  the  net  strength  of  the  plate  with  the  shearing  strength  of 
the  rivet  by  using  Rule  1.  If,  on  the  other  hand,  the  diam  is  more  than  1.7  or 
.85  of  the  thickness  of  the  plate,  as  the  case  may  be,  the  calculation  will  show  the 
crippling  strength  of  the  rivet  to  be  the  least ;  aftd  then  we  use  Rule  2. 

Rem.  1.  Butt  joints  in  double  shear,  or  with  2  covers,  being  the 
only  ones  here  considered,  and  inasmuch  as  rivets  may  always  be  used  with  a 
diam  greater  than  .85  of  the  thickness  of  the  plate,  we  may  in  practice  always  use 
Rule  2  for  such  joints ;  and,  therefore,  we  gave  it  alone. 

Item.  2.  When  using  these  rules  for  other  kinds  of  joint, 
such  as  laps,  or  butts  with  single  covers,  remember  that  the  rivets  in  such  are  in 
single  shear;  and,  therefore,  we  can  use  Rule  2  above  (for  crippling)  only  when 
the  diam  is  either  1.7  or  more  times  the  thickness  of  plate.  If  less,  use 
Rule  1  above  for  shearing*;  all  on  the  assumption  that  our  foregoing  coefs  of 
crippling  and  shearing  are  used. 

But  the  coef  for  tension  must  be  changed  for  each  kind  of  these 
other  joints,  to  allow  for  the  weakening  effects  of  the  bending  shown  at  W,  Figs 
3,  as  deduced  approximately  from  experiment.  The  writer  believes  that  the  fol- 
lowing tension  units  will  give  safe  approximate  results  without  friction.  For 
double-cover  butts,  double-riveted,  38500  Ibs  per  sq  inch,  as  adopted  above. 
For  double-riveted  laps,  or  one-cover  butts,  28000.  For  single-riveted 
laps,  or  one-cover  butts,  24000.  But,  as  before  remarked,  no  great  certainty  is 
attainable  in  riveting. 

Rein.  3.  A  joint  may  fail  by  crippling  without  the  facts  being 
known  or  even  suspected,  for  it  does  not  imply  that  anything  breaks,  but 
merely  that  the  joint  has  stretched ;  and  this  might  not  be  detected  even  on 
a  slight  inspection  of  it.  Still  it  might,  and  probably  often  has  sufficed  to  endanger, 
and  even  destroy  both  bridges  and  roofs  by  generating  strains  where  none  were 
provided  for. 


LOADS/ON    PROPS.  661 

LOADS  ON  PROPS. 


A  load  resting  on  two  props  either  at  its  ends  or  otherwise.  When  a 
load  c  of  any  shape  whatever,  rests  in  any  position  upon  two  props  x  and  z,  the 
portions  of  its  wt  borne  by  the  respective  props  will  be  to  each 
other  inversely  as  the  horizontal  distances  ox,  oz,  from  the 
cen  of  grav  o  of  the  load,  to  the  props.  Thus  if  o  z  is  two,  three  I 
or  four  times  as  great  as  o  x,  then  will  x  bear  two,  three,  or  four 
times  as  much  of  the  entire  wt  of  the  load  c  as  z  does.  There- 
fore to  find  bow  much  each  prop  bears,  first  find 
the  cen  of  grav  c  of  the  entire  load ;  and  its  hor  dist  (say  o  x)  from 
either  one  oT  the  props,  (say  x.) 

rph  The  entire  hor  dist  x  z    .   Entire  wt   •  .    ru  t        .    Wt  borne  by  the 

•**  between  the  two  props     •   of  load      *  *•  *        *•  other  prop  z. 

And  this  wt  taken  from  the  entire  load  leaves  the  wt  borne  by  x. 

This  all  amounts  to  the  same  as  if  we  consider  x  z  to  be  the  clear  span 
of  a  beam  without  wt,  and  supporting  a  load  equal  to  c,  concentrated  at  the  cen 
of  grav  of  c.  See  Bern,  p  478. 

Conversely,  to  place  two  props  x  and  z  so  that  each  may  bear  a  given  por- 
tion of  the  entire  load  c,  take  any  two  hor  dists  o  x  and  o  z  from  the  cen  of  grav 
c,  inversely  as  the  two  portions  of  the  load  to  be  borne  by  each.  Thus  if  x  is 
to  bear  two-thirds  of  the  wt,  make  o  z  equal  to  two-thirds  of  x  z. 


SYPHON ;  continued  from  p  583. 

At  Blue  Ridge  Tunnel,  Virginia,  Col.  C.  Crozet  constructed  a  drainage 
syphon  1792  ft  long  of  cast  iron  faucet  pipes  3  ins  bore,  9  ft  long.  Its  summit  was 

9  ft  above  the  surface  of  the  water  to  be  drained  ;  and  its  discharge  end  was  20  ft 
below  said  surface,  thus  giving  it  a  head  of  20  ft.    At  the  summit  570  ft  from  the 
inlet,  was  an  ordinary  cast  iron  air-vessel  with  a  chamber  3  ft  high  and  15  ins 
inner  diam.    In  the  stem  connecting  it  with  the  syphon  was  a  cut-off  stop- 
cock ;  and  at  its  top  was  an  opening  6  ins  diam,  closed  by  an  air  tight  screw  lid. 
At  each  end  of  the  syphon  was  a  stopcock.    To  start  the  flow  these  end 
cocks  are  closed,  and  the  entire  syphon  and  air-vessel  are  filled  with  water  through 
the  opening  at  top  of  air-vessel.    This  opening  is  then  closed  airtight,  and  the  two 
end  cocks  afterwards  opened ;  the  cut-off'  cock  remaining  open.    The  flow  then 
begins,  and  theoretically  it  should  continue  without. diminution,  except  so 
far  as  the  head  diminishes  by  the  lowering  of  the  surface  level  of  the  pond,    lint 
in  practice  with  very  long  syphons  this  is  not  the  case,  for  air  begins  at  once 
to  disengage  itself  from  the  water,  and  to  travel  up  the  syphon  to  the  summit, 
where  it  enters  the  air-vessel,  and  rising  to  the  top  of  the  chamber  gradually 
drives  out  the  water.    If  this  is  allowed  to  continue  the  air  would  first  fill  the  en- 
tire chamber,  and  then  the  summit  of  the  syphon  itself,  where  it  would  act  as  a 
wad  completely  stopping  the  flow.    The  water-level  in  the  air  chamber 
can  be  detected  by  the  sound  made  by  tapping  against  the  outside  with  a  hammer. 

To  prevent  this  stoppage,  the  cut-off  at  the  foot  of  the  chamber  is 
closed  before  the  water  is  all  driven  out ;  and  the  lid  on  top  being  removed  the 
chamber  is  refilled  with  water,  the  lid  replaced,  and  the  cut-off  again  opened. 
The  flow  in  the  meantime  continues  uninterrupted,  but  still  gradually  diminish- 
ing notwithstanding  the  refilling  of  the  chamber;  and  after  a  number  of  refill- 
ings  it  will  cease  altogether,  and  the  whole  operation  must  then  be  repeated  by 
filling  the  whole  syphon  and  air  chamber  with  water  as  at  the  start. 

At  Col.  Crozet's  syphon  at  first  owing  to  the  porosity  of  the  joint-caulking, 
which  was  nothing  but  oakum  and  pitch,  air  entered  the  pipes  so  rapidly  as  to 
drive  all  the  water  from  the  chamber  and  thus  require  it  to  be  refilled  every  5  or 

10  minutes;  but  still  in  two  hours  the  syphon  would  run  dry.    The  joints  were 
then  thoroughly  recaulked  with  lead,  and  protected  by  a  covering  of  white  and 
red  lead  made  into  a  putty  with  Japan  varnish  and  boiled  linseed  oil.    But  even 
then  the  chamber  had  to  be  refilled  with  water  about  every  two  hours ;  and  after 
six  hours  the  syphon  ran  dry,  and  the  whole  had  to  be  refilled.    In  this  way  it 
continued  to  work. 

In  the  writer's  opinion  an  inside,  and  probably  an  outside  coating  of  the  pipes 
and  air-vessel  with  the  coal  pitch  varnish,  Art  34,  p  581,  would  effect  a  great  im- 
provement. 


662 


THE   CORD   OR    FUNICULAR   MACHINE. 


THE  COED  OE  FUNICULAE  MACHINE, 


Art.  1.    Some  allusion  to  this  subject  has  already  been  made  on    ,_0 

which  see.  Theory  requires  that  the  cord,  rope  or  string,  &c,  shall  be  ab- 
solutely flexible,  inextensible,  frictionless,  without  weight,  and  infinitely  thin ; 
and  that  such  pulleys,  posts,  pins  or  pegs,  loops  or  rings  as  may  be  used  with  the 
cord  shall  also  be  absolutely  frictionless ;  and  at  times  deyoid  of  weight  unless 
said  wt  is  included  in  one  of  the  acting  forces.  These  assumptions  cannot  of 
course  be  realized  in  practice,  which  however  will  agree  with  theory  in  propor- 
tion as  we  approximate  to  them ;  and  this  we  can  frequently  do  so  far  as  to  render 
the  theory  of  great  use.  We  know  that  all  cords  have  wt  and  thickness,  and  can 
be  stretched ;  and  that  so  far  from  being  flexible,  they  may  require  very  con- 
siderable force  to  bend  them  around  pulleys,  posts,  pins,  &c.  They  also  possess 
friction.  Also  that  all  pulleys,  pins,  sliding  rings  or  loops  &c,  have  more  or  less 
friction ;  which  when  there  are  many  of  them,  may  entirely  vitiate  all  calcu- 
lations. A  pulley  has,  in  itself,  no  advantage  over  a  smooth 
cylindrical  pin,  or  post  (as  the  case  may  be)  except  that  its  friction  being 
a  rolling*  one,  is  less  than  the  sliding*  friction  which  takes  place  between  a 
cord  and  a  pin,  <fec.  Therefore  in  what  follows  we  may  usually  (so  far  as  theory  is 
concerned)  substitute  one  for  the  other. 

Rent.  It  will  be  seen  that  the  principle  of  the  cord  serves  for  finding  the 
strains  on  the  ropes  of  the  different  systems  of  pulleys. 

Art.  2.  Assuming*  then  such  a  theoretical  cord,  pulleys,  pins,  <fec,  all  devoid 
of  friction,  the  broad  principle  of  the  cord  is  that  any  force/,  Figs  1, 
imparted  to  either  end  of  a  cord  (whether  the  cord  be  straight  as  fg  or  fc;  or 

bent  out  of  its  course  as/  e  by  any  nurn- 
ker  of  friction!6^  pulleys  or  pins  as  n  r 
^  ^  m  *»  &c»  however  placed)  is  all  trans- 

Q  f**j  mitted  along  the  entire  cord  to  its  other 

~  end,straining  it  uniformly  in  every  part. 

In  fg  or/ c  this  may  be  regarded  as  self- 
evident;  in  /  e  it  is  proved  by  experi- 
ment.  It  is  perhaps  needless  to  add  that 
a  straining  lorce  cannot  be  imparted  to 
one  end  if  there  is  not  an  equal  one  at 
the  other  end  to  react  or  strain  against  it.  See  Strain,  p  444.  Also  note,  p  252. 

If  at  any  one  of  the  pulleys,  pins,  &c,  as  m,  we  make  mo,  ma,  each  equal  to  the 
force  at  either  /  or  e,  and  complete  the  parallelogram  m  o  d  a  of  forces,  then  the 
diagonal  or  resultant  d  m  will  give  both  the  direction  and  amount  of  strain  which 
the  cord  produces  on  that  pin.  This  strain  will  differ  at  the  several  pins  ac- 
cording to  the  angles  formed  by  the  two  components ;  and  may  thus  be  greater, 
or  less,  or  equal  to  the  force  at/ or  e. 

Rem.  With  theoretical  cords,  pulleys,  pins,  &c,  the  two  components  mo,  ma 
at  any  pin  will  always  form  equal  angles  with  the  resultant  m  d;  and 
upon  this  fact  the  principle  of  the  cord  depends. 

Art.  3.  The  principle  of  the  cord  applies  also  when  as  in  Figs  2,  3,  4,  a  load 
or  third  force/  is  imparted  between  the  ends  of  the  cord  as  at  a,  by  means  of 


a  frictionless  pulley,  ring  or  loop  which  can  move  along  the  cord  with  perfect 
ease.  If  such  a  pulley,  ring  or  loop  be  first  placed  upon  the  cord  near  m  or  w,  it 
will  with  its  load  or  other  force  move  down  along  the  steepest  part  of  the  cord 
until  it  comes  to  rest  at  that  point  a  at  which  a  m  and  a  n  form  equal  angles 
b  a  c,b  a  o  with  the  direction  a  b  of  the  force/.  If  both  ends  m  n  of  the  cord  are 
at  the  same  height,  and  the  force/  a  load  and  of  course  acting  vertically, 
then  a  will  be  at  the  middle  of  the  cord;  but  if  the  ends  are  at  different 


THE   CORD 


R   MACHINE. 


663 


heights  as  in  Yig^from  either^f  them  as  n  draw  a  vertical,  asnx;  and  from 
the  other  one^Kwlny  off  tber'whole  length  w  x  of  the  cord  ;  bisect  w  z  at  e,  and 
draw  e  a  horizontal ;  owiHDe  the  required  point. 

If  we  draw  a  b  to jjhtfw  both  the  direction  and  the  amount  of  the  force/,  and 
complete  the  parallelogram  b  c  a  o  of  forces,  then  either  a  c  or  its  equal  a  o  will 
give  the  uniform  strain  which  /  produces  from  end  to  end  (m  to  n)  of  the  cord. 
From  which  it  is  plain  that  a  force  equal  to  a  c  or  a  o  may  be  considered  to  be  im- 
parted at  each  end  m,  n,  of  the  cord,  and  there  to  react  against  a  c  and  a  o,  there- 
by straining  the  cord,  uniformly  throughout. 

Rem.  1.  Precisely  the  same  result  will  follow  if  one  or  both  of  the  ends  w, 
n,  instead  of  being  fixed  of  fastened  at  those  points  as  is  supposed  in  Figs  2  and  3, 
are  continued  as  at  m.  Fig  4,  over  a  frictionless  pulley  or  pin,  and  prolonged  either 
vertically  as  m  /,  in  which  case  it  will  sustain  a  load  or  vertical  force  I  equal 
to  a  o  or  a  c ;  or  in  any  other  direction  as  m  s  or  m  e,  in  which  case  it  will  sustain 
at  s  or  6  a  pull  equal  to  a  o  or  a  c  acting  in  said  directions.  And  the  same  will 
take  place  if  one  or  both  ends  as  n,  Fig  4,  be  extended  over  or  under  any  num- 
ber of  such  pulleys  or  pins,  (no  matter  what  angles  they  form)  say  to  v.  The 
strain  at  v  will  still  be  equal  to  a  c  or  a  o,  and  will  be  uniform  from  v  to  /.  The 
strain  on  any  pulley  or  pin  may  plainly  be  found  as  in  Art  2. 

Rem.  2.  If  we  know  the  angles  b  ao,b  a  c,  and  the  force  or  resultant  a  b, 
we  can  calculate  the  strains  or  components  a  o,  ac;  or  knowing  the 
angles  and  components  we  can  calculate  the  force  a  b,  all  by  the  formulas  in  Art 
45,  p  472,  which  are  based  upon  plane  trigonometry,  which  enables  us  to  find  un- 
known parts  of  a  triangle  when»certain  other  parts  are  given. 

Rem.  3.  If  the  angle  m,  a,  n,  is  12O°,  each  component  ao,  a  c,  will  be 
equal  to  the  resultant  a  b ;  if  it  is  less  than  120°,  each  component  will  be  less  than 
the  resultant ;  and  vice  versa. 

Art.  4.  A  little  reflection  will  enable  the  student  to  see  that  this  Art  is 
merely  a  farther  illustration  of  the  preceding  ones.  In  Fig  5  a  load  *,  or  any 
equal  vertical  pull  will  strain  the 
strings  s  a  and  z  v  each  to  an  amount 
equal  the  load,because  they  both 
act  in  its  own  direction.  If 
the  parts  m  a,  g  a  of  the  cord  a  m  n  g 
a  which  passes  over  the  friction- 
less  pulley  or  pin  z  were  also  vert- 
ical, each  of  them  would  be  strained 
equal  to  one-half  the  load  s ;  but 
they  are  not  vertical,  and  if  we  make 
a  b  to  represent  s,  and  complete  the 
parallelogram  &  o  a  c,  we  shall  find  that 
the  resulting  strains  a  c,  a  o  are  each 
rather  more  than  half  of  s;  and 
they  will  increase  if  the  angles  at  a 
increase;  and  maybe  calculated 
by  the  formulas  in  Art  45,  p  472. 

For  the  foregoing  reason  the  short 
inclined  part  of  the  rope  in  Fig 
53,  p  480,  which  is  attached  to  the 

fixed  pulley,  is  actually  strained  rather  more  than  the  1  ton  with  which  it  is 
figured ;  but  in  such  arrangements  of  pulleys  the  inclination  is  generally  so  slight 
that  the  difference  of  strain  in  practice  is  disregarded. 

In  Fig  5  we  had  the  force  s  or  a  b  given  as  a  resultant,  and  from  it  we  found  its 
two  components  or  strains  a  o,  a  c.  In  Fig  &  we  have  the  two  components  h  m,  h 
g  given,  (each  evidently  equal  to  the  load  s)  and  from  them  we  deduce  the  result- 
ant h  b,  which  represents  the  strain  on  the  pulley,  and  on  the  string  z  w.  It  is 
plain  that  this  string  cannot  be  vertical  as  z  v  is,  but  must  (as  z  v  does)  adjust  it- 
self to  the  direction  of  the  force  or  resultant  which  pulls  it.  If  the  load  s  is  the 
same  in  both  Figs,  we  see  that  the  pull  on  s  w  will  be  about  twice  as  great  as  on 
z  v ;  and  the  one  along  the  cord  smgp  about  twice  as  great  as  that  along  amnga. 
This  difference  will  of  course  vary  with  the  angles. 

Rem.  1.  Let  .*,  Fig  5,  instead  of  the  load  there  represented,  be  a  suspended 
man  weighing  200ft>s,  and  holding  m  a,  g  a  together  at  a  with  both  hands ;  or  let 
m  a  and  g  a  both  hang  vertically  from  the  pulley,  and  let  him  hold  their  lower 
ends  apart  by  one  hand  at  eacli  end.  Now  if  each  part  of  the  rope  will  bear  but 
a  little  more  than  100  fts,  and  z  v  a  little  more  than  200  tbs,  he  will  be  safe  from 
falling.  But  if  he  lets  go  one  end  of  the  rope,  putting  it  into  the  hands  of  an- 


664  THE   CORD   OR   FUNICULAR   MACHINE. 

other  man  standing  by  to  hold  it ;  or  if  he  hooks  one  end  to  a  projecting  spike  in 
a  wall  close  at  hand,  the  rope  will  break,  and  he  will  fall,  because  he  at  once 
doubles  the  strain  along  both  the  ropes  a  m  n  g  a,  and  z  v. 

Rem.  2.  Fig  7  shows  a  device  by  which  boatmen  sometimes  haul  a  boat  out 
of  the  water,  and  up  on  to  the  beach,  at  a  landing,  when  it  is  too  heavy  for  their 
unaided  efforts.  The  rope  e  m  x  n  b  is  to  be  considered  as  horizontal  in  this  case. 
One  end  b  being  fastened  to  the  bow  of  the  boat,  the  rope  is  carried  past  one 
smooth  post  n,  to  another  at  m,  around  which  it  makes  a  whole  turn ;  and  a  man 
stands  at  that  end  e  to  take  in  the  slack  while  the  others,  taking  hold  of  the  rope 
midway  between  m  and  n,  pull  it  into  a  position  m  a  n  in  which,  if  the  angle 
man  exceeds  12O°,  (see  Rem  3,  Art  3)  each  component  an,  am  of  their 
force  will  exceed  said  force  itself;  and  a  strain  equal  to  one  of  these  components 
(except  so  far  as  it  may  be  reduced  by  the  friction  of  the  rope  against  the  post  n) 
will  be  transmitted  uniformly  to  the  boat  at  b,  drawing  it  a  short  distance  up  the 
beach.  The  rope  is  then  straightened  again  from  m  to  n  by  taking  in  the  slack 
at  e,  and  the  operation  is  repeated  as  often  as  necessary. 

To  find  the  strain  a  m  or  a  n,  divide  the  force  by  twice  the  nat  cosine 

of  the  angle  x  a  n,  or  x  a  m.    Or  to  find  how  many  times  the  strain  exceeds  the 

force  divide  the  distance  a  n  by  twice  the  distance 

<|H  y JV         a  x-    Here  a  x  represents  only  half  the  force,  or  half 

sr- — '™ j    ^--'"'^1         the  diagonal  or  resultant  of  a  m  and  a  n. 

/          "~j|  The  force  of  a  man  pulling  by  jerks  at  a;  or  a 

£  W  will  average  between  30  and  80  ibs. 

._     _  Art.  5.    We  will  now  dispense  with  the  fric- 

rlU  I  •       \A       tionless  pulley  required  by  the  principle  of  the 

*  Uf|       cord,  and  which  would  of  itself  roll  along  the  cord 

until  it  came  to  rest  at  a  point  which  would  cause 

equal  angles  and  consequently  uniform  strain  throughout.  We  will 
substitute  for  it  a  tight  knot  which  cannot  slide,  at  a  point  a,  Fig  8, 
which  causes  unequal  angles  bam,ba  w,  be- 
tween the  direction  a  &  of  the  force  /,  and  the  two 
parts  a  m,  a  n,  of  the  cord.  Here,  completing  the 
parallelogram  6  ca  o,  we  find  that  the  part  a  n  of  the 
cord  is  strained  to  an  amount  denoted  by  ac;  and 
that  same  strain  would  affect  any  extension  of  the 
cord  on  that  side,  like  that  to  v,  Fig  4.  The 
part  a  m  would  be  strained  equal  to  a  o ;  and  that 

T}«     Q       v    |\4  strain  would  continue  to  the  end  m,  or  to  the  end 

i-in  O,  f  pi  of  any  extension  of  that  part.    Hence  the 

strain  is  not  uniform  from  end  to  end  of 
the  cord,  as  it  would  have  been  if  the  knot  had  been 
frictionless;  in  which  case  it  would  have  slid 
along  the  cord  until  the  angles  would  be  equal. 

Rem.  1.  But  even  in  this  case  of  a  tight  knot  at  a,  there  is  always  one 
direction,  as  £as,  in  which  the  force  can  be  imparted  so  as  to  cause  equal  an- 
gles sam,san,  with  the  direction  a  s  of  the  force  t,  and  then  the  strain  will  be 
uniform  from  end  to  end,  as  if  a  frictionless  pulley  had  been  used. 

Rem.  2.  With  a  tight  knot  at  a  it  is  plain  that  a  force  may  be  imparted 
there  from  any  direction  as/a,  ta  &c.  If  the  direction  coincides  with 
either  part  a  m,  or  a  n  of  the  rope,  that  part  will  bear  all  the  strain  ; 
the  other  part  remaining  entirely  free. 

Rem.  3.  From  Rule  1,  p  27,  for  drawing  an  Ellipse,  it  will  be  seen  that  at 
whatever  point  as  a,  Fig  8,  we  apply  force  to  an  in  extensible  cord  nam  with  fixed 
ends,  that  point  will  be  in  the  circumference  of  an  ellipse,  the  foci  of  which  are 
at  the  ends  mt  n. 


CENTERS   FOR/ARCHES.  665 

CENTEM-FOB  AKCHES, 

Art.  1.  A  center'IsXiemporary  wooden  structure  (built  lying  flat,  on  a  full 
size  drawing,  on  a  fixejr  platform,  under  cover  or  not)  for  supporting  an  arch 
while  it  is  being  builtX^It  consists  of  a  number  of  trusses  or  frames,/,  f,  Fig.  1. 
placed  from  1  to  6Jf\ apart  from  cen  to  cen,  and  covered  with  a  flooring  /,  I,  of 
rough  boards  or  planks,  usually  laid  close,  and  called  the  sheeting  or  lag- 
ging,  immediately  upon  which  the  archstones  are  laid.  In  Fig  3,  the  lag- 
ging is  not  laid  close.  There  is  no  great  economy  in  placing  the  frames  very 
far  apart,  on  account  of  the  greater  required  amount  of  lagging,  the  thickness 
of  which  increases  rapidly.  For  the  thickness  of  lagging  see  Rein.  9,  Art.  8.  The 
frames  are  of  many  designs.  Thus  Figs?  and  9,  pp  250,  251,  are  often  used  for 
small  spans  (say  15  to  25  ft),  their  upper  timbers  supporting  throughout  their 
length  planks  on  edge,  with  their  upper  edges  trimmed  to  conform  to  the  curve 
of  the  arch.  Fig  14,  p  260,  cov- 
ered in  the  same  way,  is  some- 
times used  for  still  longer  spans, 
say  25  to  40  feet ;  also  Fig  28,  p 
283;  Fig  31,  p  284;  and  Fig  35,  p 
286,  for  still  longer  ones.  InFig  1, 
/,  /,  are  these  frames  as  seen  in 
place  transversely  of  the  arch 
a  a.  They  rest  by  the  ends  of 
their  chords,  c,  upon  wooden 
striking  wedges  w,  Fig  1, 
supported  by  standards  com- 
posed of  posts  p,  whose  tops  are 
connected  by  cap-pieces  o; 
and  whose  feet  rest  on  string- 
ers s;  the  whole  being  braced 
diagonally  as  shown. 

If  the  ground  is  very  firm,  and 
the  arch  light,  the  standards  may 
rest  on  it,  with  the  interposition 
of  adjusting-blocks,  n,  be- 
low the  stringer,  to  accommodate 
irregularities  of  the  surface  of  the  ground,  as  in  the  Fig.  These  blocks  should 
be  somewhat  double- wedge-shaped,  so  that  by  driving  them  the  standard  may 
be  raised  at  any  point  in  case  it  should  settle  a  little  into  the  ground.  But  for 
heavy  arches  the  standards  must  rest  on  a  much  firmer  foundation,  such  as  short 
blocks  of  brickwork  sunk  a  few  feet  into  the  ground,  or  some  other  device 
adapted  to  the  case.  Frequently  projecting  offsets  or  footings,  or  at  times  re- 
cesses, are  provided  in  the  masonry  of  the  abutments  and  piers  for  this  express 
purpose ;  and  with  a  view  to  this  it  is  well  to  design  the  center  at  the  same  time 
as  the  arch.  Knowing  the  wt  of  the  arch  the 
proper  dimensions  of  the  posts  may  readily  be 
found  by  the  table  p  239.  Up  to  spans  of  50  or  60 
ft  a  single  row  of  rjpsts  (one  under  each  end  of 
each  frame)  will  suffice ;  but  for  much  larger  ones 
two  or  three  rows,  2  or  more  feet  apart  may  be- 
come expedient,  as  in  the  lower  Fig  2. 

The  striking  or  lowering-wedges 

before  alluded  to  are  for  striking  or  lowering  the 
center  after  the  completion  of  the  arch.  They 
consist  of  pairs  of  wedge-shaped  blocks,  w  w,  at  A, 
Figs  2,  of  hard  wood,  from  1  to  2  ft  long,  about  half 
as  wide,  and  a  quarter  or  more  as  thick,  (sufficient 
to  lower  the  center  from  say  2  to  6  or  more  inches, 
according  to  span  and  other  circumstances,)  rest- 
ing on  the  cap  o,  of  the  standard,  while  the  chord 
c  of  the  frame  rests  on  them.  When  the  end  of  a 
frame  is  supported  by  two  or  more  posts  p,  as  at  B, 
Fig  2,  instead  of  upon  one,  the  striking-wedges  are 
sometimes  made  as  there  shown  ;  and  where  B  v 
is  one  long  wedge  at  right  angles  to  the  abutment, 
and  acting  as  four  wedges  which  may  all  be  low- 
ered together  by  blows  against  the  end  B. 
Up  to  spans  of  60  or  80  ft,  all  the  frames  may  rest  on  but  two  wedges  like  B  t>, 


Figsl. 


f  o 


666  CENTERS   FOR   ARCHES. 

each  so  long  as  to  reach  transversely  across  the  entire  arch.  Then  all  the 
frames  can  be  lowered  at  one  operation,  as  described  near  end  of  Art  9. 

If  we  had  to  consider  only  the  friction  of  dry  wood  against  dry  wood,  the  taper 
of  these  wedges  might  be  as  steep  as  1  vert  to  3  hor,  without  any  danger  of  their 
sliding  upon  each  other  of  their  own  accord ;  and  they  would  then  require  very 
moderate  blows  to  start  them,  or  even  to  entirely  separate  them,  when  the  center 
had  finally  to  be  lowered.  But  it  is  of  the  utmost  importance,  especially  in  large 
arches,  that  the  centers  should  be  lowered  very  slowly,  otherwise 
the  momentum  acquired  by  so  heavy  a  body  as  the  arcli  in  descending  suddenly 
even  but  2  or  3  ins,  might  possibly  affect  its  shape,  or  even  its  safety. 

Therefore  the  wedges  should  not  have  a  taper  steeper  than  about  1  in  6  or  8  for 
arches  of  less  than  about  50  ft  span ;  or  than  1  in  8  or  10  for  larger  spans.  Vertical 
lines  at  equal  dists  apart  should  be  drawn  on  the  long  sides  of  the  wedges  as  a 
guide  for  lowering  them  all  to  the  same  extent  at  a  time ;  and  this  should  not  ex- 
ceed in  all  about  half  an  inch  a  day  in  intervals  of  about  an  eighth  of  an  inch,  for 
50  ft  spans ;  or  about  .1  to  .25  of  an  inch  per  day  in  all,  for  spans  over  100  ft. 
Slowness  is  especially  to  be  recommended  in  brick  arches,  not  only  because 
their  greater  number  of  joints  exposes  them  to  greater  derangement  of  shape, 
but  because  even  good  brick  has  much  less  than  the  average  crushing  strength 
of  good  granite,  limestone,  or  sandstone,  and  therefore  is  far  more  liable  than 
they  to  crack,  or  even  to  crush  (as  the  writer  has  seen)  when  the  strains  are 
thrown  almost  entirely  upon  their  edges,  as  described,  in  Art  3.  For  more  on  brick 
arches,  see  Art  9. 

At  Gloucester  Bridge,  England,  of  first  class  cut  stone,  span  150  ft,  rise 
35  ft,  the  centers  were  entirely  struck  within  the  very  short  space  of  3  hours ;  and 
the  crown  of  the  arch  descended  10  ins!  At  Crrosveiior  Bridge,  England, 
of  first  class  cut  stone,  span  200  ft,  rise  42  ft,  such  care  was  taken  in  easing  the 
centers  that  the  crown  of  the  arch  settled  but  2.5  ins.  This  case  however  was 


sagging  or  settlement,  consisted 
essentially  of  vertical  and  in- 
clined posts  or  struts,  see  Fig  3, 
footing  on  four  temporary  piers 
of  masonry,  7  or  8  feet  thick,  built 
in  the  river,  parallel  to  the  abut- 
ments, and  as  long  as  they.  These 
piers  supported  six  frames  (or 
rather  six  series)  about  7  ft  apart 
cen  to  cen,  of  such  struts,  footing 
on  cast  iron  shoes.  Fig  3  shows 
half  of  one  series.  Each  frame 
or  series  consisted  of  four  fan-like 
sets  of  posts,  all  in  the  same  ver- 
tical plane.  The  long  horizontal  pieces  seen  extending  from  side  to  side  of  the 
arch  were  bolted  to  the  struts  to  increase  their  stiffness ;  and  other  pieces  for  the 
same  purpose  united  the  six  series  transversely.  Here  each  strut  sustains  its  own 
share  of  the  weight  of  the  archstones,  and  transfers  it  directly  to  the  unyielding 
foundation  of  the  pier ;  whereas  in  the  usual  trussed  centers,  the  entire  load  rests 
upon  the  frames,  and  is  finally  transferred  to  the  comparatively  unstable  support 
of  the  posts  at  their  ends. 

The  tops p  of  the  posts  of  a  series  varied  about  from  5  to  8  ft  apart  cen  to  cen ; 
and  were  connected  by  a  continuous  curved  rib,  rr,  of  two  thicknesses  of  4  inch 
plank,  bent  to  conform  approximately  to  the  curve  of  the  arch.  On  this  rib  were 
placed  pairs  of  striking-wedges  w  like  Fig  2,  about  16  ins  long,  10  to  12  ins  wide,  and 
tapering  1.5  ins,  so  near  together  (varying  about  from  2.5  to  3.5  ft  cen  to  cen)  that 
there  was  a  pair  under  each  joint  of  the  archstones,  a  a.  On  these  wedges,  and  ex- 
tending over  all  six  of  the  frames,  were  the  lagging  pieces  /,  4.5  ins  thick. 

This  peculiar  arrangement  of  the  striking-wedges  and  Ia0 
ging  has,  in  large  spans,  great  advantages  over  the  usual  one  of  placing  them  only 
at  the  ends  of  the  frames.  In  the  last  the  entire  center  and  the  entire  arch  are 
lowered  together,  without  giving  an  opportunity  to  rectify  any  slight  derange- 
ments of  shape  or  inequality  of  bearing  that  may  have  occurred  in  the  arch  during 
its  construction,  This  center,  designed  by  Mr.  Trubshaw,  admits  of  lowering 
either  the  whole  equally,  or  any  one  part  a  little  more  or  less  than  the  others. 
He  had  much  experience  in  large  arches,  and  stated  that  during  the  striking  he 
found  that  he  had  an  arch  under  better  control,  or  could  humor  it  better,  by  keep- 
ing the  haunches  a  little  down,  and  the  crown  a  little  up,  until  near  the  end  of 
the  operation. 


CENTERS   FPK7ARCHES.  667 

atom.  1.  Instead  of  piers  ofinasonry  for  supporting  the  feet  of  the 
posts,  wooden  cribs  or  pihj»*iiiay  often  be  used  if  the  arch  is  over  water. 

The  priii  cip  I  ejw suitor  ting;  even  trussed  frames  by  struts 
at  points  of  the  chi>ra  as  far  £rom  the  abutments  as  circumstances  will  admit  of 
(in  addition  to  thQse  at  the/very  ends)  should  always  be  applied  when  possible, 
in  order  to  reduce  their/sagging  to  a  minimum.  Stops  or  offsets  in  the 
masonry  of  the  abujbments  and  piers  may  be  provided  for  receiving  the  feet 
of  such  struts,  when  tHey  are  inclined. 

Rem.  2.  Screws  may  be  used  instead  of  wedges  for  lowering  centers.  At 
the  Pont  d' Alma^Paris,  ellipse  of  141.4  ft  span,  and  28.2  ft  rise,  the  frames  were  sup- 
ported by  woodfen  pistons  or  plungers,  the  feet  of  which  rested  on  sand  con- 
fined in  pi  sc  to- iron  cylinders  1  ft  in  diam  and  height,  and  having  near 
the  bottom  of  each  a  plug  which  could  be  withdrawn  and  replaced  at  pleasure, 
thus  regulating  the  outflow  of  the  sand  and  the  descent  of  the  center.  This  de- 
vice succeeded  perfectly,  and  is  well  worthy  of  adoption  under  arches  exceeding 
about  60  ft  span.  When  much  larger  than  this  the  driving  of  the  wedges  on 
striking  requires  heavy  blows,  and  becomes  a  somewhat  awkward  operation,  re- 
quiring at  times  a  battering-ram,  even  when  the  wedges  are  lubricated.  In  rail- 
road cuttings  crossed  by  bridges,  the  earth  under  the  arch  has  been 
made  to  serve  as  a  center,  by  dressing  its  surface  to  the  proper  curve,  and  then 
embedding  in  it  curved  timbers  a  few  feet  apart,  and  extending  from  abut  to  abut, 
for  supporting  the  close  plank  lagging. 

Rem.  3.  All  centers  must  yield  or  settle  more  or  less  under  the  wt 
of  the  arch,  especially  when  supported  only  near  their  ends;  and  since  the  arch 
itself  also  settles  somewhat  not  only  when  the  centers  are  struck,  but  for  some 
time  after,  it  is  advisable  to  make  them  at  first  a  little  higher  than  the  finished 
arch  is  intended  to  be.  This  extra  height,  when  the  supports  are  at  the  ends, 
may  be  from  2  to  4  ins  per  100  ft  of  span  for  cut  stone  arches  (according  to  time 
of  striking,  character  of  masonry,  workmanship,  etc.),  and  about  twice  as  much  in 
brick  ones. 

Rem.  4.  The  proper  time  for  striking  centers  is  a  disputed 
point  among  engineers,  some  contending  that  it  should  be  done  as  soon  as  the 
arch  is  finished  and  sufficiently  backed  up;  and  others  that  the  mortar  should 
first  be  given  time  to  harden.  It  is  the  writer's  opinion  that  inasmuch  as  in 
cut-stone  arches  the  mortar  joints  should  be  very  thin  ;  and  in  which,  in  fact,  the 
mortar  is  at  best  of  very  little  service,  it  is  of  no  importance  when  they  are  struck  ; 
but  that  in  brick  or  rubble,  the  numerous  joints  of  both  of  which  require  much 
mortar,  (which  for  hardness  should  consist  largely  of  cement,)  3  or  4  mouths,  or 
longer,  if  possible,  should  be  allowed  it  to  harden  sufficiently  to  prevent  undue 
compression  and  consequent  settlement  when  the  centers  are  struck.  The  con- 
tinuance of  the  centers  need  not  interfere  with  traffic  over  the  bridge. 

Art.  2.  The  pressure  of  archstones  against  a  center  is  very  trifling  until  after 
the  arch  is  built  up  so  far  on  each  side  that  the  joints  form  angles  of  25°  or  30° 
with  the  horizontal.  Theoretical  discussions  on  this  pressure  make  no  allowance 
for  accidental  jarrings  in  laying  the  archstones,  or  by  the  accumulation  of  material 
ready  for  use,  laborers  working  on  it,  &c.  Without  going  into  any  detail,  we  merely 
advise  on  the  score  of  safety  not  to  assume  it  at  less  than  about  the  following  pro- 
portions or  ratios  to  the  weight  of  the  entire  arch,  namely,  in  a  semicircular  arch 
.47;  rise  .35  span,  .61;  rise  .25  span,  .79;  rise  .2  span,  .86;  rise  .167  span,  or  less,  1, 
or  equal  to  the  wt  of  the  arch.  This  gives  the  pressure  of  a  semicircular  arch 
upon  its 'centers  rather  less  than  half  its  wt.  Ihe  wt  of  the  centers 
themselves  when  supported  only  near  the  ends  must  be  considered  as  part 
of  the  load  borne  by  them. 

Art.  3.  We  have  seen  that  as  an  arch  a  a  a  is  being  gradually  built  upward  on 
both  sides,  after  passing  the  points  e,  e,  Fig  4,  where  its  joints  form  angles  a  s  e,  of 
about  30°  with  the  horizontal  a  a,  the  arch  begins  to  press  more  and  more 
upon  the  centers  ;  thereby  tending  to  flatten  them  at  the  haunches,  as  shown  at  h 
in  the  dotted  line ;  and  consequently  to  raise  them  at  the  crown,  as  shown  at  c. 
But  as  the  building  goes  on  still  higher,  the  added  stones  press  much  more  heavily 
upon  the  centers  than  those  below  had  done,  and  thereby  tend  to  a  final  derange- 
ment of  the  centers  just  the  reverse  of  that  caused  by  the  lower  ones ;  namely  to 
depress  them  at  the  crown  a,  as  at  o;  and  consequently  to  raise  the  haunches  as 
at  n;  and  this  the  more  because  the  upper  stones  actually  tend  to  lift  or  ease  the 
lower  ones  from  the  lagging.  In  some  cases  where  this  tendency  has  been  in- 
creased by  forcing*  the  keystones  into  place  by  too  hard  driving,  the  lagging 
under  the  haunches  could  be  drawn  out  without  any  trouble  before  the  centers 
were  eased  at  all.  On  striking  the  centers  this  tendency  to  sink  at  crown  and 


668 


CENTERS    FOR   ARCHES. 


rise  at  haunches  is  very  apt  to  exhibit  itself  more  or  less  dangerously  in  the  arch- 
stones  themselves,  as  in  Fig  5,  causing  those  near  the  crown  to  press  very  hard 
together  at  the  extrados,  and  to  separate  from  each  other  at  the  intrados ;"  while 
near  the  haunches  the  reverse  takes  place.  Hence  the  angles  of  the  stones  are 
frequently  split  and  spawled  off  near  c  and  h  by  this  unequal  pressure.  These 


Rg  4. 

derangements  are  of  course  much  more  likely  to  be  serious  in  high  arches  than 
in  flat  ones,  especially  if  their  spandrels  are  not  sufficiently  built  up  befoee 
lowering  the  centers. 

In  the  Grosvenor  bridge,  before  alluded  to,  of  200  ft  span,  this  dangerous  excess 
of  pressure  near  c  and  h  was  prevented  by  covering  the  skewback  joint  of  the 
springing  course  at  each  abutment  with  a  wedge  of  lead  1.5  ins  thick  at  the  iu- 
trados  of  the  arch,  and  running  out  to  nothing  at  the  extrados.  Beside  this  a 
strip  9  ins  wide  of  sheet  lead  was  laid  along  the  intrados  edy;e  of  every  joint  until 
reaching  that  point  at  which  it  was  judged  that  the  line  of  pressure  would  pass 
from  the  intrados  to  the  extrados ;  after  which  similar  strips  were  laid  along  the 
extrados  edges  of  the  joints,  up  to  the  crown.  Hence  when  the  centers  were 
struck,  this  excess  of  pressure  merely  compressed  the  lead,  and  was  thus  enabled 
to  distribute  itself  more  evenly  over  the  entire  depth  of  the  joints.  See  Trans 
Inst  Civ  Eng  London,  vol  i. 

At  the  bridge  at  Wen  illy,  France  (of  5  elliptic  arches  of  120  ft  span, 
and  30  ft  rise),  the  centers  were  so  radically  defective  in  design  that  the  arches 
sank  13.25  ins  at  crown  during  the  time  of  building;  and  10.5  ins  more  during 
and  immediately  after  the  striking ;  or  say  2  ft  in  all.  Their  construction  made 
the  striking  very  tedious  and  hazardous ;  greatly  endangering  the  lives  of  the 
workmen  and  the  existence  of  the  arches.  Some  of  the  joints  at  the  extrados 
at  the  haunches  opened  an  inch  each ;  and  those  at  the  intrados  of  the  crown  .25 
of  an  inch.  By  the  exercise  of  great  care  and  humoring  in  lowering  the  centers, 
these  openings  were  much  reduced. 

Rem.  1.  Chamfering  the  edges  of  the  archstones  diminishes 
the  danger  of  their  spawling  off  from  unequal  pressure ;  as  does  also  the  scrap- 
ing out  of  the  mortar  of  the  Joints  for  an  inch  or  two  in  depth  be- 
fore striking  the  centers. 

Rem.  2.  It  is  evident  that  in  order  to  prevent,  or  at  least  to  diminish  the 
alternate  derangements  of  the  center,  those  of  its  web  members  which  at  first 
acted  as  struts  near  the  haunches,  Fig.  4,  to  prevent  them  from  sinking  as  at 
A,  must  afterwards  act  as  ties  to  prevent  them  from  rising  as  at  n;  while  those 

which  at  first  acted  as  ties  near 
the  crown  a,  to  prevent  it  from 
rising  as  at  c,  must  afterwards 
act  as  struts  to  prevent  it  from 
sinking  as  at  o.  In  other  words, 
the  principle  of  counter- 
bracing  must  be  attended 
to  as  well  in  a  frame  or  truss 
for  a  center,  as  in  one  for  a 
bridge.  If  the  web  members 
are  on  the  Warren  or  simple 
triangle  system,  as  in  Fig  23, 
p  271,  this  may  be  effected  by 
making  each  member  a  tie- 
strut;  or  the  Pratt,  or  the 
Howe  system,  Fig  35,  p  286, 
may  be  used. 
Art.  4.  From  the  foregoing  it  is  plain  that  a  simple  unbraced  wooden 


R   ARCHES. 


669 


arcli.  or  curvedrife'Jg<<ra  account  of  its  great  flexibility,  about  as  unfit  a  form 
as  could  be  cJ^BSSrTfofa  center,  except  for  very  small  spans,  where  a  great  propor- 
tional deptnof  lib  can  be  readily  secured.  Still  the  writer  has  seen  it  used  for  a 
cut-steitfe  semicircular  arch  of  35  ft  span,  with  archstones  2  ft  deep.  Fig  6  shows 
one  rib  r  r,  and  the  arch,  a  a,  drawn  to  a  scale.  Each  rib  consisted  of  two  thicknesses 
of  2  inch  plank  in  lengths  of  about  6.5  ft,  treenailed  together  so  as  to  break  joint, 
as  at  B.  Each  piece  of  plank  was  12  ins  deep  at  middle,  and  8  ins  at  each  end ; 
the  top  edge  being  cut  to  suit  the  curve  of  the  arch.  The  treenails  were  1.25  ins 
in  diam ;  and  12  of  them  showed  to  each  length.  These  ribs  were  placed  17  ins 
apart  from  cen  to  cen,  and  steadied  together  by  a  bridging  piece  of  inch  board,  13 
ins  long,  at  each  joint  of  the  planks,  or  about  3.25  ft  apart.  Headway  for  traffic 
being  necessary  under  the  arch,  there  were  no  chords  to  unite  the  opposite  feet 
of  the  ribs.  The  ribs  were  covered  with  close  board  lagging,  which  also  assisted 
in  steadying  them  together  transversely.  As  the  arch  approached  about  two- 
thirds  of  its  height  on  each  side,  the  ribs  began  to  sink  at  the  haunches,  as  at  A, 
Fig  4 ;  and  to  rise  at  the  crown,  as  at  c.  This  was  rectified  by  loading  the  crown 
with  stone  to  be  used  in  completing  the  arch ;  which  was  then  finished  without 
further  trouble. 

A  still  more  striking1  example  of  the  use  of  a  simple  unbraced 
wooden  rib,  was  in  the  old  National  Turnpike  bridge  over  Wills  Creek,  Virginia. 
This  bridge,  of  which  one  arch  with 
its  center  is  shown  in  Fig  7  drawn 
to  a  scale,  consisted  of  two  elliptic 
cut  stone  arches  26.5  ft  wide  across 
roadway,  and  of  60  ft  span,  and  15 

ft  rise.    The  archstones  were  3  ft       . . . 

deep  at  crown,  and  4  ft  deep  at      Ft/  60 1  F- 

skewbacks.     Each  frame  of     .-r-Hfi  1^*^*7 

the  center  was  a  simple  rib  6    -r-H  | ^^      1  MJ  '  * 

ins  thick,  composed  of  three  thick- 
nesses of  2  inch  oak  plank  in  different  lengths  (about  7  to  15  ft)  to  suit  the  curve, 
and  at  the  same  time  to  preserve  a  width  of  about  16  ins  at  the  middle  of  each 
length,  and  12  ins  at  each  of  its  ends.  The  thicknesses  were  well  treenailed  to- 
gether, breaking  joint  and  showing  from  10  to  16  treenails  to  a  length. 

Here,  as  in  Fig  6,  there  were  no  chords,  owing  to  the  violence  of  the  floods  in 
the  creek.  These  ribs  were  placed  18  ins  from  cen  to  cen,  and  steadied  against 
one  another  by  a  board  bridging-piece  1  ft  long,  at  every  5  ft.  These  were  of 
course  assisted  by  the  lagging. 

When  the  archstones  had  approached  to  within  about  12  ft  of  each  other  near 
the  middle  of  the  span,  the  sinking  at  the  crown,  and  the  rising  at  the  haunches 
had  become  so  alarming  that  pieces  or  12  X  12  oak,  00,  were  hastily  inserted  at 
intervals,  and  well  wedged  against  the  archstones  at  their  ends.  The  arch  was 
then  finished  in  sections  between  these  timbers,  which  were  removed  one  by  one 
as  this  was  done. 

lie  in.  1.  Such  instances  of  partial  failure  are  very  instructive. 
It  is  indeed  by  such,  rather  than  by  theoretical  deductions,  that  the  proper  dimen- 
sions are  arrived  at  in  a  vast  number  of  cases  pertaining  to  engineering,  ma- 
chinery, &c.*  Thus  we  might  with  entire  confidence  of  no  serious  mishap,  apply 
ribs  of  the  foregoing  dimensions  to  spans  only  half  as  great. 

Hem.  2.  Assuming  the  rib-planks  to  be  12  ins  wide,  it  would,  as  a  matter  of 
detail,  be  better  to  make  them  about  10  ins  wide  at  the  ends  instead  of  the  8  ins 
in  Fig  6  making  top  curve  2  ins.  To  secure  this,  their  lengths,  depending  on  the 
radius  of  the  rib,  must  not  exceed  those  in  the  following  table : 


Bad 
of  Arch. 

Greatest  Length. 

Bad 
of  Arch. 

Greatest  Length. 

Feet. 

Feet  and  Ins, 

Feet. 

Feet  and  Ins. 

5 

2      "      5 

30 

6      "      4 

10 

3       "      4 

35 

7       "      0 

15 

4       "       2 

40 

7       "      6 

20 

5       "      0 

45 

7       "     10 

25 

5       "       9 

50 

8       "      2 

*  The  young  engineer  should  make  and  preserve  full  notes  in  detail  of  all  such  as  may  fall  withiu 
his  notice  ;  and  if  the  professional  journals  would  do  the  same  thing  in  regard  to  failures  which  are 
constantly  occurring,  they  would  greatly  increase  the  value  of  their  papers. 

43 


670 


CENTERS    FOR   ARCHES. 


(5 


If  cut  \}A  times  as  long  as  this  table,  they  will  be  very  approximately  8  ins 
wide  at  ends  ;  or  each  will  on  top  curve  4  ins. 

Art.  5.    In  cases  where  all  possible  headway  is  essential 

during  the  building  of  the  arch,  as  in  the  two  foregoing  ones,  the  writer  would 

suggest  the  expedient  rudely  illustrated  by 
Fig  8;  namely  to  place  the  centers 
above  the  arch,  instead  of  below 
it ;  and  after  the  arch  is  completed  in  sec- 
tions, a  a,  instead  of  lowering*  the  cen- 
ters, to  take  them  apart.  The  cen- 
ters might  resemble  in  principle  Fig  35U, 
p  287. 

Fig  8  is  a  transverse  section  through  part 
of  the  center,  and  of  the  arch  a  a.  Here 
re,  re,  re,  are  frames  of  the  center  say  5  or  6 
ft  apart ;  and  of  any  depth  and  construction 

whatever  that  may  be  necessary  to  insure  absolute  safety ;  and  1 1  is  the  lagging. 
Having  built  the  arch  from  abutment  to  abutment  in  a  series  of  sections  a,  a,  a,  ne- 
cessarily separated  say  a  foot  or  more  by  the  deep  frames,  we  may  take  the  centers 
apart,  and  then  fill  in  the  narrow  intermediate  sections  upon  a  lagging  suspended 
by  iron  rods  from  the  already  completed  sections.  Good  concrete  might  be  used 
for  these  narrow  sections.  In  some  cases  it  might  be  well  to  use  deep  plate- 
iron  ribs  of  I  section,  resting  the  lagging  on  the  lower  flange.  Part  of  the 
web  might  be  left  remaining  embedded  in  the  masonry ;  and  the  upper  part  and 
both  flanges  removed  after  the  arch  is  finished. 

Art.  6.  Centers  with  hor  chords  c  c  Fig  9  are  objectionable  (notwith- 
standing their  strength)  in  large  spans  of  great  rise,  as  on  right  side  of  the  Fig,  on 

account  of  the  excessive  length 
required  for  the  web  members; 
and  hence  it  will  in  such  cases 
usually  be  found  expedient  to 
adopt  something  analogous  to 
what  is  shown  on  the  left  hand 
of  the  Fig.  Here  a  truss/,  shorter 
and  shallower  than  that  on  the 
right  hand,  is  substituted  for  the 
latter.  At  its  ends  provision  must 
be  made  for  supporting  not  only 
itself  but  the  archstones  below 
it.  As  the  pressure  of  these  low- 
er archstones  is  comparatively 
small,  this  may  usually  be  effected 
by  resting  the  end  of  the  frame 
/  upon  another  and  shallower  frame  o  a.  This  may  in  large  spans  be  aided  by 
either  inclined  or  vertical  struts,  either  single  or  braced  together ;  or  as  the  trestles 
on  p  307.  Sometimes  one  shallow  truss  like  /  is  sustained  upon  another  truss 
throughout  its  entire  length.  The  striking-wedges  for  these  various  supports  may 
be  placed  at  either  their  tops  or  their  feet,  as  may  be  most  convenient. 

Art.  7.  For  flat  arches  of  1O  feet  clear  span,  a  mere  board  o  o 
Fig  10,  12  ins  deep,  by  1.5  ins  thick,  with  another  piece  c  of  the  same  thickness 
on  top  of  it,  trimmed  to  the  curve,  and  con- 
fined to  o  o  by  nailing  on  two  cleats  of  nar- 
row board,  will  answer  every  purpose,  with 
intervals  of  18  ins  from  cen  to  cen.  If  the 
upper  piece  also  is  as  much  as  12  ins  deep  at 
its  center,  the  clear  span  may  be  extended 
to  15  ft. 

For  spans  of  1O  to  15  ft,  and  of  any 
rise,  two  thicknesses  of  plank  from  1  to  2  ins 
thick  according  to  span ;  8  to  12  ins  wide  at 
middle  of  each  piece,  in  lengths  as  per  table,  Bern  2,  Art  4,  well  nailed  or 
spiked  together,  according  to  span,  breaking  joint  as  in  Fig  6,  will  answer  for 
distances  of  2  to  3  ft  apart  cen  to  cen.  For  greater  dists  apart  increase  the  thick- 
ness of  the  planks  proportionally. 

If  the  centers  have  to  be  moved  from  place  to  place,  to  serve 
for  other  arches,  then,  to  preserve  them  from  injury  in  handling,  their  feet  should 
be  united  by  nailing  on  one  or  both  sides  of  each  frame  a  chord  piece  of  about  1 


STEES  FOR  ARCHES. 


671 


inch  board ;  and  also  a/vertical  piece  or  pieces  of  the  same  size  from  the  center 
of  the  chord  to  the  tori  of  the  frame. 

Even  when  thejy  are  not  to  be  moved,  the  chord  pieces  are  useful 
even  in  so  small  spans/inasmuch  as  they  render  the  striking  easier,  by  not  allow- 
ing the  feet  of  the  ribs]  to  give  trouble  by  spreading  outward  and  pressing  against 
the  abutments.  / 

For  spans  of  lj»  to  SO  ft,  and  for  any  rise  not  less  than  one  sixth  of  the 
span,  the  following  dimensions,  varying  with  the  span,  may  be  used  for  distances 
apart  of  3  ft  from  cfen  to  cen. 
See  Fig  11.  For  the  bow  &, 
two  thicknesses  of  1  to  2  inch 
plank  from  9  to  12  ins  wido 
at  the  middle;  and  from  7  to 
10  ins  at  each  end,  well  spiked 
together  breaking  joint  as  at  B, 
Fig  6.  For  the  chord  c,  two 
thicknesses  of  plank  of  same 
size  as  the  bow  at  its  middle; 
placed  on  outsides  of  bow,  and 
well  spiked  to  its  ends.  A 
vertical  v,  in  one  piece  as 
wide  as  a  bow  plank,  and  twice 

as  thick.  Its  top  is  placed  under  the  bow,  and  is  confined  to  it  by  two  pieces,  o,  o, 
of  bow  plank  twice  as  long  as  the  bow  plank  is  deep,  and  spiked  to  both  v  and  the 
bow.  The  foot  of  v  passes  between  the  two  thicknesses  of  the  chord  c,  and  is 
spiked  to  them.  Two  oblique  tie-struts,  s,  each  of  two  pieces  of  bow 
plank,  outside  of  the  bow  and  vertical  v;  footing  against  each  other;  and  spiked 
to  bow  and  v.  These  with  v  divide  the  bow  into  4  parts. 

Rein.  1.  The  above  dimensions  are  suitable  to  a  rise  of  one  sixth.  If  the 
rise  is  one  fourth,  the  thickness  only  of  the  planks  may  be  reduced  one  third 
part ;  and  for  a  rise  of  one  third  or  more,  we  may  reduce  to  one  half. 

Rom.  2.  If  in  the  larger  of  these  spans  the  struts  s  should  show  any  incli- 
nation to  bend  sideways,  nail  on  some  pieces  t  from  frame  to  frame.  Also  in  the 
larger  ones  with  rises  exceeding  one  third,  insert  four  double  struts  s,  instead 
of  two ;  thus  dividing  the  bow  into  6  parts,  as  at  left  side  of  Fig.  11.  For  spans  of 
25  to  35  ft,  add  also  two  struts  like  a  «,  of  same  size  as  v. 

Art.  8.  For  spans  greater  than  about  SO  ft,  the  writer  believes 
that  aa  a  general  rule  (liable  to  modifications  according  to  the  judgment  of  the 
engineer  in  charge)  the  following  ideas  will  lead  to  safe  practice.  Namely,  to 
adopt  a  bowstring  truss  with  a  simple  Warren  or  triangular  web,  as  at  /  on  the 
left  side  of  Fig  9.  The  bow  to  rest  on  the  chord,  and  each  to  be  of  a  single  thick- 
ness. The  web  members  (especially  in  large  spans)  to  be  also  of  single  thickness, 
and  placed  below  the  bow,  resting  on  the  chords,  and  well  strapped  to  both,  so  as 
to  act  as  either  ties  or  struts.  In  smaller  spans  the  web  members  may  each  be  in 
two  thicknesses,  one  bolted  or  treenailed  to  each  side  of  the  bow  and  chord.  Other 
modes  will  suggest  themselves;  but  we  have  not  space  for  such  details. 

Or  a  web  of  the  Howe,  or  of  the  Pratt  system,  as  on  the  right  side  of  Fig  9  may  be 
used.  But  in  reference  to  both  of  these  it  may  be  remarked  that  the  use  of 
long  iron  rods  in  centers  of  large  spans  is  highly  objectionable,  owing 
to  the  different  rates  of  expansion  between  iron  and  wood.  Therefore  if  these 
systems  are  used,  all  the  members  should  be  of  wood.  The  lattice  may  be  used. 

Even  when  the  rise  of  the  arch  exceeds  .25  of  the  span,  it  is  better  not  to  let 
that  of  the  centers  exceed  that  limit;  but  adopt  the  expedient  shown  at 
the  left  side  of  Fig  9,  with  a  rise  of  about  one  sixth  of  the  span. 

Rem.  1.  To  fix  on  the  number  of  web  triangles  in  a  Warren 
truss  or  frame  for  a  center,  find  the  square  root  of  the  span,  and  to  it  add  one 
tenth  of  the  span.  Divide  their  sum  by  2,  and  call  the  quotient  n.  Divide  the 
span  by  n.  If  this  quotient  is  a  whole  number  use  it;  or  if  the  quotient  is  partly 
decimal,  use  the  whole  number  nearest  to  it,  as  a  distance  in  feet  to  be  stepped  off 
along  the  chord ;  thus  dividing  the  chord  into  a  number  of  equal  parts.  All  the 
points  thus  found  on  the  chord,  are  the  places  for  the  feet  of  the  triangles. 
Next,  from  half  way  between  each  two  of  these  points,  draw  vertical  lines  to  the 
bow.  The  points  thus  found  along  the  bow,  are  the  places  of  the  tops  of  the 
triangles.  This  rule  will  be  used  in  connection  with  the  following  Table  of  Areas 
of  Bows,  as  the  two  are  dependent  on  each  other. 

In  large  arches  the  timber  of  the  bow  should  not  be  wasted  by 
trimming  its  upper  edges  to  the  curve  of  the  arch,  but  should  be  left  straight ;  and 
separate  pieces  so  trimmed,  like  c  in  Fig.  10,  should  be  spiked  on  top  of  them. 


672 


CENTERS    FOR   ARCHES. 


The  transverse  area  of  the  bow,  in  square  inches,  may  be  taken  from 
the  following  table  ;  and  may  in  practice  be  assumed  to  be  uniform  throughout 
its  entire  length  ;  which  in  fact  it  is  quite  approximately.  See  Rem  2. 

TABLE  FOR  BOWSTRING   CENTERS. 

Table  of  areas  in  square  inches  at  the  crown  of  each  Bow,  of  properly 
trussed  Bowstring  frames  for  centers  of  stone  or  brick  arches.  The  frames  to 
be  placed  5  feet  apart  from  cen  to  cen.  With  these  areas,  the  combined  weights 
of  arch,  center  (of  oak),  and  lagging,  will  in  no  case  in  the  table  strain  the  Bow 
at  crown  of  the  greatest  spans  quite  1000  Ibs  per  square  inch  ;•  diminishing  grad- 
ually to  600  or  700  Ibs  in  the  smallest  spans,  which  are  more  liable  to. casualties. 
The" depths  of  the  archstones  may  be  taken  fully  equal  to  those  in  our  table,  p 
345.  Although  centers  of  moderate  span  are  usually  made  of  white  or  yellow 
pine,  spruce,  or  hemlock,  all  of  which  are  considerably  lighter  than  oak,  we  have 
for  safety  assumed  them  to  be  of  oak,  in  preparing  our  table. 

For  spans  of  from  1O  to  2O  feet  use  the  same  sizes  as  for  20  feet. 


Original. 
Rise  in  parts  of  the  Span. 
o5     .4     .35    .3     .25     .2     .15     .1 

Span 

in  feet. 

Areas  of  transverse  section  of  Bow, 

in  square  inches. 

20 

14 

17 

19 

21 

24 

29 

38 

59 

25 

18 

22 

25 

28 

33 

40 

53 

80 

30 

23 

28 

32 

37 

43 

51 

71 

103 

35 

28 

34 

40 

45 

54 

64 

87 

125 

40 

34 

41 

48 

55 

65 

77 

106 

150 

45 

40 

49 

57 

65 

76 

92 

126 

175 

50 

47 

57 

66 

76 

89 

107 

146 

203 

55 

53 

64 

75 

87 

102 

121 

166 

233 

60 

GO 

73 

85 

99 

115 

135 

187 

263 

65 

68 

81 

95 

110 

129 

151 

209 

294 

70 

75 

90 

105 

122 

143 

168 

233 

325 

75 

83 

99 

115 

133 

157 

184 

256 

357 

80 

91 

108 

125 

145 

171 

201 

279 

390 

85 

99 

117 

136 

157 

185 

218 

302 

423 

90 

108 

127 

147 

169 

199 

235 

325 

457 

95 

115 

136 

158 

181 

214 

252 

348 

490 

100 

123 

146 

169 

194 

229 

270 

372  • 

524 

110 

133 

166 

191 

219 

260 

307 

420 

592 

120 

155 

187 

213 

246 

291 

345 

470 

660 

130 

172 

208 

237 

274 

323 

384 

520 

140 

190 

230 

263 

303 

357 

424 

572 

150 

209 

252 

289 

333 

393 

466 

160 

229 

276 

315 

365 

430 

509 

170 

250 

299 

343 

399 

469 

180 

272 

323 

373 

435 

511 

190 

294 

347 

403 

472 

200 

318 

372 

435 

509 

Rent.  2.  The  square  root  of  any  of  these  areas  gives  in  inches  the  side  of 
a  square  bow  of  that  area.  The  distances  apart  of  the  triangles  which  form 
the  web  of  the  frame,  having  first  been  found  by  Rem  1  (for  said  Rem  and  this 
table  are  dependent  on  each  other),  the  above  areas  for  bows  5  ft  apart  from4  cen 
to  cen,  suffice  not  only  to  resist  the  pressure  along  the  bow,  but  also,  as  square 
beams,  to  sustain  with  a  safety  in  no  case  less  than  about  5,  the  load  of  arch- 
stones  resting  upon  them  between  the  adjacent  tops  of  two  triangles ;  and  with 
very  trifling  deflections.  It  is  therefore  unnecessary  to  deepen  the  ribs  for  that 
purpose ;  although  it  may  be  done  (preserving  the  same  area)  in  case  consider- 
ations of  detail  should  render  it  desirable. 

As  before  suggested,  it  will  generally  be  best,  in  spans  exceeding  30  or  40  ft,  to 
give  the  bow  a  rise  not  exceeding  about  one  fifth  or  one  sixth  of  the  span  ;  and 
to  support  the  frames  as  at/,  Fig  9. 

The  size  of  the  chord  may  be  the  same  as  that  of  the  bow ;  and  like  it 
uniform  from  end  to  end ;  care  however  being  taken  that  it  be  not  materially 
weakened  by  footing  the  bow  upon  its  ends ;  or  (when  too  long  for  single  tim- 
bers) by  tfee  splicing  necessary  to  prevent  its  being  stretched  or  pulled  apart  by 


CENTER&^FOB   ARCHES. 


673 


the  thrust  of  the  bow.  When,Xowever,  the  chord  can  be  placed  at,  or  a  little 
below  the  springs  of  the  arch/all  danger  of  this  kind  may  be  avoided  by  simply 
wedging  its  ends  well  againsac  the  faces  of  the  abutments. 

As  to  the  size  of /Oie  web  members,  when  a  bowstring  truss  is 
fully  loaded  on  tonxof  the  bow,  (as  is  approximately  the  case  with  a  center 
and  its  archstones,)  tke  strains  on  the  web  members  are  quite  insignificant,  and 
arise  chiefly  from  the  weight  of  the  center  itself;  but  while  it  is  being:  so 
loaded,  they  are  not  only  greater,  but  are  constantly  changing,  not  only  in 
amount,  but  also  in  character — being  at  one  period  compressive,  and  at  another 
tensile. 

Hence  it  would  be  very  tedious  to  calculate  the  dimensions  of  the  web  members. 
Fortunately  the  necessity  for  doing  so  is  in  a  great  measure  obviated  by  the  fact 
that  a  center  being  but  a  temporary  structure,  the  timber  composing  it  is  not  ulti- 
mately wasted  if  a  greater  quantity  of  it  is  used  than  is  absolutely  required. 
Moreover  facility  of  workmanship  is  secured  by  not  having  to  employ  timbers 
of  many  different  sizes. 

Hence  the  writer  will  venture  to  suggest,  entirely  as  a  rule  of  thumb,  to  give 
each  web  member  half  the  transverse  area  of  the  bow, 
taking  care  to  make  each  of  them  a  tie-strut. 

Rein.  3.  As  to  details  of  joints,  we  refer  to  the  Figs  on  pages  292, 
294 ;  merely  suggesting  here  the  use  of  long  and  wide  iron  shoes  where  timbers 
are  subjected  to  great  pressure  sideways. 

Rein.  4.  To  prevent  the  thrust  of  the  bow  when  its  rise  is  small,  from  split- 
ting off  the  ends  of  the  chords,  the  two  may  be  united  by  many  more  bolts  than 
are  employed  in  roof  trusses,  &c,  where  only  one  is  generally  placed  near  each  end 
of  the  chord.  But  they  may  when  required  be  inserted  at  intervals  extending  to 
many  feet  from  the  ends.  They  should  have  strong  large  washers ;  and  may  have 
about  the  same  inclination  as  the  shortest  web  member. 

Another  way  of  securing  the  same  end  in  smaller  spans,  is  by  completely  en- 
casing the  two  sides  of  the  bow  and  chord,  to  a  distance  of  a  few  feet  from  their 
ends,  in  short  pieces  of  board  or  plank  spiked  to  both  of  them,  and  having  about 
the  same  inclination  as  just  suggested  for  bolts. 

Rem.  5.  Build  up  both  sides  of  the  arch  at  once,  in  order  to  strain  the  cen- 
ters as  little  as  possible. 

Rem.  6.  When  a  bridge  consists  of  more  than  one  arch,  and  they  are  to  be 
built  one  at  a  time,  there  must  be  at  least  two  centers;  for  a  center  must  not 
be  struck  until  the  contiguous  arches  on  both  sides  are  finished,  for  fear  of  over- 
turning the  outer  unsupported  pier.  Therefore  if  there  are  but  two  arches^they 
must  be  built  at  once,  requiring  two  centers.  •» 

Rem.  7.  Always  use  supports  either  vertical  or  inclined  (and  pro- 
vided with  striking- wedges)  under  the  frames,  and  intermediate  of  the  end  sup- 
ports, when  possible ;  even  if  they  can  extend  out  but  a  few  feet  from  the  abut- 
ments, as  at  the  left  side  of  Fig  9. 

Rem.  8.  The  weight  of  large  centers  and  their  lagging  is  greater 
for  flat  arches  than  for  high  ones  of  the  same  span ;  and  also  approaches  nearer 
to  that  of  the  supported  arch. 

Rem.  9.  Thickness  of  lagging.  The  following  table  gives  thicknesses 
which  will  not  bend  more  than  an  eighth  of  an  inch  under  the  weight  of  any 
probable  archstones  adapted  to  the  respective  spans ;  and  generally  not 
so  much. 

TABLE  OF   L.AGG1NG.— Original. 


Distance  apart 

Span  of  center  in  feet. 

of  frames, 
in  the  clear. 

10. 

20. 

50. 

100. 

150.          20O. 

Feet. 

Thickn 
Ins. 

ess  of  close 
Ins. 

lagging  n 
Ins. 

>t  to  bend 
Ins. 

nore  than 
Ins. 

y8inch. 
Ins. 

6 

®A 

3% 

4K 

4% 

5 

5% 

5 

2H 

m 

3% 

3J^ 

4 

4 

1«2 

2y 

2V-4 

2% 

JKg 

3 

3 

jlx 

w| 

1% 

1% 

2 

2 

2 

5 

4 

1 

V/8 

m 

1$ 

With  thicknesses  three  quarters  as  great  as  these,  the  bending  may  reach 
a  full  quarter  inch ;  which  may  be  allowed  in  dists  apart  of  3  or  more  ft. 

Rem.  1O.  Centers  are  framed,  or  put  together,  (like  iron  bridges)  on  a 
firm,  level  temporary  floor  or  platform,  on  which  a  full-size  drawing  of  a  frame  is 


674 


CENTERS    FOR    ARCHES. 


first  made.    As  eac'a  frame  is  finished,  it  is  removed  to  its  place  on  the  piers  or 
abuts. 


from  35  to  50  ft  high.  It  contains  about  15400  cub  yds  of  masonry.*  Each  center 
consisted  of  7  frames  or  trusses  of  hemlock  timber,  of  the  Bowstring  pattern,  with 
lattice  (Fig  33,  p  285)  web-members;  and  as  nearly  as  may  be,  of  the  same  span  arid 
rise  as  the  arches.  They  were  placed  4.5  ft  apart  from  center  to  center;  and  were 

supported  near  each  end  /,  Fig  13 
(a  transverse  section  to  scale)  by 
a  hemlock  post  p,  12  ins  square. 
The  bow  was  of  two  thicknesses 
bb  of  hemlock  plank,  6  ins  apart 
clear,  in  lengths  of  6  ft,  with  their 
upper  edges  cut  to  suit  the  curve 
of  the  arch.  Each  piece  was  4  ins 
thick,  by  13.5  ins  deep  at  its  middle, 
and  12  ins  at  its  ends.  These  pieces 
did  not  break  joint;  but  at  each 
joint  were  four  %  inch  bolts,  with 
nuts  and  washers,  uniting  them 
with  chocks  or  filling-in  pieces. 
The  bow,  bb,  footed  on  top  of  the  ends  of  the  chords  /;  and  the  angle  formed  by 
their  meeting  (seen  only  in  a  side  view)  was  (for  about  2.5  ft  horizontal  and  5.5  ft 
vertical)  filled  up  solid  with  vertical  pieces,  to  afford  a  firmer  base  for  resting  the 
frame  on  n ;  beyond  which  it  extends  (in  a  side  view)  about  18  ins. 

The  chords  /were  of  two  thicknesses  of  4X  12  hemlock  plank,  6  ins  apart 
clear,  and  most  of  them  in  two  or  three  lengths;  breaking  joint,  and  with  two  % 
inch  bolts,  with  nuts  and  washers,  at  each  joint,  for  bolting  them  together,  and  to 
filling-in  pieces.  The  wel>  members  of  each  frame  were  26  lattices,  o,  of 
3  X  12  inch  hemlock,  crossing  each  other  about  at  right  angles,  at  intervals  of  about 
3.5  ft  from  center  to  center,  and  passing  between  the  two  thicknesses  b  b  of  the  bow, 
and//  of  the  chords.  A  few  of  the  lattices  were  in  two  lengths,  and  the  joints  were 
not  at  the  crossings.  The  lattices  were  connected  at  each  crossing  by  two  hard  wood 
treenails  9  ins  long,  and  2  ins  diam ;  and  one  such,  18  ins  long,  passed  through  the 
intersection  of  each  end  of  a  lattice  with  a  bow  or  chord.  The  first  lattice  foots 
aboi^t  4  ft  from  the  end  of  a  chord.  They  do  not  extend  above  the  top  of  the  bow. 
All  the  spaces  between  the  two  thicknesses  of  bow  or  chord,  where  not  occupied  by 
the  ends  of  lattices,  were  completely  filled  by  chocks,  well  spiked.  , 

Each  frame  contained  about  360  cub  ft  of  timber;  and  weighed  about  5 
tons.  They  were  very  flexible  laterally  until  in  place,  and  braced  together  by  4 
transverse  horizontal  planks  spiked  to  their  chords ;  and  by  5  others  above  them, 
spiked  to  the  lattices. 

Until  the  keystones  were  placed,  all  the  joints  of  the  frames  continued  tight,  under 
the  pressure  from  the  arch,  and  from  the  unfinished  backing  to  the  height  of  about 
14  ft  above  the  springing  line ;  but  after  the  keystones  were  set,  all  the  joints  of  the 
chords  alone  opened  from  .25  to  .75  of  an  inch  ;  and  at  the  same  time  the  lagging  un- 
der the  haunches  of  the  arches  became  slightly  separated  from  the  soffit  of  the  masonry. 
Each  center  sank  but  a  full  inch  at  the  middle,  under  the  pressure  from 
the  arch  and  14  ft  of  backing. 

The  portion  of  the  bridge  above  the  piers  was  about  two  thirds  completed  before 
the  centers  were  struck. 

There  was  one  wedge  w,  w,  (32.5  ft  long,  of  12  X  12  inch  oafr)  under  each 
end  of  a  center.  It  was  trimmed  to  form  7  smaller  ones  w,  w,  each  4.5  ft  long,  and 
tapering  7  ins ;  one  under  each  end  of  each  frame  /.  They  played  between  tapered 
blocks  a,  a,  of  oak,  2  ft  long,  1  ft  wide,  let  1  inch  into  the  cap  c,  or  into  the  piece  », 
on  which  last  the  frames  /,/,  rested.  The  sliding  surfaces  were  well  lubricated  with 
tallow  when  put  in  place. 

The  weflgfes  were  struck  with  ease,  at  one  end  of  a  center  at  a  time,  by  an 
oak  log  battering-ram  18  ft  long,  and  nearly  a  ft  in  diam,  suspended  by  ropes,  and 
swung  and  guided  by  4  men.  They  generally  yielded  and  moved  several  inches  at 
the  second  blow  with  a  3  or  4  ft  swing.  Although  each  wedge  was  loosened  entirely 
within  2  or  3  minutes,  thus  lowering  the  centers  very  suddenly,  yet  on  account  of  the 

•X-  This  bridge,  finished  without  accident,  at  the  end  of  1882,  reflects  much  credit  on  William  Lorenz 
Esq,  Ch.  Eng;  on  Mr.  Charles  W.  Buchholz,  Assistant  in  Charge  ;  and  on  the  skilful  and  energetic 
contractors,  William  &  James  Nolan,  of  Reading,  Penna.  These  last  most  cordially  assisted  the 
writer  in  making  observations  during  the  entire  progress  of  the  work. 


FOR   ARCHES. 


675 


, 
U 


good  charact^of  the  rodsonry,  not  the  slightest  crack  of  a  mortar  joint  could  after- 
wards be^fected  iaXny  part  of  the  work.  After  three  days  the  average  sinking  of 
the  kp5mones  waMmly  .35  of  an  inch  ;  the  least  was  1^;  and  the  greatest  %  of  an 
inch.  The  heads  and  feet  of  the  posts  p  compressed  the  hemlock  caps  c,  and  the 
sills,  about  %  of  an  inch  each,  showing  that  for  arches  of  this  size  the  caps  and  sills 
had  better  be  of  some  harder  wood,  as  yellow  pine  or  oak;  although  probably  the 
compression  was  facilitated  by  the  large  mortices,  3  by  12  ins,  and  6  ins  deep. 

Art.  IO.  Brick  Arches.  Since  even  good  brick  fit  for  large  arches  has 
far  less  crushing  strength  than  good  granite  or  limestone,  and  is  inferior  even  to 
good  sandstone,  while  its  weight  does  not  differ  very  materially  from  stone,  it  is 
plain  that  it  cannot  be  used  in  arches  of  as  great  span  as  stone  can.  Some  of 
those  already  built,  and  which  have  stood  for  many  years,  have  a  theoretical  co- 
efficient of  safety  of  but  about  3  ;  whereas  the  authorities  direct  us  not  to  trust  even 
stone  with  more  than  one-twentieth  of  its  crushing  load.  This  last,  however,  ap- 
pears to  the  writer  to  be  one  of  those  hasty  assumptions  which,  when  once  ad- 
mitted into  professional  books,  are  difficult  to  be  got  rid  of.  It  is  his  opinion  that 
with  good  cement,  and  proper  care  in  striking  the  centers,  one-tenth  of  the  ulti- 
mate strength  is  sufficiently  secure  against  even  the  abnormal  strains  caused  by 
the  settling  at  crown,  and  rising  at  the  haunches  when  the  centers  are  struck.  It 
is  useless  to  attempt  to  fix  limits  of  safety  for  bad  materials  poorly  put  together. 
Item.  1.  The  common  practice  of  building  brick  arches  in  a  series  of  con- 
centric rings,  as  at  a  c  e  e,  Fig  12,  with  no  other  bond  between  them  than 

that  afforded  by  the  mortar,  is  censured  by 
authorities,  on  the  ground  that  the  line  of 
pressure  in  passing  from  the  extrados  to 
the  intrados  tends  to  separate  the  rings, 
and  thus  weaken  the  arch  by,  as  it  were, 
splitting  it  longitudinally.  The  reason 
for  using  these  rings,  instead  of  making 
the  radial  joints  continuous  throughout 
the  depth  m  n  of  the  arch,  as  at  b,  is  to 
avoid  the  thick  mortar-joints  at  the  back  of 
the  arch,  and  shown  in  the  Fig.  If  the 
center  of  an  arch  built  as  at  b  be  struck 
too  soon,  the  soft  mortar  in  these  thick 

joints  will  be  so  much  compressed  as  to  cause  great  settlement  at  the  crown, 
throwing  the  arch  out  of  shape,  and  creating  such  inequality  of  pressure  as 
might  even  lead  to  its  fall,  especially  if  flat.  As  a  compromise  between  rings 
and  continuous  joints,  they  are  sometimes  employed  together,  so  as  to  get  rid  of 
some  of  the  long  radial  joints  ;  and  at  the  same  time  to  break  at  intervals 
the  continuity  of  the  rings.  Thus  in  Fig  12.  which  is  supposed  to  be  brick-and- 
a-half  deep,  beginning  at  the  abutment  a,  we  may  lay  half-brick  rings  as  far  as 
say  to  e  o  e;  then  cutting1  away  the  brick  o  to  the  line  e  e,  we  may  lay  from. 
e  e  to  m  n  a  block  of  bricks  with  continuous  radial  joints,  the  same  as  at  b;  and 
then  start  again  with  three  rings;  and  so  on  alternately.  A  still  better,  but 
more  expensive,  mode  would  be  to  fill  ee,mn  with  a  regular  cut-stone  voussoir. 
The  proper  intervals  for  changing  from  rings  to  blocks  will  depend  upon  the 
number  of  the  rings  and  the  depth  c  a  of  the  arch  ;  reference  being  also  had  to 
reducing  the  amount  of  brick  cutting  as  much  as  possible. 

These  points  can  be  best  decided  on  from  a  drawing  of  a  portion  of  the  arch 
on  a  scale  of  3  or  4  ins  to  a  foot.  Generally  the  rings  are  made  only  half-brick,  or 
about  4  to  4.5  ins  thick,  as  at  a  c;  and  in  Brunei's  Maidenhead  viaduct  of  two  ellip- 
tic brick  arches  of  128  ft  span,  and  24.25  ft  rise  ;  the  boldest  brick  arches  yet  at- 
tempted ;  but  which  have  been  estimated  to  have  a  co-efficient  of  safety  of  but 
three  against  crushing  at  the  crown. 

So  many  othersof  from  70  to  100  ft  span  have  been  successfully  built  entirely  in 
rings  of  either  half  or  whole  brick  thick,  as  to  justify  us  in  attaching  but  little  weight 
to  the  above  theoretical  objection,  provided  first  class  cement  be  used,  and  time 
allowed  it  to  become  nearly  or  quite  as  hard  as  the  bricks  themselves,  before 
striking  the  centers.  Under  such  circumstances  we  should  not  object  to  a  series 
of  rings  even  1.5  bricks  thick,  laid  alternately  header  and  stretcher,  as  at  6. 

If  the  bricks  were  voussoir-shaped,  that  is,  a  little  thicker  at  one 
end  than  the  other,  then  rings  a  whole-brick  thick  could  be  used  without  any  in- 
crease in  thickness  of  mortar-joint  at  the  back  of  each  ring.  Still  with  more 
than  one  ring,  the  radial  joints  would  not  be  continuous,  as  at  b,  but  broken  as  at 
ac.  Such  bricks  however  would  be  more  expensive  to  make;  and  moreover,  in 
order  fully  to  answer  the  intended  purpose,  they  would  have  to  be  made  of  many 
patterns,  so  as  to  conform  to  the  many  radii  used  in  arches;  and  even  to  the 
radii  of  the  different  rings,  when  the  depth  of  the  arch  required  several  of  them. 


676 


TO    MEASURE    ANGLES    BY    A    2-FOOT    RULE. 


To  make  voussoir-shaped  bricks  that  shall  insure  continuity  of  radial  joints  with 
uniform  thickness  of  mortar,  in  deep  arches,  is  therefore  (commercially  speaking) 
impossible;  and  we  must  depend  on  good  cement  to  overcome  the  difficulty. 

Item.  2.  Wet  the  bricks  before  laying.     See  last  paragraph  of  p  497. 

Item.  3.  When  the  ends  or  faces  of  a  brick  arch  are  to  be  finished  with  cnt- 
Stone  voussoirs,  these  had  better  not  be  inserted  until  some  time  after  the 
completion  of  the  brickwork,  the  hardening  of  the  mortar,  and  a  partial  easing 
of  the  centers ;  lest  they  be  cracked  or  spawled  by  the  unequal  settlements  of  them- 
selves and  the  bricks.  For  more  on  brick  arches  see  p  344. 

To  Measure  Angles  by  a  2  Ft  Kule,  etc, 

The  four  fingers  of  the  hand,  held  at  right  angles  to  the  arm,  and  at 

arras-length  from  the  eye,  cover  about  7  degrees,  Aud  7°  corresponds  to  about  12. 2  ft  in  100  ft;  or  to 
36.6  ft  iu  100  yds  ;  or  to  645  ft  in  a  mile  ;  or  in  the  same  proportion  as  the  distance. 

The  following:  Table 

may  sometimes  be  found  useful  for  the  rough  measurement  of  angles,  either  on  a  drawing,  or  be- 
tween distant  objects  in  the  field.  If  the  inner  edges  of  a  common  two-foot  rule  be  opened  to  the  ex- 
tent shown  in  the  column  of  inches,  its  edges  will  be  inclined  at  the  angles  shown  in  the  columns  of 
angles.  Since  an  opening  of  %  of  an  inch  up  to  19  inches  or  about  105°,  corresponds  to  from  about 
%  to  1°,  no  great  accuracy  is  to  be  expected ;  and  beyond  105°  still  less ;  the  liability  to  error  in- 
creasing very  rapidly  as  the  opening  becomes  greater.  Thus,  the  last  J^  inch  corresponds  to  about  12°. 
As  to  the  table  itself,  angles  for  openings  intermediate  of  those  therein  given,  may  be  calculated  to 
the  nearest  minute  or  two,  by  simple  proportion,  up  to  23  inches  of  opening,  or  about  147°. 

Table  of  Angles  corresponding:  to  openings  of  a  2-foot  rule. 

(Original.) 
D,  degrees ;  M,  minutes.  Correct. 


Ins. 

D.  M. 

Ins. 

D.  M. 

Ins. 

D.  M. 

Ins. 

D.  M. 

Ins. 

D.  M. 

Ins. 

D.  M. 

M 

1  12 

4>4 

20  24 

8^  • 

40  13 

l2Ji 

61  23 

16^ 

85  14 

MX 

115  5 

1  48 

21 

40  51 

62  5 

86  3 

116  12 

M 

2  24 

X 

21  37 

M 

41  29 

% 

62  47 

H 

86  52 

% 

117  20 

3  00 

22  13 

42  7 

63  28 

87  41 

118  30 

% 

3  36 

K 

22  50 

% 

42  46 

*A 

64  11 

% 

88  31 

H 

119  40 

4  11 

23  27 

43  24 

64  53 

89  21 

120  52 

i 

4  47 

5 

24  3 

9 

44  3 

L3 

65  35 

17 

90  12 

21 

122  6 

5  2:3 

24  39 

44  42 

66  18 

91  3 

123  20 

M 

5  58 

34 

25  16 

K 

45  21 

y* 

67  1 

34 

91  54 

34 

124  36 

6  34 

25  53 

45  59 

67  44 

92  46 

125  54 

M 

7  10 

M 

26  30 

% 

46  38 

M 

68  28 

34 

93  38 

34 

127  14 

7  46  - 

27  7 

47  17 

69  12 

94  31 

128  35 

H 

8  22 

H 

27  44 

K 

47  56 

H 

69  55 

M 

95  24 

H 

129  59 

8  58 

28  21 

48  35 

70  38 

96  17 

131  25 

2 

9  34 

6 

28  58 

10 

49  15 

14 

71  22 

18 

97  11 

22 

132  53 

10  10 

29  35 

49  54 

72  6 

98  5 

134  24 

v\ 

10  46 

H 

30  11 

M 

50  34 

34 

72  51 

H 

99  00 

H 

135  58 

11  22 

30  49 

51  13 

73  36 

99  55 

137  35 

% 

11  58 

H 

31  26 

H 

51  53 

y* 

74  21 

y* 

100  51 

H 

139  16 

12  34 

32  3 

52  33 

75  6 

101  48 

141  1 

H 

13  10 

M 

32  40 

% 

53  13 

H 

75  51 

*A 

102  45 

H 

142  51 

13  46 

33  17 

53  53 

76  36 

103  43 

1M  46 

3 

14  22 

7 

33  54 

il 

54  34 

15 

77  22 

19 

104  41 

23 

146  48 

14  58 

34  33 

55  14 

78  8 

105  40 

148  58 

34 

15  34 

M 

35  10 

J* 

55  55 

M 

78  54 

34 

06  39 

X 

151  17 

16  10 

35  47 

56  35 

79  40 

07  40 

153  48 

X 

16  46 

X 

36  25 

% 

57  16 

M 

80  27 

x 

08  4] 

M 

156  34 

17  22 

37  3 

57  57 

81  14 

09  43 

159  43 

*A 

17  59 

M 

37  41 

% 

58  38 

H 

82  2 

H 

10  46 

H 

163  27 

18  35 

38  19 

59  19 

82  49 

11  49 

168  18 

4 

19  12 

8 

38  57 

12 

60  00 

L6 

83  37 

20 

112  53 

24 

180  00 

19  48 

39  85 

60  41 

84  26 

113  58 

Or  this  table  may  be  used  thus.  From  any  point  measure  12  ft 
towards  each  object,  and  place  marks.  Measure  the  dist  in  ft  between  these 
marks.  Suppose  the  first  cols  in  the  table  to  be  ft  instead  of  ins ;  then  opposite 
the  dist  in  ft  will  be  the  angle.  One-eighth  of  a  ft  is  1.5  ins. 

The  following  is  a  good  way  to  measure  an  angle.  Measure 
100  or  any  other  number  of  ft  towards  each  object,  and  place  marks.  Measure  the 
dist  between  the  marks.  Then 

As  dist  measured     .  1    .  .  Half  the  dist  .  nat  sine  of 
toward  one  object   •-*-..  between  marks  •  Half  the  angle. 

Find  this  nat  sine  in  the  table  of  nat  sines,  take  out  the  corresponding  angle, 
and  multiply  it  by  2.  See  near  foot  of  p  41. 


TO   FIND 


To 


FERENCES   OF   CIRCLES. 


677 


Ircuinljjrof  circles  when  the  diam  contains 
decimals. 


Role.    Find  the/dircumf  for  the  whole  number  by  table,  p.  18.    Then  for 
the  decimal  paryuse  the  following  table : 


Diam. 

Circ.  / 

Diam. 

Circ. 

Diam. 

Circ. 

i 
Diam. 

Circ. 

Diam. 

Circ. 

.1 

.314159 

.01 

.031416 

.001 

.003142 

.0001 

.000314 

.00001 

.000031 

.2 

.628319 

.02 

.062832 

.002 

.006283 

.0002 

.000628 

.00002 

.000063 

.3 

.942478 

.03 

.094248 

.003 

.009425 

.0003 

.000942 

.00003 

.000094 

.4 

1.256637 

.04 

.125664 

.004 

.012566 

.0004 

.001257 

.00004 

.000126 

.5 

1.570796 

.05 

.157080 

.005 

.015708 

.0005 

.001571 

.00005 

.000157 

.6 

1.884956 

.06 

.188496 

.006 

.018850 

.0006 

.001885 

.00006 

.000188 

.7 

2.199115 

.07 

.219911 

.007 

.021991 

.0007 

.002199 

.00007 

.000220 

.8 

2.513274 

.08 

.251327 

.008 

.025133 

.0008 

.002513 

.00008 

.000251 

.9 

2.827433 

.09 

.282743 

.009 

.028274 

.0009 

.002827 

.00009 

.000283 

Example.    What  is  the  circumf  of  a  circle  whose  diam  is  67.35824  ins,  or 
feet,  Ac.  ? 

Here,  first  by  table,  p  18,  we  have  for  diam  67.  Circumf  =  210.487 


Then  by  above  table 


.3 
.05 

.008 
.0002 
.00004 


.942478 
=  .157080 
=  .025133 
=  .000628 
=  .000126 


Circumf  required  =  211.612445 

Rem.  This  mode  is  correct  within  an  error  of  less  than  1  in  the  third,  fourth, 
or  fifth  decimal,  according  to  the  number  of  decimals  in  the  first  circumf,  taken 
from  table,  p  18.  Thus  the  true  circumf  (found  approximately  to  be 
211.612)  will  not  be  as  great  as  211.613,  nor  as  small  as  211.611. 


Many  things 


Abrasiortoy  streams,  563,  570. 
Abutment-piers,  347. 
Abutments,  to  proportion,  345. 
Acceleration  of  gravity,  172,  449,  587. 
Acre,  square  and  circular,  76. 
Acres  drained  by  pipes,  569. 

required  per  mile  for  R  R,  390. 
Action  and  reaction,  449. 
Adhesion  of  cement,  503. 

of  glue,  620. 

of  mortar,  497. 

of  nails  and  spikes,  383. 
Adjustment  of  compass,  164. 

of  hand-level,  167. 

of  plumb  level,  167. 

of  slope  instrument,  167. 

of  spirit  level,  154. 

of  the  box-sextant,  164. 

of  theodolite,  162. 

of  transit,  160. 
Air,  519. 

buoyancy  of,  033. 

compressed,  in  cyls,  327,  631. 

lock,  327,  631. 

pressure  of,  in  a  diving-bell,  520. 

quantity  of,  for  ventilation,  519. 

valve  for  water-pipes,  579. 

vessel,  615. 

Alioth,  (star,)  100.    Alligation,  72. 
Angle  iron,  373. 
Angle-blocks  of  Howe  truss,  283. 

complement  and  supplement  of,  62. 

exterior  and  interior,  Ac,  62. 

limiting  of  resistance,  487. 

of  friction  or  repose,  453,'*1485,  598. 

of  maximum  pressure,  335. 

on  sloping  ground,  40,  41. 

salient,  and  re-entering,  62. 

to  meas  by  a  tape  line,  footnote,  41, 42. 

"      "     by  a  2  ft  rule,  676. 

"      "     by  hand,  67G. 

"     "     by  sextant,  41. 

'     without  any  instrument,  41. 
Angles  of  deflection,  &c,  416. 

of  depression,  40. 
Angular  velocity,  447. 
Animal  power,  605. 


INDEX. 

contained   in   the   Index  may  be   found   in  the 
Glossary. 


Anthracite,  wt  of,  76,  384. 
Apothecaries'  weight,  74. 
Appendix,  630. 
Application,  point  of,  447. 
Aqueducts,  flow  in,  562.  Kutter,  650. 

Pittsburg,  596. 
Arches,  braced,  274,  287.  BrieJe,  675. 

cast-iron,  Chesinut  St  bridge,  288. 
"  "  Severn  Valley  R  R,  288. 
"  "  Whipple's,  288. 

existing,  343. 

Centers  for,  665. 

hor.  pres.  equal  throughout,  493. 

iron,  in  roofs,  two  examples  of,  289. 

large,  rubble,  344,  506. 

line  of  pressure  or  thrust,  348,  493. 
"     "         "         to  find,  493. 

of  corrugated  iron,  371. 

pressures  on  key,  342,  468,  491. 

principles  of,  341,  479,  491.         J675. 

stone  and  brick  for  culverts,  &c,  341, 
"        "        "    tables  of,  345,  351,  &c. 

wooden,  rule  for,  307. 
Archstones,  tables  of,  343,  345. 

to  find  depth  of,  341. 
Arcs,  cen  of  grav  of,  442. 

circular,  tables  of  length,  21,  23. 

having  span  and  rise,  to  find  rad,  16. 

in  common  use,  table,  434. 

large,  to  draw,  17,  67. 
Arithmetic,  69. 

decimal,  71. 

Artificial  stone,  Coignet's,  507. 
Ashlar  masonry,  cost  of,  312,  630. 
Atmosphere,  519. 
Avoirdupois  or  commercial  wt,  74. 
Axis  of  flotation,  635. 

Bag-scoop,  or  bag-spoon,  330. 
Bailing  by  bucket,  day's  work  at,  606, 
Ballast  for  railroads,  414. 
Balloon,  principle  of,  533. 
Balls,  weight  of,  362,  366,  377. 
Barometer,  levelling  by,  167. 

levelling,  tables  for,  169, 171, 
Batter,  347. 

Beams)  box,  Fairbairn's,  214,  217. 
679 


680 


INDEX. 


Beams,  channel,  211,  213,  640. 
closed  and  open,  644. 
Cooper  &  Hewitt's,  213,  214. 
constants  for  breaking  loads,  185. 
continuous,  641.    Of  concrete,  507. 
cylindrical,  186, 193. 
deck,  211 . 

deflections  of,  191, 196. 
deflections  of  ^-Q  of  the  span,  201. 
examined   on   the  principle   of    the 

lever,  477. 

exposed  to  both  transverse  and  lon- 
gitudinal strains,  190. 
general    facts    respecting    strengths 

of,  186. 

Hodgkinson's,  208. 
hollow,  193. 
inclined,  188. 
limit  of  elasticity,  197. 
loads  within  elastic  limit,  185, 198. 
moments  of  rupture  and  resistance, 

217. 
parts  may  be  cut  away  without  loss 

of  strength,  187. 
plate,  214. 

rolled  I,  210,  212,  213. 
"          as  pillars,  638. 
"          as  short  bridges,  304. 
shearing  of,  181,  642. 
Btone,  185,  203. 
strength  of,  183,  &c. 

"       of  some  experimental 

beams,  209. 

table  of   constants   for  safe    deflec- 
tions, 199. 
"     of  loads  not  to  bend  more  than 

-$1 -Q  of  the  span,  204,  205. 
"     of  safe  loads,  191, 203. 
tables  of  breaking  loads,  206,  207. 
to  find  break'g  load  at  any  point,  188. 
"    "    center  breaking  load,  186. 
"     "    safe  dimensions,  189. 
"    "    the  breadth  of  a  beam,  189. 
"    "    the  depth        "        "       189. 
"    "    uniform  load,  187. 
to  splice,  291. 
Trenton,  213. 
triangular,  187, 188. 
Searing  and  reverse  bearing  not  alike, 

93. 

Bearing  power  of  soils,  314. 
Bell-joints  or  faucets  of  pipes,  574. 
Bends  in  water-pipes,  effects  of,  539, 548. 
Beton,  504. 

Coignet's,  507. 


Blake's  stone-crusher,  505. 
Board  measure,  table  of,  357. 
Boats,  cost,  604. 
Bodies,  falling,  587.    Table  of  vels,  552. 

floating,  533,  635. 

mass  of,  456. 

regular,  the  solidities,  &c,  of,  38. 
Body,  defined,  444. 
Boiling  ^vater  to  meas  hts  by,  170. 
Bollman  truss,  269,  282. 
Bolts  and  nuts,  374;  to  find  diam,  180; 

copper,  375. 

strength  of,  upset  and  not  upset,  375. 
Boring  in  soils  by  augur,  313,  636. 

test-holes,  wells,  313,  636. 
Borrotv-pits,te  measure  beforehand,30. 
Bottoms  to  bear  diff  vels,  563,  570. 
Bowstring  truss,  270,  286. 
Box  beams,  Fairbairn's,  214. 

drains,  355. 

sextant,  163. 

Bracing,  horizontal  diagonal,  291,  304. 
Brass,  rolled,  weight  of,  376. 

sheets,  weight  of,  367. 

wire,         "        "  368. 
Brick  Arches,  344,  675. 
Bricklaying,  a  day's  work  at,  498. 
Bricks,  and  mortar,  496. 

crushing  strength,  175,  498. 

English  rod  of,  499. 

number  in  a  sq  ft  of  wall,  498. 

paving  with,  498. 

prices  in  Philada,  498. 

proportions  of  brick  and  mortar,  496. 

tensile  strength,  499. 

to  render  impervious  to  water,  499. 

weight  of,  384,  498. 
Bridges,  arches  of  existing,  343. 

Bollman,  269,  282.    Burr,  289. 

bowstring,  270,  286. 

braced  arch,  274,  287. 

coefs  of  safety,  298. 

camber,  302.     Centers  for,  665. 

cast-iron  arches,  288. 

Chestnut  St,  Philada,  288. 

cub  yds  of  masonry  in ;  tables,  351, 

353,  354. 
-  deflections,  302. 

depth  of  archstones,  341,  345. 

Fink,  281,  305. 

floor  girders,  215,  291,  296. 

greatest  load  on,  297. 

Howe,  283,  284. 

iron,  cost  of,  300. 

lattice,  285. 


Bridges,  Moseley, 

Pratt,  284, 

raising  of,  303. 

spandrel  walls  of,  346. 

stone,  341. 

stone,  drainage  o^foadways  of,  356. 

suspension, 

swing,  strain's  on,  275. 

to  proportion  stone  abutments,  345. 

Town,  285. 

turning,  friction  rollers  for,  431. 

Warren,  254,  279. 

weight  of,  295.    WissahicJton,  674. 

Whipple's  cast-iron,  N.  York,  288. 

width  and  headway,  288,  306,  307. 
British  imperial  measures,  77. 
Brokerage,  73. 
Brunlee's  iron  piles,  326. 
Babble-glass,  to  replace,  162. 
Btickled  plates,  369. 
Buildings,  cost  per  cub  ft,  631. 
Buoyancy  of  air,  533. 

of  liquids,  533,  635. 
Buttresses,  340. 

Cables,  number  of  wires  in,  369. 

suspension,  588,  597. 
Caissons,  316. 

for  East  River  susp.  bridge,  328. 
Camber  of  trusses,  302. 
Canals,  boats,  cost,  604. 

flow  in,  564. 

leakage  of,  521. 

traction  on,  604. 
Cantilevers,  219,  275,  643. 
Cars,  axles  of,  414. 

weight  of,  413.    Wheels,  413. 
Cart,  weight  of,  436,  608. 
Castings,  to  judge  the  wt  of  by  the 

patterns,  362. 
Cast-iron,  weight  of,  362. 

pi  pes,  363,  364. 
Ceilings,  weight  of,  248. 
Cement,  adhesion  of,  503. 

brick-dust,  496. 

effect  of  freezing  on,  502. 
"      in  preserving  metals,  500. 

for  pointing,  501. 

for  rough  casting,  500. 

for    stopping  crtcks  around   chim- 
neys, 512,  514. 

hydraulic,  500. 

"          cost,  500. 

mortar,  503,  508. 

Portland,  how  made,  500. 


681 

Cement,  protection  from  moisture,  500. 

quantity  reqd  for  mortar,  500. 

restoration  by  reburuing,  500. 

setting  of,  501. 

strength  of,  502,  503.     Wt.  385,  500. 
Centers  lor  arches,  665. 
Center  of  buoyancy,  635. 

of  force  or  pressure,  333,  482. 

of  force  of  water,  526,  635. 

of  gravity,  442,  481. 

of  gyration,  495,  617,  622. 

of  oscillation,  173. 

of  percussion,  173. 

Centrifugal  and  centripetal  force,  494. 
Chain.,  Gunter's,  74,  98. 
Chaining,  deductions  on  sloping 

ground,  40,  98. 

Chains,  wt  and  strength  of,  381. 
Chairs,  railroad,  390. 

sleeve,  392. 

"        modified,  393. 

Wilson's,  stop,  391. 
Channel-iron,  as  pillars,  236,  G37, 640. 

wtof  and  strength  as  beams,  211, 213. 
Channels,  flow  in,  562,  &c. 
Charing  Cross  Railw'y,  girders  on,215. 
Chord  of  1°  of  earth's  great  circle,  22. 
Chords,  long,  419. 

of  a  truss  defined,  243. 

of  flat  iron  bars,  293. 

principle  of,  246. 

table  of  for  protracting,  608. 
Circles,  16,  62,  66.    By  decimals,  677. 

earth's  great,  22. 

table  of,  18. 

to  describe  large  ones,  17,  67, 434. 
Circular  arcs,  17,  21,  23,  434. 

inch,  76. 

lunes,  25. 

ordi nates,  to  calculate,  20. 

rings,  17,  22,  32. 

sectors,  22. 

segments,  areas  of,  24. 

spindle,  solidity  of,  39. 

tables,  of,  21,  23,  434. 

"   in  common  use,  434. 

xouss,  areas  of,  25. 
Cisterns,  432,  434,  532. 
Civil,  or  clock  time,  80. 
Clay,  swelling  of,  314. 
Clinometer,  or  slope  instrument,  167. 
Cloth,  tracing,  152. 
Coal  and  coke,  wts  of,  76,  384. 

corrosive  fumes  from,  370,  511. 

per  H.  P.,  495. 


682 


INDEX. 


Coffer-dams,  317. 

earthen,  317. 

of  cribs,  318,  319. 

of  two  enclosures  of  cribs,  319. 
Cog-wheels,  power  of  a  train  of,  479. 
Cohesion,  tables  of,  177, 179, 180. 
Coiff  net's  beton,  507. 
Coins,  foreign,  value  of,  81. 
Cold,  effect  of,  on  iron,  180. 
Colors,  152. 
Columns.     See  Pillars. 
Combination,  72. 
Commission,  73. 
Compass,  to  adjust,  164. 

variation  of,  165. 
Compensation  -water,  579. 
Composition  and  res  of  forces,  457. 
Compound  levers,  479. 
Compressed  air  in  cyls,  327,  631. 
Concrete,  504.     As  Beams,  507. 

at  Croton  darn,  &c,  505. 

cost,  507. 

large  arches  built  of  it,  344,  506. 

mixing  of,  506. 

modes  of  using  under  water,  506. 

proportion  of  void  and  solid,  504. 

ramming  of,  505. 

strength  of,  505,  507 

weight  of  concrete,  507. 
Cones,  32. 

Conical  screw-pan,  for  excavating,  326. 
Conoid,  solidity  of,  38. 
Constants  for  deflections  of  -%\^j  of  the 
span,  201. 

of  rupture,  185, 195.   Of  Elas.,  177. 

to  find  for  beams,  183,  Ac. 
Continuous  beams,  641. 
Contour  lines,  147. 
Contracted  vein,  552,  554. 
Contrary  flexure,  point  of,  641. 
Cooper  and  Hewitt  beams,  213,  214. 
Copper,  cost,  367, 368. 

for  roofs,  377. 

pipes,  weight  of,  365,  378. 

weight  of,  367,  376. 
Cord,  or  funicular  machine,  463,  662. 
Corrugated  iron,  370, 371. 

cost  of,  371. 
Cost  of  buildings,  631. 

cements,  500.    Of  concrete,  507. 

of  dredging,  330. 

earthwork,  435. 

hauling,  607. 

iron,  364. 

lumber,  361.    Of  R  R,  416, 


Cost  of  masonry,  312,  630.    Shops,  415. 
Counterbracing,  245,  252,  275, 306  Rm. 
Counterforts,  340. 
Couples,  483. 
Couplings  for  tubes,  365. 
Creeping  of  rails,  391. 
Creosote,  358. 
Crescent  truss,  270. 
Crib  dams,  cost  of,  586. 

foundations,  315. 
Cribs,  315. 

Cross-hairs,  to  replace,  162. 
Cross-ties,  414. 

Crowds,  weight  of,  297,  595,  footnote. 
Crusher,  stone,  505. 
Crushing  loads,  tablesof,  174,175, 176. 
Cube  roots,  rules  for,  60.    Table  of,  48. 
Cubic  foot,  what  equal  to,  76. 

inch,  what  equal  to,  76. 

or  solid  measure,  76. 

yards  in  a  cutting  2  ft  wide,  427. 
Culverts,  tables  of  cub  yds  in,  351,  353, 
354. 

to  find  lengths  of,  350. 
Curvature  of  the  earth,  table  of,  42. 
Curves,  railroad,  table  of,  416,  633. 
Cuttings,  level,  table  of,  420. 

to  prepare  a  table  of,  418. 
Cycloid,  29. 
Cylinders,  31. 

brick,  327. 

for  foundations,  324,  327,  631. 

friction  of,  323. 

riveted,  strength  as  beams.  193. 

screw,  324. 

strength  of,  193,  531. 

table  of  contents  of,  46,  47,  77. 

with  piles  inside,  329. 
Cylindric  beams,  hollow,  strength  of, 
193. 

cast-iron  beams,  table  of  loads,  207. 

ungulas,  31,  630. 
Cyma,  to  draw,  67. 

Dams,  583, 528. 

coffer,  318. 

discharge  over,  558,  561. 

earthen,  317,  578. 

in  California,  531,  578. 

one  at  Poona,  &30. 

rise  of  water  produced  by,  587. 

sluices  in,  686. 

stone,  528, 529,  Ac. 

tremblings  in,  586. 
Dam  walls,  high,  in  France,  529. 


SDEX. 


683 


rJtJ&srtwvet^tl) 


bailing,  606. 


Day's  wort 

at  bricklayi 

at  dresfjitfgstone,  312. 

at-dflfling,  311. 

at  excavating,  435. 

at  hauling  by  a  rope,  606. 

at  plastering,  509.    At  tin  roof,  379. 

at  treadwbeel,  tympan,  606. 

with  a  gin,  winch,  &c,  606. 
Decagons,  15.     Decimals,  71. 
Inflection  angles,  «fcc,  416. 
Deflections  of  beams,  191,  196. 

"   bridges,  302. 
Degree,  length  of,  in  a  mile,  22. 

of  long  and  lat,  74.  Dew  point,  520. 
Diagonal  horizontal  bracing,  291. 

of  a  truss,  to  find  length  of,  69,  303. 
Dialling,  150. 
Discharge,  canals,  561. 

pipes,  538.    In  sewers,  566,  652. 

under  water,  554,  note. 
Discount,  73. 
Distances  by  sound,  173. 
Distributing  reservoirs,  579. 
Diving-bell,  pressure  in,  520. 
Diving-dress,  cost  of,  329. 
Dodecaedrons,  38. 
Dodecagons,  15. 
Dollar,  74. 
Draft,  of  vessels.  534,  635. 

on  roads  and  canals,  603,  607. 
Drainage  of  roadways,  of  bridges,  356. 
Drains,  box,  355. 

pipes,  discharge  by,  568,  569. 

should  be  well  fouhded  under  high 
embankt,  355. 

terra  cotta,  569. 

Drawbridge,  strains  on,  275. 
Drawing  materials,  152. 
Dredged  material,  wt  of,  330,  385,  386. 
Dredging,  329. 

by  bag-spoon,  330. 

by  screw-pan,  326. 

cost,  330. 

Drilling  in  rock,  311,  324,  636. 
Drop  timbers,  585. 
Dry  drains,  355. 

measure.  77. 
Dynamics  defined,  444,  459,  note. 

E  and  Wline,  to  run,  93. 
Earth,  bearing  power  of,  314. 

embankment,  settlement  of,  630. 

friction  on,  340,  603.  [&c. 

natural  slope  of,  338.  Pressure  of,  331, 


Earth,  table  of  curvature,  42. 
Earthwork,  cost  of,  with  tables,  435. 
Elasticity,  limit  of,  in  beams,  197. 

limit  of  in  steel  and  iron,  176,  185. 

modulus  of,  defined,  177. 
"        "  table  of,  632. 
Elevation  of  outer  rail,  419. 
Ellipse,  25,  630;  to  draw,  27. 
Ellipsoid,  solidity  of,  38. 
Elliptic  arcs,  table  of,  27. 

segments,  areas  of,  27. 
Elongation  of  north  star,  99. 
Embankments,  settlement  of,  630. 
Energy,  459. 
Equality  of  moments,  476. 
Equation  of  payments,  72. 
Equilibrium,  stable,  &c.  Note,  481, 635. 
Establishment  of  port.  See  Tides,  626. 
Evaporation,  521. 

of  locomotives,  434. 
Excavation,  cost  of,  435. 

level  cuttings,  tables  of,  420. 

100  ft  long,  arid  2  ft  wide,  table,  427. 
Expansion  links,  rockers,  rollers,  295. 

of  iron  bridges,  245. 

of  solids  by  heat,  310. 
Eye-bars  and  pins,  293. 

Fairbairn  beams,  214. 

on  the  safety  of  iron  bridges,  298. 

table  of  box  girders,  217. 
Falling  bodies,  587.    Vels  of,  552. 
False-works,  defined,  303,  619. 
Fascines,  328. 

Faucet,  or  bell-joints  of  pipes,  574. 
Fellowship,  73. 
Fences,  415. 

Ferrules  for  service  pipes,  574. 
Figure,  defined,  61 . 

irregular,  to  find  the  areas  of,  16. 

or  maps,  to  enlarge  or  reduce,  69. 
Filtration,  leakage,  &c,  521. 
Fink  truss,  264,  281,  297,  305. 

roof,  weight  of,  298.  300. 
Fireplugs,  or  hydrants,  576. 
Fisher's  rail-joint,  392. 
Fifih-j>lates,  391. 
Flexure,  point  of,  contrary,  641. 
moating  bodies,  533,  635. 
Floor  girders  for  bridges,   215,   291, 

296. 
Foot,  and  decimals  of  inches,  table,  75. 

cubic,  of  substances,  wt  of,  384. 
"       what  equal  to,  76. 

spherical,  what  equal  to,  76,  77. 


684 


INDEX. 


Force,  can  be  fully  imparted  only  at 
right  angles,  452. 

centre  of,  333,  482,  526,  635. 

centrifugal,  494. 

conip  and  res  of,  457. 

couples,  483. 

denned,  445. 

equality  of  moments,  476. 

great,  is  imparted  gradually,  453, 456. 

impartation  of,  447. 

in  rigid  bodies,  443. 

living,  or  vis  viva,  446,  455. 

moving,  448. 

nothing  can  destroy  but  other  force, 
445. 

of  a  pile  driver,  321. 

parallel,  481. 

parallelogram  of,  458. 

parallelepiped  of,  471. 

point  of  application,  447. 

polygon  of,  467. 

"        "    resultant  of,  466,  &c. 

working,  defined,  448,  454. 
Forces  in  different  planes,  470. 
Foreign  coins,  value  of,  81. 
Foundations,  313. 

by  plenum  process,  327,  631. 

by  vacuum  process,  326. 

in  caissons,  316. 

on  artificial  islands,  328. 

on  cribs,  315. 

on  cylinders  of  iron,  brick,  &c,  327. 

on  fascines,  328. 

on  grillage,  634. 

on  piles  enclosed  in  a  cylinder,  329. 

on  random  stone,  314.  . 

on  random  stone  and  piles,  315. 

on  sand  piles,  328. 

safe  loads  on,  314. 

testing  for,  313. 
Fractions,  vulgar,  69. 
Francis,  «7.^B.,hydraulic  expts,  558, 560. 
French  weights  and  measures,  78. 
Friction,  172 ;  456,  near  top ;  597. 

angle  of,  453,  485,  598. 

at  backs  of  walls.    Kern  3,  332. 

head,  535,  543. 

hydraulic  press,  632. 

in  pipes,  551. 

journal,  601. 

masonry  on  wood  and  earth,  340,  603. 

of  piles,  323. 

Parry's  friction  rollers,  429. 

pivot,  600. 

power  consumed  by,  601. 


Friction  rollers,  176,  295,  429,  602. 

rollers  for  drawbridges,  431. 

rolling  and  axle,  602. 

tables  of,  599,  600.    Wall,  336. 
Frictional  stability,  486,  494. 
Frogs,  switches,  &c,  397. 

J.  Wood's  self-acting,  400. 

Pennsylvania  Steel  Go's,  398,  400. 

tables  of,  401,  402, 

spring-rail  frog,  400. 
Frustums  of  prisms,  solidity,  30. 
Fuel  for  locomotives,  411. 

per  H.  P.,  495. 
Fumes,  corrosive,  370,  511. 
Funicular  machine,  463,  662. 

g,  449. 

Gain  of  power,  474. 
Galvanized  iron,  370. 
Galvanizing,  370. 
Gates  for  water  pipes,  572. 
Gauge,  Am  and  Birm,  367. 

Stubs,  368. 
Geometry,  practical,  or    drawing    of 

figures,  61. 

Gin,  day's  work  with,  606. 
Girders,  for  bridge  floors,  215,  291,  296. 

Hodgkinson's,  208. 

plate  and  box,  214,  217. 

rolled  I,  210. 
Glass  and  glazing,  514. 

strength  of,  175, 180, 185,  515. 
Glossary  of  terms,  615. 
Glue,  adhesion  of,  620. 
Gold  and  silver,  74. 
Gordon's  rules  for  iro"n  pillars,  221,  &c. 

table  of  pillars  by  his  rules,  232,  &c. 
Grades,  tables  of,  388,  389,  629. 

hydraulic,  536. 

wt  of  trains  on,  412. 
Gravity,  acceleration  of,  172,  449,  587. 

center  of,  442,  481. 

"  to  find  mechanically,  443. 

on  inclined  planes,  172,  486. 

specific,  383. 

table  of,  384. 
Great  Bear,  101. 
Grillage,  634. 
Grout,  497. 

Gudgeons,  to  find  diams,  620. 
Guide  rails,  398. 

Gunter's  chain,  74,  98.  [622. 

Gyration,  rad.  and  cen.  of,  173,  495, 617, 

Hand  level,  Locke's,  166. 


685 


Hand)  to  meas 
Hauling  by  hoj 

by  men,  606. 

effect  of  width  of  ti^s,  608. 
Head  of  water,  535VVirtual,  571. 

theoretical,  table',  552. 
Heads  and  nut^oT  bolts,  374,  375. 
Headway,  ofyforidges,  288  note.    307. 
Heat)  expansion  of  solids  by,  310. 
Heights,  to  find  by  barometer,  167. 
to  find  by  boiling  water,  170. 
"       "   reflection,  44. 
"       "   shadow,  43. 
"       "  trigonometry,  40,  &c. 
Hemp  ropes,  382. 
Heptagons,  15. 
Hexaedrons,  38. 
Hexagon,  area  of,  15. 

to  draw,  67. 
Hodgkinson  on  pillars,  221. 

beams,  208. 

Hollow  beams,  strength  of,  187, 193. 
Horse,  traction  of,  603,  605. 

amount  of  work  by  a  gin,  606. 

"        "        "    in  pumping,  606.  • 
power,  571,  605. 
"       coal  per,  495. 
"       of  a  running  stream,  571. 
"       of  falling  water,  571. 
walks,  diameter  of,  605. 
weight  of,  605. 
Howe  truss,  283. 

"      table  of,  284. 
Hydrants,  or  fireplugs,  576. 
Hydraulics,  534. 

acres  drained  by  pipes,  569. 

adjutages,  554. 

air  valves,  579. 

bends,  resistance  of,  539,  548. 

"  "        table  of,  550. 

bursting  pressure  in  pipes.  531,  536. 
cases  of  incomplete  contraction,  557. 
city  pipe  systems,  580. 
Clegg's  dam,  singular  effect  at,  561. 
coeffs  of  Lesbros  and  Poncelet,  555. 
disch  in  open  channels,  561.  564. 
"    from  one  reservoir  into  another, 

556. 

"     openings,  551. 
"     over  weirs,  558. 
"     through  pipes,  537. 
"  "        •    "    tables  of,  539, 544. 

"  "       short  tubes.  543.  553. 

«  "       thin  partition,  554. 

"          "      triangular  notches,561. 


Hydraulics,  floating  mills,  571. 
flow  through  pipes,  534. 
"    affected  by  material  of  pipes,  537. 
"    Weisbach's  rule  for.  543. 
Francis',  Mr.,  experiments,  558,  560. 
friction  head,  535.     Fireplugs,  576. 
friction  in  pipes  in  pumping,  551. 
head,  defined,-  535. 
'•     required  for  a  pipe,  542. 
"  "        for  bends ;  table,  550. 

horse-powers  of  falling  water,  571. 

'•  "     of  running  water,  571. 

hydraulic  grade  line,  536. 
"         mean  depth,  565. 
"         radius,  565,  650. 
Kutter's  formula,  650. 
obstructions  by  piers,  570. 
piezometers,  537. 

pressure  of  running  water,  571.  [544. 
pipes,  discharge,  tables  of,  539,  540, 
"  "  "    Weisbach's,  544. 

"  "          through,  537. 

"      general  laws  of  flow,  534. 
"      old  formulas  said  to  be  defect- 
ive, 543. 

"      resistance  to  pumping,  551. 
"      to  find  diam,  542. 
Rennie's  expts  with  bends,  550. 
reservoirs,  577. 

square  roots  of.  5th  powers,  548. 
velocity  head,  5^5. 

"        in  regular  channels,  562. 

"        in  rivers,  563. 

"        in  sewers  and  drain  pipes, 

568,  569,  652. 
"        of  falling  bodies,table  of,  552. 

of  flow  in  pipes,  537,  544. 
vena  contracta,  552,  554. 
Hydraulic  cements,  500. 
grade,  536. 
press,  632. 
radius,  565,  650. 
ram,  571. 
Hydrostatics,  521. 

buoyancy  of  liquids,  533,  635. 
centre  of  pressure,  482,  526,  635. 
compressibility  of  liquids,  516, 534. 
draught  of  vessels,  534,  635 
pressure  against  various  surfaces,522. 
**        in  pipes,  532,  536. 
"        independentofquantity,522. 
"        of  water,  521. 
"        table  of,  523. 
'."        transmission  of,  526. 
"       walls  to  resist,  528. 


44 


686 


INDEX. 


Hydrostatics,  pressure,  walls  to  resist, 
tables  of,  528,  530. 

I  beams,  210. 

as  pillars,  tables  of,  638,  639. 
safe  loads,  tables  of,  212,  213. 
Ice,  516. 

may  lift  piles,  324. 
strength  of,  175. 
Icosaedron^  38. 
Imperial  British  measures,  77. 
Impulse  or  impact  defined,  448. 

experiments  on,  448. 
Inch,  circular,  76. 

cubic,  what  equal  to,  76. 
of  rain,  519. 

spherical,  what  equal  to,  77. 
Inches  in  decimals  of  a  foot,  75. 
Inclined  plane,  172,  484. 
acceleration,  172. 
ropes  for,  381. 
stability  on,  493. 
table  of  pres  on,  486. 
Inertia,  446. 

moment  of,  173,  195. 

"  in  beams,  195;  647. 

Insurance,  73. 
Interest,  73. 

Iron  buckled  'plates,  369.    Angle,  373. 
cast,  wt  of  square  and  round,  362. 
corroded  by  coal  fumes,  370. 
cost,  364.     Chains,  381. 
effect  of  cold  on,  180. 
"      "  mortar  on,  500,  592. 
"      "  water  on,  324,  note,  517. 
elastic  limit  of,  176, 178. 
paints  for,  513. 
porosity,  532. 

proper  kind  for  sea  water,  324. 
rolled,  wt  of  square  and  round,  355. 
ropes  of,  380.    Star,  373. 
sheet,  how  put  on  roofs,  370. 
strength,  bolts,  375. 
"          crushing,  176. 
"         of  beams.    See  Beams. 
"         pillars,  221,  638. 
"          shearing,  181. 
"  "        table  of,  656. 

"         tensile,  178. 
"          torsional,  182. 
"         transverse,  185. 
stretching  of,  178. 
weight  of  angle,  T,  star,  &c,  373. 
"      of  bolts,  376. 
"      of  chains,  381. 


Iron,  weight  of  corrugated,  370. 
weight  of  flat  bars,  372. 

"      of  pillars,  224,  234,  &c. 
"      of  sheet,  367. 
"      of  wire,  368,  369. 

Jet,  water,  325. 

Joints,  Alex.  W.  Rae's,  396. 

bell,  or  faucet,  in  pipes,  574. 

C.  E.  Smith's  inverted  j.,  396. 

figures  of  iron,  268. 
"       "  wooden,  294. 

Fisher's,  392. 

fish-plates,  391,  &c. 
"        "       compensating,  391. 

flexible,  for  water  pipes,  575. 

for  rails,  390. 

J.  Button  Steele's,  394. 

Fritz  &  Sayre's,  395. 

Pettie's,  392. 

Phoenix  sleeve,  392. 
"        suspended,  396. 

ring,  of  C.  &  Amboy  R  R,  393. 

riveted,  653. 

Wilson's  stop-chair,  391. 
Journal  friction,  601. 
Journals,  621. 
Jumper  and  hand-drill,  311. 

Keystones,  depth  of,  341. 

table  of  existing,  343. 

pressure  on,  342,  468,  491. 
Knees  in  water  pipes,  effect  of,  550. 
Knife  edges,  strength  of,  176. 
Knot,  74.    Kinetics,  459. 
Kutter's  formula,  650. 

Land  measure,  76. 

numb  of  acres  drained  by  pipes,  569. 

reqd  per  mile  for  R.  R.,  390. 

surveying,  90. 
Laths,  plastering,  510. 

shingling,  512. 

slating,  511. 
Latitude,  74. 
Lattice  truss,  285. 
Lead,  balls,  377. 

for  roofs,  377. 

pipes,  wt,  377. 

"     strength  of,  533. 

req  in  laying  pipes,  574. 

strength  of,  176-178. 

white,  512. 

wt  of  bars  and  sheets,  376. 
Leakage  of  canals,  &c,  521. 


SDEX. 


687 


Leakage  of  reseprtfirs,  578, 521. 
Level,  book  i^rm.  for,  156. 

cuttiagfMables  of,  420. 

hand,  Locke's,  166. 

meaning  of  the  word,  42. 

plumb,  to  adjust,  167. 

the  spirit,  or  engineer's,  152. 
Levelling,  by  barometer,  167. 

by  boiling  water,  170. 
leverage  and  moments,  473,  645. 
Levers,  473. 

beams  regarded  as,  477. 

compound,  479. 
Lime,  496.    Effect  on  timber,  497. 

air-slacking,  496. 

from  coral,  497. 

weight,  386,  496. 

Limit  of  elasticity  in  beams,  185,  197. 
Line  of  pressure,  or  thrust  of  an  arch, 

348,  493. 
Lines,  61. 
Link,  expansion,  295. 

of  Gunter's  chain,  74. 
Liquid  measure,  76. 
Liquids,  buoyancy  of,  533,  635. 

compressibility  of,  516,  534. 

transmission  of  pressure,  526. 
Load,  on  a  bridge  or  floor,  297,  595. 

on  a  roof,  300.    On  props,  661. 

safe,  on  earth,  314.     Of  sand,  510. 
Locke's  hand-level,  166. 
Lock  Ken  viaduct,  288,  326. 
Lock-nut,  washers,  375, 
Locomotives,  annual  expense  of  run- 
ning, 412. 

cost,  413. 

dimensions  of,  413. 

evaporation,  434. 

fuel,  411. 

tractive  power  of,  411. 

wt  of  trains  on  grades,  412. 
Logarithms,  613.    Long  Meas,  74. 
Longitude,  length  of  degrees  of,  75. 
Lorenz  safety  stvitch,  408. 
Lumber,  board  measure,  357. 

price  in  Philada,  361. 
Lune,  circular,  area  of,  25. 

Man,  weight  of,  297,  595. 

work  by  pump,  treadwheel,  &c,  606. 

work  of  hauling  by  a  rope,  606. 
Maps,  to  reduce  or  enlarge,  69. 
Masonry,  cost  of,  312,  630. 

cub  yds  in  culverts,  351  to  355. 

cub  yds  in  wing  walls,  353. 


Masonry,  weight  of,  386,  529. 
Mass,  defined,  456. 

Materials,  crushing    strength,   table, 
174,  &c. 

shearing  and  torsion,  181. 

strength  of,  174. 

"         "  tensile,  177  to  180. 
"         "  transverse,  185. 

weight  of,  384,  &c.    See  Weight. 
Matter,  defined,  444.    Quantity,  455. 
Measures,  73,  &c. 

to  find  size  of  commercial  by  the 

weight  of  water,  516. 
Measuring  on  sloping  ground,  40,  98. 
Mechanics,  443. 
Melting  point,  310. 
Mensuration,  13. 
Meridian  line,  to  find,  99. 
Metacenterf  635. 
Metal  roofing,  268,  377,  378. 
Metals,  crushing  strength,  176". 

expansion  and  melting  by  heat,  310, 

shearing  strength,  181. 

sheet,  wt  of,  367-376. 

tensile  strengths,  178. 

transverse  strengths,  185. 
Metre,  79. 
Mile,  sea,  and  land,  74. 

scale,  89; 

Mills,  floating,  571. 
Minutes  and  sec,  in  dec  of  a  deg,  17. 
Mitchell's  screw-pile,  324. 
Models,  compared  with  structures,  641. 
Modulus  of  Rup.,  195.    Of  Bias.,  177. 
Moment  of  inertia,  173,  195,  647. 

of  resistance,  217, 646. 

of  rupture,  217,  645.  Of  stability,  489 
Moments,  equality  of,  476. 

and  leverages,  473. 
Momentum,  448. 
Money,  74-81. 
Monkey-switch,  407. 
Mortar,  bricks,  496. 
Mortar,  adhesion  of,  497. 

cement-mortar,  503,  508. 

frozen,  502.     Grout,  497. 

with  brickdust,  496.     Pointing,  501. 

proportion  of,  in  brickwork,  496. 
"  "        rubble,  386. 

protects  iron,  592.   Decays  wood,  497. 

should  not  be  depended  on  against 
sliding,  332,  453. 

strength,  497. 

Moseley  on  strength  of  beams,  19U. 
Moseley's,  W.  H.,  bridges,  289. 


688 


INDEX. 


Motion  defined,  444. 

accelerated,  retarded,  &c,  449. 

quantity  of,  448. 
Mud,  weight  of,  330,  385,  386. 

accumulation  in  reservoirs,  578. 
MuskratS)  577. 

Nails,  weight  of,  383. 

adhesion  and  shearing  strength  of, 
383. 

shingling,  512. 

slating,  511. 

Needle  of  compass,  164. 
Neutral  axis,  195,  246,  645. 
Nonagons,  15. 

North  and  South  line,  to  find,  99. 
North  star,  100. 

diagram  of  its  positions,  101. 
Nuts,  weight  of,  374. 

Obliques  in  a  truss,  defined,  243. 

"    to  find  length  of,  69,  303. 
Obstaeles9  to  pass,  97. 
Obstructions  by  piers,  570. 
Octaedrons,  38. 

Octagons,  area  of,  15 ;  to  draw,  67. 
Ordinates  for  bending  rails,  418. 

of  R.  R.  curves,  20,  416,  633. 
Oscillation,  center  of,  173. 
Ovals,  to  draw,  68. 
Overfalls,  or  weirs,  discharge  by,  559. 

tables  of  discharge,  560,  561. 

fainting  house,  512. 

cost,  513. 

Panel  of  a  truss,  defined,  244. 
"         "     length  of,  244. 

panel-point  defined,  244. 

to  find  diagonal  for,  69,  303. 
Paper,  151. 

profile  and  tracing,  152. 
Parabola,  28. 

to  draw,  29. 

Paraboloid,  solidity,  &c.  of,  38. 
Parallel  forces,  481. 
Parallelograms,  13,  64. 

of  forces,  458. 
Parallelopipeds,  30. 

of  forces,  471. 
Patterns,  362. 
Paving,  with  bricks,  498. 

Belgian,  cost  of,  313. 
Pencils,  152. 
Pendulums,  172. 
Pentagons,  15., 


Percussion,  center  of,  173. 
Perimeter,  wet,  Rule  1,  564,  650. 
Permutation,  72. 
Perpendiculars,  to  draw,  65. 
Phcenix  I  beams,  199,  201,  210. 

"        as  pillars,  638. 
pillars  for  trestles,  308. 
rolled  segment  pillars,  234. 
Pierce's  Well  JSorer,  636. 
Pierre  perdue,  or  random  stone,  314. 
Piers,  abutment,  347. 
cub  yds,  in,  356. 
obstructions  by,  570. 
Piezometers,  537. 
Piles,  315,  320,  323,  534. 
blunt  ended,  323. 
Brunei's,  324. 
Brunlee's,  326. 

driver,  Shaw's  gunpowder,  321. 
driving  by  short  ropes,  607. 
"        by  treadwheel,  321. 
French  rule  for  safety,  322. 
friction  of,  323. 
hollow,  326. 
ice  around,  324. 
inside  of  a  cylinder,  329. 
iron,  324. 

Mitchell's  screw,  324. 
preserving  heads  from  splitting,  324. 
sand,  328. 

Sanders'  rule  for  driving,  322. 
sheet,  321. 
shoes,  for,  323. 

Trautwine's  rule  for  driving,  322. 
to  withdraw,  324. 
wooden,  cost,  320. 
Pillars,  iron,  strength  of,  221. 

table  of  hollow  cast  round,  224,  232. 
"  "         "     square,  222, 232. 

"  "       wrt  round,  228,  232. 

"  "       wrt  square,  232. 

"        solid  cast  round,  226,  232. 
solid  cast  square,  227,  232. 
"        solid  wrt  round,  230,  232. 
"        solid  wrt  square,  231,  232. 
L,  -h  H ,  &c,  235  to  236,  637  to  840. 
Phoenix  rolled  segment,  234. 

"     iron,  various ;  strength  of,  233. 
pinned,  hinged,  or  jointed,  233. 
safety,  coefficient  of,  225. 
swelled  at  center,  237.    Steel,  238. 
wooden,  strength  of,  238. 
wrt  I  beams,  as  pillars,  638. 
Pins  for  eyebars,  293. 
Pipes,  air-valves  for,  579. 


INDEX, 


689 


Pipes,  Ball's  ipaffaud  cement,  577. 
t  of,  539,  548. 

branches,  575. 

contents,  46,  540,  541. 

copper  and  brass,  wt  of,  '365. 

cost  of  laying,  574. 

cub  ft  in  1  ft  length,  541. 

flow  affected  by  material,  537. 

flow  through.    See  Hydraulics. 

gallons  in  1  ft  length,  46. 

giitta  percha,  577. 

iron,  wt  of  cast,  363,  364. 
"        "     wrought,  364. 

knees,  effect  of,  550. 

miles  of,  in  Philada,  577. 

of  bituminized  paper,  577. 

of  bored  logs,  577. 

of  lead,  strength,  533. 
wt,  377. 

pressure  in,  531,  536. 

resistance  to  pumping,  551. 

service,  533,  573. 

ferrules  for,  574. 

steam  warming,  363. 

swellings  in,  effect  of,  551. 

terra  cotta,  569. 

thickness  to  resist  pressure,  531. 

to  mend,  575. 

to  prevent  concretions  in,  581. 

valves  or  gates  for,  572. 

Ward's  flexible  joint,  575. 

water,  must  be  closed  slowly,  533, 573. 

wedges  for,  instead  of  lead,  574. 

wt  of  lead  for  laying,  574. 

wt  of  water  in  1  ft  lengths,  540. 

WyckofTs  patent,  577. 
Pipe-drains,  568,  569. 
Pivot  friction,  600. 
Plane,  inclined,  172,  484. 

"        ropes  for,  381. 

stability  on,  488,  493. 
"        table  of  pres  on,  486. 
Plane  trigonometry,  39. 
Planck,  thickness  to  bear  pres  of  water, 

317. 

Plastering,  509. 
Plate-iron  beams,  214. 
Plates,  buckled,  369. 
Plenum  process  for  foundations,  327,631. 
Plumb-level,  to  adjust,  167. 
Point  of  contrary  flexure,  641. 
Pointing  mortar,  501. 
Polygons,  15.    To  draw,  67. 

of  forces,  467. 

to  reduce  to  a  triangle,  68. 


Poona  dam,  530. 
Porosity  of  cast  iron,  532. 
Port,  establishment  of.    Tides,  626. 
Portland  cement,  385,  500. 
Potts,  Drf  process,  326, 631. 
Powder,  310. 

quantity  in  blasting,  311. 
Power,  animal,  605. 
defined,  449,  474. 
fifth,  546. 
gain  of,  474. 

locomotive,  411.    Water,  571. 
Pratt,  truss,  284,  285,  306. 
Preserving  timber,  358,  362,  497. 
Press,  hyd,  friction  of,  632. 
Pressure,  center  of,  333,  482,  526,  635. 
in  pipes,  531,  536. 
line  of,  in  arches,  348,  493. 
of  air  in  a  diving-bell,  520. 
of  water,  522,  526.    Of  earth,  331. 
table  of,  523.     Of  wind,  520. 
of  running  water,  571. 
Prismoidal  formula,  33. 
Prismoids,  solidity  of,  33. 
Prisms,  and  frustums  of,  30. 
Profile  paper,  152. 
Progression,  71,  72. 
Proportion,  71.    Props,  661. 
Protracting  by  chords,  147. 

table  of  chords  for,  608. 
Pulleys,  479,  662. 
Pumping,  by  hand,  &c,  433,  606. 

engines,  cost  of,  433. 
Purlins,  247,  267,  294,  300. 
Pyramids,  32. 
frustums  of,  33. 

Quarrying,  cost  of,  311,  440. 

Radii  and  ordinates  of  curves,  416. 
Radius,  hyd  mean,  565,  650. 

«        of  gyrations,  173,  495,  617,  622. 
Railroads,  409. 

annual  expenses,  &c,  409. 

ballast  for,  414. 

cars,  413. 

cost  of,  414, 415. 

locomotives,  411. 

shops,  cost  of,  415.    Ties,  414. 

water-stations,  432. 
Kails,  415. 

creeping  of,  391. 

elevation  of  outer,  419. 

guide,  398. 

joints,  chairs,  &c.    See  Joints. 


690 


INDEX. 


Jin  Us,  ordinates  for  bending,  418. 

welded  together,  do  not  creep,  391. 
Main,  518.    Reaching  sewers,  566. 
Raised  tie-bar,  strains  by,  262. 
Raising  of  bridges,  303. 
Ram,  hydraulic,  571. 
Random  stone  foundations,  314. 
Reaction,  strain,  444,  449. 
Reflection,  to  meas  heights  by,  44. 
Refraction  and  curvature,  table,  42. 
Regular  bodies,  the,  38. 
Reservoirs,  577.    Evap  from,  521. 

compensating,  5T9. 

discharge  from  one  into  another,  556. 

distributing,  579.    Storing,  578. 

leakage  of,  578.    3fud  in,  578. 
Resistance,  moment  of,  217,  646. 

"  angle  of,  485,  487,  598. 

Retaining-walls,  bulging  of,  332. 

offsetted,  333.    On  Piles,  534. 

surcharged,  334,  337. 

table  of  contents,  341. 

tables  of,  334,  338,  528  to  530. 

theory  of,  334. 

transformation  of,  339. 

with  curved  profiles,  340. 

with  offsetted,  or  stepped  backs,  333. 
Revetments,  340. 
Rhumb  line,  93. 

Right  angle,  to  lay  off  by  a  triangle,  42. 
Rings,  area  of,  22.     Solidity  of,  32. 

joint  for  rails,  393. 

tightening,  268. 

to  find  breadth,  17. 
Riprap,  314. 

Rise  of  roofs,  effect  of,  on  their  wt,  301. 
Rivers,  flow  in,  563. 
Riveted  joints  and  rivets,  653, 
Rivets,  No.  in  100  Ibs,  653. 

shearing  strengths,  table,  656. 
Roads,  greatest  slope  for,  389. 

traction  on,  603,  605,  607. 
Rock,  excavation,  cost  of,  440. 

increase  of  bulk  when  quarried,  440, 
630. 

quarrying,  311,  440. 
Rocker,  expansion,  295. 
Rod  of  brickwork,  English,  499. 
Rollers,  expansion,  295. 

"    loads  on,  176. 

friction,  176,  43*1,  602. 

Parry's,  429. 
Rolling  friction,  602. 
Roofs,  arched  iron,  289.  Bowstring,  270. 

cost  of,  300.    Copper,  377. 


Roofs,  coverings,  wt  of,  301. 

crescent,  270. 

Fink,  264. 

weight  of,  298,  300. 

how  covered  with  sheet  iron,  370. 
"         "  "  corrugated  iron,  370. 

iron,  details  of,  268.    Lead,  377. 

least  pitch  for  metal  coverings,  378. 
"         "        "    slate         "  511. 

simple,  247,  &c.    Tin,  378. 

strains  on  trusses,  247  to  268,  298. 

weight  allowed  on,  300. 

wind  on,   Tredgold's  allowance  for, 
520. 

wooden,  details  of,  294. 

wt  of,  263,  298  to  301. 

wt  of,  affected  by  rise,  301. 
Roots,  fifth,  546. 

square  and  cube,  table,  48. 

square  of  5th  powers,  548. 

to  calculate,  60. 
Ropes,  382. 

wire,  380. 

wire,  Roebling's  notes  on,  380. 
Rot,  timber,  360. 
Rubble  masonry,  cost,  312,  630. 

proportion  ot  mortar  in,  386. 

quarry,  loose,  440,  608. 
Rule  of  three,  71. 

2  foot,  to  measure  angles  by,  676. 
Rupture,  constant  for,  185, 195. 

moment  of,  217,  645. 

modulus  of,  195. 
Russian  weights  and  measures,  80. 

Safety,  in  pillars,  225  ;  in  bridges,  298. 
Sand,  singular  fact  in  pres  of,  340. 

voids  in,  503.     Wt  of,  387,  503. 

piles,  328.     Load  of,  510. 

pump,  328,  636. 
Scales,  platform,  409. 
Scarfs  for  timbers,  291. 
Schuylltill  River  bridge  of  cast-iron 

arches,  288. 

Scour  of  streams,  563,  570,  571. 
Screw,' American  standard  proportions, 
374.     The  Screw,  479. 

cylinders,  324. 

piles,  324. 

wrench,  629. 
Seamless  brass  and  copper  tubes,  365. 
Seaworms,  414. 

Secants,  cosecants,  &c,  to  find,  101. 
Sector  of  a  circle,  22.  C  of  Gr,  442. 
Sediment  in  reservoirs,  578. 


INDEX. 


691 


Segment,  circulas^area  of,  22. 
e  of,  24. 

spherical,  curved  surf  of,  37. 

"  solidity  of,  34. 

Sellers9  turntable,  429. 
Service  pipes,  377,  533,  573. 
Settlement  of  embankment,  630. 
Severn  Valley  R.  R.  arch  bridges,  288. 
Sewers,  tables  of,  568,  569,  652. 

velocities  in,  568,  652. 
Sextant,  box  or  pocket,  163. 
Shafting,  iron,  strength  of,  182. 

for  tunnels,  627. 
Shearing  strength  of  woods,  181,  642. 

of  nails,  383.    Of  iron  &c,  181. 

of  rivets,  table  of,  656. 
Sheet-piles,  321. 
Sheet-metals,  wt  of,  367. 
Shell,  spherical,  38. 

weight  of,  362,  366. 
Shingling,  512. 
Shops,  railroad,  cost  of,  415. 
Shrinkage  of  embankment,  630. 
Similar  figures,  &c,  defined,  61. 
Sines  and  tangents  explained,  62. 

table  of,  102. 
Single  rule  of  three,  71. 
Skeleton  diagram,  247. 
Sleeve-chair,  392.    Slating,  510. 
Slope  instrument,  167. 

natural,  of  earth,  338. 

of  maximum  pres,335. 

per  100  ft,  98,  388,  389,  629. 
Sluices  in  dams,  586. 
Snow,  519. 
Soils,  or  earths,  safe  loads  for,  314. 

weight  of,  385. 

Solids,  expansion  by  heat,  310. 
Sound,  173. 

Spandrel  walls,  341,  346. 
Spanish  measures  and  weights,  80. 
Specific  gravity,  383. 

"        table  of,  384. 

Spheres,  or  globes,  contents  of  one  1  ft 
diam,  77. 

contents  of  one  1  inch 
diam,  77. 

segments  of,  34,  37. 

table  of,  35. 

zones  of,  38. 
Spherical  shells,  38. 

weight  of,  362,  366. 
Spheroids,  3.8. 
Spikes,  weight  of,  382. 

adhesion  of,  and  nails,  383. 


Spindle,  circular,  solidity  of,  39. 
Splicing  of  timbers,  291. 
Sqtiare  hollow  iron  beams,  193. 
Squares  and  cubes,  and  roots,  48. 
Stability,  488. 

diff  measures  of,  490. 

frictional,  486. 

moment  of,  475,  489. 

of  an  arch,  to  find,  491. 

on  inclined  planes,  488,  493. 
Stand-pipes,  for  waterworks,  625. 

for  water  stations,  433. 
Star,  Nortli,  100. 

Alioth,  100. 
Star-iron,  373. 
Stars,  to  regulate  a  watch  by,  80. 

table  of  times,  80. 
Statics  defined,  444. 
Station  house,  cost  of,  415. 

water,  432. 
Steam-pipes,  363. 
Steel,  cost  of,  364. 

elastic,  limit  of,  176,  179. 

pillars,  238. 

strength  of,  176, 178, 185. 

wt  of,  366,  367. 
Stone,  artificial,  507. 

beams,  203. 

breaking,  414,  505. 

bridges,  341. 

broken,  swelling  of,  440,  608,  630. 
"       voids  in,  504,  630. 

cost  of  dressing,  311. 

"        quarrying,  311,  440. 

dams,  528,  530. 

expansion  of,  310. 

random,  or  pierre  perdue,  314. 

strength  of,  175, 180, 185,  203. 
Stonework,  310. 

cost  of,  312,  630. 
"    "    Bunker  Hill  Mont,  312. 

weight  of,  386,  528. 
Stop-valves,  or  gates,  572. 
Storing  reservoirs,  578. 
Strain  defined,  444,  449. 

on  trusses,  243,  &c. 

three  simple  processes  for  finding, 

253. 

Streams^  bottoms,  to  bear  diff  vels,  570, 
563. 

flow  in,  to  measure,  562. 

horse-power  of  falling,  571. 
"  "  running,  571. 

scour  in,  563,  570,  571. 

virtual  head  of,  571. 


692 


INDEX. 


Street  pipes.    See  Pipes. 
Strength  of  beams.    See  Beams. 

crushing,  174. 

of  cylinders,  193,  531. 

of  glass,  175,  180, 185,  515. 

of  iron  pillars,  221. 

of  iron  rods,  376. 

of  materials,  174. 

of  shafting,  182. 

of  stones,  175, 180, 185,  203. 

of  wooden  pillars,  238. 

shearing,  181,  642. 

tensile,  177,  &c. 

torsional,  131. 

transverse,  183,  &c. 
Stress,  445. 

Struts  to  distinguish  from  ties,  463. 
Stucco,  509. 

Surcharged  retaining-walls,  334,  337. 
Surveying,  90. 
Suspension  bridges,  588. 

aqueduct  at  Pittsburgh,  596. 

data  for  cables,  table,  588. 

Fairmount,  Philada,  595. 

Finley's,  596. 

Freyburg,  594. 

length  of  cables,  to  find,  590. 
"      of  suspending  rods,  591. 

links,  295. 

Niagara,  588,  594. 

strains  on  cables,  588,  589. 
"       on  piers,  591. 

table  of  wire  in  cables,  369,  597. 

Wheeling,  596. 
Sway  bracing,  291. 
Swing-bridge,  strains  on,  275. 
Switches,  frogs,  &c,  397. 

Lorenz,  408. 

monkey,  407. 

tumbling,  405.    Stub,  405. 

Wharton's  safety,  407. 
Swivel,  tightening,  268. 
Syphon,  582,  661. 

T-iron,  373. 

Table  of  acres  reqd  per  mile,  390. 

angles  by  a  2-ft  rule,  676. 

arcs  in  common  use,  table,  434. 

balls,  wt  of,  362,  366,  377. 

board  meas,  357.    Bolts,  376. 

camber,  302. 

chains,  381. 

chords  for  protracting,  608. 
;       circles,  18.  By  decimals,  677. 

circular  arcs,  21,  23,  434. 


Table  of  circular  segments,  24. 

constants  for  defl  3-^  span,  201. 

"         for  safe  defl,  199. 
cpntents  of  cylinders  or  pipes,  46,  47, 

540,  541. 

contents  of  retaining-walls,  341. 
Cooper  &  Hewitt's  beams,  213. 
copper  pipes,  wt  of,  365,  378. 
cubic  foot  and  inch,  76. 
cub  yds  in  a  pier,  356. 

"         «'      culverts,  351,  354. 

"         "     retaining-walls,  341. 

"         "     wing  walls,  353. 
curvature  of  the  earth,  42. 
curves  for  railroads,  416. 
deductions  in  -chaining  on  sloping 

ground,  98. 

diams,  &c,  of  circles,  18. 
elliptic  arcs,  27. 
Fairbairn  girders,  217. 
Fink  trusses,  298,  300,  305. 
friction,  599. 
grades,  388,  389,  629. 
heads  and  nuts  of  bolts,  374. 
heads,  theoretical,  552. 
Howe  bridges,  284. 
inches  in  decimals  of  a  ft,  75. 
iron  pillars,  Gordon's  rule,  224,  &c. 
iron  rods,  strength  of,  376. 
joints,  riveted,  654. 
lead  pipes,  wt  of,  377. 
level  cuttings,  420. 
levelling  by  barom  or  boiling  water, 

171. 

loads  on  grades,  412. 
logarithms,  613. 

min  and  sec  in  decimals  of  a  deg,  17. 
motion  of  stars,  81. 
polygons,  15. 
Pratt  truss,  285, 306. 
pressure  of  water,  523. 
rivets,  653. 
ropes,  380,  382. 

safe  loads  of  wooden  beams,  191. 
sines,  &c,  102. 
slopes,  98,  388,  629. 
spheres,  35. 

square  roots  of  5th  powers,  548. 
squares  and  cubes,  and  roots,  48. 
tin,  379.    Turnouts,  402. 
traverse,  82. 
Trenton  beams,  213. 
value  of  foreign  money,  81. 
vel  of  falling  bodies,  table,  552. 
walls  to  resist  earth,  334,  338. 


INDEX. 


693 


Table  of  walla  to  resist  water,  528  to  530. 

weight  of  flat  aiyi  corrug  iron,  370. 
"         iron/nd  steel  bars,  366. 
"         me^al  sheets,  367. 
"         wire,  368,  369. 

weights  and  measures,  74  to  81. 

weights  of  bridges,  296. 
Tangents  explained.  63,  66, 101. 

to  draw,  67. 
Tanks,  432,  434,  532. 
Tarns,  wt  per  box,  379. 
Tenders  of  locomotives,  413. 
Tensile  strength  of  materials,  177,  &c. 
Tension,  to  find  diams  to  resist,  180. 
Terms,  glossary  of,  615.     Teredo,  414. 
Terra  cotta  pipes,  569. 
Tetraedrons,  38. 
Theodolite,  162. 
Thermometers,  309. 
TJiin  partition  explained,  554. 
Tides,  626. 

Tie-rod  raised,  strains  by,  262. 
Ties,  railroad,  414. 

to  distinguish  from  struts,  463. 
Tightening  swivel,  268. 

ring,  268. 
Timber,  cost  of,  361. 

crushing  strength,  174. 

preserving,  358,  362,  497. 

shearing,  181,  642. 

tensile  strength,  177. 

transverse  strength,  185,  317. 

weight,  384,  &c. 
Time,  by  a  dial,  150. 

by  stars,  80. 

civil  or  clock,  80. 

reqd  to  transmit  force,  456. 
Tin,  378. 

contents  and  wt  per  box,  379. 
Tires,  wagon,  608.    Of  locos,  413. 
Torsion,  181. 
Town's  lattice  truss,  285. 
Tracing  paper  and  cloth,  152. 
Traction,  on  roads,  &c,  411,  603,  605. 
Train  of  cog-wheels,  power  of,  479. 

weight  of,  on  grades,  412. 
Transit,  the  engineer's,  157, 
Transverse  strength  of  materials,  185. 
Trapeziums,  14. 

in  earthwork  measurements,  15. 
Trapezoin,  14. 
Traverse  table,  82. 
Treadwheel,  day's  work  on,  606. 
Tremblings  in  dams,  586. 
Tremie,  506. 


Trenton  beams,  table  of,  213. 
Trestles,  307. 

of  Phoenix  segment  pillars,  308. 
Triangles,  13,  63. 

to  measure  by  trigonometry,  39. 
Triangular  walls  to  resist  water,  529. 
Trigonometry,  39. 
Troy  weight,  74. 
Trusses,  243. 

beam,  244. 

Bollman,  269,  282. 

Bowstring,  270,  286. 

braced  arch,  274,  287. 

Burr,  289. 

camber  of,  302. 

cantilevers,  219,  275,  643. 

cost,  300. 

counterbracing,  245,  252, 275,  306. 

crescent,  270. 

Fink  bridge,  281,  305. 

Fink  roof,  264,  297,  300. 

for  roofs,  247,  298  to  302. 

for  short  bridges,  304. 

horizontal  diag  bracing,  291. 

Howe,  283. 

king  and  queen  roof,  297  to  299. 

lattice,  or  Town's,  285. 

parts  of,  denned,  243.  j 

Pratt,  284,  306. 

raising  of,  303. 

roof,  wt  as  affected  by  rise,  301. 

strains,  3  simple  processes  for  find* 
ing,  253. 

suspension,  244. 

Warren,  254,  279. 

weight  of  bridge,  295. 

weight  of  roof,  263,  298  to  300. 
Tubes,  cast-iron,  table  of,  363,  364. 

couplings,  365. 

gutta  percha,  577. 

lead,  377,  533. 

seamless  brass  and  copper,  365. 

various  kinds,  577. 

welded,  364. 

wrought-iron,  table  of,  364. 
Tumbling  switch,  405. 
Tunnels,  627. 
Turnbuckle,  268,  628. 
Turnouts,  frogs,  switches,  Ac,  397. 

table  of,  402.    Laying  out  of,  401. 
Turnpikes,  max  slope,  389. 
Turntables,  cast-iron,  Sellers',  429. 

other  kinds,  430. 

Undecagons,  15. 


694 


INDEX. 


Vngulas,  31. 

Upset  rods,  strength  of,  375. 

Vacuum  process,  326. 
Valve  towers,  579. 
Valves,  air,  579. 

defined,  628. 

for  water  pipes,  572. 
Variation  of  the  compass,  165. 
Velocity ,  angular,  447 

defined,  447,449. 

in  pipes,  534. 

Kutter's  formula,  650. 

of  sound,  173. 

of  water  in  rivers,  tables,  562,  563. 
"        in  regular  channels,  tables, 

567,  568. 
"        in  sewers,  568,  652. 

of  wind,  520. 

theoretical  of  falling  bodies,  table,  552. 

to  abrade  soils,  563. 

vel  head,  535. 

virtual,  477. 

Vena  contracta,  552,  554. 
Ventilationf  air  required,  519. 
Vernier >  162. 
Vertical,  meaning  of,  61. 
Verticals,  in  a  truss,  defined,  243. 
Vessels,  draft  of,  534,  635. 
Viaduct,  Loch  Ken,  288,  326. 
Virtual  velocity,  477. 

head,  571. 
Vis  viva,  446,  455. 
Voids  in  broken  stone,  504,  630. 

in  rubble,  496,  630. 

in  sand,  503. 
Vulgar  fractions,  69. 

Walls,  high,  for  dams,  in  France,  529. 
intended  to  resist  water,  528,  534. 
"  "  ."  tables,  528,530. 

"  table  of  contents,  341. 
number  of  bricks  reqd,  496,  498. 
retaining,  331.  On  piles,  534. 

"          several    forms  with    the 
same  quantity  of  mason- 
ry, 530.  [&c. 
"          tables  of,  334,  338,  341, 528, 
"          transformation  of,  339. 
"          triangular,  529. 
spandrel,  341,  346. 
wharf,  339,  534. 
wing,  350. 

wing,  table  of  contents,  353. 
Ward's  flexible  pipe-joint,  575. 


Warren  truss,  254,  279,  &c. 
Washers,  size,  and  weight  of,  373,  374. 

lock-nut,  375. 

Wash'n  Monument  concrete,  505. 
Washes  for  outdoor's  work,  513. 

for  brickwork,  514. 

white,  514. 

Watch,  to  regulate  by  stars,  80,  note. 
Water,  515. 

boiling,  to  measure  heights  by,  170. 

buoyancy  of,  533,  635. 

cisterns,  432,  434,  532. 

column  or  stand-pipe,  433,  625. 

compensation,  579. 

cub  ft,  and  gallons  in  pipes,  46,  541. 

effects  on  iron,  324,  516. 

horse-power  of,  571. 

pressure,  522.    Table  of,  523. 
"         of  running  water,  571. 
"        in  pipes,  531,  536. 

quantity  reqd  in  cities,  580. 

resistance  to  moving  bodies,  571. 

stations  on  railroads,  432. 

under  pres,  oozes  through  iron,  532. 

walls,  to  resist,  528,  534. 

wheels,  571. 

wt  of,  for  finding  the  size  of  commer- 
cial measures,  516. 

wt  of,  in  1  ft  length  of  pipes,  540. 
Water-jet,  325. 
Web  members  of  a  truss,  243. 
Wedges,  33,  481. 

instead  of  lead  in  pipe  laying,  574. 
Weight  allowed  on  a  bridge,  297. 

allowed  on  a  roof,  300. 

cars,  413. 

carts,  436,  608. 

cement,  385,  500. 

crowds,  297,  595,  footnote. 

defined,  444. 

iron,  angle,  T,  channel,  star,  373. 
"     corrugated,  370. 
"     flat  bars,  372. 
"     pillars,  224,  &c. 
"     square  and  round,  366. 

lime,  496. 

masonry,  386,  529. 

nails,  383. 

of  a  cub  ft  of  substances,  384. 

of  a  horse,  605. 

of  a  roof  truss,  as  affected  by  its 
rise,  301. 

of  balls,  362,  366,  377. 

of  bolts,  nuts,  washers,  374. 

of  bricks,  384,  498. 


695 


Weight  of  bridj 

of  ceilings  and  floors,  248. 

of  dredged  materials, : 

of  earths,  385. 

of  patterns,  362. 

of  pipes,  brass  anXcopper,  365. 

,  364. 
"  lead,  377. 

roof  covering,  301,  511.    Rails,  415. 

roofs,  iron,  263,  298,  299,  300. 

sheet  metals,  367,  376. 

shells,  362,  366. 

snow,  519.    Sand,  387,  503. 

spikes,  382. 

tenders,  413. 

timber,  384,  &c. 

water,  515. 
"        in  1  ft  length  of  pipes,  540. 

wire,  368,  369. 

Weights  and  measures,  73. 
Weirs,  558. 

Weisbach's  rule  for  flow  in  pipes,  543. 
Wells,  contents  of,  47. 

boring,  636. 

lining  of,  47. 

Wet  perimeter,  rule  1,  p  564,  650. 
Wharf  walls,  339,  534. 
Wharton's  safety  switch,  407. 
Wheelbarrow  ft,  in  earthwork,  438. 
Wlieels,  cog,  power  of  a  train  of,  479. 

of  cars,  413. 

tires,  608. 

water,  571. 

wheel  and  axle,  477. 
Whipple,  Mr.,  243,  note. 


Whipple  bridges,  arched,  288. 
Whitewash,  514. 
Winch,  day's  work  at,  606. 
Wind,  520. 

mills  for  water  stations,  433. 
Wing-walls,  cub  yds  in,  353. 
Wire  gauges,  367.    Fences,  415. 

rope,  wt  and  strength,  380. 
"     Mr.  Roebling's  notes  on,  380. 

strength  of,  369.    Rigging,  381. 

table  of  weight,  368,  369. 
Wires ,  to  find  the  number  of  in  a  cable, 

369. 
Wood,  crushing  strength,  174. 

shearing,  "         181. 

table  of,  642. 

tensile,  "         177. 

transverse,  "         185,  191. 

Wooden  jnllars,  rules  for,  238. 

table  of,  239  to  242. 
Work  defined,  448. 

quantity  of,  454,  455. 

unit  of,  449. 

unit  of  rate  of,  449. 

Zinc  or  spelter,  378,  388. 

in  sheets  for  roofing,  379. 

paint,  512. 

paint  does  not  adhere  well  to,  370. 

price  of,  379,  note. 

strength  of,  176,  180. 

vessels  for    water,  said    to   be    un- 
healthy, 379. 
Zones,  circular,  25. 

spherical,  38. 


Jtemarh.  —  Many  things  not  mentioned  in  this  Index  will  be  found 
in  the  Glossary. 


THE  END. 


INDEX  TO  ADVERTISEMENTS. 


PAGE 

HELLER  &   BRIGHTLY, 699 

CAMBRIA  IRON  AND  STEEL  WORKS, 700 

J.  G.  BRILL  &  Co.,      .  • 700 

D.  P.  DIETERICH, 701 

GEO.  J.  BURKHARDT'S  SONS,      . 702 

WILSON  BROTHERS  &  Co., 702 

WM.  SELLERS  &  Co., 703 

CARNEGIE  BROS.  &  Co.,  LIMITED, 704 

JAMES  BEGGS  &  Co., 704 

TATHAM  &  BROTHERS, 705 

VANDERBILT  &  HOPKINS, 706 

H.  B.  SMITH  MACHINE  Co., 706 

PENNSYLVANIA  STEEL  Co., 707 

SAMUEL  J.  CRESWELL, 708 

FRENCH'S  PAINT,  PLASTER,  AND  CEMENT  DEPOT,  .  .  708 
VERONA  TOOL  WORKS,. METCALF,  PAUL  &  Co.,  .  .  .  709 
CRESCENT  STEEL  WORKS,  MILLER,  METCALF  &  PARKIN,  709 
THE  PHILADELPHIA  BRIDGE  WORKS,  COFRODE  &  SAYLOR,  710 

BORGNER    &    O'BRIEN, 7ia 

COOPER,  HEWITT  &  Co., 711 

WM.  B.  SCAIFE  &  SONS, 712 

CHESTER  STEEL  CASTINGS  Co., 712 

ELBA  IRON  &  BOLT  Co.,  LIMITED, 713 

THE  PEERLESS  BRICK  Co., 714 

THE  PHCENIX  IRON  Co., 714 


697 


RUMENTS  OF   PRECISION.) 


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OF 

Lead  Pipe,  Tin-lined  !ron  Pipe,  Block- 
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'  WEIGHTS  OF  SHEET  LEAD. 


Weight  per 
sq  are  foot. 

™X| 

Weight  per 

Thick- 

Weight  per 

Thick-    ! 

Weight  per 

Th  ck- 
ness. 

3      " 
4      " 

.042  in. 
.051  " 
.068  " 

5  fts. 
6   " 
7   " 

.085  in. 
.102  '• 
.119  " 

8  fts. 
9  " 

10   " 

.136  in. 
.153  " 
.170  " 

i**'; 

16   " 

.•jo:;  in. 
.2:57  '• 

.271    " 

WEIGHTS  OF  LEAD  PIPE. 

2i 

a 

if 

I 
• 

a 

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s 

£ 

a  . 

i£ 

Weight  per  fo«t 
aud  rod. 

a  ff 
|§ 

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Weight  per  foot 

and  rud. 

I 

0 

Weight  per  foot 
aud  rod. 

11 

u." 

%  in. 

7  fts.  per,rod. 
10  oz.  per  foot. 

6 

8 

%  i 

-, 

2%  fts 

3"^   '" 

per  foot. 

22 
25 

l^in. 

3  Its.  per  foot. 
3%  fts.    " 

14 

16 

" 

1  tl 

" 

12 

% 

11. 

16     " 

per  rod. 

* 

" 

4% 

19 

" 

1/4 

fts.   " 

16 

1  M     " 

per  foot. 

II 

4 

6              " 

25 

19 

r. 

l^in. 

14 

fcin- 

9 

"    per  rod. 

7 

• 

2)4    " 

' 

K 

4}^             ' 

17 

%  ft 

.  per  foot. 

9 

3      " 

21 

5                ' 

19 

1  " 

11 

v: 

23 

fts.  " 

M 

16 

I  in. 

4%    " 
21%  fts.  per  rod. 

'K 

1%'in. 

H 
4 

27 
13 

" 

2 

41          « 

19 

2      " 

per  foot. 

11 

5 

17 

3 

25 

H 

21 

Kin. 

12ft 

<(.  per  rod. 

8 

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3J4     ' 

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17 

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8J<2                  ' 

27 

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1    ' 

per  foot. 

9 

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4        ' 

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2  in. 

4% 

15 

1  V<5  ' 

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24 

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18 

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2        ' 

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22 

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9               " 

27 

LEAD  PIPE  OF  LARGER  CALIBRE. 


Indies. 

I  Thick. 

^  Thick. 

£  Thick. 

&  Thick. 

3 

ft. 
17 
20 
22 
25 

31 

oz, 
0 
0 
0 
0 

0 

ft.            oz. 
14                0 
16               0 
18               8 
21                0 

ft.             oz. 
11               0 
12                0 
15                0 
16                0 
18                0 
20                0 

ft,           oz. 
8               0 
9                0 
9                8 
12                8 
14               0 

\^i  inch,  2  fts.  per  foot. 
2        "      3  fts. 

%  inch,  4J^.  6J^,  and  8oz 
^     "      6.  7H,  and  10  <>/ 
^     "      8  and  10  oz. 

LEAD  WASTE  PIPE. 

3  inch,  31A  and  5  ft<.  per  foot.         4^  inch,  <•  and  8  fts.  per  foot. 
4     "       5,  6.  and  8  IDs.  per  foot.       5         "       8,  10,  and  12  fts.  per  ft. 
BLOCK-TIN  PIPE. 

per  ft.     *4  inch,  10  and  12  oz.  per  foot         1J£  inch,  2  and  '2l/>  fts.  per  foot. 
1       "       15  aud  18  oz.         "               2        "       2^  and  3  fts. 
Itf  "      1H  and  1'4  ft«.    ' 

SHOT-AMERICAN  STANDARD  SIZES. 


No. 

Diameter 
iu  inches. 

CHILLED 
DROP  SHOT. 

DROP  SHOT. 

No. 

Diameter 
iu  inches. 

CHILLED 

DROP  SHOT. 

Duoi-  SHOT 

No.  of  shot     !     No.  of  shot 

No.  of  shot 

No.  of  shot 

to  the  oz. 

to  tfce  oz. 

to  the  oz. 

to  theoz. 

n 

•05 

2385 

2326 

2 

•15 

88 

H; 

11 

•06 

1380 

1346 

1 

•16 

73 

71 

10 

•07 

868 

848 

B 

•17 

61 

59 

9 

•08 

585 

568 

BB 

•18 

52 

50 

8 

09 

409 

399 

BUR 

•19 

43 

42 

7 

•10 

299 

291 

T 

•20 

36 

6 

•u 

223 

218 

TT 

•21 

31 

5 

•12 

172                         1«8 

F 

•22 

27 

4 

•13 

136                         132 

FF 

•23 

24 

3 

•14 

109                        106 

COMPRESSED  BUCK  SHOT. 


No.  |  Diameter  in  inches. 

No.  of  balk  to  the  ft.  |     No. 

Diameter  in  inches. 

No.  of  balls  to  the 

Ib. 

3 
2 
1 
0 

•25 
•27 
•30 
•82 

288 
225 
172 
140                  | 

00 
000 
Balls 
" 

•34 
•36 
•38 
•44 

113 
100 
83 
50 

E.    W.    VANDERBILT.  E.    M.    HOPKINS. 

VANDERBILT  &  HOPKINS, 

RAILROAD  TIES, 

CAR  AND  RAILROAD  LUMBER, 

White  anl  Yellow  Pine,  M,  Gnm  anf  Cypress, 

120  LIBERTY  STREET, 

NEW   YORK. 


Boards,  Plank  and  Dimension  Lumber  Sawed  to  Order. 
General  Railroad  Supplies,  and  Roofing  Slate. 


[OUR   NEW   STYLB   IO-INCH   MOULDER.] 

WOOD-CUTTING  MACHINERY 

OF  SUPERIOR  QUALITY, 

For  Car  Shops,  Planing  Mills,  and  Sash  and  Door  Factories. 

Correspondence  Solicited.    Cable  Address,  "Machine,  Phila." 

H.  B.  SMITH  MACHINE  CO., 

926  MARKET  ST.,  PHILADELPHIA,  PA.,  U.  S.  A. 

706 


STEEL  CO,, 

/No.  ^208  South  Fourth  Street, 
PHILADELPHIA,  PA., 

Manufacturers  of 

STEEL  RAILS 

AND  STEEL  FORCINGS. 


STEEL  RAILROAD  FROGrS 

In  Several  Improved  Patterns. 
IMPROVED  SAFETY  SWITCHES, 

RAILROAD-CROSSING  FROGS, 

SWITCH  STANDS  AND  SIGNALS, 

INTERLOCKING  APPARATUS, 

by  which  Switches  are  operated  from  convenient  stations  with  a 
great  saving  of  expense,  and  securing  Freedom  from  Accidents. 


WORKS  AT  STEELTON,  PENNA. 


NEW  YOMK  AGENT, 

STEPHEN  W.  BALDWIN,  160  Broadway,  N.  T. 

707 


SAMUEL  J.  CRESWELL, 

ARCHITECTURAL  AND  ORNAMENTAL 

-*IRON  WORKS*- 

Twenty-Third  and  Cherry  Sts,, 


PHILADELPHIA. 


Fronts  for  Buildings,  Girders,  Columns,  "Wrought 
Iron  Beams.  Sidewalk  Lights,  Stairs,  Railings, 
Orestings,  Stable  Fixtures,  and  Lamp  Posts. 

FRENCH'S 

PAINT,  PLASTER,  AND  CEMENT 


-^  DEPOT,  gh 

York  Avenue  and  Callowhill  Street, 
PHILADELPHIA,  PA. 


S* MANUFACTURERS    AND    IMPORTERS.- 


FOREIGN  AND  DOMESTIC 

PAINTS,  PLASTERS,  AND  CEMENTS 


OF  ALL  GRADES. 
708 


WORKS, 

MtETCALF,  PAUL  &  co. 

McCance's  Block,  Serena  Aye,  &  Liberty  St.,  PittsWi,  Pa. 

MAKE  A  SPECIALTY  OF  SOLID  STEEL 

RAILROAD  TRACK 
TOOLS. 

ALSO,  SOLE  MANUFACTURERS  OF 

THE  PATENT  VERONA  NUT  LOCK, 

BRANCH  HOUSE, 

No.  22  West  Lake  Street,  Chicago. 

ST-2 SEND  FOB  CATALOGUE. =^ 

CRESCENT  STEEL  WORKS. 

ESTABLISHED   1865. 


MILLER,METCALF&  PARKIN 

OFFICE,  No.  81  WOOD  STREET, 

Works,  49th  and  50th  Streets  and  A,  V,  E,  E,, 

PITTSBURGH,  PA. 


Fine  Tool  Steel,   Drill   Rods,   Needle  Wire,   Cold 
Rolled  Strips,  Clock  Spring-  Sheets,  etc.  etc. 


PHILADELPHIA,     ....     1232  MARKET  STKEET. 

NEW  YOKK,     ......       178J  WATEK  STKEET. 

CHICAGO,     ....     22  &  24  WEST  LAKE  STREET. 

709 


DIE-FORGED  EYE-BARS.  MACHINE  RIVETING. 

THE  PHILADELPHIA  BRIDGE  WORKS, 

SHOPS  AT  POTTSTOWN,  PA. 

JOSEPH    H.    COFRODK.  FRANCIS   H.  SAYLOR. 

COFRODE  &  SAYLOR, 

CIVIL  ENGINEERS  AND  BRIDGE  BUILDERS, 

OFFPCE,  No,  257  SOUTH  FOURTH  STREET, 

PHILADELPHIA. 


Contractors  for  the  Construction  and  Erection  of 

IRON  or  WOODEN  BRIDGES,  VIADUCTS,  TURN-TABLES,  ROOFS,  and  BUILDINGS. 


Plans  and  Prices  Furnished  on  Application,    Specifications  Solicited, 

BRIDGE  RODS  WITH  UPSET  ENDS.  CASTINGS  FOR  WOODEN  BRIDGES. 

Superior  Quality.  Best  Workmanship. 

BORGNER  &  O'BRIEN, 

Twenty -Third  Street,  above  Race  Street, 
PHILADELPHIA,  PA,  U.S.  A, 

MANUFACTURERS  OP 


FIRE  BRICK 


AND 


CLAY  RETORTS 


For  Heating  and  Melting  Furnaces  of  Every  Description, 

Particnlar  Attention  Given  to  Soecial  Snanes. 

710 


EDWARD  CpsffR,^  JAMES  HALL, 

t65'        EDWIN  F.  BEDELL. 


COOPER,  HEWITT  &  CO., 

No.   17  BURLING   SLIP,  NEW  YORK. 

TRENTON  IRON  WORKS,  TRENTON,  N.  J. 

PEQUEST  IRON  WORKS,  OXFORD,  N.  J. 

RINGWOOD  IRON  WORKS,  RINGWOOD,  N.  J. 

DURHAM  IRON  WORKS,  RIEGELSVILLE,  PA. 


IRON  ORE,  PIG  IRON, 

ROLLED  IRON  BEAMS, 
CHANNELS,  ANGLES,  AND  TEES, 

WELDLESS,  DIE-FORGED  EYE-BARS,  RAILS, 
MERCHANT  IRON,  BRAZIER  AND  WIRE  RODS, 
STAPLES,  RIVETS,  CHAINS,  BRIDGES,  ROOFS, 
AND  OTHER  IRON  STRUCTURES. 


IRON  AND  STEEL  WIRE 

OF  ALL  KINDS. 

A  Specialty  is  made  of  Superior  Qualities 

of  Wire  straightened  and  cut  to  lengths. 

711 


WM.  B.  SCAIFE  &  SONS, 

OFFICE,  NO.  119  FIRST  AVENUE, 

PITTSBURGH,  PA., 
DESIGN,  MANUFACTURE  AATD  ERECT 

IRON  MILL  BUILDINGS, 

AND  ALL  KINDS  OF 

IRON   ROOF   FRAMES, 
CORRUGATED  IRON  FOR  ROOFS  AND  SIDING, 

SHEET  AND  PLATE  IRON-WORK, 
CaldwelPs  Patent  Hollow-Shaft  Spiral  Conveyer, 

For  Grain,  Goal,  Sawdust,  Cottonseed,  Flaxseed,  etc. 
<^« — —SEND    FOR    CATALOGUES    AND    PRICE-LISTS. <*-^* 

From  1-4  to  15,000  Ibs. 
Weight. 

True 'to  pattern,  sound  and 
solid,  of  unequalled  strength, 
toughness  and  durability.    An 
invaluable  substitute  for  forg- 
ings,  or  for  cast-iron  requiring 
three-fold  strength. 
Gearing  of   all    kinds,   Shoes,   Dies,    Hammer- Heads,    Cross- 
Heads    for   Locomotives,    etc.      15,000  Crank   Shafts  and  10,000 
Gear  Wheels  of  this  steel  now  running  prove  its  superiority  over 
other  Steel  Castings. 

SPECIALTIES: 

CRANK  SHAFTS,  CROSS-HEADS,  AND  GEARINGS. 


STEEL 
CASTINGS 


STEEL  CASTINGS  OP  EVERY  DESCRIPTION, 


Please  send  for  Circulars.     Address 

CHESTER  STEEL  CASTINGS  CO., 

Works,  Chester,  Pa. 

OFFICE,  407  LIBRARY  ST.,  PHILADELPHIA. 

712 


ELBA  iHOlrt  BOLT  CO.,  Limited, 

V^  MANUFACTURERS  OF 

MERCHANT  BAR  IRON, 

SKELP  IRON,  SPLICE  BARS,  RAILWAY  TRACK 

BOLTS,  CAR,  BRIDGE,  AND  MACHINERY 

BOLTS,  NUTS,  ETC. 


We  invite    the    attention  of   RAILROAD    MEN    especially   to 
our  make  of 

Splice  Bars  and  Track  Bolts. 

Using  the  best  brands  of  REFINED  IRON,  and  paying  close  at- 
tention to  the  finish  of  our  manufactures,  we  are  enabled  to  offer 
our  patrons 

BOLTS,  NUTS,  SPLICE  BARS,  ETC., 

of  excellent  quality.  Our  works  have  been  enlarged  within  a  few 
years;  all  orders  are  now  executed  with  promptness;  all  our  work 
guaranteed. 

SEND  FOR  PRICE-LISTS  AND  INFORMATION  TO 

ELBA  IRON  &  BOLT  Co.,  LIMITED, 

PITTSBURGH,  PA. 

LOVEJOY  &  DRAKE,  Agents, 

49  Cortlandt  Street,  Netv 

713 


THE  PEERLESS  BRICK  COMPANY, 

OP  PHILADELPHIA, 

MANUFACTURE  AND  KEEP  IN  STOCK 

ARCHITECTURAL    SHAPES,    300  KINDS. 

ALSO 

BED  Pressed  Fronts.     Extra  fine  in  color  and  quality. 

BUFF,  solid  rich  color — beautiful.     One  of  the  finest  bricks  made. 

DRAB,  handsomer  and  more  durable  than  stone. 

BROWN,  very  strong  and  superior  to  brown  stone. 

GRAY,  a  very  desirable  shade. 

All  the  above  are  solid  colors  throughout.  Special  Shapes  made  to 
order.  Particular  attention  paid  to  making  and  fitting  ARCHES  of  all 
kinds  and  sizes  (from  plain  and  moulded  bricks)  from  drawings  furnished. 

BLACK,  Velvety  jet  face.    The  only  black  brick  fit  for  a  fine  building, 
producing  a  beautiful  effect,  and  free  from  the  glossy  and  greasy  look 
of  other  black  or  dipped  bricks. 
DIAPERING  AND  ORNAMENTAL  Bricks  made  in  the  above  colors. 

For  further  descriptions  see  pages  466  to  469  of  this  book. 

Illustrated  Catalogue  and  Price-List  sent  free  on  application. 

Office, 208  S,  Seventh  St, (W.Washington  Square.) 

SAMUEL  HART,  Pres.  JOSEPH  WOOD,  JR.,  Treas. 

THE  PH(ENIX  IRON  COMPANY 

No.   41O    WALNUT   STREET, 
PHILADELPHIA,  PA. 


Manufacture  Rolled  Beams,  Channels,  Angles,  Tee, 
Shape,  and  Bar  Iron  of  all  sizes. 

Roof  Trusses,  Girders,  and  Joists  for  Fire-Proof 
Buildings  framed  and  fitted  as  per  plans. 

PHCENIX  Wrought-Iron  Columns  of  all  sizes. 
DIE-FORGED  WELDLESS  EYE-BARS  a  specialty. 


Designs  and  Estimates  furnished  upon  application. 
714 


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AUCHIWCI.O8S.—  The    Practical  Application  of  toe  Slide- 

J\_  Valve  and  Link-Motion  to  Stationary,  Portable,  Locomotive,  and  Marine 
Engines,  with  New  and  Simple  Methods  for  proportioning  the  Parts.  By 
WILLIAM  S.  AUCHINCLOSS,  C.E.  Seventh  Edition.  Revised  and  Enlarged. 
8vo,  cloth.  $3.00. 

BIL.GRAM.—  Slide-  Valve  Gears.    A  new  graphical  method 
for  Analyzing  the  Action  of  Slide-Valves,  moved  by  eccentrics,  link- 
motion,  and  cnt-oft  gears,    By  HUGO  BILGRAM,  M.E.    16mo,  cloth.    $1.OO. 

/COOPER.—  A  Treatise  on  the  Use  of  Belting  for  the  Traiis- 

vy  mission  of  Power.  With  numerous  illustrations  of  approved  and  actual 
methods  of  arranging  Main  Driving  and  Quarter  Twist  Belts,  and  of  Belt 
Fastenings.  Examples  and  Rules  in  great  number  for  exhibiting  and  calcu- 
lating the  size  and  driving  power  of  Belts.  Plain,  Particular,  and  Practical 
Directions  for  the  Treatment,  Care,  and  Management  of  Belts.  Descriptions 
of  many  varieties  of  Beltings,  together  with  chapters  on  the  Transmission  of 
ower  by  Ropes  ;  by  Iron  and  Wood  Fractional  Gearing;  on  the  Strength  of 


Belting  Leather  ;  and  on  the  Experimental  Investigations  of  Morin,  Briggs. 
s.    By  JOHN  H.  COOPER,  M.  E.    1  vol.,  demi  octavo,  cloth.    S3.5O. 


Po 

Belt 

and  others. 

DANBY.—  Practical  Guide  to  the  Determination  of  Miner- 
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T.  W.  DANBY,  M.A.,  F.G.S.    $2.5O. 

DRAKE.—  A    Systematic   Treatise,  Historical,  Etiological, 
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DWYER.—  The  Immigrant  Builder;  or,  Practical  Hints  to 
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tion.   Demy  8vo,  cloth.    $1.5O. 

/"I  ENTRY.—  The    House    Sparrow    at    Home    and    Abroad.. 

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jO  IRARO.—  Herpetologry  of  the    United    States    Exploring 

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tion  ; 


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ROBERT  GRIMSHAW.  Quarto,  cloth.  $2.OO. 

f  1  RIMSMAW.— Saws.    The  History,  development .  Action, 

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-Modern  Milling.      Being    the  substance  of 

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HARTMAN  and  MECHENER'S  Coiichology.-€oiichologia 
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JM  and  Engineer's  Reference  Book  By  SAMUKL  NICHOLS,  Foreman  Boiler- 
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XTYSTROM.— A  New  Treatise  011    Elements  of    Mechanics, 

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panied with  an  Appendix  on  Duodenal  Arithmetic  and  Metrology.  By  JOHN 
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By  JOHN  W.  NYSTROM.  C.E.  Svo,  cloth.  $1.5O. 

OVERMAN. -Mechanics    for    the    Millwright.    Engineer, 
Machinist.  Civil   Engineer,  and  Architect.    By  FREDERICK   OVERMAN. 
12mo,  cloth.     l.")0  illustrations.    $1.50. 

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2 


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RIDDRtL..— The  Carpenteraiid  Joiner  Modernized.  Third 
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parts,  circumferences  of  circles,  length  of  rafters  and  braces,  board  measure, 
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4to,  cloth.  S7.5O. 

J )II>I>EL,L,.— The  ITew  Element*  of  Hand  Railing.    Revised 

XX  edition,  containing  forty-one  plates,  thirteen  of  which  are  now  for  the 
first  time  presented,  together  with  the  accompanying  letter-press  description. 
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1)OPER.— Questions  and  Answers  for  Engineers.  This  little 
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^l.OA>.-tity  uit<l    Suburban  Art-hit*  t  ture.      In  which  are 

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..ntile  Building.    By.--  -  ith  131 

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^  I.  O  \  >.— Constructive     Arehiteetnre.      A    Guide    for    the 

O  Ider  and  Carpenter:  -  for  the  construc- 
tion of  Roofs.  Domes,  ai  iples  of  the 
fttf  Orders  ef  Arckiltdbtrt :  selected  from  Grecian  and 
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SPAX«i.— A    Practical    Treatise    011     I.i^lun in-    Protection. 
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.— A    New   Method    of   Calculating    the    Cubic 
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together  with  Directions  for  estimating  the  cost  of  Earthwork.    By  J 
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enlarged,    bvo,  cloth.    £2.OO. 


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rr"RAFTWIXE*S  Civil  Engineer's  Pocket- Book  of  3Iensnra- 

I        n.  Trigonometry  Ilydrauli 

•''  P?n-C1? 
and  Iron  Roof  and  E  "idges  and  Cnily 

^nsion  Br:  ads.  Tun:  lonns. 

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^\^HITE.-Tli€    Elements  of  Theoretical   and    Oescriptive 

\  >  .mv   for  the  ose  of  Colleges  and  Academies.    By  CHARLES  J. 

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